Finite Group Databasebeta

← #72278519

Subgroups — #72278519

Order 2048.

At least 6032 conjugacy classes of subgroups; only the 1528 normal subgroups are shown.

The subgroup lattice has 507 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72278511C211yesmaximal, characteristic, complemented
2#72278500C211yesmaximal, characteristic, complemented
3C2×#534011C211yesmaximal, characteristic, complemented
4#72278479C212yesmaximal, complemented
5#72278468C212yesmaximal, complemented
6#72278461C212yesmaximal, complemented
7#72278451C212yesmaximal, complemented
8#72278441C214yesmaximal, complemented
9#72278432C214yesmaximal, complemented
10#72278420C218yesmaximal, complemented
11#72240399C211yesmaximal, characteristic, complemented
12C2×#482360C211yesmaximal, characteristic, complemented
13#72240342C211yesmaximal, characteristic, complemented
14C2×#6346631C211yesmaximal, characteristic
15C2×#62773C2211yescharacteristic
16C22×#2535C2211yescharacteristic
17C2×#20722C2211yescharacteristic
18#6346631C4116yescomplemented
19C2×#18212C2212yes—
20C2×#18210C2212yes—
21C2×#18169C2212yes—
22C2×#18162C2212yes—
23C2×#18158C2214yes—
24C2×#17364C2211yescharacteristic
25C2×#17362C2211yescharacteristic
26#534011C2214yes—
27#534011C2214yescomplemented
28C2×#14577C2211yescharacteristic, complemented
29C2×#14572C2211yescharacteristic, complemented
30C2×#14248C2211yescharacteristic, complemented
31C2×#14171C2211yescharacteristic, complemented
32C2×#13960C2211yescharacteristic
33C2×C4×#386C2211yescharacteristic
34C2×#11875C2214yes—
35C2×#11797C2212yescomplemented
36C2×#11796C2212yescomplemented
37C2×#11774C2214yescomplemented
38#363909C2212yes—
39#363909C2212yes—
40#362751C2212yescomplemented
41#361664C2212yescomplemented
42#358863C2214yescomplemented
43#186019C2212yes—
44#186019C2212yes—
45#185993C2214yescomplemented
46#185948C2212yescomplemented
47#185945C2212yescomplemented
48#137364C2218yescomplemented
49#137363C2218yescomplemented
50#137356C2218yescomplemented
51#137355C2218yescomplemented
52#137316C2218yescomplemented
53#137315C2218yescomplemented
54#137308C2218yescomplemented
55#137307C2218yescomplemented
56#127617C2214yescomplemented
57#127590C2214yescomplemented
58#127561C2212yescomplemented
59#127561C2212yescomplemented
60#126987C2214yescomplemented
61#126971C2212yescomplemented
62#126971C2212yescomplemented
63#126947C2214yescomplemented
64#62773C2×C418yescomplemented
65C2×#3490C2312yes—
66C2×#3501C2311yescharacteristic
67C2×#3496C2311yescharacteristic
68C2×#3459C2312yes—
69C23×C4×D4D411yescharacteristic
70C23×C4×D4C2311yescharacteristic
71C24×C42D411yescharacteristic
72C2×#2535C2×C418yes—
73C2×#2535D418yes—
74C2×#2535D412yes—
75C2×#2535D412yes—
76C2×#2535C2312yes—
77C2×#2535C2312yes—
78C22×#397D411yescharacteristic
79C22×#397C2311yescharacteristic
80C22×#392D411yescharacteristic
81C22×#392C2311yescharacteristic
82C22×#386D411yescharacteristic
83C22×#386D411yescharacteristic
84C22×#382D411yescharacteristic
85C22×#380D411yescharacteristic
86C22×C4×C22⋊C4D411yescharacteristic
87C23×#95D411yescharacteristic
88C2×#1819C2312yes—
89C2×#1799C2312yes—
90C2×C4×#99C2312yes—
91#18212C2×C4116yescomplemented
92#18210C2×C4116yescomplemented
93#18169C2×C4116yescomplemented
94#18162C2×C4116yescomplemented
95#18158C2×C4132yescomplemented
96C2×#1608C2314yes—
97C2×#1607C2314yes—
98C2×#1596C2314yes—
99C2×#1595C2314yes—
100#17364C2×C418yescomplemented
101C2×#1466C2312yes—
102C2×#1465C2312yes—
103C2×#1462C2312yes—
104C2×#1462C2312yes—
105C2×#1460C2312yes—
106C2×#1459C2312yes—
107#14577C2314yes—
108#14572C2314yes—
109#14248C2314yes—
110#14171C2314yes—
111#13960D412yes—
112#13960D412yes—
113#13960C2312yes—
114#13960C2312yes—
115C2×#1441D411yescharacteristic
116C2×#1441D411yescharacteristic
117C2×#1439C2311yescharacteristic
118C2×#1439C2311yescharacteristic
119C4×#386D412yes—
120C4×#386D412yes—
121C4×#386C2312yes—
122C4×#386C2312yes—
123C2×C22⋊#95C2311yescharacteristic
124C2×C22⋊#95C2311yescharacteristic
125C2×#1344C2318yes—
126C2×#1343C2318yes—
127#11390C2314yes—
128#11390C2314yes—
129#10995C2314yes—
130#10995C2314yes—
131#8405C2314yes—
132#8405C2314yes—
133#8403C2314yes—
134#8403C2314yes—
135#7250C2318yes—
136#7250C2318yes—
137#7187C2318yes—
138#7187C2318yes—
139C24×D4C2×D411yescharacteristic
140C24×D4C2×D411yescharacteristic
141C25×C4C2×D411yescharacteristic
142C25×C4C2×D411yescharacteristic
143C25×C4C2×D411yescharacteristic
144C25×C4C2×D411yescharacteristic
145#3501C22×C414yescomplemented
146#3496C22×C414yescomplemented
147#3490C22×C418yes—
148#3459C22×C418yescomplemented
149C2×#543C2412yes—
150C2×#534C2412yes—
151C22×C4×D4C22⋊C418yescomplemented
152C22×C4×D4C22×C414yes—
153C22×C4×D4C2×D412yes—
154C22×C4×D4C2×D412yes—
155C22×C4×D4C2×D414yes—
156C22×C4×D4C2×D411yescharacteristic
157C22×C4×D4C2×D411yescharacteristic
158C22×C4×D4C2×D411yescharacteristic
159C22×C4×D4C2×D411yescharacteristic
160C22×C4×D4C2×D411yescharacteristic
161C22×C4×D4C2×D411yescharacteristic
162C22×C4×D4#4114yes—
163C22×C4×D4#4114yes—
164C22×C4×D4C2411yescharacteristic
165C22×C4×D4C2411yescharacteristic
166C23×C4⋊C4C2×D411yescharacteristic
167C23×C4⋊C4C2×D411yescharacteristic
168C23×C4⋊C4C2×D411yescharacteristic
169C23×C4⋊C4C2×D411yescharacteristic
170C23×C22⋊C4C2×D411yescharacteristic
171C23×C22⋊C4C2×D411yescharacteristic
172C23×C22⋊C4C2×D411yescharacteristic
173C23×C22⋊C4C2411yescharacteristic
174C23×C42C2×D411yescharacteristic
175C23×C42C2×D412yes—
176C23×C42C2×D412yes—
177#2535C22⋊C4116yescomplemented
178#2535C22⋊C4116yescomplemented
179#2535C22×C418yes—
180#2535C22×C418yes—
181#2535C2×D418yes—
182#2535C2×D418yes—
183C2×#397C22⋊C418yescomplemented
184C2×#397C22×C414yes—
185C2×#397C2×D414yes—
186C2×#397C2×D412yes—
187C2×#397C2×D412yes—
188C2×#397C2×D412yes—
189C2×#397C2412yes—
190C2×#392C22⋊C418yescomplemented
191C2×#392C22×C414yes—
192C2×#392C2×D414yes—
193C2×#392C2×D412yes—
194C2×#392C2×D412yes—
195C2×#392C2×D412yes—
196C2×#392C2412yes—
197C2×#386C2×D412yes—
198C2×#386C2×D412yes—
199C2×#386C2×D412yes—
200C2×#386C2×D412yes—
201C2×#382C22⋊C418yescomplemented
202C2×#382C2×D412yes—
203C2×#382C2×D412yes—
204C2×#380C2×D412yes—
205C2×#380C2×D412yes—
206C2×C4×C22⋊C4C2×D412yes—
207C2×C4×C22⋊C4C2×D412yes—
208C22×#95C2×D411yescharacteristic
209C22×#95C2×D411yescharacteristic
210C22×#95C2×D411yescharacteristic
211C22×#95C2×D411yescharacteristic
212C22×#95C2×D412yes—
213C22×#95C2×D412yes—
214C2×#342C2414yes—
215#1608C22×C4116yescomplemented
216#1607C22×C4116yescomplemented
217#1596C22×C4116yescomplemented
218#1595C22×C4116yescomplemented
219#1466C22×C418yescomplemented
220#1462C22×C418yes—
221#1460C22×C418yescomplemented
222C2×#323C2414yes—
223C2×#323C2414yes—
224#1441C2×D412yes—
225#1441C2×D412yes—
226#1441C2×D412yes—
227#1441C2×D412yes—
228#1439C2×D412yes—
229#1439C2×D412yes—
230#1439C2412yes—
231#1439C2412yes—
232C22⋊#95C2×D412yes—
233C22⋊#95C2×D412yes—
234C22⋊#95C2412yes—
235C22⋊#95C2412yes—
236C26#12011yescharacteristic
237C26C22×D411yescharacteristic
238C23×D4C2×C22⋊C414yes—
239C23×D4C4×D414yes—
240C23×D4#12011yescharacteristic
241C23×D4#12011yescharacteristic
242C23×D4#12114yes—
243C23×D4C22×D411yescharacteristic
244C23×D4C22×D411yescharacteristic
245C23×D4C22×D412yes—
246C24×C4C4×D414yes—
247C24×C4C4×D414yes—
248C24×C4#12011yescharacteristic
249C24×C4#12011yescharacteristic
250C24×C4#12011yescharacteristic
251C24×C4#12011yescharacteristic
252C24×C4#12011yescharacteristic
253C24×C4#12011yescharacteristic
254C24×C4#12011yescharacteristic
255C24×C4#12114yes—
256C24×C4#12114yes—
257C24×C4C22×D411yescharacteristic
258C24×C4C22×D411yescharacteristic
259C24×C4C22×D411yescharacteristic
260C24×C4C22×D411yescharacteristic
261C24×C4C22×D411yescharacteristic
262C24×C4C22×D411yescharacteristic
263C24×C4C22×D412yes—
264C24×C4C22×D412yes—
265#543C23×C414yes—
266#534C23×C414yes—
267C2×C4×D4C2×C22⋊C418yes—
268C2×C4×D4C2×C22⋊C418yes—
269C2×C4×D4C2×C22⋊C414yes—
270C2×C4×D4C2×C22⋊C414yes—
271C2×C4×D4C4×D414yes—
272C2×C4×D4C4×D414yes—
273C2×C4×D4C4×D414yes—
274C2×C4×D4C4×D414yes—
275C2×C4×D4#12114yes—
276C2×C4×D4#12114yes—
277C2×C4×D4#12114yes—
278C2×C4×D4#12114yes—
279C2×C4×D4#12314yes—
280C2×C4×D4#12314yes—
281C2×C4×D4C23×C412yes—
282C2×C4×D4C23×C412yes—
283C2×C4×D4C22×D412yes—
284C2×C4×D4C22×D412yes—
285C2×C4×D4C2×#4112yes—
286C2×C4×D4C2×#4112yes—
287C2×C4×D4C2×#4112yes—
288C2×C4×D4C2×#4112yes—
289C22×C4⋊C4C2×C22⋊C414yes—
290C22×C4⋊C4C4×D414yes—
291C22×C4⋊C4#12011yescharacteristic
292C22×C4⋊C4#12011yescharacteristic
293C22×C4⋊C4#12114yes—
294C22×C4⋊C4C22×D412yes—
295C22×C4⋊C4C22×D411yescharacteristic
296C22×C4⋊C4C22×D411yescharacteristic
297C22×C4⋊C4C22×D412yes—
298C22×C4⋊C4C22×D412yes—
299C22×C22⋊C4C2×C22⋊C414yes—
300C22×C22⋊C4#12011yescharacteristic
301C22×C22⋊C4#12011yescharacteristic
302C22×C22⋊C4#12314yes—
303C22×C22⋊C4C23×C412yes—
304C22×C22⋊C4C22×D412yes—
305C22×C22⋊C4C22×D412yes—
306C22×C22⋊C4C22×D411yescharacteristic
307C22×C22⋊C4C22×D411yescharacteristic
308C22×C22⋊C4C22×D411yescharacteristic
309C22×C22⋊C4C22×D412yes—
310C22×C22⋊C4C22×D411yescharacteristic
311C22×C22⋊C4C22×D411yescharacteristic
312C22×C22⋊C4C2×#4112yes—
313C22×C22⋊C4C2×#4112yes—
314C22×C22⋊C4C2511yescharacteristic
315C22×C42C2×C22⋊C414yes—
316C22×C42#12011yescharacteristic
317C22×C42#12011yescharacteristic
318C22×C42#12314yes—
319C22×C42C22×D411yescharacteristic
320C22×C42C22×D411yescharacteristic
321#397C2×C22⋊C418yes—
322#397C2×C22⋊C418yes—
323#397C2×C22⋊C418yes—
324#397C23×C414yes—
325#397C22×D414yes—
326#392C2×C22⋊C418yes—
327#392C2×C22⋊C418yes—
328#392C2×C22⋊C418yes—
329#392C23×C414yes—
330#392C22×D414yes—
331#386AGL(1,ℤ/8)18yescomplemented
332#382C2×C22⋊C418yes—
333#382C2×C22⋊C418yes—
334C2×#95C2×C22⋊C414yes—
335C2×#95C2×C22⋊C414yes—
336C2×#95C22×D412yes—
337C2×#95C22×D412yes—
338C2×#95C22×D412yes—
339C2×#95C22×D412yes—
340#323C23×C418yes—
341C25#38614yes—
342C25C2×C4×D412yes—
343C25C2×#12011yescharacteristic
344C25C2×#12011yescharacteristic
345C25C2×#12011yescharacteristic
346C25C2×#12011yescharacteristic
347C25C2×#12112yes—
348C25C23×D411yescharacteristic
349C22×D4#38614yes—
350C22×D4#38614yes—
351C22×D4C22×C22⋊C414yes—
352C22×D4C2×C4×D412yes—
353C22×D4C2×C4×D412yes—
354C22×D4C2×#12112yes—
355C22×D4C2×#12112yes—
356C22×D4C2×AGL(1,ℤ/8)14yes—
357C22×D4#57514yes—
358C23×C4#38614yes—
359C23×C4#38614yes—
360C23×C4#38614yes—
361C23×C4#38614yes—
362C23×C4#38614yes—
363C23×C4#38614yes—
364C23×C4#38614yes—
365C23×C4B(4,𝔽2)11yescharacteristic
366C23×C4B(4,𝔽2)11yescharacteristic
367C23×C4B(4,𝔽2)11yescharacteristic
368C23×C4B(4,𝔽2)11yescharacteristic
369C23×C4B(4,𝔽2)14yes—
370C23×C4C22×C22⋊C412yes—
371C23×C4C2×C4×D412yes—
372C23×C4C2×C4×D412yes—
373C23×C4C2×C4×D412yes—
374C23×C4C2×C4×D412yes—
375C23×C4C2×C4×D412yes—
376C23×C4C2×#12011yescharacteristic
377C23×C4C2×#12011yescharacteristic
378C23×C4C2×#12011yescharacteristic
379C23×C4C2×#12011yescharacteristic
380C23×C4C2×#12011yescharacteristic
381C23×C4C2×#12011yescharacteristic
382C23×C4C2×#12011yescharacteristic
383C23×C4C2×#12011yescharacteristic
384C23×C4C2×#12112yes—
385C23×C4C2×#12112yes—
386C23×C4C2×#12112yes—
387C23×C4C2×#12112yes—
388C23×C4C2×#12112yes—
389C23×C4C2×#12312yes—
390C23×C4C2×AGL(1,ℤ/8)14yes—
391C23×C4C23×D411yescharacteristic
392C23×C4C23×D411yescharacteristic
393C2×C4⋊C4#38614yes—
394C2×C4⋊C4#38614yes—
395C2×C4⋊C4C22×C22⋊C414yes—
396C2×C4⋊C4C2×C4×D412yes—
397C2×C4⋊C4C2×C4×D412yes—
398C2×C4⋊C4C2×#12112yes—
399C2×C4⋊C4C2×#12112yes—
400C2×C22⋊C4#38614yes—
401C2×C22⋊C4#38614yes—
402C2×C22⋊C4C22×C22⋊C412yes—
403C2×C22⋊C4C22×C22⋊C412yes—
404C2×C22⋊C4C22×C22⋊C412yes—
405C2×C22⋊C4C2×C4×D412yes—
406C2×C22⋊C4C2×C4×D412yes—
407C2×C22⋊C4C2×#12112yes—
408C2×C22⋊C4C2×#12112yes—
409C2×C22⋊C4C2×#12312yes—
410C2×C22⋊C4C2×#12312yes—
411C2×C22⋊C4C2×#12312yes—
412C2×C22⋊C4C2×AGL(1,ℤ/8)14yes—
413C2×C22⋊C4C24×C411yescharacteristic
414C2×C22⋊C4C23×D411yescharacteristic
415C2×C22⋊C4C22×#4111yescharacteristic
416C2×C22⋊C4C22×#4111yescharacteristic
417C2×C42#38614yes—
418C2×C42#38614yes—
419C2×C42C22×C22⋊C412yes—
420C2×C42C22×C22⋊C412yes—
421C2×C42C2×#12312yes—
422C2×C42C2×#12312yes—
423#95C22×C22⋊C414yes—
424#95C22×C22⋊C414yes—
425C24C2×#38612yes—
426C24C2×#38612yes—
427C24C2×#38612yes—
428C24C2×#38612yes—
429C24#300312yes—
430C24#300512yes—
431C24C2×B(4,𝔽2)12yes—
432C24C2×B(4,𝔽2)11yescharacteristic
433C24C2×B(4,𝔽2)11yescharacteristic
434C24C22×C4×D411yescharacteristic
435C24C22×#12011yescharacteristic
436C24C22×#12111yescharacteristic
437C24C22×AGL(1,ℤ/8)12yes—
438C24C2×#57512yes—
439C2×D4C4×AGL(1,ℤ/8)18yes—
440C2×D4#303618yes—
441C22×C4#188914yes—
442C22×C4#188914yes—
443C22×C4#188914yes—
444C22×C4#188914yes—
445C22×C4C2×#38612yes—
446C22×C4C2×#38612yes—
447C22×C4C2×#38612yes—
448C22×C4C2×#38612yes—
449C22×C4C2×#38612yes—
450C22×C4C2×#38612yes—
451C22×C4C2×#38612yes—
452C22×C4C2×#38612yes—
453C22×C4#300312yes—
454C22×C4#300512yes—
455C22×C4C2×B(4,𝔽2)11yescharacteristic
456C22×C4C2×B(4,𝔽2)11yescharacteristic
457C22×C4C2×B(4,𝔽2)11yescharacteristic
458C22×C4C2×B(4,𝔽2)11yescharacteristic
459C22×C4C2×B(4,𝔽2)12yes—
460C22×C4C2×B(4,𝔽2)12yes—
461C22×C4C23×C22⋊C411yescharacteristic
462C22×C4C22×C4×D411yescharacteristic
463C22×C4C22×#12111yescharacteristic
464C22×C4C22×#12311yescharacteristic
465C23#2092514yes—
466C23C2×#188912yes—
467C23C2×#188912yes—
468C23#3354612yes—
469C23#3354612yes—
470C23#3355112yes—
471C23#3355212yes—
472C23C22×#38611yescharacteristic
473C23C2×C4×AGL(1,ℤ/8)12yes—
474C23C2×#300311yescharacteristic
475C23C2×#300511yescharacteristic
476C23#6038711yescharacteristic
477C23#6038711yescharacteristic
478C23C22×B(4,𝔽2)11yescharacteristic
479C23C22×B(4,𝔽2)11yescharacteristic
480C23C2×#303612yes—
481C2×C4#2092514yes—
482C2×C4C2×#188912yes—
483C2×C4C2×#188912yes—
484C2×C4C2×#188912yes—
485C2×C4C2×#188912yes—
486C22#49457114yes—
487C22#49460014yes—
488C22#634659611yescharacteristic
489C22#634659611yescharacteristic
490C22#634659811yescharacteristic
491C22#634659811yescharacteristic
492C22C2×#2092511yescharacteristic
493C22#637143911yescharacteristic
494C22C22×#188911yescharacteristic
495C22C2×#3354611yescharacteristic
496C22C2×#3354611yescharacteristic
497C22C2×#3355111yescharacteristic
498C22C2×#3355211yescharacteristic
499C2#7223104611yescharacteristic
500C2C2×#49460011yescharacteristic
501C2#7224030711yescharacteristic
502C2#7227815511yescharacteristic
503C2#7227820711yescharacteristic
504C2C2×#49457111yescharacteristic
505C2#7227834511yescharacteristic
subgroupG/H
center, socleC23C22×B(4,𝔽2)
commutatorC2×C22⋊C4C24×C4
Frattini, cosocleC22×C22⋊C4C25
prime radicalC2×#6346631C2
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ C23×C4×D4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #72278511C2semidirect
2. #72278500C2semidirect
3. C2×#534011C2semidirect
4. #72278479C2semidirect
5. #72278468C2semidirect
6. #72278461C2semidirect
7. #72278451C2semidirect
8. #72278441C2semidirect
9. #72278432C2semidirect
10. #72278420C2semidirect
11. #72240399C2semidirect
12. C2×#482360C2semidirect
13. #72240342C2semidirect
18. #6346631C4semidirect
27. #534011C22semidirect
28. C2×#14577C22semidirect
29. C2×#14572C22semidirect
30. C2×#14248C22semidirect
31. C2×#14171C22semidirect
35. C2×#11797C22semidirect
36. C2×#11796C22semidirect
37. C2×#11774C22semidirect
40. #362751C22semidirect
41. #361664C22semidirect
42. #358863C22semidirect
45. #185993C22semidirect
46. #185948C22semidirect
47. #185945C22semidirect
48. #137364C22semidirect
49. #137363C22semidirect
50. #137356C22semidirect
51. #137355C22semidirect
52. #137316C22semidirect
53. #137315C22semidirect
54. #137308C22semidirect
55. #137307C22semidirect
56. #127617C22semidirect
57. #127590C22semidirect
58. #127561C22semidirect
59. #127561C22semidirect
60. #126987C22semidirect
61. #126971C22semidirect
62. #126971C22semidirect
63. #126947C22semidirect
64. #62773C2×C4semidirect
91. #18212C2×C4semidirect
92. #18210C2×C4semidirect
93. #18169C2×C4semidirect
94. #18162C2×C4semidirect
95. #18158C2×C4semidirect
100. #17364C2×C4semidirect
145. #3501C22×C4semidirect
146. #3496C22×C4semidirect
148. #3459C22×C4semidirect
151. C22×C4×D4C22⋊C4semidirect
177. #2535C22⋊C4semidirect
178. #2535C22⋊C4semidirect
183. C2×#397C22⋊C4semidirect
190. C2×#392C22⋊C4semidirect
201. C2×#382C22⋊C4semidirect
215. #1608C22×C4semidirect
216. #1607C22×C4semidirect
217. #1596C22×C4semidirect
218. #1595C22×C4semidirect
219. #1466C22×C4semidirect
221. #1460C22×C4semidirect
331. #386AGL(1,ℤ/8)semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#722785112
#722785002
C2×#5340112
#722784792
#722784682
#722784612
#722784512
#722784412
#722784322
#722784202
#722403992
C2×#4823602
#722403422
C2×#63466312