Finite Group Databasebeta

← #72252823

Subgroups — #72252823

Order 5184.

2596 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 2596 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×C22⋊A4C311yesmaximal, characteristic
2#13180116C311yesmaximal, characteristic, complemented
3C22×#88907C312yesmaximal, complemented
4#13180073C316yesmaximal, complemented
5#13180070C313yesmaximal, complemented
6#13049240—41nomaximal
7A4×#1094—42nomaximal
8C2×C32×C22⋊A4—31no—
9C2×#88907—32no—
10C26×C32C3211yescharacteristic
11C22×C3×C22⋊A4—33no—
12C22×C3×C22⋊A4C3212yes—
13C22×C3×C22⋊A4C3211yescharacteristic
14C22⋊(C22⋊(C22⋊C9))C3213yescomplemented
15C22×C22⋊(C22⋊C9)C3216yescomplemented
16C32×C22⋊A4A411yescharacteristic
17C22×C32×A4—42no—
18C22×C32×A4—121no—
19#88907—42no—
20#88907—41no—
21#88907—122no—
22#88907—121no—
23#88907—123no—
24#88907A412yescomplemented
25C22×#1094—42no—
26C22×#1094—42no—
27C22×#1094—122no—
28#88881—46no—
29#88880—412no—
30(C22⋊A4)⋊C9—43no—
31A4×C22⋊C9—46no—
32C3×#1094—161no—
33A4×3−1+2—162no—
34C25×C32—31no—
35C25×C32—32no—
36C25×C32—91no—
37C25×C32—92no—
38C25×C32—93no—
39C2×C3×C22⋊A4—39no—
40C2×C3×C22⋊A4—32no—
41C2×C3×C22⋊A4—31no—
42C2×C22⋊(C22⋊C9)—36no—
43C2×C32×A4—122no—
44C2×C32×A4—123no—
45C2×#1094—122no—
46C2×#1094—122no—
47C2×#1094—126no—
48C26×C3—31no—
49C26×C3C3311yescharacteristic
50C22×C22⋊A4—32no—
51C22×C22⋊A43−1+213yescomplemented
52C24×C32—31no—
53C24×C32—32no—
54C24×C32—33no—
55C24×C32—31no—
56C24×C32—32no—
57C24×C32—92no—
58C24×C32—91no—
59C24×C32—93no—
60C24×C32—96no—
61C24×C32—93no—
62C24×C32—96no—
63C24×C32—92no—
64C24×C32—92no—
65C24×C32—92no—
66C24×C32—92no—
67C24×C32—91no—
68C24×C32—93no—
69C24×C32—912no—
70C24×C32—96no—
71C24×C32—93no—
72C24×C32—96no—
73C24×C32—93no—
74C24×C32—96no—
75C24×C32C3×A412yes—
76C24×C32C3×A411yescharacteristic
77C3×C22⋊A4—33no—
78C3×C22⋊A4C3×A412yes—
79C3×C22⋊A4C3×A411yescharacteristic
80C22×C3×A4—42no—
81C22×C3×A4—44no—
82C22×C3×A4—121no—
83C22×C3×A4—122no—
84C22×C3×A4—126no—
85C22×C3×A4—129no—
86C22⋊(C22⋊C9)—46no—
87C22⋊(C22⋊C9)—43no—
88C22⋊(C22⋊C9)—126no—
89C22⋊(C22⋊C9)—123no—
90C22⋊(C22⋊C9)—129no—
91C22⋊(C22⋊C9)C3×A416yescomplemented
92C22×C22⋊C9—46no—
93C22×C22⋊C9—46no—
94C22×C22⋊C9—126no—
95C22×C33—161no—
96C32×A4—42no—
97C32×A4—121no—
98C22×3−1+2—162no—
99#1094—42no—
100#1094—42no—
101#1094—122no—
102#1094—166no—
103#1094—162no—
104#1094—161no—
105#1094—482no—
106#1094—481no—
107#1094—483no—
108C3×C22⋊C9—163no—
109C9⋊A4—1612no—
110C9×A4—166no—
111C25×C3—33no—
112C25×C3—31no—
113C25×C3—32no—
114C25×C3—92no—
115C25×C3—91no—
116C25×C3—92no—
117C25×C3—93no—
118C25×C3—93no—
119C25×C3—96no—
120C25×C3—99no—
121C2×C22⋊A4—36no—
122C2×C22⋊A4—33no—
123C3×3−1+2—641noSylow
124C23×C32—33no—
125C23×C32—32no—
126C23×C32—36no—
127C23×C32—32no—
128C23×C32—32no—
129C23×C32—91no—
130C23×C32—92no—
131C23×C32—91no—
132C23×C32—92no—
133C23×C32—93no—
134C23×C32—96no—
135C23×C32—96no—
136C23×C32—93no—
137C23×C32—96no—
138C23×C32—93no—
139C23×C32—96no—
140C23×C32—93no—
141C23×C32—96no—
142C23×C32—93no—
143C23×C32—96no—
144C23×C32—96no—
145C23×C32—96no—
146C23×C32—96no—
147C23×C32—93no—
148C23×C32—93no—
149C23×C32—96no—
150C23×C32—912no—
151C23×C32—912no—
152C23×C32—912no—
153C23×C32—96no—
154C23×C32—96no—
155C23×C32—912no—
156C23×C32—91no—
157C23×C32—92no—
158C2×C3×A4—122no—
159C2×C3×A4—124no—
160C2×C3×A4—123no—
161C2×C3×A4—126no—
162C2×C3×A4—1218no—
163C2×C3×A4—1227no—
164C2×C22⋊C9—126no—
165C2×C22⋊C9—126no—
166C2×C22⋊C9—1218no—
167C26C3×3−1+211yescharacteristic, complemented, Sylow
168C2×C33—481no—
169C2×3−1+2—482no—
170C24×C3—33no—
171C24×C3—31no—
172C24×C3—32no—
173C24×C3—33no—
174C24×C3—32no—
175C24×C3—31no—
176C24×C3—32no—
177C24×C3—31no—
178C24×C3—92no—
179C24×C3—91no—
180C24×C3—92no—
181C24×C3—93no—
182C24×C3—91no—
183C24×C3—92no—
184C24×C3—92no—
185C24×C3—92no—
186C24×C3—92no—
187C24×C3—96no—
188C24×C3—96no—
189C24×C3—93no—
190C24×C3—99no—
191C24×C3—96no—
192C24×C3—96no—
193C24×C3—93no—
194C24×C3—99no—
195C24×C3—96no—
196C24×C3—93no—
197C24×C3—96no—
198C24×C3—99no—
199C24×C3—93no—
200C24×C3—96no—
201C24×C3—918no—
202C24×C3—918no—
203C24×C3—918no—
204C24×C3—91no—
205C24×C3—92no—
206C24×C3—93no—
207C24×C3—912no—
208C24×C3—96no—
209C24×C3—93no—
210C24×C3—96no—
211C24×C3—96no—
212C24×C3—99no—
213C24×C3—936no—
214C24×C3—99no—
215C24×C3—93no—
216C24×C3—918no—
217C24×C3—918no—
218C24×C3C32×A412yes—
219C24×C3C32×A411yescharacteristic
220C22⋊A4—32no—
221C22⋊A4#109413yescomplemented
222C22×A4—46no—
223C22×A4—123no—
224C22×A4—126no—
225C22×A4—124no—
226C22×C32—31no—
227C22×C32—32no—
228C22×C32—31no—
229C22×C32—32no—
230C22×C32—33no—
231C22×C32—92no—
232C22×C32—91no—
233C22×C32—92no—
234C22×C32—92no—
235C22×C32—92no—
236C22×C32—93no—
237C22×C32—96no—
238C22×C32—93no—
239C22×C32—96no—
240C22×C32—93no—
241C22×C32—96no—
242C22×C32—93no—
243C22×C32—912no—
244C22×C32—96no—
245C22×C32—93no—
246C22×C32—96no—
247C22×C32—92no—
248C22×C32—91no—
249C22×C32—161no—
250C22×C32—162no—
251C22×C32—483no—
252C22×C32A4211yescharacteristic
253C22×C32A4212yes—
254C3×A4—42no—
255C3×A4—44no—
256C3×A4—121no—
257C3×A4—122no—
258C3×A4—126no—
259C3×A4—129no—
260C22×C9—166no—
261C22⋊C9—46no—
262C22⋊C9—46no—
263C22⋊C9—126no—
264C22⋊C9—166no—
265C22⋊C9—163no—
266C22⋊C9—486no—
267C22⋊C9—483no—
268C22⋊C9—489no—
269C25—32no—
270C25—31no—
271C25—91no—
272C25—92no—
273C25—93no—
274C33—161no—
2753−1+2—162no—
2763−1+2—646no—
2773−1+2—641no—
278C3×C9—643no—
279C23×C3—36no—
280C23×C3—39no—
281C23×C3—33no—
282C23×C3—36no—
283C23×C3—32no—
284C23×C3—32no—
285C23×C3—32no—
286C23×C3—92no—
287C23×C3—92no—
288C23×C3—96no—
289C23×C3—92no—
290C23×C3—91no—
291C23×C3—93no—
292C23×C3—92no—
293C23×C3—91no—
294C23×C3—96no—
295C23×C3—93no—
296C23×C3—96no—
297C23×C3—96no—
298C23×C3—93no—
299C23×C3—93no—
300C23×C3—96no—
301C23×C3—99no—
302C23×C3—93no—
303C23×C3—96no—
304C23×C3—96no—
305C23×C3—99no—
306C23×C3—93no—
307C23×C3—96no—
308C23×C3—93no—
309C23×C3—99no—
310C23×C3—99no—
311C23×C3—918no—
312C23×C3—96no—
313C23×C3—918no—
314C23×C3—918no—
315C23×C3—918no—
316C23×C3—918no—
317C23×C3—93no—
318C23×C3—96no—
319C23×C3—96no—
320C23×C3—96no—
321C23×C3—93no—
322C23×C3—96no—
323C23×C3—96no—
324C23×C3—93no—
325C23×C3—912no—
326C23×C3—912no—
327C23×C3—912no—
328C23×C3—99no—
329C23×C3—918no—
330C23×C3—918no—
331C23×C3—918no—
332C23×C3—99no—
333C23×C3—936no—
334C23×C3—96no—
335C23×C3—918no—
336C23×C3—99no—
337C23×C3—918no—
338C23×C3—936no—
339C23×C3—936no—
340C23×C3—918no—
341C23×C3—918no—
342C23×C3—912no—
343C23×C3—96no—
344C23×C3—936no—
345C23×C3—92no—
346C23×C3—91no—
347C2×A4—126no—
348C2×A4—129no—
349C2×A4—1218no—
350C2×A4—1212no—
351C2×C32—31no—
352C2×C32—32no—
353C2×C32—91no—
354C2×C32—92no—
355C2×C32—93no—
356C2×C32—481no—
357C2×C32—482no—
358C2×C32—489no—
359C2×C9—486no—
360C24—31no—
361C24—32no—
362C24—33no—
363C24—32no—
364C24—31no—
365C24—92no—
366C24—91no—
367C24—92no—
368C24—92no—
369C24—92no—
370C24—93no—
371C24—96no—
372C24—93no—
373C24—96no—
374C24—96no—
375C24—93no—
376C24—92no—
377C24—91no—
378C24—93no—
379C24—912no—
380C24—96no—
381C24—93no—
382C24—96no—
383C24A4×3−1+212yescomplemented
384C24C3×#109411yescharacteristic, complemented
385C22×C3—32no—
386C22×C3—33no—
387C22×C3—31no—
388C22×C3—32no—
389C22×C3—31no—
390C22×C3—33no—
391C22×C3—32no—
392C22×C3—31no—
393C22×C3—92no—
394C22×C3—91no—
395C22×C3—92no—
396C22×C3—93no—
397C22×C3—91no—
398C22×C3—92no—
399C22×C3—92no—
400C22×C3—92no—
401C22×C3—92no—
402C22×C3—96no—
403C22×C3—99no—
404C22×C3—96no—
405C22×C3—93no—
406C22×C3—96no—
407C22×C3—99no—
408C22×C3—96no—
409C22×C3—93no—
410C22×C3—96no—
411C22×C3—99no—
412C22×C3—96no—
413C22×C3—93no—
414C22×C3—93no—
415C22×C3—96no—
416C22×C3—918no—
417C22×C3—918no—
418C22×C3—918no—
419C22×C3—99no—
420C22×C3—936no—
421C22×C3—99no—
422C22×C3—96no—
423C22×C3—918no—
424C22×C3—918no—
425C22×C3—93no—
426C22×C3—93no—
427C22×C3—96no—
428C22×C3—912no—
429C22×C3—96no—
430C22×C3—93no—
431C22×C3—92no—
432C22×C3—91no—
433C22×C3—163no—
434C22×C3—482no—
435C22×C3C3×A4212yes—
436C22×C3C3×A4211yescharacteristic
437A4—46no—
438A4—123no—
439A4—126no—
440A4—124no—
441C32—161no—
442C32—162no—
443C32—483no—
444C32#1059678211yescharacteristic
445C9—166no—
446C9—643no—
447C23—33no—
448C23—36no—
449C23—32no—
450C23—32no—
451C23—32no—
452C23—92no—
453C23—91no—
454C23—92no—
455C23—91no—
456C23—96no—
457C23—93no—
458C23—96no—
459C23—96no—
460C23—93no—
461C23—93no—
462C23—96no—
463C23—93no—
464C23—96no—
465C23—93no—
466C23—96no—
467C23—96no—
468C23—96no—
469C23—93no—
470C23—96no—
471C23—96no—
472C23—93no—
473C23—912no—
474C23—912no—
475C23—912no—
476C23—96no—
477C23—96no—
478C23—912no—
479C23—92no—
480C23—91no—
481C2×C3—33no—
482C2×C3—32no—
483C2×C3—31no—
484C2×C3—92no—
485C2×C3—93no—
486C2×C3—96no—
487C2×C3—99no—
488C2×C3—92no—
489C2×C3—91no—
490C2×C3—93no—
491C2×C3—483no—
492C2×C3—486no—
493C22—31no—
494C22—32no—
495C22—32no—
496C22—31no—
497C22—33no—
498C22—91no—
499C22—92no—
500C22—92no—
501C22—92no—
502C22—92no—
503C22—96no—
504C22—93no—
505C22—96no—
506C22—93no—
507C22—93no—
508C22—96no—
509C22—93no—
510C22—912no—
511C22—96no—
512C22—93no—
513C22—96no—
514C22—92no—
515C22—91no—
516C22A4×#109412yescomplemented
517C22#1304924011yescharacteristic, complemented
518C3—31no—
519C3—163no—
520C3—482no—
521C3C3×#1059678211yescharacteristic
522C2—32no—
523C2—31no—
524C2—91no—
525C2—92no—
526C2—93no—
subgroupG/H
center, FrattiniC3C3×#10596782
commutator, socle, cosocleC26×C3C33
FittingC26×C32C32
prime radicalC22×C32×C22⋊A4C3
hypercenterC32#10596782
hypocenterC26C3×3−1+2
LCSG ▷ C26×C3 ▷ C26
UCSC1 ◁ C3 ◁ C32

Complements

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

NH ≅ G/Ntype
2. #13180116C3semidirect
3. C22×#88907C3semidirect
4. #13180073C3semidirect
5. #13180070C3semidirect
14. C22⋊(C22⋊(C22⋊C9))C32semidirect
15. C22×C22⋊(C22⋊C9)C32semidirect
24. #88907A4semidirect
51. C22×C22⋊A43−1+2semidirect
91. C22⋊(C22⋊C9)C3×A4semidirect
167. C26C3×3−1+2semidirect
221. C22⋊A4#1094semidirect
383. C24A4×3−1+2semidirect
384. C24C3×#1094semidirect
516. C22A4×#1094semidirect
517. C22#13049240semidirect

Maximal subgroups

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

subgroupindex
C22×C32×C22⋊A43
#131801163
C22×#889073
#131800733
#131800703
#130492404
A4×#10944