Finite Group Databasebeta

← #118120

Subgroups — #118120

Order 512.

710 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1805C211yesmaximal, characteristic
2C2×#1801C211yesmaximal, characteristic
3C2×#1570C211yesmaximal, characteristic
4C2×#1479C211yesmaximal, characteristic
5C2×#1301C211yesmaximal, characteristic
6C2×#1286C211yesmaximal, characteristic
7C2×#1286C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#103—21no—
15C22×#101—21no—
16C2×#382C2211yescharacteristic
17#1805—22no—
18#1805—22no—
19#1801—22no—
20#1801—22no—
21C2×#343—21no—
22C2×#340—21no—
23#1570—22no—
24#1570—22no—
25C2×#334—22no—
26#1479—22no—
27#1479C414yescomplemented
28C2×Q8⋊C8—22no—
29#1301—22no—
30#1286—22no—
31#1286—22no—
32C23×Q8D411yescharacteristic
33C22×M16—21no—
34C22×M16—21no—
35C23×C8—21no—
36C2×C4×Q8—21no—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8C2×C412yes—
47C2×C4×Q8Q812yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D411yescharacteristic
51C22×C42—21no—
52C22×C42C2311yescharacteristic
53#431—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431C2×C412yes—
59#431C2×C412yes—
60#431D412yes—
61#423—21no—
62#423—21no—
63#423—21no—
64#423—21no—
65#423—21no—
66#423—21no—
67#423D412yes—
68#423D412yes—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8D412yes—
81C2×#103—24no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#101—24no—
85C2×#101—42no—
86C2×#101—42no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8D412yes—
91#382—21no—
92#382—21no—
93#382—21no—
94#382D412yes—
95C2×#95—21no—
96C2×#95—21no—
97#343—24no—
98#340—24no—
99#334—28no—
100Q8⋊C8—28no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—21no—
112C22×Q8C4⋊C412yes—
113C22×Q8SD1612yes—
114C22×Q8SD1612yes—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—41no—
118C23×C4C2×D411yescharacteristic
119C23×C4#4111yescharacteristic
120C23×C4#4111yescharacteristic
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—41no—
126C2×M16—41no—
127C2×M16—41no—
128C2×M16—41no—
129C2×M16—41no—
130C2×M16—41no—
131C2×M16—41no—
132C2×M16—41no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—41no—
136C22×C8—41no—
137C22×C8—41no—
138C22×C8—41no—
139C22×C8—41no—
140C22×C8—41no—
141C22×C8—41no—
142C22×C8—41no—
143C22×C8—41no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—41no—
153C4×Q8—41no—
154C4×Q8—41no—
155C4×Q8—41no—
156C4×Q8—41no—
157C4×Q8—41no—
158C4×Q8—41no—
159C4×Q8—41no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4C4⋊C412yes—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—41no—
181C2×C42—41no—
182C2×C42—41no—
183C2×C42—41no—
184C2×C42C22×C411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×Q811yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—41no—
206C4⋊C8—41no—
207C4⋊C8—41no—
208C4⋊C8—41no—
209C4⋊C8—41no—
210C4⋊C8—41no—
211C4⋊C8—41no—
212#103—48no—
213#101—48no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—41no—
218C4×C8—22no—
219C4×C8—22no—
220C4×C8—41no—
221C4×C8C22⋊C414yes—
222#95—22no—
223#95—22no—
224#95—41no—
225#95—41no—
226C24#12311yescharacteristic
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8#10614yes—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—81no—
272C22×C4—81no—
273C22×C4C2×C4⋊C411yescharacteristic
274C22×C4C4×D411yescharacteristic
275C22×C4C4×D411yescharacteristic
276C22×C4#12011yescharacteristic
277C22×C4#12011yescharacteristic
278C22×C4#12111yescharacteristic
279C22×C4#12111yescharacteristic
280C22×C4#12111yescharacteristic
281C22×C4#12111yescharacteristic
282C22×C4C22⋊Q811yescharacteristic
283C22×C4C22⋊Q811yescharacteristic
284C22×C4#12311yescharacteristic
285C22×C4#12311yescharacteristic
286C22×C4#12311yescharacteristic
287C22×C4#12411yescharacteristic
288C22×C4#12411yescharacteristic
289C22×C4#12611yescharacteristic
290C22×C4#12611yescharacteristic
291C22×C4C2×SD1611yescharacteristic
292C22×C4C2×SD1611yescharacteristic
293C22×C4#13511yescharacteristic
294C22×C4#13511yescharacteristic
295C22×C4AGL(1,ℤ/8)11yescharacteristic
296C22×C4AGL(1,ℤ/8)11yescharacteristic
297C22×C4#13711yescharacteristic
298C22×C4#13711yescharacteristic
299M16—42no—
300M16—42no—
301M16—42no—
302M16—42no—
303M16—81no—
304M16—81no—
305M16—81no—
306M16—81no—
307C2×C8—24no—
308C2×C8—44no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—82no—
316C2×C8—81no—
317C2×C8—81no—
318C2×C8—81no—
319C2×C8—81no—
320C2×C8—81no—
321C2×C8—81no—
322C2×C8—81no—
323C2×C8—81no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—81no—
335C4⋊C4—81no—
336C42—42no—
337C42—42no—
338C42—42no—
339C42—42no—
340C42C2×C22⋊C411yescharacteristic
341C42#11711yescharacteristic
342C42C4×D411yescharacteristic
343C42C4×D411yescharacteristic
344C42#12011yescharacteristic
345C42#12111yescharacteristic
346C42#12111yescharacteristic
347C42#12111yescharacteristic
348C42#12111yescharacteristic
349C42C22⋊Q811yescharacteristic
350C42C22⋊Q811yescharacteristic
351C42C22⋊Q811yescharacteristic
352C42C22⋊Q811yescharacteristic
353C42#12311yescharacteristic
354C42#12411yescharacteristic
355C42#12711yescharacteristic
356C23—22no—
357C23—22no—
358C23#38511yescharacteristic
359C23#39411yescharacteristic
360C23#39711yescharacteristic
361C23#48111yescharacteristic
362C23#48211yescharacteristic
363C23#48311yescharacteristic
364C23#48511yescharacteristic
365Q8—24no—
366Q8—24no—
367Q8—42no—
368Q8—42no—
369Q8—42no—
370Q8—42no—
371Q8—42no—
372Q8—42no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—162no—
400C2×C4—162no—
401C2×C4#38011yescharacteristic
402C2×C4#38511yescharacteristic
403C2×C4#38611yescharacteristic
404C2×C4#38811yescharacteristic
405C2×C4#38811yescharacteristic
406C2×C4#39011yescharacteristic
407C2×C4#39211yescharacteristic
408C2×C4#39311yescharacteristic
409C2×C4#39411yescharacteristic
410C2×C4#39611yescharacteristic
411C2×C4#39611yescharacteristic
412C2×C4#39911yescharacteristic
413C2×C4C2×#10611yescharacteristic
414C2×C4#42711yescharacteristic
415C2×C4#42811yescharacteristic
416C2×C4#42811yescharacteristic
417C2×C4#44811yescharacteristic
418C2×C4#44811yescharacteristic
419C2×C4#44911yescharacteristic
420C2×C4#44911yescharacteristic
421C2×C4#45011yescharacteristic
422C2×C4#45011yescharacteristic
423C2×C4#45211yescharacteristic
424C2×C4#45211yescharacteristic
425C2×C4#46511yescharacteristic
426C2×C4#46511yescharacteristic
427C2×C4#46511yescharacteristic
428C2×C4#46511yescharacteristic
429C2×C4#46811yescharacteristic
430C2×C4#46811yescharacteristic
431C2×C4#46911yescharacteristic
432C2×C4#47011yescharacteristic
433C2×C4#48111yescharacteristic
434C2×C4#48211yescharacteristic
435C2×C4#48311yescharacteristic
436C2×C4#48511yescharacteristic
437C8—44no—
438C8—82no—
439C8—82no—
440C8—82no—
441C8—82no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22#143911yescharacteristic
451C22#185311yescharacteristic
452C22#185411yescharacteristic
453C22#203411yescharacteristic
454C22#203511yescharacteristic
455C22#207711yescharacteristic
456C22#207711yescharacteristic
457C22#208012yes—
458C22#208212yes—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—44no—
468C4—82no—
469C4—162no—
470C4#180711yescharacteristic
471C4#180911yescharacteristic
472C4#185311yescharacteristic
473C4#185411yescharacteristic
474C4#200011yescharacteristic
475C4#200011yescharacteristic
476C4#203411yescharacteristic
477C4#203511yescharacteristic
478C2—21no—
479C2—21no—
480C2—21no—
481C2—21no—
482C2#1064312yes—
483C2#1064412yes—
484C2#1108611yescharacteristic
485C2#1109511yescharacteristic
486C2#1113411yescharacteristic
subgroupG/H
center, socleC23#397
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
27. #1479C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18052
C2×#18012
C2×#15702
C2×#14792
C2×#13012
C2×#12862