Finite Group Databasebeta

← #10828645

Subgroups — #10828645

Order 768.

1030 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3492C211yesmaximal, characteristic
2#82915C212yesmaximal, complemented
3#82157C211yesmaximal, characteristic, complemented
4#71901C211yesmaximal, characteristic, complemented
5#71900C211yesmaximal, characteristic, complemented
6#71877C211yesmaximal, characteristic, complemented
7#71876C211yesmaximal, characteristic, complemented
8#71712C212yesmaximal, complemented
9#71665C212yesmaximal, complemented
10#71664C212yesmaximal, complemented
11#71562C211yesmaximal, characteristic, complemented
12#24079—31nomaximal, Sylow
13C3×#554—22no—
14C3×#553—22no—
15C3×#552—22no—
16C3×#547—22no—
17C3×#544C2211yescharacteristic
18C3×#539—22no—
19C3×#537C2211yescharacteristic
20C3×#525—22no—
21C3×#525C2212yes—
22C3×#520C2212yes—
23C22×C3×C4⋊C4C2211yescharacteristic
24C2×#926C2211yescharacteristic
25C2×#926C2211yescharacteristic
26C2×#837C2212yes—
27C2×#837C2212yes—
28#5232C2212yes—
29#5232C2212yes—
30#5210—22no—
31#5207C2212yes—
32#5045—22no—
33#5045—22no—
34#5035—22no—
35#5035—22no—
36#4995C2212yes—
37#4995C2212yes—
38#4977C2211yescharacteristic
39#4977C2211yescharacteristic
40#4975C2211yescharacteristic
41#4975C2211yescharacteristic
42#4972—22no—
43#4532C2212yes—
44#4532C2212yes—
45#4530C2211yescharacteristic
46#4530C2211yescharacteristic
47#4530C2212yes—
48#3492D311yescharacteristic
49#3093—32no—
50#2317—31no—
51#2061—31no—
52#2060—31no—
53#2040—31no—
54#2039—31no—
55#1897—32no—
56#1860—32no—
57#1859—32no—
58#1821—31no—
59C2×C3×#41—22no—
60C2×C3×#41C2312yes—
61C23×C3×C4D412yes—
62C23×C3×C4C2311yescharacteristic
63C3×C4⋊Q8—22no—
64C3×#126—24no—
65C3×#125—24no—
66C3×#125—22no—
67C3×#123—24no—
68C3×#123—24no—
69C3×#123—22no—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—24no—
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8—22no—
74C3×C22⋊Q8D412yes—
75C3×C22⋊Q8D412yes—
76C3×#121—22no—
77C3×#121—22no—
78C3×#121—22no—
79C3×#121D412yes—
80C3×#121D412yes—
81C3×#120—22no—
82C3×C4×Q8—22no—
83C3×C4×D4—24no—
84C3×C4×D4—22no—
85C3×C4×D4—24no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×#117—22no—
89C3×#117—22no—
90C3×#117C2312yes—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4D412yes—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×#217—21no—
98C2×#217—21no—
99#926—22no—
100#926—22no—
101#926—22no—
102#926—22no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—22no—
107C2×Dic3⋊C4—22no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C2×#208C2312yes—
112C2×#208C2312yes—
113#837—24no—
114#837—24no—
115#835—22no—
116#835—22no—
117#832—22no—
118#832—22no—
119C3⋊#95—22no—
120C3⋊#95—22no—
121#830—22no—
122#830—22no—
123#808—22no—
124#808—22no—
125#807—22no—
126#807—22no—
127#554—22no—
128#553—22no—
129#552—22no—
130#547—22no—
131#544C2×D311yescharacteristic
132#539—22no—
133#537C2×D311yescharacteristic
134#525—22no—
135#525C2×D312yes—
136#520C2×D312yes—
137C2×#117—31no—
138C2×#117—31no—
139C22×C4⋊C4C2×D311yescharacteristic
140#485—62no—
141#485—62no—
142#482—62no—
143#482—62no—
144#428—32no—
145#428—32no—
146C2×#104—32no—
147C2×#104—32no—
148#417—32no—
149#417—32no—
150#407—32no—
151#389—62no—
152#385—62no—
153#384—31no—
154#384—31no—
155#382—31no—
156#382—31no—
157#343—31no—
158#343—31no—
159#343—32no—
160#339—32no—
161#339—32no—
162C24×C3C2×D411yescharacteristic
163C3×#41—22no—
164C3×#41—22no—
165C3×#41—22no—
166C3×#41—22no—
167C3×#41—84no—
168C2×C3×Q8—22no—
169C2×C3×Q8C2×D412yes—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4C2×D412yes—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—24no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—82no—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2411yescharacteristic
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4C2×D412yes—
203C3×C4⋊C4C2×D412yes—
204C3×C22⋊C4—24no—
205C3×C22⋊C4—24no—
206C3×C22⋊C4—22no—
207C3×C22⋊C4—22no—
208C3×C22⋊C4—22no—
209C3×C22⋊C4—22no—
210C3×C22⋊C4—22no—
211C3×C42—24no—
212C3×C42—22no—
213#217—24no—
214#217—41no—
215#217—41no—
216C4⋊Dic3—24no—
217C4⋊Dic3—22no—
218C4⋊Dic3—22no—
219Dic3⋊C4—82no—
220Dic3⋊C4—82no—
221C4×Dic3—21no—
222C4×Dic3—21no—
223C4×Dic3—42no—
224C4×Dic3#4112yescomplemented
225C4×Dic3#4112yescomplemented
226#208—22no—
227#208—22no—
228#208—22no—
229#208—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×#41—22no—
233C2×#41C22×D312yes—
234C23×C4#6712yes—
235C23×C4C22×D311yescharacteristic
236C2×M16—32no—
237C2×M16—32no—
238C4⋊Q8—22no—
239#126—24no—
240#125—24no—
241#125—22no—
242#123—24no—
243#123—24no—
244#123—22no—
245C22⋊Q8—22no—
246C22⋊Q8—24no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249C22⋊Q8#6712yes—
250C22⋊Q8#6712yes—
251#121—22no—
252#121—22no—
253#121—22no—
254#121#6712yes—
255#121#6712yes—
256#120—22no—
257C4×Q8—22no—
258C4×D4—24no—
259C4×D4—22no—
260C4×D4—24no—
261C4×D4—22no—
262C4×D4—22no—
263#117—22no—
264#117—22no—
265#117—62no—
266#117—62no—
267#117—62no—
268#117—62no—
269#117C22×D312yes—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—62no—
277C2×C4⋊C4—62no—
278C2×C4⋊C4#6712yes—
279C2×C4⋊C4C22×D311yescharacteristic
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C22⋊C4—61no—
282C2×C22⋊C4—61no—
283C2×C42—31no—
284C2×C42—31no—
285C2×C42—61no—
286#107—62no—
287#107—62no—
288#106—62no—
289#106—62no—
290#104—64no—
291#104—64no—
292#103—62no—
293#103—62no—
294#102—62no—
295#102—62no—
296C22⋊C8—62no—
297C22⋊C8—62no—
298#95—62no—
299#95—62no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—22no—
303C23×C3—41no—
304C23×C3—41no—
305C23×C3#12011yescharacteristic
306C23×C3#12011yescharacteristic
307C23×C3C22×D411yescharacteristic
308C3×Q8—22no—
309C3×Q8—22no—
310C3×Q8—82no—
311C3×D4—22no—
312C3×D4—22no—
313C3×D4—44no—
314C3×D4—44no—
315C3×D4—42no—
316C3×D4—42no—
317C3×D4—82no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—44no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—82no—
335C2×C3×C4—84no—
336C2×C3×C4#12012yes—
337C2×C3×C4C22×D412yes—
338C2×C3×C4C2×#4111yescharacteristic
339C2×C3×C4C2×#4111yescharacteristic
340C2×C3×C42−1+412yes—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C3⋊C8—42no—
352C3⋊C8—42no—
353C24C2×#6711yescharacteristic
354#41—22no—
355#41—22no—
356#41—22no—
357#41—22no—
358#41—84no—
359C2×Q8—22no—
360C2×Q8C2×#6712yes—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4C2×#6712yes—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—61no—
374C22×C4—61no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—82no—
378C22×C4C2×#6711yescharacteristic
379C22×C4C2×#6712yes—
380C22×C4C2×#6712yes—
381C22×C4C2×#6712yes—
382C22×C4C2×#6712yes—
383C22×C4C23×D311yescharacteristic
384M16—62no—
385M16—62no—
386M16—62no—
387M16—62no—
388C2×C8—62no—
389C2×C8—62no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—64no—
400C4⋊C4—62no—
401C4⋊C4—62no—
402C4⋊C4—242no—
403C4⋊C4—242no—
404C4⋊C4C2×#6712yes—
405C4⋊C4C2×#6712yes—
406C22⋊C4—24no—
407C22⋊C4—24no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—64no—
414C22⋊C4—121no—
415C22⋊C4—121no—
416C42—24no—
417C42—22no—
418C42—32no—
419C42—32no—
420C42—61no—
421C42—61no—
422C42—122no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3C2×#12011yescharacteristic
437C22×C3#52511yescharacteristic
438C22×C3#53611yescharacteristic
439C3×C4—22no—
440C3×C4—42no—
441C3×C4—42no—
442C3×C4—42no—
443C3×C4—42no—
444C3×C4—42no—
445C3×C4#54712yes—
446C3×C4#54712yes—
447Dic3—42no—
448Dic3—42no—
449Dic3—81no—
450Dic3—81no—
451C23—21no—
452C23—21no—
453C23—22no—
454C23—41no—
455C23—41no—
456C23#95311yescharacteristic
457C23#95311yescharacteristic
458C23C22×#6711yescharacteristic
459Q8—22no—
460Q8—22no—
461Q8—82no—
462D4—22no—
463D4—22no—
464D4—44no—
465D4—44no—
466D4—42no—
467D4—42no—
468D4—82no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—82no—
488C2×C4—84no—
489C2×C4—121no—
490C2×C4—121no—
491C2×C4—121no—
492C2×C4—121no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—121no—
496C2×C4—121no—
497C2×C4#95312yes—
498C2×C4D3×#4111yescharacteristic
499C2×C4D3×#4111yescharacteristic
500C2×C4#101012yes—
501C2×C4C22×#6712yes—
502C8—122no—
503C8—122no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—41no—
507C2×C3—41no—
508C2×C3—82no—
509C2×C3#240611yescharacteristic
510C2×C3#301911yescharacteristic
511C2×C3#302011yescharacteristic
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22#582511yescharacteristic
526C22#582811yescharacteristic
527C22C2×#95311yescharacteristic
528C4—22no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—42no—
533C4—42no—
534C4—122no—
535C4—122no—
536C4—241no—
537C4—241no—
538C4#582412yes—
539C4#582412yes—
540C3#2407911yescharacteristic, complemented, Sylow
541C2—21no—
542C2—21no—
543C2—41no—
544C2—41no—
545C2—82no—
546C2#8409011yescharacteristic
547C2#8410411yescharacteristic
548C2#8411611yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3492C2
socleC22×C3C2×#120
hypercenter#3492D3
hypocenterC3#24079
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3492

Complements

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

NH ≅ G/Ntype
2. #82915C2semidirect
3. #82157C2semidirect
4. #71901C2semidirect
5. #71900C2semidirect
6. #71877C2semidirect
7. #71876C2semidirect
8. #71712C2semidirect
9. #71665C2semidirect
10. #71664C2semidirect
11. #71562C2semidirect
224. C4×Dic3#41semidirect
225. C4×Dic3#41semidirect
540. C3#24079semidirect

Maximal subgroups

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

subgroupindex
C3×#34922
#829152
#821572
#719012
#719002
#718772
#718762
#717122
#716652
#716642
#715622
#240793