Finite Group Databasebeta

← #10828634

Subgroups — #10828634

Order 768.

1038 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3493C211yesmaximal, characteristic
2#82915C212yesmaximal, complemented
3#82224C211yesmaximal, characteristic, complemented
4#71901C211yesmaximal, characteristic, complemented
5#71900C211yesmaximal, characteristic, complemented
6#71877C211yesmaximal, characteristic, complemented
7#71876C211yesmaximal, characteristic, complemented
8#71711C212yesmaximal, complemented
9#71665C212yesmaximal, complemented
10#71664C212yesmaximal, complemented
11#71589C211yesmaximal, characteristic, complemented
12#24078—31nomaximal, Sylow
13C3×#556—22no—
14C3×#555—22no—
15C3×#553—22no—
16C3×#552—22no—
17C3×#544C2211yescharacteristic
18C3×#538—22no—
19C3×#537C2211yescharacteristic
20C3×#529—22no—
21C3×#525C2212yes—
22C3×#520C2212yes—
23C2×C3×#117C2211yescharacteristic
24C2×#926C2211yescharacteristic
25C22×C4×Dic3C2211yescharacteristic
26C2×#837C2212yes—
27C2×#837C2212yes—
28#5232C2212yes—
29#5232C2212yes—
30#5207C2212yes—
31#5045—22no—
32#5045—22no—
33#5035—22no—
34#5035—22no—
35#4995C2212yes—
36#4995C2212yes—
37#4977C2211yescharacteristic
38#4977C2211yescharacteristic
39#4975C2211yescharacteristic
40#4975C2211yescharacteristic
41#4942—22no—
42C4×Dic3⋊C4—22no—
43#4532C2212yes—
44#4532C2212yes—
45#4530C2211yescharacteristic
46#4530C2211yescharacteristic
47#4530C2212yes—
48#3493D311yescharacteristic
49#3093—32no—
50#2285—31no—
51#2061—31no—
52#2060—31no—
53#2040—31no—
54#2039—31no—
55#1896—32no—
56#1860—32no—
57#1859—32no—
58#1822—31no—
59C2×C3×#41C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×Dic3—21no—
62C3×C4⋊Q8—22no—
63C3×#126—24no—
64C3×#126—24no—
65C3×#125—22no—
66C3×#125—22no—
67C3×#124—22no—
68C3×#124—24no—
69C3×#123—24no—
70C3×#123—22no—
71C3×C22⋊Q8—22no—
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8D412yes—
74C3×C22⋊Q8D412yes—
75C3×#121—24no—
76C3×#121—22no—
77C3×#121—22no—
78C3×#121D412yes—
79C3×#121D412yes—
80C3×#120—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×C4×D4—24no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×#117—24no—
89C3×#117—22no—
90C3×#117—22no—
91C3×#117—22no—
92C3×#117C2312yes—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4D412yes—
97C2×C3×C42D412yes—
98C2×#217—21no—
99#926—22no—
100#926—22no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×Dic3⋊C4—22no—
104C2×Dic3⋊C4—22no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—22no—
107C2×C4×Dic3—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#556—22no—
128#555—22no—
129#553—22no—
130#552—22no—
131#544C2×D311yescharacteristic
132#538—22no—
133#537C2×D311yescharacteristic
134#529—22no—
135#525C2×D312yes—
136#520C2×D312yes—
137C2×#117—31no—
138C2×#117C2×D311yescharacteristic
139C22×C42—31no—
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#384—31no—
152#384—31no—
153#382—31no—
154#382—31no—
155#380—62no—
156C4×C4⋊C4—62no—
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—
167C2×C3×Q8—22no—
168C2×C3×Q8C2×D412yes—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4C2×D412yes—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—24no—
178C22×C3×C4—22no—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2411yescharacteristic
185C22×Dic3—21no—
186C22×Dic3—24no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—42no—
191C22×Dic3—42no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4C2×D412yes—
201C3×C4⋊C4C2×D412yes—
202C3×C22⋊C4—24no—
203C3×C22⋊C4—24no—
204C3×C22⋊C4—24no—
205C3×C22⋊C4—22no—
206C3×C22⋊C4—22no—
207C3×C22⋊C4—22no—
208C3×C22⋊C4—22no—
209C3×C22⋊C4—22no—
210C3×C42—24no—
211C3×C42—22no—
212C3×C42—22no—
213C3×C42—22no—
214#217—42no—
215C4⋊Dic3—22no—
216C4⋊Dic3—22no—
217Dic3⋊C4—44no—
218Dic3⋊C4—44no—
219C4×Dic3—24no—
220C4×Dic3—21no—
221C4×Dic3—21no—
222C4×Dic3—41no—
223C4×Dic3—41no—
224C4×Dic3#4112yes—
225C4×Dic3#4112yes—
226#208—22no—
227#208—22no—
228#208—22no—
229#208—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×#41C22×D312yes—
233C23×C4—61no—
234C23×C4C22×D311yescharacteristic
235C2×M16—32no—
236C2×M16—32no—
237C4⋊Q8—22no—
238#126—24no—
239#126—24no—
240#125—22no—
241#125—22no—
242#124—22no—
243#124—24no—
244#123—24no—
245#123—22no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C22⋊Q8#6712yes—
249C22⋊Q8#6712yes—
250#121—24no—
251#121—22no—
252#121—22no—
253#121#6712yes—
254#121#6712yes—
255#120—22no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—24no—
261C4×D4—22no—
262C4×D4—22no—
263#117—24no—
264#117—22no—
265#117—22no—
266#117—22no—
267#117—62no—
268#117—62no—
269#117C22×D312yes—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4C22×D311yescharacteristic
276C2×C4⋊C4C22×D311yescharacteristic
277C2×C22⋊C4—61no—
278C2×C22⋊C4#6712yes—
279C2×C42—31no—
280C2×C42—31no—
281C2×C42—62no—
282C2×C42—62no—
283C2×C42—61no—
284C2×C42—61no—
285C2×C42#6712yes—
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#12012yes—
306C23×C3C22×D411yescharacteristic
307C3×Q8—22no—
308C3×Q8—22no—
309C3×Q8—82no—
310C3×D4—22no—
311C3×D4—22no—
312C3×D4—42no—
313C3×D4—42no—
314C3×D4—82no—
315C3×D4—82no—
316C3×D4—82no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—22no—
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—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—84no—
333C2×C3×C4#12011yescharacteristic
334C2×C3×C4#12011yescharacteristic
335C2×C3×C4C22×D412yes—
336C2×C3×C4C2×#4111yescharacteristic
337C2×C3×C4C2×#4111yescharacteristic
338C2×C3×C42−1+412yes—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—48no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—41no—
349C2×Dic3—41no—
350C3⋊C8—42no—
351C3⋊C8—42no—
352C24C2×#6711yescharacteristic
353#41—22no—
354#41—22no—
355#41—22no—
356#41—22no—
357C2×Q8—22no—
358C2×Q8C2×#6712yes—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4C2×#6712yes—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—61no—
370C22×C4—64no—
371C22×C4—61no—
372C22×C4—61no—
373C22×C4—61no—
374C22×C4—122no—
375C22×C4—122no—
376C22×C4C2×#6711yescharacteristic
377C22×C4C2×#6712yes—
378C22×C4C2×#6712yes—
379C22×C4C2×#6712yes—
380C22×C4C2×#6712yes—
381C22×C4C23×D311yescharacteristic
382M16—62no—
383M16—62no—
384M16—62no—
385M16—62no—
386C2×C8—62no—
387C2×C8—62no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—62no—
397C4⋊C4—62no—
398C4⋊C4—124no—
399C4⋊C4—124no—
400C4⋊C4C2×#6712yes—
401C4⋊C4C2×#6712yes—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—122no—
411C42—24no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—32no—
416C42—32no—
417C42—64no—
418C42—61no—
419C42—61no—
420C42—121no—
421C42—121no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—22no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3C2×#12011yescharacteristic
433C22×C3#52511yescharacteristic
434C22×C3#53611yescharacteristic
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—42no—
438C3×C4—42no—
439C3×C4—42no—
440C3×C4—42no—
441C3×C4—42no—
442C3×C4#54712yes—
443C3×C4#54712yes—
444Dic3—24no—
445Dic3—42no—
446Dic3—81no—
447Dic3—81no—
448C23—21no—
449C23—21no—
450C23—22no—
451C23—41no—
452C23—41no—
453C23#95312yes—
454C23C22×#6711yescharacteristic
455Q8—22no—
456Q8—22no—
457Q8—82no—
458D4—22no—
459D4—22no—
460D4—42no—
461D4—42no—
462D4—82no—
463D4—82no—
464D4—82no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—84no—
485C2×C4—128no—
486C2×C4—121no—
487C2×C4—121no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—121no—
491C2×C4—121no—
492C2×C4#95311yescharacteristic
493C2×C4#95311yescharacteristic
494C2×C4D3×#4111yescharacteristic
495C2×C4D3×#4111yescharacteristic
496C2×C4#101012yes—
497C2×C4C22×#6712yes—
498C8—122no—
499C8—122no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—42no—
503C2×C3—82no—
504C2×C3#240611yescharacteristic
505C2×C3#301811yescharacteristic
506C2×C3#301811yescharacteristic
507C22—21no—
508C22—21no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22#582511yescharacteristic
518C22#582811yescharacteristic
519C22C2×#95311yescharacteristic
520C4—21no—
521C4—21no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—64no—
528C4—122no—
529C4—241no—
530C4—241no—
531C4#582412yes—
532C4#582412yes—
533C3#2407811yescharacteristic, complemented, Sylow
534C2—21no—
535C2—21no—
536C2—42no—
537C2—82no—
538C2#8409011yescharacteristic
539C2#8412111yescharacteristic
540C2#8412111yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3493C2
socleC22×C3C2×#120
hypercenter#3493D3
hypocenterC3#24078
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3493

Complements

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

NH ≅ G/Ntype
2. #82915C2semidirect
3. #82224C2semidirect
4. #71901C2semidirect
5. #71900C2semidirect
6. #71877C2semidirect
7. #71876C2semidirect
8. #71711C2semidirect
9. #71665C2semidirect
10. #71664C2semidirect
11. #71589C2semidirect
533. C3#24078semidirect

Maximal subgroups

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

subgroupindex
C3×#34932
#829152
#822242
#719012
#719002
#718772
#718762
#717112
#716652
#716642
#715892
#240783