Finite Group Databasebeta

← #10784418

Subgroups — #10784418

Order 768.

670 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81610C211yesmaximal, characteristic, complemented
2#81608C211yesmaximal, characteristic, complemented
3#81604C211yesmaximal, characteristic, complemented
4#81573C211yesmaximal, characteristic
5#81571C211yesmaximal, characteristic
6#81569C211yesmaximal, characteristic
7C3×#1711C211yesmaximal, characteristic, complemented
8#71227C211yesmaximal, characteristic, complemented
9#71226C211yesmaximal, characteristic, complemented
10#71222C211yesmaximal, characteristic, complemented
11#71195C211yesmaximal, characteristic, complemented
12#71193C211yesmaximal, characteristic, complemented
13#71190C211yesmaximal, characteristic, complemented
14#68065C211yesmaximal, characteristic, complemented
15#68049C211yesmaximal, characteristic, complemented
16#19744—31nomaximal, Sylow
17#5728C2211yescharacteristic
18#5728C2211yescharacteristic
19D3×C4⋊Q8C2211yescharacteristic
20#5716C2211yescharacteristic
21C3×#487C2211yescharacteristic
22C3×#487C2211yescharacteristic
23C3×#487C2211yescharacteristic
24C3×#486C2211yescharacteristic
25C3×#486C2211yescharacteristic
26C3×#486C2211yescharacteristic
27#5093C2211yescharacteristic
28#5093C2211yescharacteristic
29#5092C2211yescharacteristic
30#5080C2211yescharacteristic
31#5080C2211yescharacteristic
32#5079C2211yescharacteristic
33#5079C2211yescharacteristic
34#5078C2211yescharacteristic
35#5078C2211yescharacteristic
36#5076C2211yescharacteristic
37#5070C2211yescharacteristic
38#5070C2211yescharacteristic
39#4814—22no—
40#4811—22no—
41#4809—22no—
42#4796—22no—
43#4795—22no—
44#4793—22no—
45#4784—22no—
46#4783—22no—
47#4774—22no—
48#4767—22no—
49#4766—22no—
50#4759—22no—
51#4699C2211yescharacteristic
52#4699C2211yescharacteristic
53#4685C2211yescharacteristic
54#4685C2211yescharacteristic
55#4685C2211yescharacteristic
56#4682C2211yescharacteristic
57#4682C2211yescharacteristic
58#4682C2211yescharacteristic
59#4682C2211yescharacteristic
60C3×C82C2211yescharacteristic
61#4454C2211yescharacteristic
62#4454C2211yescharacteristic
63#4454C2211yescharacteristic
64#3231—31no—
65#3230—31no—
66#3228—31no—
67#3226—31no—
68#3210—31no—
69#3209—31no—
70#1723—31no—
71#1723—31no—
72#1722—31no—
73#1720—31no—
74#1719—31no—
75#1718—31no—
76#1713—31no—
77#1711D311yescharacteristic, complemented
78#1454—31no—
79C2×D3×Q8—21no—
80C2×#237—21no—
81C3×C4⋊Q8C2311yescharacteristic
82C3×C4⋊Q8C2311yescharacteristic
83C3×C4⋊Q8C2311yescharacteristic
84C3×#127C2311yescharacteristic
85#946—21no—
86#946—21no—
87Q8×Dic3—21no—
88Q8×Dic3—21no—
89#944—21no—
90C2×#216—22no—
91#938—21no—
92#938—21no—
93#936—21no—
94D4×Dic3—21no—
95D4×Dic3—21no—
96C2×#214—22no—
97C2×Dic12—22no—
98C2×Dic12—22no—
99#897—21no—
100#897—21no—
101#892—21no—
102#892—21no—
103D3×C4⋊C4—21no—
104D3×C4⋊C4—21no—
105#890—21no—
106#890—21no—
107#888—21no—
108#888—21no—
109#887—21no—
110#887—21no—
111C42×D3C2311yescharacteristic
112#869C2311yescharacteristic
113#869C2311yescharacteristic
114#869C2311yescharacteristic
115#869C2311yescharacteristic
116C3×#103—22no—
117C3×#103—22no—
118C3×#103—22no—
119C3×#102—22no—
120C3×#102—22no—
121C3×#102—22no—
122C3×C4×C8C2311yescharacteristic
123C3×C4×C8C2311yescharacteristic
124C3×C4×C8C2311yescharacteristic
125#835—22no—
126#835—22no—
127#832—22no—
128#832—22no—
129(C2×D3)⋊C8D412yes—
130(C2×D3)⋊C8D412yes—
131(C2×D3)⋊C8D412yes—
132#818—22no—
133#818—22no—
134#818—22no—
135#818—22no—
136Dic3⋊C8D412yes—
137Dic3⋊C8D412yes—
138Dic3⋊C8D412yes—
139#810—22no—
140#810—22no—
141#808—22no—
142#808—22no—
143#807—22no—
144#807—22no—
145C4×C3⋊C8C2311yescharacteristic
146C4×C3⋊C8C2311yescharacteristic
147C4×C3⋊C8C2311yescharacteristic
148#533—31no—
149#533—31no—
150#532—31no—
151C2×C4⋊Q8—31no—
152#500—31no—
153#500—31no—
154#500—31no—
155#500—31no—
156#500—31no—
157#500—31no—
158#498—31no—
159#498—31no—
160#494—31no—
161#494—31no—
162#494—31no—
163#492—31no—
164#487—31no—
165#487—31no—
166#487—31no—
167#487—31no—
168#487C2×D311yescharacteristic
169#487C2×D311yescharacteristic
170#487C2×D311yescharacteristic
171#486—31no—
172#486—31no—
173#486C2×D311yescharacteristic
174#486C2×D311yescharacteristic
175#486C2×D311yescharacteristic
176#485—62no—
177#484—62no—
178#482—62no—
179#480—62no—
180#479—62no—
181#478—62no—
182#477—62no—
183#474—62no—
184#464—62no—
185#462—62no—
186#452—62no—
187#451—62no—
188#431—31no—
189#431—31no—
190#431—31no—
191#322—31no—
192#322—31no—
193#322—31no—
194C82C2×D311yescharacteristic
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×#67—41no—
202C22×Dic3—41no—
203D3×Q8—82no—
204#237—82no—
205C2×C4×D3—41no—
206C2×C4×D3C2×D411yescharacteristic
207C2×C4×D3C2×D411yescharacteristic
208C2×C4×D3C2×D411yescharacteristic
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—41no—
214C2×C3×C8C2×D412yes—
215C2×C3×C8C2×D412yes—
216C2×C3×C8C2×D412yes—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C42C2411yescharacteristic
224#217—41no—
225#217—41no—
226#216—82no—
227#214—82no—
228(C2×D3)⋊C4—41no—
229(C2×D3)⋊C4—41no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—41no—
239C4⋊Dic3—41no—
240Dic3⋊C4—41no—
241Dic3⋊C4—41no—
242Dic3⋊C4—41no—
243Dic3⋊C4—41no—
244C4×Dic3—41no—
245C4×Dic3—41no—
246C4×Dic3C2×D411yescharacteristic
247C4×Dic3C2×D411yescharacteristic
248C4×Dic3C2×D411yescharacteristic
249C2×C3⋊C8C2×D412yes—
250C2×C3⋊C8C2×D412yes—
251C2×C3⋊C8C2×D412yes—
252Dic12—82no—
253Dic12—82no—
254C2×#41—61no—
255C22×Q8—61no—
256C2×Q16—62no—
257C2×Q16—62no—
258C2×Q16—62no—
259C2×SD16—62no—
260C4⋊Q8—31no—
261C4⋊Q8—31no—
262C4⋊Q8—31no—
263C4⋊Q8—31no—
264C4⋊Q8—61no—
265C4⋊Q8—61no—
266C4⋊Q8—61no—
267C4⋊Q8C22×D311yescharacteristic
268C4⋊Q8C22×D311yescharacteristic
269C4⋊Q8C22×D311yescharacteristic
270#127C22×D311yescharacteristic
271#125—61no—
272#125—61no—
273#124—61no—
274C22⋊Q8—61no—
275C22⋊Q8—61no—
276C22⋊Q8—61no—
277C22⋊Q8—61no—
278#121—61no—
279#121—61no—
280C4×Q8—61no—
281C4×Q8—61no—
282C4×Q8—61no—
283C4×Q8—61no—
284C4×D4—61no—
285C4×D4—61no—
286#117—61no—
287#117—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C42—31no—
291#107—62no—
292#107—62no—
293#107—62no—
294#107—62no—
295#107—62no—
296#107—62no—
297#106—62no—
298#106—62no—
299C4⋊C8—32no—
300C4⋊C8—32no—
301C4⋊C8—32no—
302#103—22no—
303#103—22no—
304#103—22no—
305#103—62no—
306#103—62no—
307#103—62no—
308#103—62no—
309#102—22no—
310#102—22no—
311#102—22no—
312#102—62no—
313#102—62no—
314C22⋊C8—32no—
315C22⋊C8—32no—
316C22⋊C8—32no—
317C4×C8—31no—
318C4×C8—31no—
319C4×C8—31no—
320C4×C8C22×D311yescharacteristic
321C4×C8C22×D311yescharacteristic
322C4×C8C22×D311yescharacteristic
323C23×C3—41no—
324C22×D3#12711yescharacteristic
325C3×Q8—42no—
326C3×Q8—42no—
327C3×Q8—42no—
328C3×D4—42no—
329C3×D4—42no—
330C3×D4—42no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C22×D411yescharacteristic
337#67—82no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342C2×Dic3—82no—
343C2×Dic3#12711yescharacteristic
344C2×Dic3#12711yescharacteristic
345C2×Dic3#12711yescharacteristic
346C4×D3—22no—
347C4×D3—22no—
348C4×D3—22no—
349C4×D3—82no—
350Dic6—42no—
351Dic6—42no—
352Dic6—42no—
353Dic6—42no—
354Dic6—82no—
355C3×C8—22no—
356C3×C8—22no—
357C3×C8—22no—
358C3⋊C8—22no—
359C3⋊C8—22no—
360C3⋊C8—22no—
361#41—242no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—61no—
366C2×Q8—61no—
367C2×Q8—61no—
368C2×Q8—61no—
369C2×Q8—121no—
370C2×Q8—242no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—121no—
375C22×C4—31no—
376C22×C4—31no—
377C22×C4—31no—
378C22×C4—121no—
379C22×C4—121no—
380Q16—242no—
381Q16—242no—
382Q16—242no—
383SD16—242no—
384C2×C8—32no—
385C2×C8—32no—
386C2×C8—32no—
387C2×C8D3×D412yes—
388C2×C8D3×D412yes—
389C2×C8D3×D412yes—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—61no—
397C4⋊C4—61no—
398C4⋊C4—61no—
399C4⋊C4—61no—
400C4⋊C4—61no—
401C4⋊C4—61no—
402C4⋊C4—61no—
403C4⋊C4—61no—
404C4⋊C4—121no—
405C4⋊C4—121no—
406C4⋊C4—121no—
407C4⋊C4—121no—
408C4⋊C4—121no—
409C4⋊C4—121no—
410C22⋊C4—121no—
411C22⋊C4—121no—
412C22⋊C4—121no—
413C22⋊C4—121no—
414C42—31no—
415C42—31no—
416C42—31no—
417C42—121no—
418C42—121no—
419C42C23×D311yescharacteristic
420C22×C3—81no—
421C22×C3—81no—
422C22×C3—81no—
423C22×C3C2×#12711yescharacteristic
424C2×D3—21no—
425C2×D3—21no—
426C2×D3—21no—
427C3×C4—81no—
428C3×C4—81no—
429C3×C4—81no—
430C3×C4#57612yes—
431C3×C4#57812yes—
432C3×C4#57812yes—
433Dic3—21no—
434Dic3—21no—
435Dic3—21no—
436Dic3—81no—
437Dic3—81no—
438Dic3—81no—
439Dic3—81no—
440C23—31no—
441C23—41no—
442Q8—42no—
443Q8—42no—
444Q8—42no—
445Q8—122no—
446Q8—122no—
447Q8—122no—
448Q8—122no—
449Q8—242no—
450D4—42no—
451D4—42no—
452D4—42no—
453D4—242no—
454C2×C4—31no—
455C2×C4—31no—
456C2×C4—31no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—242no—
468C2×C4—242no—
469C2×C4C2×D3×D411yescharacteristic
470C2×C4C2×D3×D411yescharacteristic
471C2×C4C2×D3×D411yescharacteristic
472C8—22no—
473C8—22no—
474C8—22no—
475C8—62no—
476C8—62no—
477C8—62no—
478C2×C3—161no—
479C2×C3#315511yescharacteristic
480C2×C3#315611yescharacteristic
481C2×C3#315611yescharacteristic
482D3—41no—
483C22—61no—
484C22—61no—
485C22—61no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22D3×#12711yescharacteristic
490C4—61no—
491C4—61no—
492C4—61no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4—241no—
497C4—241no—
498C4—241no—
499C4—241no—
500C4#576912yes—
501C4#577112yes—
502C4#577112yes—
503C3#1974411yescharacteristic, complemented, Sylow
504C2—121no—
505C2—161no—
506C2#8182311yescharacteristic
507C2#8182411yescharacteristic
508C2#8182411yescharacteristic
subgroupG/H
centerC22D3×#127
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1711C2
socleC22×C3C2×#127
prime radical#5716C22
hypercenter#1711D3
hypocenterC3#19744
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1711

Complements

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

NH ≅ G/Ntype
1. #81610C2semidirect
2. #81608C2semidirect
3. #81604C2semidirect
7. C3×#1711C2semidirect
8. #71227C2semidirect
9. #71226C2semidirect
10. #71222C2semidirect
11. #71195C2semidirect
12. #71193C2semidirect
13. #71190C2semidirect
14. #68065C2semidirect
15. #68049C2semidirect
77. #1711D3semidirect
503. C3#19744semidirect

Maximal subgroups

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

subgroupindex
#816102
#816082
#816042
#815732
#815712
#815692
C3×#17112
#712272
#712262
#712222
#711952
#711932
#711902
#680652
#680492
#197443