Finite Group Databasebeta

← #10784422

Subgroups — #10784422

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#81944C211yesmaximal, characteristic, complemented
2#81943C211yesmaximal, characteristic, complemented
3#81802C211yesmaximal, characteristic, complemented
4#81767C211yesmaximal, characteristic
5#81574C211yesmaximal, characteristic
6#81571C211yesmaximal, characteristic
7C3×#1722C211yesmaximal, characteristic, complemented
8#71367C211yesmaximal, characteristic, complemented
9#71362C211yesmaximal, characteristic, complemented
10#71302C211yesmaximal, characteristic, complemented
11#71261C211yesmaximal, characteristic, complemented
12#71192C211yesmaximal, characteristic, complemented
13#71183C211yesmaximal, characteristic, complemented
14#68096C211yesmaximal, characteristic, complemented
15#68074C211yesmaximal, characteristic, complemented
16#19744—31nomaximal, Sylow
17#5728C2211yescharacteristic
18#5728C2211yescharacteristic
19D3×C4⋊Q8C2211yescharacteristic
20#5716C2211yescharacteristic
21C3×#500C2211yescharacteristic
22C3×#500C2211yescharacteristic
23C3×#494C2211yescharacteristic
24C3×#492C2211yescharacteristic
25C3×#486C2211yescharacteristic
26C3×#486C2211yescharacteristic
27#5186—22no—
28#5182—22no—
29#5173—22no—
30#5166—22no—
31#5093C2211yescharacteristic
32#5093C2211yescharacteristic
33#5092C2211yescharacteristic
34#5092C2211yescharacteristic
35#5080C2211yescharacteristic
36#5079C2211yescharacteristic
37#5078C2211yescharacteristic
38#5078C2211yescharacteristic
39#5078C2211yescharacteristic
40#5070C2211yescharacteristic
41#5070C2211yescharacteristic
42#5070C2211yescharacteristic
43#4888—22no—
44#4884—22no—
45#4877—22no—
46#4875—22no—
47#4779—22no—
48#4774—22no—
49#4763—22no—
50#4759—22no—
51#4685C2211yescharacteristic
52#4685C2211yescharacteristic
53#4685C2211yescharacteristic
54#4682C2211yescharacteristic
55#4681C2211yescharacteristic
56#4681C2211yescharacteristic
57#4681C2211yescharacteristic
58#4681C2211yescharacteristic
59#4680C2211yescharacteristic
60C3×#322C2211yescharacteristic
61#4455C2211yescharacteristic
62#4454C2211yescharacteristic
63C8×C3⋊C8C2211yescharacteristic
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#1722D311yescharacteristic, complemented
73#1720—31no—
74#1719—31no—
75#1718—31no—
76#1713—31no—
77#1711—31no—
78#1454—31no—
79C2×D3×Q8—21no—
80C2×#237—21no—
81C2×C3×Q16—22no—
82C2×C3×SD16—22no—
83C3×C4⋊Q8C2311yescharacteristic
84C3×C4⋊Q8C2311yescharacteristic
85C3×C4⋊Q8C2311yescharacteristic
86C3×#127C2311yescharacteristic
87#946—21no—
88#946—21no—
89Q8×Dic3—21no—
90Q8×Dic3—21no—
91#944—21no—
92C2×#216—22no—
93C2×#216—22no—
94#938—21no—
95#938—21no—
96#936—21no—
97D4×Dic3—21no—
98D4×Dic3—21no—
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×#107—22no—
117C3×#107—22no—
118C3×#102—22no—
119C3×#102—22no—
120C3×C4×C8C2311yescharacteristic
121C3×C4×C8C2311yescharacteristic
122C3×C4×C8C2311yescharacteristic
123#835—22no—
124#832—22no—
125#832—22no—
126#832—22no—
127(C2×D3)⋊C8D412yes—
128(C2×D3)⋊C8D412yes—
129(C2×D3)⋊C8D412yes—
130#818—22no—
131#817—22no—
132#816—22no—
133#816—22no—
134#816—22no—
135#816—22no—
136Dic3⋊C8D412yes—
137Dic3⋊C8D412yes—
138Dic3⋊C8D412yes—
139#810—22no—
140#810—22no—
141#808—22no—
142#807—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#500C2×D311yescharacteristic
157#500C2×D311yescharacteristic
158#498—31no—
159#498—31no—
160#494—31no—
161#494—31no—
162#494C2×D311yescharacteristic
163#492C2×D311yescharacteristic
164#487—31no—
165#487—31no—
166#487—31no—
167#487—31no—
168#487—31no—
169#487—31no—
170#487—31no—
171#486—31no—
172#486—31no—
173#486—31no—
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#322C2×D311yescharacteristic
194C82—31no—
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—
214C3×Q16—82no—
215C3×SD16—82no—
216C2×C3×C8C2×D412yes—
217C2×C3×C8C2×D412yes—
218C2×C3×C8C2×D412yes—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C42C2411yescharacteristic
226#217—41no—
227#217—41no—
228#216—82no—
229#216—82no—
230(C2×D3)⋊C4—41no—
231(C2×D3)⋊C4—41no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—41no—
241C4⋊Dic3—41no—
242Dic3⋊C4—41no—
243Dic3⋊C4—41no—
244Dic3⋊C4—41no—
245Dic3⋊C4—41no—
246C4×Dic3—41no—
247C4×Dic3—41no—
248C4×Dic3C2×D411yescharacteristic
249C4×Dic3C2×D411yescharacteristic
250C4×Dic3C2×D411yescharacteristic
251C2×C3⋊C8C2×D412yes—
252C2×C3⋊C8C2×D412yes—
253C2×C3⋊C8C2×D412yes—
254C2×#41—61no—
255C22×Q8—61no—
256C2×Q16—22no—
257C2×Q16—62no—
258C2×Q16—62no—
259C2×SD16—22no—
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—22no—
292#107—22no—
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—62no—
303#103—62no—
304#103—62no—
305#103—62no—
306#103—62no—
307#103—62no—
308#103—62no—
309#102—22no—
310#102—22no—
311#102—62no—
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—82no—
381Q16—242no—
382Q16—242no—
383SD16—82no—
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#575712yes—
501C4#577912yes—
502C4#577912yes—
503C3#1974411yescharacteristic, complemented, Sylow
504C2—121no—
505C2—161no—
506C2#8178411yescharacteristic
507C2#8187311yescharacteristic
508C2#8187311yescharacteristic
subgroupG/H
centerC22D3×#127
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1722C2
socleC22×C3C2×#127
prime radical#5716C22
hypercenter#1722D3
hypocenterC3#19744
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1722

Complements

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

NH ≅ G/Ntype
1. #81944C2semidirect
2. #81943C2semidirect
3. #81802C2semidirect
7. C3×#1722C2semidirect
8. #71367C2semidirect
9. #71362C2semidirect
10. #71302C2semidirect
11. #71261C2semidirect
12. #71192C2semidirect
13. #71183C2semidirect
14. #68096C2semidirect
15. #68074C2semidirect
72. #1722D3semidirect
503. C3#19744semidirect

Maximal subgroups

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

subgroupindex
#819442
#819432
#818022
#817672
#815742
#815712
C3×#17222
#713672
#713622
#713022
#712612
#711922
#711832
#680962
#680742
#197443