Finite Group Databasebeta

← #10950859

Subgroups — #10950859

Order 768.

556 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79598C211yesmaximal, characteristic, complemented
2#79583C211yesmaximal, characteristic, complemented
3C3×#2177C211yesmaximal, characteristic, complemented
4#71469C211yesmaximal, characteristic, complemented
5#71466C211yesmaximal, characteristic, complemented
6#70885C211yesmaximal, characteristic, complemented
7#70884C211yesmaximal, characteristic, complemented
8#69347C211yesmaximal, characteristic, complemented
9#69341C211yesmaximal, characteristic, complemented
10#69196C211yesmaximal, characteristic, complemented
11#69195C211yesmaximal, characteristic, complemented
12#69186C211yesmaximal, characteristic, complemented
13#69185C211yesmaximal, characteristic, complemented
14#69077C211yesmaximal, characteristic, complemented
15D3×#346C211yesmaximal, characteristic, complemented
16#33746—31nomaximal, Sylow
17#5557—21no—
18#5548—21no—
19C2×C3×D16C2211yescharacteristic
20C3×C4×D8C2211yescharacteristic
21C3×C4×D8C2211yescharacteristic
22#5157C2211yescharacteristic
23#5157C2211yescharacteristic
24#5155C2211yescharacteristic
25#5151C2211yescharacteristic
26C2×#827C2211yescharacteristic
27C2×#826C2211yescharacteristic
28C4×#214—21no—
29C4×#213—21no—
30C2×#800C2211yescharacteristic
31C2×D3×C16C2211yescharacteristic
32C2×D3×C16C2211yescharacteristic
33#4887C2211yescharacteristic
34#4883C2211yescharacteristic
35#4878C2211yescharacteristic
36#4874C2211yescharacteristic
37#4778—21no—
38#4775—21no—
39#4768—21no—
40#4760—21no—
41C4×Dic12C2211yescharacteristic
42C4×D24C2211yescharacteristic
43C4×D3×C8C2211yescharacteristic
44C3×#367C2211yescharacteristic
45C3×#357C2211yescharacteristic
46C3×#357C2211yescharacteristic
47C3×#346C2211yescharacteristic
48#4562C2211yescharacteristic
49#4562C2211yescharacteristic
50#4509C2211yescharacteristic
51#4504C2211yescharacteristic
52#4503C2211yescharacteristic
53#4502C2211yescharacteristic
54#4502C2211yescharacteristic
55#4492C2211yescharacteristic
56#4491C2211yescharacteristic
57#4490C2211yescharacteristic
58#4489C2211yescharacteristic
59#4461C2211yescharacteristic
60#3169—31no—
61#3168—31no—
62#2255—31no—
63#2250—31no—
64#2244—31no—
65#2243—31no—
66#2217—31no—
67#2217—31no—
68#2215—31no—
69#2215—31no—
70#2177—31no—
71#2177D311yescharacteristic, complemented
72#2175—31no—
73#2175—31no—
74C2×#346—31no—
75C2×#235—41no—
76C2×C3×D8C2311yescharacteristic
77C2×C3×D8C2311yescharacteristic
78C3×C4×D4—21no—
79C3×C4×D4—21no—
80#940—21no—
81#938—21no—
82#938—21no—
83#936—21no—
84C2×#214—21no—
85C2×#213—21no—
86C4×#67—41no—
87C4×#67—41no—
88#926—41no—
89C2×Dic12C2311yescharacteristic
90C2×D24C2311yescharacteristic
91C2×D3×C8D411yescharacteristic
92C2×D3×C8C2311yescharacteristic
93#897—21no—
94#895—21no—
95#890—21no—
96#888—21no—
97#885—41no—
98#884—41no—
99#883—41no—
100#879—41no—
101C4×D12—21no—
102C42×D3D411yescharacteristic
103C4×Dic6—21no—
104C3×D16—22no—
105C2×C3×C16C2311yescharacteristic
106C2×C3×C16C2311yescharacteristic
107C3×#107C2311yescharacteristic
108C3×#107C2311yescharacteristic
109C3×#102—21no—
110C3×#102—21no—
111C3×C4×C8C2311yescharacteristic
112#832—21no—
113#832—21no—
114#827—22no—
115#826—22no—
116#821—21no—
117#818C2311yescharacteristic
118#818C2311yescharacteristic
119#816—21no—
120C8×Dic3D411yescharacteristic
121C8×Dic3C2311yescharacteristic
122C2×C3⋊C16C2311yescharacteristic
123C2×C3⋊C16C2311yescharacteristic
124#810—21no—
125#809—21no—
126#808—21no—
127#807—21no—
128C4×C3⋊C8D411yescharacteristic
129#800—22no—
130D3×C16—22no—
131D3×C16—22no—
132#532—61no—
133#529—61no—
134C2×SD32—31no—
135C2×SD32—31no—
136C2×D16—31no—
137C2×D16C2×D311yescharacteristic
138C22×C16—31no—
139C22×C16—31no—
140#500—31no—
141#499—31no—
142#495—31no—
143#493—31no—
144#488—61no—
145#486—61no—
146#467—31no—
147#466—31no—
148#466—31no—
149#466—31no—
150#466—61no—
151#465—61no—
152C4×Q16—31no—
153C4×SD16—61no—
154C4×D8—31no—
155C4×D8—61no—
156C4×D8C2×D311yescharacteristic
157C4×D8C2×D311yescharacteristic
158C2×C4×C8—31no—
159#367—31no—
160#367C2×D311yescharacteristic
161#366—31no—
162#366—31no—
163#358—31no—
164#358—31no—
165#357—31no—
166#357—31no—
167#357—31no—
168#357—31no—
169#357C2×D311yescharacteristic
170#357C2×D311yescharacteristic
171#346—31no—
172#346—31no—
173#346—31no—
174#346C2×D311yescharacteristic
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C22×C3×C4—41no—
178C22×C3×C4—41no—
179C2×#67—41no—
180C2×#67—41no—
181C2×#67—41no—
182C2×#67—41no—
183#235—84no—
184C2×D12—21no—
185C2×D12—41no—
186C2×C4×D3—41no—
187C2×C4×D3D811yescharacteristic
188C2×C4×D3D811yescharacteristic
189C2×C4×D3C2×D411yescharacteristic
190C2×Dic6—21no—
191C2×Dic6—41no—
192C3×D8—21no—
193C3×D8—21no—
194C3×D8—21no—
195C3×D8—21no—
196C2×C3×C8C2×D411yescharacteristic
197C2×C3×C8C2411yescharacteristic
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C22⋊C4—41no—
201C3×C22⋊C4—41no—
202C3×C42C2×D411yescharacteristic
203#217—41no—
204#217—41no—
205#217—41no—
206#217—41no—
207#214—42no—
208#213—42no—
209(C2×D3)⋊C4—41no—
210(C2×D3)⋊C4—41no—
211(C2×D3)⋊C4—41no—
212C4⋊Dic3—21no—
213C4⋊Dic3—21no—
214C4⋊Dic3—41no—
215C4⋊Dic3—41no—
216Dic3⋊C4—41no—
217Dic3⋊C4—41no—
218Dic3⋊C4—41no—
219C4×Dic3—41no—
220C4×Dic3D811yescharacteristic
221C4×Dic3D811yescharacteristic
222C4×Dic3C2×D411yescharacteristic
223C2×C3⋊C8C2×D411yescharacteristic
224C2×C3⋊C8C2×D411yescharacteristic
225Dic12—21no—
226Dic12—21no—
227D24—21no—
228D24—21no—
229D3×C8—21no—
230D3×C8—21no—
231D3×C8—22no—
232C3×C16—21no—
233C3×C16#4112yescomplemented
234C3⋊C16—21no—
235C3⋊C16#4112yescomplemented
236C2×#41—121no—
237C2×Q16—31no—
238C2×SD16—61no—
239C2×D8—31no—
240C2×D8—61no—
241C2×D8C22×D311yescharacteristic
242C2×D8C22×D311yescharacteristic
243C22×C8—31no—
244C22×C8—31no—
245C4⋊Q8—61no—
246#127—61no—
247#126—121no—
248#125—61no—
249#124—61no—
250#124—121no—
251#123—121no—
252C22⋊Q8—61no—
253#121—61no—
254#121—61no—
255#121—61no—
256#121—121no—
257C4×Q8—61no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—61no—
261C4×D4—121no—
262C4×D4—121no—
263#117—121no—
264C2×C42—31no—
265SD32—62no—
266SD32—62no—
267D16—22no—
268D16—62no—
269C2×C16—31no—
270C2×C16—31no—
271C2×C16—62no—
272C2×C16—62no—
273C2×C16C22×D311yescharacteristic
274C2×C16C22×D311yescharacteristic
275#107—31no—
276#107—31no—
277#107—61no—
278#107C22×D311yescharacteristic
279#107C22×D311yescharacteristic
280#106—61no—
281#103—61no—
282#103—61no—
283#102—21no—
284#102—21no—
285#102—61no—
286#102—61no—
287#102—61no—
288#102—61no—
289C4×C8—31no—
290C4×C8—31no—
291C4×C8—31no—
292C4×C8C22×D311yescharacteristic
293C23×C3—41no—
294C23×C3—41no—
295C22×D3—41no—
296C22×D3C2×D811yescharacteristic
297C3×D4—41no—
298C3×D4—41no—
299C3×D4—41no—
300C3×D4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—82no—
304C2×C3×C4—82no—
305C2×C3×C4C2×D811yescharacteristic
306C2×C3×C4C2×D811yescharacteristic
307C2×C3×C4C22×D411yescharacteristic
308#67—82no—
309#67—82no—
310#67—82no—
311#67—82no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3C2×D811yescharacteristic
316C2×Dic3C2×D811yescharacteristic
317C2×Dic3C2×D811yescharacteristic
318D12—41no—
319D12—41no—
320D12—82no—
321C4×D3—22no—
322C4×D3—21no—
323C4×D3—21no—
324C4×D3—41no—
325C4×D3—82no—
326Dic6—41no—
327Dic6—41no—
328Dic6—82no—
329C3×C8—21no—
330C3×C8C2×#4111yescharacteristic
331C3×C8C2×#4111yescharacteristic
332C3⋊C8—21no—
333C3⋊C8#13512yes—
334#41—244no—
335C2×Q8—61no—
336C2×Q8—121no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—61no—
340C2×D4—121no—
341C2×D4—121no—
342C2×D4—121no—
343C2×D4—121no—
344C2×D4—121no—
345C22×C4—31no—
346C22×C4—31no—
347C22×C4—31no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—121no—
351Q16—61no—
352Q16—61no—
353SD16—122no—
354D8—21no—
355D8—21no—
356D8—21no—
357D8—21no—
358D8—61no—
359D8—61no—
360D8—122no—
361C2×C8—31no—
362C2×C8—31no—
363C2×C8—62no—
364C2×C8—61no—
365C2×C8—61no—
366C2×C8D3×D411yescharacteristic
367C2×C8C23×D311yescharacteristic
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—61no—
371C4⋊C4—61no—
372C4⋊C4—121no—
373C4⋊C4—121no—
374C4⋊C4—121no—
375C4⋊C4—121no—
376C4⋊C4—121no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—121no—
380C22⋊C4—121no—
381C22⋊C4—121no—
382C22⋊C4—121no—
383C22⋊C4—121no—
384C22⋊C4—121no—
385C22⋊C4—121no—
386C42—31no—
387C42—31no—
388C42—31no—
389C42—121no—
390C42D3×D411yescharacteristic
391C16—21no—
392C16—32no—
393C16—61no—
394C16#23512yescomplemented
395C22×C3—81no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—81no—
400C22×C3—81no—
401C22×C3C22×D811yescharacteristic
402C2×D3—21no—
403C2×D3—21no—
404C2×D3—81no—
405C2×D3—81no—
406C2×D3—81no—
407C2×D3#50912yes—
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—81no—
411C3×C4—81no—
412C3×C4#53211yescharacteristic
413C3×C4C2×#13511yescharacteristic
414Dic3—21no—
415Dic3—21no—
416Dic3—21no—
417Dic3—81no—
418Dic3—81no—
419Dic3—81no—
420C23—31no—
421C23—41no—
422C23—41no—
423C23—121no—
424Q8—121no—
425Q8—121no—
426Q8—242no—
427D4—41no—
428D4—41no—
429D4—41no—
430D4—41no—
431D4—121no—
432D4—121no—
433D4—242no—
434D4—242no—
435D4—242no—
436D4—242no—
437D4—242no—
438C2×C4—31no—
439C2×C4—31no—
440C2×C4—31no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—62no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—121no—
449C2×C4—121no—
450C2×C4—121no—
451C2×C4—121no—
452C2×C4—242no—
453C2×C4D3×D811yescharacteristic
454C2×C4D3×D811yescharacteristic
455C2×C4C2×D3×D411yescharacteristic
456C8—21no—
457C8—32no—
458C8—61no—
459C8C2×#23511yescharacteristic
460C8D3×#4111yescharacteristic
461C2×C3—161no—
462C2×C3—161no—
463C2×C3#316911yescharacteristic
464C2×C3C2×#50911yescharacteristic
465C2×C3#341411yescharacteristic
466D3—22no—
467D3—161no—
468C22—32no—
469C22—61no—
470C22—61no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22—241no—
478C22—241no—
479C22—241no—
480C22C2×D3×D811yescharacteristic
481C4—21no—
482C4—21no—
483C4—61no—
484C4—61no—
485C4—61no—
486C4—81no—
487C4—81no—
488C4—241no—
489C4—241no—
490C4—241no—
491C4#554811yescharacteristic
492C4D3×#13511yescharacteristic
493C3#3374611yescharacteristic, complemented, Sylow
494C2—62no—
495C2—161no—
496C2—161no—
497C2—481no—
498C2#7959811yescharacteristic
499C2D3×#50911yescharacteristic
500C2#8203511yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2177C2
socleC22×C3C22×D8
hypercenter#2177D3
hypocenterC3#33746
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2177

Complements

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

NH ≅ G/Ntype
1. #79598C2semidirect
2. #79583C2semidirect
3. C3×#2177C2semidirect
4. #71469C2semidirect
5. #71466C2semidirect
6. #70885C2semidirect
7. #70884C2semidirect
8. #69347C2semidirect
9. #69341C2semidirect
10. #69196C2semidirect
11. #69195C2semidirect
12. #69186C2semidirect
13. #69185C2semidirect
14. #69077C2semidirect
15. D3×#346C2semidirect
71. #2177D3semidirect
233. C3×C16#41semidirect
235. C3⋊C16#41semidirect
394. C16#235semidirect
493. C3#33746semidirect

Maximal subgroups

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

subgroupindex
#795982
#795832
C3×#21772
#714692
#714662
#708852
#708842
#693472
#693412
#691962
#691952
#691862
#691852
#690772
D3×#3462
#337463