Finite Group Databasebeta

← #10950858

Subgroups — #10950858

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#80740C211yesmaximal, characteristic, complemented
2#80697C211yesmaximal, characteristic, complemented
3C3×#2217C211yesmaximal, characteristic, complemented
4#71481C211yesmaximal, characteristic, complemented
5#71474C211yesmaximal, characteristic, complemented
6#71413C211yesmaximal, characteristic, complemented
7C2×#4502C211yesmaximal, characteristic, complemented
8#71082C211yesmaximal, characteristic, complemented
9#71071C211yesmaximal, characteristic, complemented
10#69341C211yesmaximal, characteristic, complemented
11#69339C211yesmaximal, characteristic, complemented
12#69217C211yesmaximal, characteristic, complemented
13#69208C211yesmaximal, characteristic, complemented
14#69186C211yesmaximal, characteristic, complemented
15#69177C211yesmaximal, characteristic, complemented
16#33746—31nomaximal, Sylow
17#5624—21no—
18#5596—21no—
19C2×C3×SD32C2211yescharacteristic
20C22×C3×C16C2211yescharacteristic
21C3×#467C2211yescharacteristic
22C3×#466C2211yescharacteristic
23#5191C2211yescharacteristic
24Dic3×Q16C2211yescharacteristic
25C2×#828C2211yescharacteristic
26#5151C2211yescharacteristic
27Dic3×D8C2211yescharacteristic
28C2×#826C2211yescharacteristic
29#5115C2211yescharacteristic
30#5115C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32C22×C3⋊C16C2211yescharacteristic
33#5048—21no—
34#5038—21no—
35C2×D48C2211yescharacteristic
36#4878C2211yescharacteristic
37#4874C2211yescharacteristic
38#4872C2211yescharacteristic
39#4872C2211yescharacteristic
40#4800—21no—
41#4788—21no—
42#4760—21no—
43#4756—21no—
44C3×#367C2211yescharacteristic
45C3×#358C2211yescharacteristic
46C3×#357C2211yescharacteristic
47#4564C2211yescharacteristic
48#4562C2211yescharacteristic
49#4509C2211yescharacteristic
50#4509C2211yescharacteristic
51#4504C2211yescharacteristic
52#4502C2211yescharacteristic
53#4502C2211yescharacteristic
54#4502C2211yescharacteristic
55#4502C2211yescharacteristic
56#4491C2211yescharacteristic
57#4491C2211yescharacteristic
58#4490C2211yescharacteristic
59#4489C2211yescharacteristic
60#3169—31no—
61#3168—31no—
62#2255—31no—
63#2250—31no—
64#2244—31no—
65#2243—31no—
66#2217—31no—
67#2217D311yescharacteristic, complemented
68#2215—31no—
69#2215—31no—
70#2177—31no—
71#2177—31no—
72#2175—31no—
73#2175—31no—
74C2×#346—31no—
75C2×#237—41no—
76C2×C3×Q16C2311yescharacteristic
77C2×C3×D8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C3×C22⋊Q8—21no—
80C3×#121—21no—
81#947—21no—
82Q8×Dic3—21no—
83C2×#215—21no—
84#940—21no—
85#939—41no—
86D4×Dic3—21no—
87C2×#213—21no—
88#930—21no—
89#930—21no—
90C2×C4×Dic3D411yescharacteristic
91C22×C3⋊C8D411yescharacteristic
92C2×D24C2311yescharacteristic
93C2×D24C2311yescharacteristic
94#898—41no—
95#893—21no—
96#893—21no—
97#892—41no—
98#890—21no—
99#888—21no—
100#886—41no—
101#885—41no—
102#67⋊C4—41no—
103#67⋊C4—41no—
104C3×SD32—22no—
105C2×C3×C16—22no—
106C2×C3×C16C2311yescharacteristic
107C2×C3×C16C2311yescharacteristic
108C3×#107C2311yescharacteristic
109C3×#107C2311yescharacteristic
110C3×#103—21no—
111C3×#102—21no—
112#835—21no—
113#832—21no—
114#828—22no—
115#826—22no—
116#821—21no—
117#821—21no—
118#818C2311yescharacteristic
119#818C2311yescharacteristic
120C8×Dic3D411yescharacteristic
121C8×Dic3D411yescharacteristic
122C8×Dic3C2311yescharacteristic
123C8×Dic3C2311yescharacteristic
124C2×C3⋊C16—22no—
125C2×C3⋊C16C2311yescharacteristic
126C2×C3⋊C16C2311yescharacteristic
127#809—21no—
128#809—21no—
129#808—21no—
130#807—21no—
131D48—22no—
132#532—61no—
133#529—61no—
134C2×SD32—31no—
135C2×SD32C2×D311yescharacteristic
136C2×D16—31no—
137C2×D16—31no—
138C22×C16—31no—
139C22×C16C2×D311yescharacteristic
140#500—31no—
141#499—31no—
142#495—31no—
143#493—31no—
144#488—61no—
145#486—61no—
146#467C2×D311yescharacteristic
147#466—31no—
148#466—31no—
149#466—61no—
150#466C2×D311yescharacteristic
151#465—61no—
152C4×Q16—31no—
153C4×SD16—61no—
154C4×D8—31no—
155C4×D8—31no—
156C4×D8—31no—
157C4×D8—61no—
158C2×C4×C8—31no—
159#367—31no—
160#367C2×D311yescharacteristic
161#366—31no—
162#366—31no—
163#358—31no—
164#358C2×D311yescharacteristic
165#357—31no—
166#357—31no—
167#357—31no—
168#357—31no—
169#357—31no—
170#357C2×D311yescharacteristic
171#346—31no—
172#346—31no—
173#346—31no—
174#346—31no—
175C2×C3×Q8—21no—
176C2×C3×D4—21no—
177C2×C3×D4—41no—
178C22×C3×C4C2×D411yescharacteristic
179C2×#67—41no—
180C2×#67—41no—
181C2×#67—41no—
182C2×#67—41no—
183C22×Dic3—41no—
184C22×Dic3D811yescharacteristic
185C22×Dic3D811yescharacteristic
186#237—84no—
187C2×D12—21no—
188C2×D12—21no—
189C2×C4×D3—41no—
190C2×C4×D3—41no—
191C2×Dic6—41no—
192C3×Q16—21no—
193C3×Q16—21no—
194C3×D8—21no—
195C3×D8—21no—
196C2×C3×C8—21no—
197C2×C3×C8—21no—
198C2×C3×C8C2×D411yescharacteristic
199C2×C3×C8C2411yescharacteristic
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—41no—
203C3×C22⋊C4—41no—
204C3×C22⋊C4—41no—
205#217—41no—
206#215—42no—
207#213—42no—
208(C2×D3)⋊C4—41no—
209(C2×D3)⋊C4—41no—
210(C2×D3)⋊C4—41no—
211(C2×D3)⋊C4—41no—
212(C2×D3)⋊C4—41no—
213(C2×D3)⋊C4—41no—
214C4⋊Dic3—21no—
215C4⋊Dic3—21no—
216C4⋊Dic3—41no—
217C4⋊Dic3—41no—
218Dic3⋊C4—41no—
219Dic3⋊C4—41no—
220C4×Dic3—41no—
221C4×Dic3D811yescharacteristic
222C4×Dic3D811yescharacteristic
223C4×Dic3C2×D411yescharacteristic
224C4×Dic3C2×D411yescharacteristic
225C2×C3⋊C8—22no—
226C2×C3⋊C8C2×D411yescharacteristic
227C2×C3⋊C8C2×D411yescharacteristic
228D24—21no—
229D24—21no—
230D24—21no—
231D24—21no—
232C3×C16—21no—
233C3×C16#4112yescomplemented
234C3⋊C16—21no—
235C3⋊C16#4112yescomplemented
236C2×#41—121no—
237C2×Q16C22×D311yescharacteristic
238C2×SD16—61no—
239C2×D8—31no—
240C2×D8—31no—
241C2×D8—61no—
242C2×D8C22×D311yescharacteristic
243C22×C8—31no—
244C22×C8C22×D311yescharacteristic
245C4⋊Q8—61no—
246#127—61no—
247#126—121no—
248#125—61no—
249#124—61no—
250#124—121no—
251#123—121no—
252C22⋊Q8—21no—
253#121—21no—
254#121—61no—
255#121—61no—
256#121—121no—
257C4×Q8—61no—
258C4×D4—61no—
259C4×D4—61no—
260C4×D4—61no—
261C4×D4—121no—
262C4×D4—121no—
263#117—121no—
264C2×C42—31no—
265SD32—22no—
266SD32—62no—
267D16—62no—
268D16—62no—
269C2×C16—22no—
270C2×C16—31no—
271C2×C16—31no—
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—21no—
282#103—61no—
283#102—21no—
284#102—61no—
285#102—61no—
286#102—61no—
287#102—61no—
288#102—61no—
289C4×C8—31no—
290C4×C8—31no—
291C4×C8—31no—
292C4×C8—31no—
293C23×C3—41no—
294C23×C3C2×D811yescharacteristic
295C22×D3—41no—
296C22×D3—41no—
297C3×Q8—41no—
298C3×Q8—41no—
299C3×D4—41no—
300C3×D4—41no—
301C3×D4—82no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4C2×D811yescharacteristic
308C2×C3×C4C22×D411yescharacteristic
309#67—82no—
310#67—82no—
311#67—82no—
312#67—82no—
313C2×Dic3—22no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—82no—
318C2×Dic3C2×D811yescharacteristic
319C2×Dic3C2×D811yescharacteristic
320C2×Dic3C2×D811yescharacteristic
321C2×Dic3C2×D811yescharacteristic
322D12—41no—
323D12—41no—
324D12—41no—
325D12—41no—
326C4×D3—82no—
327C4×D3—82no—
328Dic6—82no—
329C3×C8—21no—
330C3×C8C2×#4111yescharacteristic
331C3×C8C2×#4111yescharacteristic
332C3⋊C8—21no—
333C3⋊C8#13512yes—
334#41—244no—
335C2×Q8—21no—
336C2×Q8—121no—
337C2×D4—21no—
338C2×D4—41no—
339C2×D4—61no—
340C2×D4—61no—
341C2×D4—121no—
342C2×D4—121no—
343C2×D4—121no—
344C2×D4—121no—
345C22×C4—31no—
346C22×C4—31no—
347C22×C4—121no—
348C22×C4—121no—
349C22×C4—121no—
350C22×C4D3×D411yescharacteristic
351Q16—21no—
352Q16—21no—
353SD16—122no—
354D8—21no—
355D8—21no—
356D8—61no—
357D8—61no—
358D8—61no—
359D8—61no—
360D8—122no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—31no—
364C2×C8—31no—
365C2×C8—62no—
366C2×C8D3×D411yescharacteristic
367C2×C8C23×D311yescharacteristic
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—41no—
371C4⋊C4—61no—
372C4⋊C4—61no—
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—31no—
390C42—121no—
391C16—21no—
392C16—32no—
393C16—61no—
394C16#23712yescomplemented
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—81no—
400C22×C3#50912yes—
401C22×C3C22×D811yescharacteristic
402C2×D3—81no—
403C2×D3—81no—
404C2×D3—81no—
405C2×D3—81no—
406C2×D3—81no—
407C2×D3—81no—
408C3×C4—21no—
409C3×C4—81no—
410C3×C4—81no—
411C3×C4—81no—
412C3×C4#53211yescharacteristic
413C3×C4C2×#13511yescharacteristic
414Dic3—21no—
415Dic3—21no—
416Dic3—21no—
417Dic3—21no—
418Dic3—81no—
419Dic3—81no—
420C23—41no—
421C23—121no—
422C23—121no—
423C23D3×D811yescharacteristic
424Q8—41no—
425Q8—41no—
426Q8—242no—
427D4—41no—
428D4—41no—
429D4—82no—
430D4—121no—
431D4—121no—
432D4—121no—
433D4—121no—
434D4—242no—
435D4—242no—
436D4—242no—
437D4—242no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—31no—
441C2×C4—31no—
442C2×C4—31no—
443C2×C4—31no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—62no—
448C2×C4—121no—
449C2×C4—121no—
450C2×C4—121no—
451C2×C4—242no—
452C2×C4—242no—
453C2×C4—242no—
454C2×C4D3×D811yescharacteristic
455C2×C4C2×D3×D411yescharacteristic
456C8—21no—
457C8—32no—
458C8—61no—
459C8C2×#23711yescharacteristic
460C8D3×#4111yescharacteristic
461C2×C3—22no—
462C2×C3—161no—
463C2×C3#316911yescharacteristic
464C2×C3C2×#50911yescharacteristic
465C2×C3#341411yescharacteristic
466D3—161no—
467D3—161no—
468C22—21no—
469C22—21no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—241no—
474C22—241no—
475C22—241no—
476C22—241no—
477C22—241no—
478C22—241no—
479C22#491412yes—
480C22C2×D3×D811yescharacteristic
481C4—21no—
482C4—61no—
483C4—61no—
484C4—61no—
485C4—61no—
486C4—81no—
487C4—81no—
488C4—81no—
489C4—241no—
490C4—241no—
491C4#559611yescharacteristic
492C4D3×#13511yescharacteristic
493C3#3374611yescharacteristic, complemented, Sylow
494C2—22no—
495C2—161no—
496C2—481no—
497C2—481no—
498C2#8069711yescharacteristic
499C2C2×#491411yescharacteristic
500C2#8203011yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2217C2
socleC22×C3C22×D8
hypercenter#2217D3
hypocenterC3#33746
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2217

Complements

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

NH ≅ G/Ntype
1. #80740C2semidirect
2. #80697C2semidirect
3. C3×#2217C2semidirect
4. #71481C2semidirect
5. #71474C2semidirect
6. #71413C2semidirect
7. C2×#4502C2semidirect
8. #71082C2semidirect
9. #71071C2semidirect
10. #69341C2semidirect
11. #69339C2semidirect
12. #69217C2semidirect
13. #69208C2semidirect
14. #69186C2semidirect
15. #69177C2semidirect
67. #2217D3semidirect
233. C3×C16#41semidirect
235. C3⋊C16#41semidirect
394. C16#237semidirect
493. C3#33746semidirect

Maximal subgroups

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

subgroupindex
#807402
#806972
C3×#22172
#714812
#714742
#714132
C2×#45022
#710822
#710712
#693412
#693392
#692172
#692082
#691862
#691772
#337463