Finite Group Databasebeta

← #10950846

Subgroups — #10950846

Order 768.

496 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79728C211yesmaximal, characteristic, complemented
2#79727C211yesmaximal, characteristic
3C3×#2176C211yesmaximal, characteristic, complemented
4#71504C211yesmaximal, characteristic, complemented
5#71501C211yesmaximal, characteristic, complemented
6C4×#829C211yesmaximal, characteristic, complemented
7C4×#828C211yesmaximal, characteristic, complemented
8#69347C211yesmaximal, characteristic
9#69341C211yesmaximal, characteristic, complemented
10#69230C211yesmaximal, characteristic
11#69229C211yesmaximal, characteristic, complemented
12#69216C211yesmaximal, characteristic, complemented
13#69215C211yesmaximal, characteristic, complemented
14#69077C211yesmaximal, characteristic, complemented
15#69072C211yesmaximal, characteristic, complemented
16#33745—31nomaximal, Sylow
17#5583—21no—
18#5582—21no—
19C2×C3×Q32C2211yescharacteristic
20C3×C4×Q16C2211yescharacteristic
21C3×C4×Q16C2211yescharacteristic
22#5191C2211yescharacteristic
23#5188C2211yescharacteristic
24#5188C2211yescharacteristic
25#5183C2211yescharacteristic
26C2×#829C2211yescharacteristic
27C2×#828C2211yescharacteristic
28#5030—21no—
29#5026—21no—
30C2×#800C2211yescharacteristic
31C2×#798C2211yescharacteristic
32C2×D3×C16C2211yescharacteristic
33#4887C2211yescharacteristic
34#4883C2211yescharacteristic
35#4878C2211yescharacteristic
36#4874C2211yescharacteristic
37#4816—21no—
38#4812—21no—
39#4797—21no—
40#4794—21no—
41C4×Dic12C2211yescharacteristic
42C4×D24C2211yescharacteristic
43#4689C2211yescharacteristic
44C3×#367C2211yescharacteristic
45C3×#358C2211yescharacteristic
46C3×#358C2211yescharacteristic
47C3×#346C2211yescharacteristic
48#4564C2211yescharacteristic
49#4564C2211yescharacteristic
50#4509C2211yescharacteristic
51#4504C2211yescharacteristic
52#4503C2211yescharacteristic
53#4502C2211yescharacteristic
54Dic3×C16C2211yescharacteristic
55#4492C2211yescharacteristic
56#4491C2211yescharacteristic
57#4490C2211yescharacteristic
58#4489C2211yescharacteristic
59C4×C3⋊C16C2211yescharacteristic
60#3177—31no—
61#3175—31no—
62#2255—31no—
63#2251—31no—
64#2242—31no—
65#2241—31no—
66#2220—31no—
67#2219—31no—
68#2217—31no—
69#2216—31no—
70#2176D311yescharacteristic, complemented
71#2175—31no—
72C4×Q32—31no—
73C4×SD32—31no—
74#2108—31no—
75C2×C3×Q16C2311yescharacteristic
76C2×C3×Q16C2311yescharacteristic
77C3×C4×Q8—21no—
78C3×C4×Q8—21no—
79#947—21no—
80#946—21no—
81#946—21no—
82#944—21no—
83C2×#216—21no—
84C2×#215—21no—
85C2×Dic12C2311yescharacteristic
86C2×D24C2311yescharacteristic
87C2×ΓL(1,𝔽25)D411yescharacteristic
88C2×D3×C8C2311yescharacteristic
89#898—41no—
90#897—21no—
91#896—41no—
92#895—21no—
93#894—41no—
94#890—21no—
95#889—41no—
96#888—21no—
97#876—41no—
98#876—41no—
99#875—41no—
100C4×D12—21no—
101#872D411yescharacteristic
102#870—41no—
103C4×Dic6—21no—
104C3×Q32—22no—
105C2×C3×C16C2311yescharacteristic
106C2×C3×C16C2311yescharacteristic
107C3×#107C2311yescharacteristic
108C3×#107C2311yescharacteristic
109C3×#103—21no—
110C3×#103—21no—
111C3×C4×C8C2311yescharacteristic
112#835—21no—
113#835—21no—
114#829—22no—
115#828—22no—
116#821—21no—
117#818C2311yescharacteristic
118#818C2311yescharacteristic
119#816—21no—
120#815D411yescharacteristic
121C8×Dic3C2311yescharacteristic
122C2×C3⋊C16C2311yescharacteristic
123C2×C3⋊C16C2311yescharacteristic
124#810—21no—
125#809—21no—
126#808—21no—
127#807—21no—
128#803D411yescharacteristic
129#800—22no—
130#798—22no—
131D3×C16—22no—
132#542—61no—
133#541—61no—
134C2×Q32—31no—
135C2×Q32C2×D311yescharacteristic
136C2×SD32—31no—
137C2×SD32—31no—
138C2×M32—31no—
139C22×C16—31no—
140#500—31no—
141#499—31no—
142#495—31no—
143#494—31no—
144#491—61no—
145#489—61no—
146#470—61no—
147#468—61no—
148#467—31no—
149#467—31no—
150#467—31no—
151#466—31no—
152#441—61no—
153#440—61no—
154C4×Q16—31no—
155C4×Q16C2×D311yescharacteristic
156C4×Q16C2×D311yescharacteristic
157C4×D8—31no—
158#405—31no—
159#367—31no—
160#367C2×D311yescharacteristic
161#366—31no—
162#366—31no—
163#358—31no—
164#358—31no—
165#358—31no—
166#358—31no—
167#358C2×D311yescharacteristic
168#358C2×D311yescharacteristic
169#357—31no—
170#357—31no—
171#346—31no—
172#346C2×D311yescharacteristic
173C4×C16—31no—
174C4×C16—31no—
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177C2×D12—21no—
178C2×D12—41no—
179C2×C4×D3—41no—
180C2×C4×D3C2×D411yescharacteristic
181C2×Dic6—21no—
182C2×Dic6—41no—
183C3×Q16—21no—
184C3×Q16—21no—
185C3×Q16—21no—
186C3×Q16—21no—
187C2×C3×C8C2×D411yescharacteristic
188C2×C3×C8C2411yescharacteristic
189C3×C4⋊C4—21no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—41no—
192C3×C4⋊C4—41no—
193C3×C42—41no—
194C3×C42—41no—
195C3×C42C2×D411yescharacteristic
196#216—42no—
197#215—42no—
198(C2×D3)⋊C4—41no—
199(C2×D3)⋊C4—41no—
200(C2×D3)⋊C4—41no—
201(C2×D3)⋊C4—41no—
202(C2×D3)⋊C4—41no—
203(C2×D3)⋊C4—41no—
204(C2×D3)⋊C4—41no—
205(C2×D3)⋊C4D811yescharacteristic
206(C2×D3)⋊C4D811yescharacteristic
207C4⋊Dic3—21no—
208C4⋊Dic3—21no—
209C4⋊Dic3—41no—
210C4⋊Dic3—41no—
211Dic3⋊C4—41no—
212Dic3⋊C4—41no—
213Dic3⋊C4—41no—
214Dic3⋊C4—41no—
215Dic3⋊C4—41no—
216Dic3⋊C4—41no—
217Dic3⋊C4—41no—
218Dic3⋊C4D811yescharacteristic
219Dic3⋊C4D811yescharacteristic
220C4×Dic3—41no—
221C4×Dic3C2×D411yescharacteristic
222C2×C3⋊C8C2×D411yescharacteristic
223C2×C3⋊C8C2×D411yescharacteristic
224Dic12—21no—
225Dic12—21no—
226D24—21no—
227D24—21no—
228ΓL(1,𝔽25)—22no—
229D3×C8—21no—
230D3×C8—21no—
231C3×C16—21no—
232C3×C16#4112yes—
233C3⋊C16—21no—
234C3⋊C16#4112yes—
235C2×Q16—31no—
236C2×Q16—61no—
237C2×Q16C22×D311yescharacteristic
238C2×Q16C22×D311yescharacteristic
239C2×SD16—61no—
240C2×D8—31no—
241C2×M16—31no—
242C22×C8—31no—
243C4⋊Q8—61no—
244C4⋊Q8—61no—
245#126—121no—
246#126—121no—
247#126—121no—
248#125—61no—
249#125—121no—
250#125—121no—
251#124—61no—
252#124—121no—
253#123—121no—
254C22⋊Q8—61no—
255C22⋊Q8—61no—
256C22⋊Q8—61no—
257C22⋊Q8—121no—
258#121—61no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—61no—
262C4×D4—61no—
263#117—31no—
264Q32—22no—
265Q32—62no—
266SD32—62no—
267SD32—62no—
268M32—62no—
269C2×C16—31no—
270C2×C16—31no—
271C2×C16—62no—
272C2×C16C22×D311yescharacteristic
273C2×C16C22×D311yescharacteristic
274#107—31no—
275#107—31no—
276#107—61no—
277#107C22×D311yescharacteristic
278#107C22×D311yescharacteristic
279#106—61no—
280#103—21no—
281#103—21no—
282#103—61no—
283#103—61no—
284#103—61no—
285#103—61no—
286#102—61no—
287#102—61no—
288#97—31no—
289#97—31no—
290C4×C8—31no—
291C4×C8C22×D311yescharacteristic
292C22×D3—41no—
293C22×D3C2×D811yescharacteristic
294C3×Q8—41no—
295C3×Q8—41no—
296C3×Q8—41no—
297C3×Q8—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4C2×D811yescharacteristic
303C2×C3×C4C2×D811yescharacteristic
304C2×C3×C4C22×D411yescharacteristic
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3C2×D811yescharacteristic
309C2×Dic3C2×D811yescharacteristic
310C2×Dic3C2×D811yescharacteristic
311D12—41no—
312D12—41no—
313D12—161no—
314C4×D3—21no—
315C4×D3—21no—
316C4×D3—161no—
317Dic6—41no—
318Dic6—41no—
319Dic6—161no—
320C3×C8—21no—
321C3×C8C2×#4111yescharacteristic
322C3×C8C2×#4111yescharacteristic
323C3⋊C8—21no—
324C3⋊C8#13512yes—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—61no—
328C2×Q8—121no—
329C2×D4—61no—
330C2×D4—121no—
331C22×C4—31no—
332C22×C4—121no—
333Q16—21no—
334Q16—21no—
335Q16—21no—
336Q16—21no—
337Q16—61no—
338Q16—61no—
339Q16—122no—
340SD16—122no—
341D8—61no—
342D8—61no—
343M16—62no—
344C2×C8—31no—
345C2×C8—31no—
346C2×C8—61no—
347C2×C8—61no—
348C2×C8D3×D411yescharacteristic
349C2×C8C23×D311yescharacteristic
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—31no—
353C4⋊C4—31no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—61no—
357C4⋊C4—61no—
358C4⋊C4—121no—
359C4⋊C4—121no—
360C4⋊C4—121no—
361C4⋊C4—121no—
362C4⋊C4—121no—
363C4⋊C4—121no—
364C4⋊C4—121no—
365C4⋊C4—121no—
366C4⋊C4—121no—
367C22⋊C4—31no—
368C22⋊C4—31no—
369C22⋊C4—121no—
370C22⋊C4—121no—
371C22⋊C4—121no—
372C22⋊C4—121no—
373C22⋊C4—121no—
374C22⋊C4—121no—
375C22⋊C4—121no—
376C42—31no—
377C42—41no—
378C42—41no—
379C42—121no—
380C42D3×D411yescharacteristic
381C16—21no—
382C16—32no—
383C16—61no—
384C16#23512yes—
385C22×C3C22×D811yescharacteristic
386C2×D3—21no—
387C2×D3—21no—
388C2×D3—21no—
389C2×D3—81no—
390C2×D3—81no—
391C2×D3—81no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—81no—
395C3×C4—81no—
396C3×C4—81no—
397C3×C4—81no—
398C3×C4#53211yescharacteristic
399C3×C4C2×#13511yescharacteristic
400Dic3—21no—
401Dic3—21no—
402Dic3—21no—
403Dic3—81no—
404Dic3—81no—
405Dic3—81no—
406C23—31no—
407C23—121no—
408Q8—41no—
409Q8—41no—
410Q8—41no—
411Q8—41no—
412Q8—121no—
413Q8—121no—
414Q8—481no—
415D4—121no—
416D4—121no—
417D4—481no—
418C2×C4—31no—
419C2×C4—31no—
420C2×C4—31no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—61no—
426C2×C4—61no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—481no—
431C2×C4D3×D811yescharacteristic
432C2×C4D3×D811yescharacteristic
433C2×C4C2×D3×D411yescharacteristic
434C8—21no—
435C8—32no—
436C8—61no—
437C8C2×#23511yescharacteristic
438C8D3×#4111yescharacteristic
439C2×C3#316911yescharacteristic
440C2×C3#341611yescharacteristic
441C2×C3#341711yescharacteristic
442D3—41no—
443D3—161no—
444C22—61no—
445C22—61no—
446C22—61no—
447C22—241no—
448C22—241no—
449C22—241no—
450C22C2×D3×D811yescharacteristic
451C4—21no—
452C4—21no—
453C4—61no—
454C4—61no—
455C4—61no—
456C4—81no—
457C4—81no—
458C4—81no—
459C4—81no—
460C4—241no—
461C4—241no—
462C4—241no—
463C4#554811yescharacteristic
464C4D3×#13511yescharacteristic
465C3#3374511yescharacteristic, complemented, Sylow
466C2—121no—
467C2—481no—
468C2#7959811yescharacteristic
469C2#8204011yescharacteristic
470C2#8204111yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2176C2
socleC22×C3C22×D8
prime radical#4883C22
hypercenter#2176D3
hypocenterC3#33745
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2176

Complements

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

NH ≅ G/Ntype
1. #79728C2semidirect
3. C3×#2176C2semidirect
4. #71504C2semidirect
5. #71501C2semidirect
6. C4×#829C2semidirect
7. C4×#828C2semidirect
9. #69341C2semidirect
11. #69229C2semidirect
12. #69216C2semidirect
13. #69215C2semidirect
14. #69077C2semidirect
15. #69072C2semidirect
70. #2176D3semidirect
465. C3#33745semidirect

Maximal subgroups

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

subgroupindex
#797282
#797272
C3×#21762
#715042
#715012
C4×#8292
C4×#8282
#693472
#693412
#692302
#692292
#692162
#692152
#690772
#690722
#337453