Finite Group Databasebeta

← #10950855

Subgroups — #10950855

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#81935C211yesmaximal, characteristic
2#81793C211yesmaximal, characteristic, complemented
3C3×#2241C211yesmaximal, characteristic, complemented
4#71386C211yesmaximal, characteristic, complemented
5#71314C211yesmaximal, characteristic, complemented
6#69230C211yesmaximal, characteristic
7#69229C211yesmaximal, characteristic, complemented
8#69228C211yesmaximal, characteristic
9#69227C211yesmaximal, characteristic, complemented
10#69209C211yesmaximal, characteristic, complemented
11#69208C211yesmaximal, characteristic, complemented
12#69207C211yesmaximal, characteristic, complemented
13#69206C211yesmaximal, characteristic, complemented
14#69059C211yesmaximal, characteristic, complemented
15#69051C211yesmaximal, characteristic, complemented
16#33745—31nomaximal, Sylow
17#5733—21no—
18#5731—21no—
19C3×#500C2211yescharacteristic
20C3×#494C2211yescharacteristic
21#5189—21no—
22#5188C2211yescharacteristic
23#5188C2211yescharacteristic
24#5184—21no—
25Dic3×Q16C2211yescharacteristic
26Dic3×Q16C2211yescharacteristic
27C2×#829C2211yescharacteristic
28C2×#829C2211yescharacteristic
29C2×#828C2211yescharacteristic
30C2×#828C2211yescharacteristic
31#5089—21no—
32#5082—21no—
33C2×#798C2211yescharacteristic
34C2×D3×C16C2211yescharacteristic
35#4889—21no—
36#4887C2211yescharacteristic
37#4886—21no—
38#4883C2211yescharacteristic
39#4873C2211yescharacteristic
40#4872C2211yescharacteristic
41#4696C2211yescharacteristic
42#4689C2211yescharacteristic
43#4683C2211yescharacteristic
44C3×#358C2211yescharacteristic
45C3×#358C2211yescharacteristic
46C3×#358C2211yescharacteristic
47C3×#358C2211yescharacteristic
48C3×C4×C16C2211yescharacteristic
49#4509C2211yescharacteristic
50#4509C2211yescharacteristic
51#4503C2211yescharacteristic
52#4503C2211yescharacteristic
53#4502C2211yescharacteristic
54Dic3×C16C2211yescharacteristic
55#4490C2211yescharacteristic
56#4490C2211yescharacteristic
57#4489C2211yescharacteristic
58#4489C2211yescharacteristic
59#4461C2211yescharacteristic
60#3177—31no—
61#3175—31no—
62#2255—31no—
63#2251—31no—
64#2242—31no—
65#2241D311yescharacteristic, complemented
66#2220—31no—
67#2219—31no—
68#2217—31no—
69#2216—31no—
70#2176—31no—
71#2175—31no—
72C4×Q32—31no—
73C4×SD32—31no—
74#2108—31no—
75C2×C3×Q16—21no—
76C2×C3×Q16C2311yescharacteristic
77C2×C3×Q16C2311yescharacteristic
78C3×C4⋊Q8—21no—
79C3×C4⋊Q8—21no—
80#947—41no—
81#946—21no—
82#946—21no—
83#946—41no—
84Q8×Dic3—21no—
85Q8×Dic3—21no—
86C2×Dic12C2311yescharacteristic
87C2×D24C2311yescharacteristic
88C2×#204—21no—
89C2×ΓL(1,𝔽25)D411yescharacteristic
90C2×D3×C8C2311yescharacteristic
91#898—41no—
92#898—41no—
93#898—41no—
94#897—21no—
95#895—21no—
96#894—41no—
97#893—21no—
98#889—41no—
99#889—41no—
100#887—21no—
101#875—21no—
102#872D411yescharacteristic
103#870—21no—
104C2×C3×C16C2311yescharacteristic
105C2×C3×C16C2311yescharacteristic
106C3×#107—21no—
107C3×#107C2311yescharacteristic
108C3×#107C2311yescharacteristic
109C3×C4×C8C2311yescharacteristic
110#835—21no—
111#835—21no—
112#835—21no—
113#835—21no—
114#829—22no—
115#829—22no—
116#828—22no—
117#828—22no—
118#818C2311yescharacteristic
119#818C2311yescharacteristic
120#817—21no—
121#815D411yescharacteristic
122C8×Dic3C2311yescharacteristic
123C2×C3⋊C16C2311yescharacteristic
124C2×C3⋊C16C2311yescharacteristic
125#810—21no—
126#810—21no—
127#809—21no—
128#809—21no—
129#803D411yescharacteristic
130#798—22no—
131D3×C16—22no—
132#542—61no—
133#541—61no—
134C2×Q32—31no—
135C2×Q32—31no—
136C2×SD32—31no—
137C2×SD32—31no—
138C2×M32—31no—
139C22×C16—31no—
140#500C2×D311yescharacteristic
141#499—31no—
142#495—31no—
143#494C2×D311yescharacteristic
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×Q16—31no—
156C4×Q16—31no—
157C4×D8—31no—
158#405—31no—
159#367—31no—
160#367—31no—
161#366—31no—
162#366—31no—
163#358—31no—
164#358—31no—
165#358C2×D311yescharacteristic
166#358C2×D311yescharacteristic
167#358C2×D311yescharacteristic
168#358C2×D311yescharacteristic
169#357—31no—
170#357—31no—
171#346—31no—
172#346—31no—
173C4×C16—31no—
174C4×C16C2×D311yescharacteristic
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177C2×C3×Q8—41no—
178C2×D12—21no—
179C2×D12—41no—
180C2×C4×D3—41no—
181C2×C4×D3C2×D411yescharacteristic
182C2×Dic6—21no—
183C3×Q16—21no—
184C3×Q16—21no—
185C3×Q16—21no—
186C3×Q16—21no—
187C3×Q16—42no—
188C2×C3×C8C2×D411yescharacteristic
189C2×C3×C8C2411yescharacteristic
190C3×C4⋊C4—21no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—41no—
193C3×C4⋊C4—41no—
194C3×C4⋊C4—41no—
195C3×C42C2×D411yescharacteristic
196(C2×D3)⋊C4—41no—
197(C2×D3)⋊C4—41no—
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)⋊C4D811yescharacteristic
204(C2×D3)⋊C4D811yescharacteristic
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207C4⋊Dic3—41no—
208C4⋊Dic3—41no—
209C4⋊Dic3—41no—
210Dic3⋊C4—41no—
211Dic3⋊C4—41no—
212Dic3⋊C4—41no—
213Dic3⋊C4—41no—
214Dic3⋊C4—41no—
215Dic3⋊C4D811yescharacteristic
216Dic3⋊C4D811yescharacteristic
217C4×Dic3—41no—
218C4×Dic3—41no—
219C4×Dic3—41no—
220C4×Dic3C2×D411yescharacteristic
221C2×C3⋊C8C2×D411yescharacteristic
222C2×C3⋊C8C2×D411yescharacteristic
223Dic12—21no—
224Dic12—21no—
225D24—21no—
226D24—21no—
227#204—42no—
228ΓL(1,𝔽25)—22no—
229D3×C8—21no—
230D3×C8—21no—
231C3×C16—21no—
232C3×C16—21no—
233C3⋊C16#4112yes—
234C3⋊C16#4112yes—
235C2×Q16—21no—
236C2×Q16—31no—
237C2×Q16C22×D311yescharacteristic
238C2×Q16C22×D311yescharacteristic
239C2×SD16—61no—
240C2×D8—31no—
241C2×M16—31no—
242C22×C8—31no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
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—61no—
260C4×Q8—61no—
261C4×Q8—61no—
262C4×D4—61no—
263#117—31no—
264Q32—62no—
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—21no—
275#107—31no—
276#107—31no—
277#107C22×D311yescharacteristic
278#107C22×D311yescharacteristic
279#106—61no—
280#103—61no—
281#103—61no—
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—
298C3×Q8—161no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4C2×D811yescharacteristic
304C2×C3×C4C2×D811yescharacteristic
305C2×C3×C4C22×D411yescharacteristic
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3C2×D811yescharacteristic
310C2×Dic3C2×D811yescharacteristic
311C2×Dic3C2×D811yescharacteristic
312D12—41no—
313D12—41no—
314D12—161no—
315C4×D3—21no—
316C4×D3—21no—
317C4×D3—161no—
318Dic6—41no—
319Dic6—41no—
320C3×C8#13512yes—
321C3×C8C2×#4111yescharacteristic
322C3×C8C2×#4111yescharacteristic
323C3⋊C8—21no—
324C3⋊C8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—41no—
328C2×Q8—61no—
329C2×D4—61no—
330C2×D4—121no—
331C22×C4—31no—
332C22×C4—121no—
333Q16—21no—
334Q16—21no—
335Q16—21no—
336Q16—21no—
337Q16—42no—
338Q16—61no—
339Q16—61no—
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—41no—
357C4⋊C4—61no—
358C4⋊C4—61no—
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—121no—
378C42—121no—
379C42—121no—
380C42D3×D411yescharacteristic
381C16—21no—
382C16—21no—
383C16—32no—
384C16—32no—
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—161no—
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—61no—
435C8—61no—
436C8#91612yes—
437C8D3×#4111yescharacteristic
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#566811yescharacteristic
464C4C2×#91611yescharacteristic
465C3#3374511yescharacteristic, complemented, Sylow
466C2—121no—
467C2—481no—
468C2#8155711yescharacteristic
469C2#8202411yescharacteristic
470C2#8202811yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2241C2
socleC22×C3C22×D8
prime radical#4883C22
hypercenter#2241D3
hypocenterC3#33745
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2241

Complements

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

NH ≅ G/Ntype
2. #81793C2semidirect
3. C3×#2241C2semidirect
4. #71386C2semidirect
5. #71314C2semidirect
7. #69229C2semidirect
9. #69227C2semidirect
10. #69209C2semidirect
11. #69208C2semidirect
12. #69207C2semidirect
13. #69206C2semidirect
14. #69059C2semidirect
15. #69051C2semidirect
65. #2241D3semidirect
465. C3#33745semidirect

Maximal subgroups

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

subgroupindex
#819352
#817932
C3×#22412
#713862
#713142
#692302
#692292
#692282
#692272
#692092
#692082
#692072
#692062
#690592
#690512
#337453