Finite Group Databasebeta

← #10950840

Subgroups — #10950840

Order 768.

520 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81935C211yesmaximal, characteristic, complemented
2#81776C211yesmaximal, characteristic, complemented
3C3×#2240C211yesmaximal, characteristic, complemented
4#71386C211yesmaximal, characteristic, complemented
5#71292C211yesmaximal, characteristic, complemented
6#69198C211yesmaximal, characteristic, complemented
7#69197C211yesmaximal, characteristic, complemented
8#69196C211yesmaximal, characteristic, complemented
9#69195C211yesmaximal, characteristic, complemented
10#69177C211yesmaximal, characteristic, complemented
11#69176C211yesmaximal, characteristic, complemented
12#69175C211yesmaximal, characteristic, complemented
13#69174C211yesmaximal, characteristic, complemented
14#69059C211yesmaximal, characteristic, complemented
15#69051C211yesmaximal, characteristic, complemented
16#33744—31nomaximal, Sylow
17#5733—21no—
18#5719—21no—
19C3×#500C2211yescharacteristic
20C3×#493C2211yescharacteristic
21#5159—21no—
22#5157C2211yescharacteristic
23#5157C2211yescharacteristic
24#5152—21no—
25Dic3×D8C2211yescharacteristic
26Dic3×D8C2211yescharacteristic
27C2×#827C2211yescharacteristic
28C2×#827C2211yescharacteristic
29C2×#826C2211yescharacteristic
30C2×#826C2211yescharacteristic
31#5089—21no—
32#5071—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×#357C2211yescharacteristic
45C3×#357C2211yescharacteristic
46C3×#357C2211yescharacteristic
47C3×#357C2211yescharacteristic
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#3176—31no—
61#3175—31no—
62#2255—31no—
63#2250—31no—
64#2242—31no—
65#2240D311yescharacteristic, complemented
66#2220—31no—
67#2218—31no—
68#2217—31no—
69#2215—31no—
70#2177—31no—
71#2175—31no—
72C4×SD32—31no—
73C4×D16—31no—
74#2108—31no—
75C2×C3×D8—21no—
76C2×C3×D8C2311yescharacteristic
77C2×C3×D8C2311yescharacteristic
78C3×C4⋊Q8—21no—
79C3×#127—21no—
80#947—41no—
81#946—41no—
82#939—41no—
83#939—41no—
84#938—21no—
85#938—21no—
86#935—41no—
87#935—41no—
88D4×Dic3—21no—
89D4×Dic3—21no—
90C2×Dic12C2311yescharacteristic
91C2×D24C2311yescharacteristic
92C2×#204—21no—
93C2×ΓL(1,𝔽25)D411yescharacteristic
94C2×D3×C8C2311yescharacteristic
95#898—41no—
96#897—21no—
97#895—21no—
98#894—41no—
99#893—21no—
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#832—21no—
111#832—21no—
112#832—21no—
113#832—21no—
114#827—22no—
115#827—22no—
116#826—22no—
117#826—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#540—61no—
134C2×SD32—31no—
135C2×SD32—31no—
136C2×D16—31no—
137C2×D16—31no—
138C2×M32—31no—
139C22×C16—31no—
140#500C2×D311yescharacteristic
141#499—31no—
142#495—31no—
143#493C2×D311yescharacteristic
144#490—61no—
145#489—61no—
146#469—61no—
147#468—61no—
148#467—31no—
149#466—31no—
150#466—31no—
151#466—31no—
152#442—61no—
153#440—61no—
154C4×Q16—31no—
155C4×D8—31no—
156C4×D8—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#357—31no—
166#357—31no—
167#357C2×D311yescharacteristic
168#357C2×D311yescharacteristic
169#357C2×D311yescharacteristic
170#357C2×D311yescharacteristic
171#346—31no—
172#346—31no—
173C4×C16—31no—
174C4×C16C2×D311yescharacteristic
175C2×C3×Q8—41no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4—41no—
179C2×C3×D4—41no—
180C2×#67—41no—
181C2×#67—41no—
182C22×Dic3—41no—
183C22×Dic3—41no—
184C2×D12—21no—
185C2×D12—41no—
186C2×C4×D3—41no—
187C2×C4×D3C2×D411yescharacteristic
188C2×Dic6—21no—
189C3×D8—21no—
190C3×D8—21no—
191C3×D8—21no—
192C3×D8—21no—
193C3×D8—42no—
194C2×C3×C8C2×D411yescharacteristic
195C2×C3×C8C2411yescharacteristic
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—41no—
199C3×C42C2×D411yescharacteristic
200#217—41no—
201#217—41no—
202#217—41no—
203#217—41no—
204(C2×D3)⋊C4—41no—
205(C2×D3)⋊C4—41no—
206(C2×D3)⋊C4—41no—
207(C2×D3)⋊C4—41no—
208(C2×D3)⋊C4—41no—
209(C2×D3)⋊C4D811yescharacteristic
210(C2×D3)⋊C4D811yescharacteristic
211C4⋊Dic3—21no—
212C4⋊Dic3—21no—
213C4⋊Dic3—41no—
214Dic3⋊C4—41no—
215Dic3⋊C4—41no—
216Dic3⋊C4—41no—
217Dic3⋊C4D811yescharacteristic
218Dic3⋊C4D811yescharacteristic
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—31no—
236C2×SD16—61no—
237C2×D8—21no—
238C2×D8—31no—
239C2×D8C22×D311yescharacteristic
240C2×D8C22×D311yescharacteristic
241C2×M16—31no—
242C22×C8—31no—
243C4⋊Q8—21no—
244#127—21no—
245#126—121no—
246#125—61no—
247#124—61no—
248#124—121no—
249#123—121no—
250#123—121no—
251#123—121no—
252C22⋊Q8—61no—
253C22⋊Q8—121no—
254#121—61no—
255#121—61no—
256#121—61no—
257#121—121no—
258#121—121no—
259C4×Q8—61no—
260C4×D4—61no—
261C4×D4—61no—
262C4×D4—61no—
263#117—31no—
264SD32—62no—
265SD32—62no—
266D16—62no—
267D16—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#102—61no—
283#102—61no—
284#102—61no—
285#102—61no—
286#102—61no—
287#102—61no—
288#97—31no—
289#97—31no—
290C4×C8—31no—
291C4×C8C22×D311yescharacteristic
292C23×C3—41no—
293C23×C3—41no—
294C22×D3—41no—
295C22×D3C2×D811yescharacteristic
296C3×Q8—161no—
297C3×D4—41no—
298C3×D4—41no—
299C3×D4—41no—
300C3×D4—41no—
301C3×D4—161no—
302C3×D4—161no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4C2×D811yescharacteristic
306C2×C3×C4C2×D811yescharacteristic
307C2×C3×C4C22×D411yescharacteristic
308#67—161no—
309#67—161no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3—161no—
314C2×Dic3—161no—
315C2×Dic3C2×D811yescharacteristic
316C2×Dic3C2×D811yescharacteristic
317C2×Dic3C2×D811yescharacteristic
318D12—41no—
319D12—41no—
320D12—161no—
321C4×D3—21no—
322C4×D3—21no—
323C4×D3—161no—
324Dic6—41no—
325Dic6—41no—
326C3×C8#13512yes—
327C3×C8C2×#4111yescharacteristic
328C3×C8C2×#4111yescharacteristic
329C3⋊C8—21no—
330C3⋊C8—21no—
331C2×Q8—41no—
332C2×Q8—61no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—61no—
338C2×D4—121no—
339C2×D4—121no—
340C2×D4—121no—
341C22×C4—31no—
342C22×C4—121no—
343C22×C4—121no—
344C22×C4—121no—
345Q16—61no—
346Q16—61no—
347SD16—122no—
348D8—21no—
349D8—21no—
350D8—21no—
351D8—21no—
352D8—42no—
353D8—61no—
354D8—61no—
355M16—62no—
356C2×C8—31no—
357C2×C8—31no—
358C2×C8—61no—
359C2×C8—61no—
360C2×C8D3×D411yescharacteristic
361C2×C8C23×D311yescharacteristic
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—31no—
365C4⋊C4—31no—
366C4⋊C4—41no—
367C4⋊C4—61no—
368C4⋊C4—61no—
369C4⋊C4—121no—
370C4⋊C4—121no—
371C4⋊C4—121no—
372C4⋊C4—121no—
373C22⋊C4—31no—
374C22⋊C4—31no—
375C22⋊C4—121no—
376C22⋊C4—121no—
377C22⋊C4—121no—
378C22⋊C4—121no—
379C22⋊C4—121no—
380C22⋊C4—121no—
381C22⋊C4—121no—
382C22⋊C4—121no—
383C22⋊C4—121no—
384C42—31no—
385C42—121no—
386C42D3×D411yescharacteristic
387C16—21no—
388C16—21no—
389C16—32no—
390C16—32no—
391C22×C3—81no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3—81no—
395C22×C3—81no—
396C22×C3—81no—
397C22×C3C22×D811yescharacteristic
398C2×D3—21no—
399C2×D3—21no—
400C2×D3—21no—
401C2×D3—81no—
402C2×D3—81no—
403C2×D3—81no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—81no—
407C3×C4—81no—
408C3×C4#53211yescharacteristic
409C3×C4C2×#13511yescharacteristic
410Dic3—21no—
411Dic3—21no—
412Dic3—21no—
413Dic3—81no—
414Dic3—81no—
415Dic3—81no—
416C23—31no—
417C23—41no—
418C23—41no—
419C23—121no—
420Q8—121no—
421Q8—121no—
422Q8—161no—
423D4—41no—
424D4—41no—
425D4—41no—
426D4—41no—
427D4—121no—
428D4—121no—
429D4—161no—
430D4—161no—
431D4—481no—
432D4—481no—
433D4—481no—
434C2×C4—31no—
435C2×C4—31no—
436C2×C4—31no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—61no—
440C2×C4—61no—
441C2×C4—121no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4—481no—
445C2×C4—481no—
446C2×C4—481no—
447C2×C4D3×D811yescharacteristic
448C2×C4D3×D811yescharacteristic
449C2×C4C2×D3×D411yescharacteristic
450C8—61no—
451C8—61no—
452C8#91612yes—
453C8D3×#4111yescharacteristic
454C8D3×#4111yescharacteristic
455C2×C3—161no—
456C2×C3—161no—
457C2×C3#316911yescharacteristic
458C2×C3#341511yescharacteristic
459C2×C3#341611yescharacteristic
460D3—41no—
461D3—161no—
462C22—61no—
463C22—61no—
464C22—61no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—241no—
472C22—241no—
473C22—241no—
474C22C2×D3×D811yescharacteristic
475C4—21no—
476C4—21no—
477C4—61no—
478C4—61no—
479C4—61no—
480C4—81no—
481C4—81no—
482C4—241no—
483C4—241no—
484C4—241no—
485C4#566811yescharacteristic
486C4C2×#91611yescharacteristic
487C3#3374411yescharacteristic, complemented, Sylow
488C2—121no—
489C2—161no—
490C2—161no—
491C2—481no—
492C2#8155711yescharacteristic
493C2#8201911yescharacteristic
494C2#8202411yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2240C2
socleC22×C3C22×D8
hypercenter#2240D3
hypocenterC3#33744
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2240

Complements

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

NH ≅ G/Ntype
1. #81935C2semidirect
2. #81776C2semidirect
3. C3×#2240C2semidirect
4. #71386C2semidirect
5. #71292C2semidirect
6. #69198C2semidirect
7. #69197C2semidirect
8. #69196C2semidirect
9. #69195C2semidirect
10. #69177C2semidirect
11. #69176C2semidirect
12. #69175C2semidirect
13. #69174C2semidirect
14. #69059C2semidirect
15. #69051C2semidirect
65. #2240D3semidirect
487. C3#33744semidirect

Maximal subgroups

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

subgroupindex
#819352
#817762
C3×#22402
#713862
#712922
#691982
#691972
#691962
#691952
#691772
#691762
#691752
#691742
#690592
#690512
#337443