Finite Group Databasebeta

← #10951978

Subgroups — #10951978

Order 768.

668 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82036C212yesmaximal, complemented
2#81791C211yesmaximal, characteristic, complemented
3#79894C211yesmaximal, characteristic, complemented
4C3×#2213C211yesmaximal, characteristic, complemented
5#71517C212yesmaximal, complemented
6#71290C211yesmaximal, characteristic, complemented
7#70921C211yesmaximal, characteristic, complemented
8#69305C211yesmaximal, characteristic, complemented
9#69300C211yesmaximal, characteristic, complemented
10#69260C212yesmaximal, complemented
11#69258C212yesmaximal, complemented
12#33858—31nomaximal, Sylow
13#5769C2212yescomplemented
14#5757—21no—
15#5757C2212yescomplemented
16#5752—21no—
17C3×#509C2212yescomplemented
18C3×#493C2211yescharacteristic, complemented
19C3×#443C2211yescharacteristic, complemented
20#5194C2211yescharacteristic
21#5160—21no—
22#5160C2212yescomplemented
23C2×#826C2212yescomplemented
24C2×#826C2212yescomplemented
25#5142—21no—
26#5077—22no—
27#4916—22no—
28#4914—22no—
29#4911—22no—
30D3×D16—22no—
31#4908C2212yescomplemented
32#4907C2212yescomplemented
33#4897C2212yescomplemented
34#4822—22no—
35#4700C2211yescharacteristic
36#4700C2211yescharacteristic, complemented
37#4695C2211yescharacteristic, complemented
38C3×#364C2211yescharacteristic, complemented
39C3×#361C2212yescomplemented
40#4567C2211yescharacteristic, complemented
41#4516C2212yescomplemented
42#4515C2212yescomplemented
43#4497C2211yescharacteristic, complemented
44#4493C2211yescharacteristic, complemented
45C4×C3⋊C16C2211yescharacteristic, complemented
46#3415—32no—
47#3292—31no—
48#3292—31no—
49#2246—31no—
50#2221—32no—
51#2213—31no—
52#2213D311yescharacteristic, complemented
53#2192—32no—
54#2192—32no—
55#2178—31no—
56#2170—31no—
57#1009—22no—
58#1004—22no—
59C3×#135C2311yescharacteristic
60C2×C3×D8—21no—
61C2×C3×D8C2312yes—
62C3×#131—21no—
63C3×#127—21no—
64#949—22no—
65#932—22no—
66#932—22no—
67#919—22no—
68#914—42no—
69#912—22no—
70#912—41no—
71#911—42no—
72#911—42no—
73D3×D8—22no—
74D3×D8—22no—
75D3×D8—41no—
76#908—22no—
77#908—42no—
78#904—41no—
79#904D411yescharacteristic
80#904C2311yescharacteristic
81C2×D24—21no—
82C2×D24—41no—
83C2×D24C2312yes—
84#901—21no—
85#901C2312yes—
86#874—21no—
87C3×SD32—22no—
88C3×D16—22no—
89C3×M32C2312yes—
90C3×#108C2311yescharacteristic
91C3×#104—21no—
92C3×C4×C8C2311yescharacteristic
93#833—22no—
94#829D411yescharacteristic, complemented
95#828D411yescharacteristic, complemented
96#827D411yescharacteristic, complemented
97#826—22no—
98#826—22no—
99#826—42no—
100#826D411yescharacteristic, complemented
101#823—22no—
102#819D411yescharacteristic
103#819C2311yescharacteristic
104C2×C3⋊C16C2311yescharacteristic
105C2×C3⋊C16C2311yescharacteristic
106#805—21no—
107#805D412yes—
108#800—22no—
109D48—22no—
110#798—22no—
111D3×C16—22no—
112#576—32no—
113#576—32no—
114#576—61no—
115#576—61no—
116#509—32no—
117#509—62no—
118#509—62no—
119#509C2×D312yescomplemented
120#508—31no—
121#508—62no—
122C2×D16—32no—
123C2×D16—32no—
124C2×D16—62no—
125#504—32no—
126#493—31no—
127#493C2×D311yescharacteristic, complemented
128#472—32no—
129#472—32no—
130#472—61no—
131#472—61no—
132#453—62no—
133#453—62no—
134#443—31no—
135#443—31no—
136#443C2×D311yescharacteristic, complemented
137#368—31no—
138#364C2×D311yescharacteristic, complemented
139#361—32no—
140#361C2×D312yescomplemented
141#359—31no—
142#359—31no—
143#350—32no—
144C4×C16—31no—
145C3×#41—21no—
146C2×C3×D4—22no—
147C2×C3×D4—81no—
148C2×#67—82no—
149#239—42no—
150#237—42no—
151#237—42no—
152D3×D4—42no—
153D3×D4—42no—
154D3×D4—42no—
155D3×D4—82no—
156#235—21no—
157#235—42no—
158#235C2×D412yes—
159C2×D12—22no—
160C2×D12—42no—
161C2×D12—81no—
162C3×Q16C2×D411yescharacteristic
163C3×SD16—21no—
164C3×D8—22no—
165C3×D8—22no—
166C3×D8—41no—
167C3×D8—41no—
168C3×D8C2×D411yescharacteristic
169C3×M16—21no—
170C3×M16—41no—
171C2×C3×C8—41no—
172C2×C3×C8C2×D411yescharacteristic
173C2×C3×C8C2411yescharacteristic
174C3×C42C2×D411yescharacteristic
175#215—42no—
176#214—42no—
177#214—42no—
178#213—42no—
179#213—42no—
180#213—42no—
181#208—21no—
182#208C2×D412yes—
183Dic12D811yescharacteristic
184Dic12C2×D411yescharacteristic
185D24—22no—
186D24—22no—
187D24—42no—
188D24—41no—
189D24—41no—
190D24D811yescharacteristic
191D24C2×D411yescharacteristic
192#204—21no—
193#204—42no—
194#204D812yes—
195ΓL(1,𝔽25)—22no—
196ΓL(1,𝔽25)—41no—
197D3×C8—22no—
198D3×C8—41no—
199C3×C16—22no—
200C3⋊C16—21no—
201C3⋊C16C2×D411yescharacteristic
202C3⋊C16C2×D411yescharacteristic
203H(5,𝔽2)—62no—
204H(5,𝔽2)—62no—
205AGL(1,ℤ/8)—62no—
206AGL(1,ℤ/8)—62no—
207AGL(1,ℤ/8)—62no—
208AGL(1,ℤ/8)—62no—
209AGL(1,ℤ/8)—122no—
210AGL(1,ℤ/8)—122no—
211AGL(1,ℤ/8)—122no—
212AGL(1,ℤ/8)—122no—
213#135—31no—
214#135—31no—
215#135—62no—
216#135—62no—
217#135—121no—
218#135—121no—
219#135C22×D311yescharacteristic
220C2×D8—21no—
221C2×D8—32no—
222C2×D8—62no—
223C2×D8—62no—
224C2×D8—61no—
225C2×D8—121no—
226C2×D8—121no—
227C2×D8C22×D312yes—
228#131—21no—
229#131—32no—
230#131—61no—
231#127—21no—
232#127—61no—
233Q32—31no—
234SD32—22no—
235SD32—31no—
236SD32—31no—
237SD32—62no—
238D16—22no—
239D16—31no—
240D16—62no—
241D16—62no—
242D16—62no—
243D16—122no—
244M32—62no—
245M32C22×D312yes—
246C2×C16—31no—
247C2×C16—31no—
248C2×C16—62no—
249#108—31no—
250#108—31no—
251#108C22×D311yescharacteristic
252#104—21no—
253#104—32no—
254#104—61no—
255#100—62no—
256#100—62no—
257C4×C8C22×D311yescharacteristic
258C23×C3—42no—
259C22×D3—42no—
260C22×D3—82no—
261C22×D3—82no—
262C3×Q8—21no—
263C3×D4—21no—
264C3×D4—41no—
265C3×D4—42no—
266C3×D4—42no—
267C3×D4—162no—
268C2×C3×C4—21no—
269C2×C3×C4—41no—
270C2×C3×C4C22×D411yescharacteristic
271#67—22no—
272#67—41no—
273#67—82no—
274#67—82no—
275#67—82no—
276C2×Dic3—82no—
277D12—21no—
278D12—42no—
279D12—42no—
280D12—82no—
281D12—162no—
282D12C2×D812yes—
283C4×D3—22no—
284C4×D3—41no—
285C4×D3—82no—
286Dic6—21no—
287Dic6C2×D812yes—
288C3×C8—41no—
289C3×C8C2×D811yescharacteristic
290C3×C8C2×D811yescharacteristic
291C3×C8C22×D411yescharacteristic
292C3×C8H(5,𝔽2)11yescharacteristic
293C3⋊C8—22no—
294C3⋊C8—41no—
295#41—21no—
296#41—32no—
297#41—61no—
298#41—122no—
299#41—122no—
300#41—122no—
301#41—122no—
302C2×D4—22no—
303C2×D4—62no—
304C2×D4—81no—
305C2×D4—122no—
306C2×D4—122no—
307C2×D4—122no—
308C2×D4—122no—
309C2×D4—241no—
310C2×D4—242no—
311C2×D4—242no—
312Q16—31no—
313Q16—31no—
314Q16D3×D411yescharacteristic
315SD16—21no—
316SD16—32no—
317SD16—61no—
318SD16—122no—
319SD16—122no—
320SD16—122no—
321SD16—122no—
322D8—22no—
323D8—22no—
324D8—31no—
325D8—31no—
326D8—41no—
327D8—41no—
328D8—62no—
329D8—62no—
330D8—122no—
331D8—122no—
332D8—122no—
333D8—122no—
334D8—121no—
335D8—121no—
336D8D3×D411yescharacteristic
337M16—21no—
338M16—32no—
339M16—41no—
340M16—62no—
341M16—61no—
342M16—121no—
343C2×C8—41no—
344C2×C8—62no—
345C2×C8—121no—
346C2×C8D3×D411yescharacteristic
347C2×C8C23×D311yescharacteristic
348C42D3×D411yescharacteristic
349C16—22no—
350C16—31no—
351C16—31no—
352C16—61no—
353C22×C3—41no—
354C22×C3—82no—
355C22×C3—162no—
356C22×C3#57611yescharacteristic
357C2×D3—22no—
358C2×D3—41no—
359C2×D3—82no—
360C2×D3—162no—
361C2×D3—322no—
362C2×D3—322no—
363C3×C4—22no—
364C3×C4—41no—
365C3×C4D4211yescharacteristic
366C3×C4C22×D811yescharacteristic
367Dic3—22no—
368Dic3—41no—
369C23—42no—
370C23—122no—
371C23—242no—
372C23—242no—
373Q8—21no—
374Q8—32no—
375Q8—61no—
376D4—21no—
377D4—32no—
378D4—41no—
379D4—42no—
380D4—42no—
381D4—62no—
382D4—61no—
383D4—121no—
384D4—122no—
385D4—122no—
386D4—162no—
387D4—242no—
388D4—242no—
389D4—242no—
390D4—242no—
391D4—482no—
392C2×C4—21no—
393C2×C4—41no—
394C2×C4—62no—
395C2×C4—121no—
396C2×C4—242no—
397C2×C4—242no—
398C2×C4C2×D3×D411yescharacteristic
399C8—41no—
400C8—62no—
401C8—121no—
402C8D3×D811yescharacteristic
403C8D3×D811yescharacteristic
404C8C2×D3×D411yescharacteristic
405C8#100911yescharacteristic
406C2×C3—21no—
407C2×C3—81no—
408C2×C3—162no—
409C2×C3D4×D811yescharacteristic
410D3—42no—
411D3—81no—
412D3—162no—
413C22—41no—
414C22—62no—
415C22—82no—
416C22—121no—
417C22—162no—
418C22—242no—
419C22—482no—
420C22—962no—
421C22—962no—
422C22#577411yescharacteristic
423C4—22no—
424C4—41no—
425C4—62no—
426C4—121no—
427C4#560911yescharacteristic
428C4C2×D3×D811yescharacteristic
429C3#3385811yescharacteristic, complemented, Sylow
430C2—21no—
431C2—81no—
432C2—122no—
433C2—162no—
434C2—241no—
435C2—482no—
436C2#8035011yescharacteristic
subgroupG/H
centerC2#80350
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2213C2
socleC2×C3D4×D8
hypercenter#2213D3
hypocenterC3#33858
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2213

Complements

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

NH ≅ G/Ntype
1. #82036C2semidirect
2. #81791C2semidirect
3. #79894C2semidirect
4. C3×#2213C2semidirect
5. #71517C2semidirect
6. #71290C2semidirect
7. #70921C2semidirect
8. #69305C2semidirect
9. #69300C2semidirect
10. #69260C2semidirect
11. #69258C2semidirect
13. #5769C22semidirect
15. #5757C22semidirect
17. C3×#509C22semidirect
18. C3×#493C22semidirect
19. C3×#443C22semidirect
22. #5160C22semidirect
23. C2×#826C22semidirect
24. C2×#826C22semidirect
31. #4908C22semidirect
32. #4907C22semidirect
33. #4897C22semidirect
36. #4700C22semidirect
37. #4695C22semidirect
38. C3×#364C22semidirect
39. C3×#361C22semidirect
40. #4567C22semidirect
41. #4516C22semidirect
42. #4515C22semidirect
43. #4497C22semidirect
44. #4493C22semidirect
45. C4×C3⋊C16C22semidirect
52. #2213D3semidirect
94. #829D4semidirect
95. #828D4semidirect
96. #827D4semidirect
100. #826D4semidirect
119. #509C2×D3semidirect
127. #493C2×D3semidirect
136. #443C2×D3semidirect
138. #364C2×D3semidirect
140. #361C2×D3semidirect
429. C3#33858semidirect

Maximal subgroups

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

subgroupindex
#820362
#817912
#798942
C3×#22132
#715172
#712902
#709212
#693052
#693002
#692602
#692582
#338583