Finite Group Databasebeta

← #10951979

Subgroups — #10951979

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
1C3×#3292C211yesmaximal, characteristic, complemented
2#84007C212yesmaximal, complemented
3#79894C211yesmaximal, characteristic, complemented
4#72122C212yesmaximal, complemented
5#71295C211yesmaximal, characteristic, complemented
6#71290C211yesmaximal, characteristic, complemented
7#71094C212yesmaximal, complemented
8#71092C212yesmaximal, complemented
9#70926C211yesmaximal, characteristic, complemented
10#70923C211yesmaximal, characteristic, complemented
11#70921C211yesmaximal, characteristic, complemented
12#33858—31nomaximal, Sylow
13C3×#576—21no—
14C3×#576C2212yescomplemented
15#5769—21no—
16#5752C2212yescomplemented
17C3×#493C2211yescharacteristic, complemented
18C3×#472—21no—
19C3×#472C2212yescomplemented
20C3×#453—22no—
21C3×#443C2211yescharacteristic
22C3×#443C2211yescharacteristic, complemented
23#5194—22no—
24#5194C2211yescharacteristic
25#5193—22no—
26#5193C2212yescomplemented
27#5147—22no—
28#5147C2212yescomplemented
29C2×#826—22no—
30C2×#826C2212yescomplemented
31C2×#826C2212yescomplemented
32#5142C2212yescomplemented
33#5140C2212yescomplemented
34#4897—21no—
35#4822—22no—
36#4700C2211yescharacteristic, complemented
37#4695C2211yescharacteristic, complemented
38#4567C2211yescharacteristic, complemented
39#4563C2212yescomplemented
40#4561C2212yescomplemented
41#4497C2211yescharacteristic, complemented
42#4495C2212yescomplemented
43#4493C2211yescharacteristic, complemented
44#4463C2211yescharacteristic, complemented
45C4×C3⋊C16C2211yescharacteristic, complemented
46#3415—32no—
47#3292—31no—
48#3292D311yescharacteristic, complemented
49#2246—31no—
50#2221—32no—
51#2213—31no—
52#2213—31no—
53#2192—32no—
54#2192—32no—
55#2178—31no—
56#2170—31no—
57C3×H(5,𝔽2)—22no—
58#1009—22no—
59C3×AGL(1,ℤ/8)—22no—
60C3×AGL(1,ℤ/8)—22no—
61C3×AGL(1,ℤ/8)—42no—
62C3×AGL(1,ℤ/8)—42no—
63C3×#135—22no—
64C3×#135—41no—
65C3×#135D411yescharacteristic
66C3×#135C2311yescharacteristic
67C2×C3×D8—22no—
68C2×C3×D8—21no—
69C2×C3×D8—41no—
70C2×C3×D8C2312yes—
71C3×#131—21no—
72C3×#131C2312yes—
73C3×#127—21no—
74#949—22no—
75#919—41no—
76#914—42no—
77D3×D8—41no—
78#908—22no—
79#908—42no—
80#904—22no—
81#904C2311yescharacteristic
82C2×D24—22no—
83C2×D24—21no—
84C2×D24C2312yes—
85#901—21no—
86#874—21no—
87C3×#108D411yescharacteristic
88C3×#108C2311yescharacteristic
89C3×#104—21no—
90C3×#104D412yes—
91C3×#100—22no—
92C3×C4×C8C2311yescharacteristic
93#829D411yescharacteristic, complemented
94#828—22no—
95#828D411yescharacteristic, complemented
96#827—22no—
97#827D411yescharacteristic, complemented
98#826—22no—
99#826—22no—
100#826—22no—
101#826—22no—
102#826—42no—
103#826D411yescharacteristic, complemented
104#823—22no—
105#819C2311yescharacteristic
106#812—22no—
107#812C2312yes—
108C2×C3⋊C16—22no—
109C2×C3⋊C16C2311yescharacteristic
110C2×C3⋊C16C2311yescharacteristic
111#805—21no—
112#576—21no—
113#576—32no—
114#576—61no—
115#576C2×D312yescomplemented
116#509—32no—
117#509—32no—
118#509—62no—
119#509—62no—
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—21no—
129#472—32no—
130#472—61no—
131#472C2×D312yescomplemented
132#453—22no—
133#453—62no—
134#443—31no—
135#443C2×D311yescharacteristic
136#443C2×D311yescharacteristic, complemented
137#368—31no—
138#364—31no—
139#361—32no—
140#361—32no—
141#359—31no—
142#359—31no—
143#350—32no—
144C4×C16—31no—
145C3×#41—21no—
146C3×#41—42no—
147C3×#41—42no—
148C3×#41C2×D412yes—
149C2×C3×D4—22no—
150C2×C3×D4—42no—
151C2×C3×D4—42no—
152C2×C3×D4—81no—
153C2×C3×D4—82no—
154#239—42no—
155D3×D4—42no—
156D3×D4—82no—
157#235—21no—
158#235—42no—
159C2×D12—22no—
160C2×D12—42no—
161C2×D12—81no—
162C3×Q16D811yescharacteristic
163C3×Q16C2×D411yescharacteristic
164C3×SD16—21no—
165C3×SD16—42no—
166C3×SD16—42no—
167C3×SD16D812yes—
168C3×D8—22no—
169C3×D8—22no—
170C3×D8—42no—
171C3×D8—42no—
172C3×D8—41no—
173C3×D8—41no—
174C3×D8D811yescharacteristic
175C3×D8C2×D411yescharacteristic
176C3×M16—22no—
177C3×M16—21no—
178C3×M16—41no—
179C3×M16C2×D412yes—
180C2×C3×C8—22no—
181C2×C3×C8—41no—
182C2×C3×C8C2×D411yescharacteristic
183C2×C3×C8C2411yescharacteristic
184C3×C42C2×D411yescharacteristic
185#215—42no—
186#213—42no—
187#208—21no—
188Dic12C2×D411yescharacteristic
189D24—22no—
190D24—22no—
191D24—42no—
192D24—41no—
193D24—41no—
194D24C2×D411yescharacteristic
195#204—21no—
196#204—42no—
197ΓL(1,𝔽25)—41no—
198D3×C8—41no—
199C3⋊C16—22no—
200C3⋊C16—21no—
201C3⋊C16C2×D411yescharacteristic
202C3⋊C16C2×D411yescharacteristic
203H(5,𝔽2)—22no—
204H(5,𝔽2)—62no—
205AGL(1,ℤ/8)—22no—
206AGL(1,ℤ/8)—22no—
207AGL(1,ℤ/8)—42no—
208AGL(1,ℤ/8)—42no—
209AGL(1,ℤ/8)—62no—
210AGL(1,ℤ/8)—62no—
211AGL(1,ℤ/8)—122no—
212AGL(1,ℤ/8)—122no—
213#135—22no—
214#135—31no—
215#135—41no—
216#135—62no—
217#135—121no—
218#135#6711yescharacteristic
219#135C22×D311yescharacteristic
220C2×D8—22no—
221C2×D8—21no—
222C2×D8—32no—
223C2×D8—41no—
224C2×D8—62no—
225C2×D8—61no—
226C2×D8—121no—
227C2×D8C22×D312yes—
228#131—21no—
229#131—61no—
230#131C22×D312yes—
231#127—21no—
232#127—61no—
233Q32—31no—
234SD32—31no—
235SD32—31no—
236SD32—62no—
237SD32—62no—
238D16—31no—
239D16—62no—
240D16—62no—
241D16—62no—
242D16—62no—
243D16—122no—
244M32—32no—
245M32—62no—
246C2×C16—31no—
247C2×C16—31no—
248C2×C16—62no—
249#108—31no—
250#108#6711yescharacteristic
251#108C22×D311yescharacteristic
252#104—21no—
253#104—61no—
254#104#6712yes—
255#100—22no—
256#100—62no—
257C4×C8C22×D311yescharacteristic
258C23×C3—42no—
259C23×C3—82no—
260C22×D3—42no—
261C22×D3—82no—
262C3×Q8—21no—
263C3×Q8C2×D812yes—
264C3×D4—22no—
265C3×D4—21no—
266C3×D4—41no—
267C3×D4—42no—
268C3×D4—42no—
269C3×D4—82no—
270C3×D4—82no—
271C3×D4—162no—
272C3×D4C2×D812yes—
273C2×C3×C4—21no—
274C2×C3×C4—22no—
275C2×C3×C4—41no—
276C2×C3×C4—82no—
277C2×C3×C4C22×D411yescharacteristic
278#67—41no—
279#67—82no—
280D12—21no—
281D12—42no—
282D12—42no—
283D12—82no—
284D12—162no—
285C4×D3—41no—
286C4×D3—82no—
287Dic6—21no—
288C3×C8—22no—
289C3×C8—41no—
290C3×C8C2×D811yescharacteristic
291C3×C8C2×D811yescharacteristic
292C3×C8C22×D411yescharacteristic
293C3×C8H(5,𝔽2)11yescharacteristic
294C3⋊C8—41no—
295#41—21no—
296#41—42no—
297#41—42no—
298#41—61no—
299#41—122no—
300#41—122no—
301#41C2×#6712yes—
302C2×D4—22no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—62no—
306C2×D4—81no—
307C2×D4—82no—
308C2×D4—122no—
309C2×D4—122no—
310C2×D4—241no—
311C2×D4—242no—
312Q16—31no—
313Q16#21311yescharacteristic
314Q16D3×D411yescharacteristic
315SD16—21no—
316SD16—42no—
317SD16—42no—
318SD16—61no—
319SD16—122no—
320SD16—122no—
321SD16#21312yes—
322D8—22no—
323D8—22no—
324D8—31no—
325D8—42no—
326D8—42no—
327D8—41no—
328D8—41no—
329D8—62no—
330D8—62no—
331D8—122no—
332D8—122no—
333D8—121no—
334D8—121no—
335D8#21311yescharacteristic
336D8D3×D411yescharacteristic
337M16—22no—
338M16—21no—
339M16—41no—
340M16—61no—
341M16—121no—
342M16C2×#6712yes—
343C2×C8—22no—
344C2×C8—41no—
345C2×C8—121no—
346C2×C8C2×#6711yescharacteristic
347C2×C8C23×D311yescharacteristic
348C42C2×#6711yescharacteristic
349C16—31no—
350C16—31no—
351C16—62no—
352C16—61no—
353C22×C3—22no—
354C22×C3—41no—
355C22×C3—82no—
356C22×C3—162no—
357C22×C3—322no—
358C22×C3#57611yescharacteristic
359C2×D3—41no—
360C2×D3—82no—
361C2×D3—162no—
362C2×D3—322no—
363C3×C4—22no—
364C3×C4—22no—
365C3×C4—41no—
366C3×C4D4211yescharacteristic
367C3×C4C22×D811yescharacteristic
368Dic3—41no—
369C23—42no—
370C23—82no—
371C23—122no—
372C23—242no—
373Q8—21no—
374Q8—61no—
375Q8C2×#21312yes—
376D4—22no—
377D4—21no—
378D4—41no—
379D4—42no—
380D4—42no—
381D4—61no—
382D4—82no—
383D4—82no—
384D4—121no—
385D4—122no—
386D4—122no—
387D4—162no—
388D4—242no—
389D4—242no—
390D4—482no—
391D4C2×#21312yes—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—82no—
396C2×C4—121no—
397C2×C4—242no—
398C2×C4C22×#6711yescharacteristic
399C8—22no—
400C8—41no—
401C8—121no—
402C8C2×#21311yescharacteristic
403C8C2×#21311yescharacteristic
404C8C2×D3×D411yescharacteristic
405C8#100411yescharacteristic
406C2×C3—21no—
407C2×C3—42no—
408C2×C3—81no—
409C2×C3—162no—
410C2×C3D4×D811yescharacteristic
411D3—81no—
412D3—162no—
413C22—22no—
414C22—41no—
415C22—82no—
416C22—121no—
417C22—162no—
418C22—242no—
419C22—322no—
420C22—482no—
421C22—962no—
422C22#583511yescharacteristic
423C4—22no—
424C4—22no—
425C4—41no—
426C4—121no—
427C4C22×#21311yescharacteristic
428C4D4×#6711yescharacteristic
429C3#3385811yescharacteristic, complemented, Sylow
430C2—21no—
431C2—42no—
432C2—81no—
433C2—162no—
434C2—241no—
435C2—482no—
436C2D4×#21311yescharacteristic
subgroupG/H
centerC2D4×#213
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#3292C2
socleC2×C3D4×D8
hypercenter#3292D3
hypocenterC3#33858
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443 ◁ #3292

Complements

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

NH ≅ G/Ntype
1. C3×#3292C2semidirect
2. #84007C2semidirect
3. #79894C2semidirect
4. #72122C2semidirect
5. #71295C2semidirect
6. #71290C2semidirect
7. #71094C2semidirect
8. #71092C2semidirect
9. #70926C2semidirect
10. #70923C2semidirect
11. #70921C2semidirect
14. C3×#576C22semidirect
16. #5752C22semidirect
17. C3×#493C22semidirect
19. C3×#472C22semidirect
22. C3×#443C22semidirect
26. #5193C22semidirect
28. #5147C22semidirect
30. C2×#826C22semidirect
31. C2×#826C22semidirect
32. #5142C22semidirect
33. #5140C22semidirect
36. #4700C22semidirect
37. #4695C22semidirect
38. #4567C22semidirect
39. #4563C22semidirect
40. #4561C22semidirect
41. #4497C22semidirect
42. #4495C22semidirect
43. #4493C22semidirect
44. #4463C22semidirect
45. C4×C3⋊C16C22semidirect
48. #3292D3semidirect
93. #829D4semidirect
95. #828D4semidirect
97. #827D4semidirect
103. #826D4semidirect
115. #576C2×D3semidirect
127. #493C2×D3semidirect
131. #472C2×D3semidirect
136. #443C2×D3semidirect
429. C3#33858semidirect

Maximal subgroups

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

subgroupindex
C3×#32922
#840072
#798942
#721222
#712952
#712902
#710942
#710922
#709262
#709232
#709212
#338583