Finite Group Databasebeta

← #80060

Subgroups — #80060

Order 384.

546 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#5817C211yesmaximal, characteristic, complemented
2C22×#215C211yesmaximal, characteristic, complemented
3C2×#914C211yesmaximal, characteristic, complemented
4#5609C211yesmaximal, characteristic, complemented
5C3×#448C211yesmaximal, characteristic, complemented
6#5176C211yesmaximal, characteristic, complemented
7#5169C211yesmaximal, characteristic, complemented
8#5163C211yesmaximal, characteristic, complemented
9#5038C211yesmaximal, characteristic, complemented
10#4819C211yesmaximal, characteristic, complemented
11#4817C211yesmaximal, characteristic, complemented
12#4812C211yesmaximal, characteristic, complemented
13#4810C211yesmaximal, characteristic, complemented
14#4732C211yesmaximal, characteristic, complemented
15#4730C211yesmaximal, characteristic, complemented
16#3265—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic, complemented
18C2×#239C2211yescharacteristic, complemented
19C2×D3×D4—21no—
20C2×D3×D4C2211yescharacteristic
21C22×D12C2211yescharacteristic
22C2×C3×SD16C2211yescharacteristic, complemented
23C2×C3×SD16C2211yescharacteristic, complemented
24C3×#121C2211yescharacteristic
25#947—21no—
26#947C2211yescharacteristic, complemented
27#946—21no—
28#946C2211yescharacteristic, complemented
29Q8×Dic3C2211yescharacteristic, complemented
30C2×#215—24no—
31C2×#215C2211yescharacteristic
32C2×#215C2211yescharacteristic
33C2×#215C2211yescharacteristic, complemented
34C2×#215C2211yescharacteristic, complemented
35#940C2211yescharacteristic
36#937—21no—
37C2×#213C2211yescharacteristic, complemented
38#930—21no—
39#930C2211yescharacteristic, complemented
40C4×#67—21no—
41C4×#67C2211yescharacteristic
42C22×C3⋊C8C2211yescharacteristic, complemented
43#914—24no—
44C2×D24C2211yescharacteristic
45C2×D24C2211yescharacteristic, complemented
46C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
47#895C2211yescharacteristic
48#893C2211yescharacteristic
49#884—21no—
50#882—21no—
51C3×#103C2211yescharacteristic, complemented
52C3×#103C2211yescharacteristic, complemented
53C3×C22⋊C8C2211yescharacteristic, complemented
54#832C2211yescharacteristic, complemented
55#821C2211yescharacteristic
56#821C2211yescharacteristic, complemented
57(C2×D3)⋊C8C2211yescharacteristic, complemented
58#815C2211yescharacteristic, complemented
59Dic3⋊C8C2211yescharacteristic, complemented
60#809C2211yescharacteristic, complemented
61#807C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C22×SD16—31no—
64#548—31no—
65D42—31no—
66#496—31no—
67#468—31no—
68#466—31no—
69#463—31no—
70#460—31no—
71#450—31no—
72#449—31no—
73#448D311yescharacteristic, complemented
74#447—31no—
75#440—31no—
76#435—31no—
77C23×D3—21no—
78C23×D3—21no—
79C2×C3×Q8—21no—
80C2×C3×Q8—22no—
81C2×C3×Q8—41no—
82C2×C3×Q8C2311yescharacteristic
83C2×C3×Q8C2311yescharacteristic
84C2×C3×D4—21no—
85C2×C3×D4C2311yescharacteristic
86C22×C3×C4—21no—
87C22×C3×C4C2311yescharacteristic
88C2×#67—21no—
89C2×#67—21no—
90C2×#67—21no—
91C2×#67—21no—
92C2×#67D411yescharacteristic
93#239—22no—
94#239—41no—
95D3×D4—22no—
96D3×D4—41no—
97D3×D4—44no—
98C2×D12—22no—
99C2×D12—21no—
100C2×D12—21no—
101C2×D12—21no—
102C2×D12—41no—
103C2×D12C2311yescharacteristic
104C2×D12C2311yescharacteristic
105C2×D12C2311yescharacteristic
106C2×C4×D3—21no—
107C2×C4×D3—21no—
108C2×C4×D3C2311yescharacteristic
109C3×SD16—22no—
110C3×SD16—41no—
111C2×C3×C8C2311yescharacteristic
112C2×C3×C8C2311yescharacteristic
113C3×C4⋊C4C2311yescharacteristic
114C3×C22⋊C4—21no—
115#217D411yescharacteristic
116#215—22no—
117#215—22no—
118#215—22no—
119#215D414yescomplemented
120#213—22no—
121(C2×D3)⋊C4—21no—
122(C2×D3)⋊C4—21no—
123(C2×D3)⋊C4—21no—
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4—21no—
127(C2×D3)⋊C4D411yescharacteristic
128C4⋊Dic3—21no—
129C4⋊Dic3C2311yescharacteristic
130Dic3⋊C4—21no—
131Dic3⋊C4D411yescharacteristic
132C4×Dic3—21no—
133C4×Dic3C2311yescharacteristic
134C2×C3⋊C8—22no—
135C2×C3⋊C8C2311yescharacteristic
136C2×C3⋊C8C2311yescharacteristic
137D24—22no—
138D24—41no—
139ΓL(1,𝔽25)—22no—
140C2×#41—31no—
141C22×Q8C2×D311yescharacteristic, complemented
142C22×D4—31no—
143C22×D4—31no—
144C22×D4—61no—
145AGL(1,ℤ/8)—64no—
146C2×SD16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—64no—
151C2×SD16C2×D311yescharacteristic, complemented
152C2×SD16C2×D311yescharacteristic, complemented
153C2×D8—31no—
154C2×D8—31no—
155C2×D8—31no—
156C2×M16—31no—
157C22×C8—31no—
158#127—31no—
159#124—31no—
160#124—61no—
161C22⋊Q8—31no—
162C22⋊Q8—61no—
163#121—31no—
164#121—31no—
165#121—61no—
166#121—61no—
167#121C2×D311yescharacteristic
168#120—61no—
169#120—61no—
170C4×Q8—31no—
171C4×D4—31no—
172C4×D4—31no—
173C4×D4—61no—
174#107—31no—
175C4⋊C8—31no—
176#103C2×D311yescharacteristic, complemented
177#103C2×D311yescharacteristic, complemented
178#102—31no—
179#102—31no—
180#102—31no—
181#102—31no—
182C22⋊C8—31no—
183C22⋊C8C2×D311yescharacteristic, complemented
184#97—31no—
185C23×C3—21no—
186C23×C3C2×D411yescharacteristic
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—21no—
190C22×D3—42no—
191C22×D3—42no—
192C22×D3—42no—
193C22×D3—42no—
194C22×D3—81no—
195C22×D3—81no—
196C22×D3C2×D411yescharacteristic
197C3×Q8—21no—
198C3×Q8—21no—
199C3×Q8—21no—
200C3×Q8—42no—
201C3×Q8C2×D412yescomplemented
202C3×D4—21no—
203C3×D4—21no—
204C3×D4—42no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—42no—
211C2×C3×C4C2×D411yescharacteristic
212C2×C3×C4C2411yescharacteristic
213#67—22no—
214#67—42no—
215#67—42no—
216#67—42no—
217#67—81no—
218C2×Dic3—21no—
219C2×Dic3C2×D411yescharacteristic
220C2×Dic3C2×D411yescharacteristic
221C2×Dic3C2×D411yescharacteristic
222D12—21no—
223D12—21no—
224D12—21no—
225D12—21no—
226D12—21no—
227D12—42no—
228D12—42no—
229D12—42no—
230D12C2×D412yes—
231C4×D3—21no—
232C4×D3—21no—
233C4×D3—42no—
234C4×D3—42no—
235C3×C8—21no—
236C3×C8—21no—
237C3⋊C8—21no—
238C3⋊C8C2×D412yescomplemented
239C24—61no—
240C24—61no—
241#41—62no—
242#41—121no—
243C2×Q8—21no—
244C2×Q8—22no—
245C2×Q8—41no—
246C2×Q8C22×D311yescharacteristic
247C2×Q8C22×D311yescharacteristic
248C2×D4—21no—
249C2×D4—31no—
250C2×D4—31no—
251C2×D4—31no—
252C2×D4—31no—
253C2×D4—62no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—61no—
259C2×D4—62no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—121no—
263C2×D4—121no—
264C2×D4—124no—
265C2×D4C22×D311yescharacteristic
266C22×C4—21no—
267C22×C4—31no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4C22×D311yescharacteristic
271SD16—22no—
272SD16—34no—
273SD16—41no—
274SD16—62no—
275SD16—62no—
276SD16—62no—
277D8—62no—
278D8—62no—
279D8—121no—
280M16—62no—
281C2×C8—31no—
282C2×C8—31no—
283C2×C8—62no—
284C2×C8C22×D311yescharacteristic
285C2×C8C22×D311yescharacteristic
286C4⋊C4—31no—
287C4⋊C4—31no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4C22×D311yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—31no—
293C22⋊C4—31no—
294C22⋊C4—61no—
295C22⋊C4—61no—
296C22⋊C4—61no—
297C22⋊C4—61no—
298C22⋊C4—61no—
299C22⋊C4—61no—
300C42—31no—
301C42—61no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3AGL(1,ℤ/8)12yescomplemented
308C22×C3C22×D411yescharacteristic
309C2×D3—21no—
310C2×D3—22no—
311C2×D3—21no—
312C2×D3—21no—
313C2×D3—41no—
314C2×D3—41no—
315C2×D3—41no—
316C2×D3—41no—
317C2×D3—41no—
318C2×D3—41no—
319C2×D3—41no—
320C2×D3—41no—
321C2×D3—84no—
322C2×D3—84no—
323C3×C4—22no—
324C3×C4—21no—
325C3×C4—41no—
326C3×C4—41no—
327C3×C4C22×D411yescharacteristic
328C3×C4H(5,𝔽2)11yescharacteristic
329Dic3—21no—
330Dic3—21no—
331Dic3—21no—
332Dic3—41no—
333C23—21no—
334C23—31no—
335C23—61no—
336C23—61no—
337C23—61no—
338C23—122no—
339C23—122no—
340C23—122no—
341C23—122no—
342C23—241no—
343C23—241no—
344C23D3×D411yescharacteristic
345Q8—21no—
346Q8—21no—
347Q8—21no—
348Q8—42no—
349Q8D3×D412yescomplemented
350D4—21no—
351D4—21no—
352D4—32no—
353D4—42no—
354D4—62no—
355D4—61no—
356D4—61no—
357D4—61no—
358D4—61no—
359D4—61no—
360D4—122no—
361D4—122no—
362D4—122no—
363D4—122no—
364D4—122no—
365D4—122no—
366D4—241no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—31no—
373C2×C4—31no—
374C2×C4—31no—
375C2×C4—42no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4D3×D411yescharacteristic
382C2×C4C23×D311yescharacteristic
383C8—21no—
384C8—21no—
385C8—32no—
386C8—61no—
387C2×C3—22no—
388C2×C3—81no—
389C2×C3D4211yescharacteristic
390C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
391C2×C3#57711yescharacteristic
392D3—41no—
393D3—42no—
394D3—81no—
395D3—81no—
396C22—21no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—61no—
402C22—61no—
403C22—61no—
404C22—62no—
405C22—121no—
406C22—121no—
407C22—121no—
408C22—121no—
409C22—121no—
410C22—121no—
411C22—121no—
412C22—121no—
413C22—244no—
414C22—244no—
415C22#91412yescomplemented
416C22C2×D3×D411yescharacteristic
417C4—21no—
418C4—22no—
419C4—41no—
420C4—41no—
421C4—61no—
422C4—61no—
423C4—61no—
424C4—121no—
425C4C2×D3×D411yescharacteristic
426C4#100411yescharacteristic
427C3#326511yescharacteristic, complemented, Sylow
428C2—22no—
429C2—81no—
430C2—121no—
431C2—122no—
432C2—241no—
433C2—241no—
434C2#559011yescharacteristic
435C2C2×#91411yescharacteristic
436C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#448C2
socleC22×C3C22×D4
hypercenter#448D3
hypocenterC3#3265
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. #5817C2semidirect
2. C22×#215C2semidirect
3. C2×#914C2semidirect
4. #5609C2semidirect
5. C3×#448C2semidirect
6. #5176C2semidirect
7. #5169C2semidirect
8. #5163C2semidirect
9. #5038C2semidirect
10. #4819C2semidirect
11. #4817C2semidirect
12. #4812C2semidirect
13. #4810C2semidirect
14. #4732C2semidirect
15. #4730C2semidirect
17. C22×C3×Q8C22semidirect
18. C2×#239C22semidirect
22. C2×C3×SD16C22semidirect
23. C2×C3×SD16C22semidirect
26. #947C22semidirect
28. #946C22semidirect
29. Q8×Dic3C22semidirect
33. C2×#215C22semidirect
34. C2×#215C22semidirect
37. C2×#213C22semidirect
39. #930C22semidirect
42. C22×C3⋊C8C22semidirect
45. C2×D24C22semidirect
46. C2×ΓL(1,𝔽25)C22semidirect
51. C3×#103C22semidirect
52. C3×#103C22semidirect
53. C3×C22⋊C8C22semidirect
54. #832C22semidirect
56. #821C22semidirect
57. (C2×D3)⋊C8C22semidirect
58. #815C22semidirect
59. Dic3⋊C8C22semidirect
60. #809C22semidirect
61. #807C22semidirect
73. #448D3semidirect
119. #215D4semidirect
141. C22×Q8C2×D3semidirect
151. C2×SD16C2×D3semidirect
152. C2×SD16C2×D3semidirect
176. #103C2×D3semidirect
177. #103C2×D3semidirect
183. C22⋊C8C2×D3semidirect
201. C3×Q8C2×D4semidirect
238. C3⋊C8C2×D4semidirect
307. C22×C3AGL(1,ℤ/8)semidirect
349. Q8D3×D4semidirect
415. C22#914semidirect
427. C3#3265semidirect

Maximal subgroups

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

subgroupindex
#58172
C22×#2152
C2×#9142
#56092
C3×#4482
#51762
#51692
#51632
#50382
#48192
#48172
#48122
#48102
#47322
#47302
#32653