Finite Group Databasebeta

← #80674

Subgroups — #80674

Order 384.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#914C211yesmaximal, characteristic, complemented
2C22×D24C211yesmaximal, characteristic, complemented
3#5630C211yesmaximal, characteristic, complemented
4#5609C211yesmaximal, characteristic, complemented
5C3×#465C211yesmaximal, characteristic, complemented
6#5176C211yesmaximal, characteristic, complemented
7#5111C211yesmaximal, characteristic, complemented
8#5046C211yesmaximal, characteristic, complemented
9#5036C211yesmaximal, characteristic, complemented
10#4869C211yesmaximal, characteristic, complemented
11#4865C211yesmaximal, characteristic, complemented
12#4819C211yesmaximal, characteristic, complemented
13#4810C211yesmaximal, characteristic, complemented
14#4786C211yesmaximal, characteristic, complemented
15#4773C211yesmaximal, characteristic, complemented
16#3264—31nomaximal, Sylow
17C2×#239C2211yescharacteristic, complemented
18C2×D3×D4—21no—
19C2×D3×D4C2211yescharacteristic, complemented
20C22×D12C2211yescharacteristic
21C22×D12C2211yescharacteristic, complemented
22C2×C3×SD16C2211yescharacteristic, complemented
23C22×C3×C8C2211yescharacteristic, complemented
24C3×C22⋊Q8C2211yescharacteristic, complemented
25C3×#121C2211yescharacteristic
26#947C2211yescharacteristic, complemented
27C2×#215C2211yescharacteristic, complemented
28C2×#215C2211yescharacteristic, complemented
29#940C2211yescharacteristic, complemented
30#937—21no—
31C2×#213C2211yescharacteristic, complemented
32C2×#213C2211yescharacteristic, complemented
33C4×#67C2211yescharacteristic
34#914—24no—
35C2×D24—24no—
36C2×D24C2211yescharacteristic
37C2×D24C2211yescharacteristic
38C2×D24C2211yescharacteristic, complemented
39C2×D24C2211yescharacteristic, complemented
40C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
41#895—21no—
42#895C2211yescharacteristic, complemented
43#895C2211yescharacteristic, complemented
44#894—21no—
45#893C2211yescharacteristic, complemented
46#893C2211yescharacteristic, complemented
47#884—21no—
48#882—21no—
49#882—21no—
50D3×C22⋊C4—21no—
51C3×#106C2211yescharacteristic, complemented
52C3×#103C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic, complemented
54#830C2211yescharacteristic, complemented
55(C2×D3)⋊C8C2211yescharacteristic, complemented
56#815C2211yescharacteristic, complemented
57Dic3⋊C8C2211yescharacteristic, complemented
58#809C2211yescharacteristic, complemented
59#809C2211yescharacteristic, complemented
60#809C2211yescharacteristic, complemented
61#809C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C22×D8—31no—
64#546—31no—
65D42—31no—
66#496—31no—
67#469—31no—
68#465D311yescharacteristic, complemented
69#463—31no—
70#459—31no—
71#450—31no—
72#449—31no—
73#447—31no—
74#447—31no—
75#442—31no—
76#435—31no—
77C23×D3—21no—
78C23×D3—21no—
79C23×D3—21no—
80C2×C3×Q8C2311yescharacteristic
81C2×C3×D4—21no—
82C2×C3×D4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C2×#67D411yescharacteristic
88#239—22no—
89#239—41no—
90D3×D4—22no—
91D3×D4—41no—
92D3×D4—44no—
93C2×D12—22no—
94C2×D12—22no—
95C2×D12—21no—
96C2×D12—21no—
97C2×D12—21no—
98C2×D12—21no—
99C2×D12—21no—
100C2×D12—41no—
101C2×D12—41no—
102C2×D12C2311yescharacteristic
103C2×D12C2311yescharacteristic
104C2×D12C2311yescharacteristic
105C2×D12C2311yescharacteristic
106C2×C4×D3—21no—
107C2×C4×D3—21no—
108C2×C4×D3—21no—
109C2×C4×D3C2311yescharacteristic
110C3×SD16—22no—
111C2×C3×C8—22no—
112C2×C3×C8C2311yescharacteristic
113C2×C3×C8C2311yescharacteristic
114C3×C4⋊C4—21no—
115C3×C4⋊C4C2311yescharacteristic
116C3×C4⋊C4C2311yescharacteristic
117C3×C22⋊C4—21no—
118C3×C22⋊C4—21no—
119#217D411yescharacteristic
120#215—22no—
121#215—41no—
122#213—22no—
123#213—41no—
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4—21no—
127(C2×D3)⋊C4—21no—
128(C2×D3)⋊C4—21no—
129(C2×D3)⋊C4—21no—
130(C2×D3)⋊C4D411yescharacteristic
131Dic3⋊C4D411yescharacteristic
132C4×Dic3C2311yescharacteristic
133C2×C3⋊C8C2311yescharacteristic
134C2×C3⋊C8C2311yescharacteristic
135D24—22no—
136D24—22no—
137D24—22no—
138D24D414yescomplemented
139ΓL(1,𝔽25)—22no—
140C2×#41—31no—
141C22×D4—31no—
142C22×D4—31no—
143C22×D4—31no—
144C22×D4—61no—
145AGL(1,ℤ/8)—64no—
146C2×SD16—31no—
147C2×SD16—31no—
148C2×SD16C2×D311yescharacteristic, complemented
149C2×D8—31no—
150C2×D8—31no—
151C2×D8—31no—
152C2×D8—31no—
153C2×D8—31no—
154C2×D8—31no—
155C2×D8—64no—
156C2×M16—31no—
157C22×C8C2×D311yescharacteristic, complemented
158#127—31no—
159#124—31no—
160#123—61no—
161C22⋊Q8C2×D311yescharacteristic, complemented
162#121—31no—
163#121—31no—
164#121—61no—
165#121—61no—
166#121C2×D311yescharacteristic
167#120—61no—
168#120—61no—
169#120—61no—
170C4×D4—31no—
171C4×D4—31no—
172C4×D4—31no—
173C2×C22⋊C4—61no—
174#106C2×D311yescharacteristic, complemented
175C4⋊C8—31no—
176#103C2×D311yescharacteristic, complemented
177#102—31no—
178#102—31no—
179#102—31no—
180#102—31no—
181#102C2×D311yescharacteristic, complemented
182C22⋊C8—31no—
183C22⋊C8—31no—
184#97—31no—
185C23×C3—21no—
186C23×C3C2×D411yescharacteristic
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—21no—
190C22×D3—21no—
191C22×D3—42no—
192C22×D3—42no—
193C22×D3—42no—
194C22×D3—42no—
195C22×D3—42no—
196C22×D3—42no—
197C22×D3—81no—
198C22×D3—81no—
199C22×D3—81no—
200C22×D3C2×D411yescharacteristic
201C3×Q8—21no—
202C3×Q8—21no—
203C3×D4—21no—
204C3×D4—21no—
205C3×D4—42no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4C2×D411yescharacteristic
212C2×C3×C4C2411yescharacteristic
213#67—22no—
214#67—42no—
215#67—42no—
216#67—81no—
217C2×Dic3C2×D411yescharacteristic
218C2×Dic3C2×D411yescharacteristic
219C2×Dic3C2×D411yescharacteristic
220D12—21no—
221D12—21no—
222D12—21no—
223D12—21no—
224D12—21no—
225D12—21no—
226D12—42no—
227D12—42no—
228D12—42no—
229D12—42no—
230D12—81no—
231D12C2×D412yes—
232D12C2×D412yescomplemented
233C4×D3—21no—
234C4×D3—21no—
235C4×D3—42no—
236C4×D3—42no—
237C4×D3—81no—
238C3×C8—21no—
239C3×C8C2×D412yescomplemented
240C3⋊C8—21no—
241C3⋊C8—21no—
242C24—61no—
243C24—61no—
244C24—61no—
245#41—62no—
246#41—121no—
247C2×Q8C22×D311yescharacteristic
248C2×D4—21no—
249C2×D4—31no—
250C2×D4—31no—
251C2×D4—31no—
252C2×D4—31no—
253C2×D4—31no—
254C2×D4—62no—
255C2×D4—62no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—62no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—121no—
266C2×D4—121no—
267C2×D4—121no—
268C2×D4—124no—
269C2×D4C22×D311yescharacteristic
270C22×C4—31no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4C22×D311yescharacteristic
275SD16—22no—
276SD16—62no—
277SD16—121no—
278D8—34no—
279D8—62no—
280D8—62no—
281D8—62no—
282D8—62no—
283D8—121no—
284M16—62no—
285C2×C8—22no—
286C2×C8—31no—
287C2×C8—31no—
288C2×C8C22×D311yescharacteristic
289C2×C8C22×D311yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—31no—
292C4⋊C4C22×D311yescharacteristic
293C4⋊C4C22×D311yescharacteristic
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—31no—
297C22⋊C4—31no—
298C22⋊C4—61no—
299C22⋊C4—61no—
300C22⋊C4—61no—
301C22⋊C4—61no—
302C22⋊C4—61no—
303C22⋊C4—61no—
304C42—31no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3AGL(1,ℤ/8)12yescomplemented
311C22×C3C22×D411yescharacteristic
312C2×D3—21no—
313C2×D3—22no—
314C2×D3—22no—
315C2×D3—21no—
316C2×D3—21no—
317C2×D3—41no—
318C2×D3—41no—
319C2×D3—41no—
320C2×D3—41no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3—41no—
326C2×D3—41no—
327C2×D3—84no—
328C2×D3—84no—
329C2×D3—84no—
330C3×C4—21no—
331C3×C4—41no—
332C3×C4—41no—
333C3×C4—41no—
334C3×C4C22×D411yescharacteristic
335C3×C4H(5,𝔽2)11yescharacteristic
336Dic3—21no—
337Dic3—21no—
338Dic3—21no—
339C23—21no—
340C23—31no—
341C23—61no—
342C23—61no—
343C23—61no—
344C23—61no—
345C23—122no—
346C23—122no—
347C23—122no—
348C23—122no—
349C23—122no—
350C23—122no—
351C23—241no—
352C23—241no—
353C23—241no—
354C23D3×D411yescharacteristic
355Q8—21no—
356Q8—21no—
357D4—21no—
358D4—21no—
359D4—32no—
360D4—32no—
361D4—42no—
362D4—62no—
363D4—61no—
364D4—61no—
365D4—61no—
366D4—61no—
367D4—61no—
368D4—61no—
369D4—122no—
370D4—122no—
371D4—122no—
372D4—122no—
373D4—122no—
374D4—122no—
375D4—241no—
376D4—241no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—31no—
383C2×C4—31no—
384C2×C4—31no—
385C2×C4—61no—
386C2×C4—61no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—241no—
390C2×C4D3×D411yescharacteristic
391C2×C4C23×D311yescharacteristic
392C8—21no—
393C8—61no—
394C8—61no—
395C8D3×D412yescomplemented
396C2×C3—22no—
397C2×C3—81no—
398C2×C3D4211yescharacteristic
399C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
400C2×C3#57611yescharacteristic
401D3—41no—
402D3—42no—
403D3—42no—
404D3—81no—
405D3—81no—
406C22—21no—
407C22—21no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—61no—
412C22—61no—
413C22—61no—
414C22—62no—
415C22—62no—
416C22—121no—
417C22—121no—
418C22—121no—
419C22—121no—
420C22—121no—
421C22—121no—
422C22—121no—
423C22—121no—
424C22—121no—
425C22—121no—
426C22—244no—
427C22—244no—
428C22—244no—
429C22#91412yescomplemented
430C22C2×D3×D411yescharacteristic
431C4—21no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—61no—
436C4—61no—
437C4—61no—
438C4C2×D3×D411yescharacteristic
439C4#100911yescharacteristic
440C3#326411yescharacteristic, complemented, Sylow
441C2—22no—
442C2—81no—
443C2—121no—
444C2—122no—
445C2—122no—
446C2—241no—
447C2—241no—
448C2#560911yescharacteristic
449C2C2×#91411yescharacteristic
450C2#576911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#465C2
socleC22×C3C22×D4
hypercenter#465D3
hypocenterC3#3264
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#914C2semidirect
2. C22×D24C2semidirect
3. #5630C2semidirect
4. #5609C2semidirect
5. C3×#465C2semidirect
6. #5176C2semidirect
7. #5111C2semidirect
8. #5046C2semidirect
9. #5036C2semidirect
10. #4869C2semidirect
11. #4865C2semidirect
12. #4819C2semidirect
13. #4810C2semidirect
14. #4786C2semidirect
15. #4773C2semidirect
17. C2×#239C22semidirect
19. C2×D3×D4C22semidirect
21. C22×D12C22semidirect
22. C2×C3×SD16C22semidirect
23. C22×C3×C8C22semidirect
24. C3×C22⋊Q8C22semidirect
26. #947C22semidirect
27. C2×#215C22semidirect
28. C2×#215C22semidirect
29. #940C22semidirect
31. C2×#213C22semidirect
32. C2×#213C22semidirect
38. C2×D24C22semidirect
39. C2×D24C22semidirect
40. C2×ΓL(1,𝔽25)C22semidirect
42. #895C22semidirect
43. #895C22semidirect
45. #893C22semidirect
46. #893C22semidirect
51. C3×#106C22semidirect
52. C3×#103C22semidirect
53. C3×#102C22semidirect
54. #830C22semidirect
55. (C2×D3)⋊C8C22semidirect
56. #815C22semidirect
57. Dic3⋊C8C22semidirect
58. #809C22semidirect
59. #809C22semidirect
60. #809C22semidirect
61. #809C22semidirect
68. #465D3semidirect
138. D24D4semidirect
148. C2×SD16C2×D3semidirect
157. C22×C8C2×D3semidirect
161. C22⋊Q8C2×D3semidirect
174. #106C2×D3semidirect
176. #103C2×D3semidirect
181. #102C2×D3semidirect
232. D12C2×D4semidirect
239. C3×C8C2×D4semidirect
310. C22×C3AGL(1,ℤ/8)semidirect
395. C8D3×D4semidirect
429. C22#914semidirect
440. C3#3264semidirect

Maximal subgroups

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

subgroupindex
C2×#9142
C22×D242
#56302
#56092
C3×#4652
#51762
#51112
#50462
#50362
#48692
#48652
#48192
#48102
#47862
#47732
#32643