Finite Group Databasebeta

← #80552

Subgroups — #80552

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
2C2×D3×D8C211yesmaximal, characteristic, complemented
3#5676C211yesmaximal, characteristic, complemented
4D4×D12C211yesmaximal, characteristic, complemented
5C3×#463C211yesmaximal, characteristic, complemented
6#5169C211yesmaximal, characteristic, complemented
7#5156C211yesmaximal, characteristic, complemented
8#5058C211yesmaximal, characteristic, complemented
9#5014C211yesmaximal, characteristic, complemented
10#4843C211yesmaximal, characteristic, complemented
11#4836C211yesmaximal, characteristic, complemented
12#4810C211yesmaximal, characteristic, complemented
13#4807C211yesmaximal, characteristic, complemented
14#4782C211yesmaximal, characteristic, complemented
15#4773C211yesmaximal, characteristic, complemented
16#3264—31nomaximal, Sylow
17C2×#239C2211yescharacteristic, complemented
18C2×D3×D4C2211yescharacteristic
19C2×D3×D4C2211yescharacteristic, complemented
20C2×D3×D4C2211yescharacteristic, complemented
21C22×D12—21no—
22C2×C3×SD16C2211yescharacteristic, complemented
23C2×C3×D8C2211yescharacteristic
24C3×#124C2211yescharacteristic, complemented
25C3×C4×D4C2211yescharacteristic, complemented
26#946C2211yescharacteristic, complemented
27C2×#215C2211yescharacteristic, complemented
28C2×#215C2211yescharacteristic, complemented
29#938C2211yescharacteristic, complemented
30C2×#213C2211yescharacteristic
31C2×#213C2211yescharacteristic, complemented
32C2×#213C2211yescharacteristic, complemented
33#930—21no—
34#914—24no—
35D3×D8—24no—
36C2×D24C2211yescharacteristic, complemented
37C2×D24C2211yescharacteristic, complemented
38C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
39C2×D3×C8C2211yescharacteristic, complemented
40#895C2211yescharacteristic
41#895C2211yescharacteristic, complemented
42#884—21no—
43#883—21no—
44#882—21no—
45#882—21no—
46#882—21no—
47D3×C22⋊C4—21no—
48#874C2211yescharacteristic, complemented
49C4×D12C2211yescharacteristic
50C4×D12C2211yescharacteristic, complemented
51C3×C4⋊C8C2211yescharacteristic, complemented
52C3×#103C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic, complemented
54#832C2211yescharacteristic, complemented
55#821C2211yescharacteristic, complemented
56#821C2211yescharacteristic, complemented
57(C2×D3)⋊C8C2211yescharacteristic, complemented
58(C2×D3)⋊C8C2211yescharacteristic, complemented
59#809C2211yescharacteristic, complemented
60#808C2211yescharacteristic, complemented
61#803C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C22×D8—31no—
64#546—31no—
65D42—31no—
66#496—31no—
67#469—31no—
68#465—31no—
69#463D311yescharacteristic, complemented
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×D4C2311yescharacteristic
82C2×C3×D4C2311yescharacteristic
83C22×C3×C4—21no—
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88#239—22no—
89#239—41no—
90D3×D4—22no—
91D3×D4—22no—
92D3×D4—22no—
93D3×D4—41no—
94D3×D4—41no—
95D3×D4—41no—
96C2×D12—21no—
97C2×D12—21no—
98C2×D12—21no—
99C2×D12—21no—
100C2×D12—21no—
101C2×D12—44no—
102C2×D12D411yescharacteristic
103C2×D12C2311yescharacteristic
104C2×D12C2311yescharacteristic
105C2×D12C2311yescharacteristic
106C2×C4×D3—21no—
107C2×C4×D3—21no—
108C2×C4×D3C2311yescharacteristic
109C2×C4×D3C2311yescharacteristic
110C3×SD16—22no—
111C3×D8—22no—
112C2×C3×C8C2311yescharacteristic
113C2×C3×C8C2311yescharacteristic
114C3×C4⋊C4C2311yescharacteristic
115C3×C22⋊C4—21no—
116C3×C22⋊C4—21no—
117C3×C22⋊C4—21no—
118C3×C42C2311yescharacteristic
119#217—21no—
120#215—22no—
121#215—41no—
122#213—22no—
123#213—41no—
124#213D414yescomplemented
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4—21no—
127(C2×D3)⋊C4—21no—
128(C2×D3)⋊C4—21no—
129(C2×D3)⋊C4D411yescharacteristic
130(C2×D3)⋊C4D411yescharacteristic
131C4⋊Dic3D411yescharacteristic
132C4⋊Dic3C2311yescharacteristic
133Dic3⋊C4—21no—
134C2×C3⋊C8C2311yescharacteristic
135C2×C3⋊C8C2311yescharacteristic
136D24—22no—
137D24—22no—
138ΓL(1,𝔽25)—22no—
139D3×C8—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—64no—
155C2×D8C2×D311yescharacteristic
156C2×M16—31no—
157C22×C8—31no—
158#127—31no—
159#124C2×D311yescharacteristic, complemented
160#123—61no—
161C22⋊Q8—31no—
162#121—31no—
163#121—31no—
164#121—31no—
165#121—61no—
166#121—61no—
167#120—61no—
168#120—61no—
169#120—61no—
170C4×D4—31no—
171C4×D4—31no—
172C4×D4C2×D311yescharacteristic, complemented
173C2×C22⋊C4—61no—
174#106—31no—
175C4⋊C8C2×D311yescharacteristic, complemented
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×C3—21no—
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—21no—
190C22×D3—42no—
191C22×D3—42no—
192C22×D3—42no—
193C22×D3—42no—
194C22×D3—42no—
195C22×D3—42no—
196C22×D3—81no—
197C22×D3—81no—
198C22×D3—81no—
199C22×D3C2×D411yescharacteristic
200C22×D3C2×D411yescharacteristic
201C3×Q8—21no—
202C3×Q8—21no—
203C3×D4—21no—
204C3×D4—21no—
205C3×D4—21no—
206C3×D4C2×D412yes—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—42no—
210C2×C3×C4C2×D411yescharacteristic
211C2×C3×C4C2×D411yescharacteristic
212C2×C3×C4C2411yescharacteristic
213#67—42no—
214#67—42no—
215#67—42no—
216#67—81no—
217C2×Dic3—21no—
218C2×Dic3C2×D411yescharacteristic
219C2×Dic3C2×D411yescharacteristic
220D12—22no—
221D12—21no—
222D12—21no—
223D12—21no—
224D12—21no—
225D12—21no—
226D12—42no—
227D12—42no—
228D12—42no—
229D12—42no—
230D12—81no—
231D12C2×D412yescomplemented
232C4×D3—21no—
233C4×D3—21no—
234C4×D3—21no—
235C4×D3—21no—
236C4×D3—42no—
237C4×D3—81no—
238C3×C8—21no—
239C3×C8—21no—
240C3⋊C8—21no—
241C3⋊C8C2×D412yescomplemented
242C24—61no—
243C24—61no—
244C24—61no—
245#41—62no—
246#41—121no—
247C2×Q8C22×D311yescharacteristic
248C2×D4—31no—
249C2×D4—31no—
250C2×D4—31no—
251C2×D4—31no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—62no—
259C2×D4—62no—
260C2×D4—62no—
261C2×D4—61no—
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—124no—
265C2×D4—121no—
266C2×D4—121no—
267C2×D4—121no—
268C2×D4C22×D311yescharacteristic
269C2×D4C22×D311yescharacteristic
270C22×C4—21no—
271C22×C4—31no—
272C22×C4—31no—
273C22×C4—61no—
274C22×C4—61no—
275SD16—22no—
276SD16—62no—
277SD16—121no—
278D8—22no—
279D8—34no—
280D8—62no—
281D8—62no—
282D8—62no—
283D8—121no—
284M16—62no—
285C2×C8—31no—
286C2×C8—31no—
287C2×C8—62no—
288C2×C8C22×D311yescharacteristic
289C2×C8C22×D311yescharacteristic
290C4⋊C4—31no—
291C4⋊C4—31no—
292C4⋊C4—61no—
293C4⋊C4C22×D311yescharacteristic
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—31no—
298C22⋊C4—31no—
299C22⋊C4—61no—
300C22⋊C4—61no—
301C22⋊C4—61no—
302C22⋊C4—61no—
303C22⋊C4—61no—
304C42C22×D311yescharacteristic
305C22×C3—22no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3C22×D411yescharacteristic
312C2×D3—21no—
313C2×D3—21no—
314C2×D3—21no—
315C2×D3—22no—
316C2×D3—21no—
317C2×D3—21no—
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—84no—
327C2×D3—84no—
328C2×D3—84no—
329C2×D3AGL(1,ℤ/8)12yescomplemented
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—41no—
333C3×C4—41no—
334C3×C4C22×D411yescharacteristic
335C3×C4H(5,𝔽2)11yescharacteristic
336Dic3—21no—
337Dic3—21no—
338Dic3—41no—
339C23—21no—
340C23—21no—
341C23—31no—
342C23—31no—
343C23—61no—
344C23—61no—
345C23—61no—
346C23—122no—
347C23—122no—
348C23—122no—
349C23—122no—
350C23—122no—
351C23—122no—
352C23—241no—
353C23—241no—
354C23—241no—
355Q8—21no—
356Q8—21no—
357D4—21no—
358D4—21no—
359D4—21no—
360D4—32no—
361D4—62no—
362D4—61no—
363D4—61no—
364D4—61no—
365D4—61no—
366D4—61no—
367D4—122no—
368D4—122no—
369D4—122no—
370D4—122no—
371D4—122no—
372D4—122no—
373D4—122no—
374D4—241no—
375D4—241no—
376D4D3×D412yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—31no—
380C2×C4—31no—
381C2×C4—42no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—61no—
385C2×C4—61no—
386C2×C4—61no—
387C2×C4—122no—
388C2×C4—241no—
389C2×C4D3×D411yescharacteristic
390C2×C4D3×D411yescharacteristic
391C2×C4C23×D311yescharacteristic
392C8—21no—
393C8—21no—
394C8—32no—
395C8—61no—
396C2×C3—42no—
397C2×C3—81no—
398C2×C3D4211yescharacteristic
399C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
400C2×C3#57611yescharacteristic
401D3—22no—
402D3—41no—
403D3—42no—
404D3—81no—
405D3—81no—
406C22—22no—
407C22—32no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—61no—
414C22—61no—
415C22—61no—
416C22—61no—
417C22—61no—
418C22—62no—
419C22—121no—
420C22—121no—
421C22—121no—
422C22—121no—
423C22—121no—
424C22—121no—
425C22—121no—
426C22—121no—
427C22—244no—
428C22—244no—
429C22—244no—
430C22C2×D3×D411yescharacteristic
431C4—21no—
432C4—21no—
433C4—41no—
434C4—41no—
435C4—61no—
436C4—61no—
437C4—121no—
438C4C2×D3×D411yescharacteristic
439C4#100911yescharacteristic
440C3#326411yescharacteristic, complemented, Sylow
441C2—42no—
442C2—62no—
443C2—81no—
444C2—121no—
445C2—122no—
446C2—241no—
447C2—241no—
448C2#560911yescharacteristic
449C2#576911yescharacteristic
450C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#463C2
socleC22×C3C22×D4
hypercenter#463D3
hypocenterC3#3264
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#914C2semidirect
2. C2×D3×D8C2semidirect
3. #5676C2semidirect
4. D4×D12C2semidirect
5. C3×#463C2semidirect
6. #5169C2semidirect
7. #5156C2semidirect
8. #5058C2semidirect
9. #5014C2semidirect
10. #4843C2semidirect
11. #4836C2semidirect
12. #4810C2semidirect
13. #4807C2semidirect
14. #4782C2semidirect
15. #4773C2semidirect
17. C2×#239C22semidirect
19. C2×D3×D4C22semidirect
20. C2×D3×D4C22semidirect
22. C2×C3×SD16C22semidirect
24. C3×#124C22semidirect
25. C3×C4×D4C22semidirect
26. #946C22semidirect
27. C2×#215C22semidirect
28. C2×#215C22semidirect
29. #938C22semidirect
31. C2×#213C22semidirect
32. C2×#213C22semidirect
36. C2×D24C22semidirect
37. C2×D24C22semidirect
38. C2×ΓL(1,𝔽25)C22semidirect
39. C2×D3×C8C22semidirect
41. #895C22semidirect
48. #874C22semidirect
50. C4×D12C22semidirect
51. C3×C4⋊C8C22semidirect
52. C3×#103C22semidirect
53. C3×#102C22semidirect
54. #832C22semidirect
55. #821C22semidirect
56. #821C22semidirect
57. (C2×D3)⋊C8C22semidirect
58. (C2×D3)⋊C8C22semidirect
59. #809C22semidirect
60. #808C22semidirect
61. #803C22semidirect
69. #463D3semidirect
124. #213D4semidirect
148. C2×SD16C2×D3semidirect
159. #124C2×D3semidirect
172. C4×D4C2×D3semidirect
175. C4⋊C8C2×D3semidirect
176. #103C2×D3semidirect
181. #102C2×D3semidirect
231. D12C2×D4semidirect
241. C3⋊C8C2×D4semidirect
329. C2×D3AGL(1,ℤ/8)semidirect
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
C2×D3×D82
#56762
D4×D122
C3×#4632
#51692
#51562
#50582
#50142
#48432
#48362
#48102
#48072
#47822
#47732
#32643