Finite Group Databasebeta

← #83665

Subgroups — #83665

Order 384.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#556C211yesmaximal, characteristic, complemented
2#5550C211yesmaximal, characteristic
3#5089C211yesmaximal, characteristic, complemented
4#5084C211yesmaximal, characteristic, complemented
5#5052C211yesmaximal, characteristic, complemented
6#5048C211yesmaximal, characteristic, complemented
7#5047C211yesmaximal, characteristic, complemented
8#5046C211yesmaximal, characteristic, complemented
9#5028C211yesmaximal, characteristic, complemented
10#5026C211yesmaximal, characteristic, complemented
11#5011C211yesmaximal, characteristic, complemented
12#5003C211yesmaximal, characteristic, complemented
13#5002C211yesmaximal, characteristic, complemented
14#5001C211yesmaximal, characteristic, complemented
15#4996C211yesmaximal, characteristic, complemented
16#3311—31nomaximal, Sylow
17C2×#235C2211yescharacteristic
18C22×D12C2211yescharacteristic
19C3×C4⋊Q8C2211yescharacteristic, complemented
20C3×#126—21no—
21C3×#126—21no—
22C3×#124—21no—
23C3×C22⋊Q8C2211yescharacteristic
24C3×C22⋊Q8C2211yescharacteristic, complemented
25C3×C4×Q8—21no—
26C3×C4×Q8C2211yescharacteristic, complemented
27C3×C4×D4C2211yescharacteristic
28C3×#117C2211yescharacteristic
29C3×#117C2211yescharacteristic, complemented
30C2×#216C2211yescharacteristic, complemented
31C2×#215C2211yescharacteristic
32C2×#215C2211yescharacteristic
33#937—21no—
34#930C2211yescharacteristic
35#924C2211yescharacteristic
36C2×#208C2211yescharacteristic
37C22×C3⋊C8C2211yescharacteristic, complemented
38#894—21no—
39#884—21no—
40D3×C22⋊C4—21no—
41#875C2211yescharacteristic
42C4×D12C2211yescharacteristic
43#835C2211yescharacteristic, complemented
44#830C2211yescharacteristic
45#830C2211yescharacteristic, complemented
46#810C2211yescharacteristic, complemented
47#810C2211yescharacteristic, complemented
48#810C2211yescharacteristic, complemented
49#809C2211yescharacteristic
50#809C2211yescharacteristic
51#809C2211yescharacteristic
52#809C2211yescharacteristic
53#809C2211yescharacteristic
54#809C2211yescharacteristic
55#808C2211yescharacteristic, complemented
56#808C2211yescharacteristic, complemented
57#807C2211yescharacteristic, complemented
58C4⋊(C3⋊C8)C2211yescharacteristic, complemented
59#803C2211yescharacteristic, complemented
60#556D311yescharacteristic, complemented
61#546—31no—
62#489—31no—
63#485—31no—
64#482—31no—
65#476—31no—
66#470—31no—
67#465—31no—
68#464—31no—
69#452—31no—
70#450—31no—
71#440—31no—
72#435—31no—
73#418—31no—
74#416—31no—
75C23×D3—21no—
76C2×C3×Q8—21no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4D411yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C2×#67—21no—
82C2×#67—21no—
83#235—22no—
84#235—41no—
85C2×D12—22no—
86C2×D12—21no—
87C2×D12—41no—
88C2×D12C2311yescharacteristic
89C2×D12C2311yescharacteristic
90C2×C4×D3—21no—
91C2×C4×D3—21no—
92C2×Dic6C2311yescharacteristic
93C3×C4⋊C4—21no—
94C3×C4⋊C4—21no—
95C3×C4⋊C4—21no—
96C3×C4⋊C4—21no—
97C3×C4⋊C4D411yescharacteristic
98C3×C4⋊C4C2311yescharacteristic
99C3×C4⋊C4C2311yescharacteristic
100C3×C4⋊C4C2311yescharacteristic
101C3×C22⋊C4—21no—
102C3×C22⋊C4—21no—
103C3×C22⋊C4—21no—
104C3×C22⋊C4—21no—
105C3×C22⋊C4D411yescharacteristic
106C3×C22⋊C4D411yescharacteristic
107C3×C42—21no—
108C3×C42—21no—
109C3×C42—21no—
110C3×C42C2311yescharacteristic
111#217—21no—
112#216—41no—
113#215—22no—
114#215—41no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4—21no—
119C4⋊Dic3C2311yescharacteristic
120Dic3⋊C4—21no—
121#208—41no—
122C2×C3⋊C8—41no—
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×#41—31no—
128C22×D4—31no—
129C2×Q16—31no—
130C2×SD16—31no—
131C2×SD16—31no—
132C2×M16—31no—
133C22×C8—31no—
134C4⋊Q8C2×D311yescharacteristic, complemented
135#126—21no—
136#126—21no—
137#124—21no—
138#124—31no—
139#123—61no—
140C22⋊Q8—31no—
141C22⋊Q8C2×D311yescharacteristic
142C22⋊Q8C2×D311yescharacteristic, complemented
143#121—31no—
144#121—61no—
145#120—61no—
146C4×Q8—21no—
147C4×Q8C2×D311yescharacteristic, complemented
148C4×D4—31no—
149C4×D4C2×D311yescharacteristic
150#117C2×D311yescharacteristic
151#117C2×D311yescharacteristic, complemented
152C2×C22⋊C4—61no—
153#107—31no—
154#106—31no—
155#106—31no—
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167C22⋊C8—31no—
168C22⋊C8—31no—
169#97—31no—
170C23×C3C2×D411yescharacteristic
171C23×C3C2×D411yescharacteristic
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—42no—
175C22×D3—42no—
176C22×D3—81no—
177C3×Q8—21no—
178C3×Q8—21no—
179C3×Q8—42no—
180C3×D4—41no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2×D411yescharacteristic
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2×D411yescharacteristic
193C2×C3×C4C2411yescharacteristic
194#67—42no—
195#67—81no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198D12—21no—
199D12—21no—
200D12—21no—
201D12—42no—
202D12#4112yes—
203C4×D3—42no—
204C4×D3—81no—
205Dic6—21no—
206Dic6—21no—
207C3⋊C8—21no—
208C3⋊C8—21no—
209C3⋊C8—21no—
210C3⋊C8—21no—
211C24—61no—
212#41—62no—
213#41—121no—
214C2×Q8—21no—
215C2×Q8—31no—
216C2×Q8C22×D311yescharacteristic
217C2×D4—31no—
218C2×D4—31no—
219C2×D4—62no—
220C2×D4—61no—
221C2×D4—61no—
222C2×D4—61no—
223C2×D4—121no—
224C2×D4#6711yescharacteristic
225C22×C4—61no—
226C22×C4—61no—
227C22×C4C22×D311yescharacteristic
228C22×C4C22×D311yescharacteristic
229Q16—121no—
230SD16—62no—
231SD16—121no—
232M16—121no—
233C2×C8—31no—
234C2×C8—31no—
235C2×C8—31no—
236C2×C8—31no—
237C2×C8—121no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—31no—
243C4⋊C4—61no—
244C4⋊C4#6711yescharacteristic
245C4⋊C4C22×D311yescharacteristic
246C4⋊C4C22×D311yescharacteristic
247C4⋊C4C22×D311yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—21no—
250C22⋊C4—21no—
251C22⋊C4—21no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4#6711yescharacteristic
258C22⋊C4#6711yescharacteristic
259C42—21no—
260C42—21no—
261C42—21no—
262C42C22×D311yescharacteristic
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3C22×D411yescharacteristic
270C2×D3—22no—
271C2×D3—41no—
272C2×D3—41no—
273C2×D3—41no—
274C2×D3—41no—
275C2×D3—41no—
276C2×D3—84no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—41no—
282C3×C4—41no—
283C3×C4—41no—
284C3×C4—41no—
285C3×C4C2×#4111yescharacteristic
286C3×C4H(5,𝔽2)11yescharacteristic
287Dic3—41no—
288Dic3—41no—
289C23—61no—
290C23—61no—
291C23—122no—
292C23—122no—
293C23—241no—
294C23C2×#6711yescharacteristic
295C23C2×#6711yescharacteristic
296Q8—21no—
297Q8—21no—
298Q8—42no—
299Q8—61no—
300Q8—61no—
301D4—32no—
302D4—41no—
303D4—61no—
304D4—61no—
305D4—61no—
306D4—122no—
307D4—122no—
308D4—241no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—122no—
320C2×C4—241no—
321C2×C4C2×#6711yescharacteristic
322C2×C4C2×#6711yescharacteristic
323C2×C4C2×#6711yescharacteristic
324C2×C4C2×#6711yescharacteristic
325C2×C4C23×D311yescharacteristic
326C8—61no—
327C8—61no—
328C8—61no—
329C8—61no—
330C2×C3—41no—
331C2×C3—41no—
332C2×C3#54611yescharacteristic
333C2×C3#57511yescharacteristic
334C2×C3#57711yescharacteristic
335D3—42no—
336D3—81no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—62no—
344C22—121no—
345C22—121no—
346C22—121no—
347C22—121no—
348C22—121no—
349C22—244no—
350C22C22×#6711yescharacteristic
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—121no—
360C4—121no—
361C4D3×#4111yescharacteristic
362C4#100911yescharacteristic
363C3#331111yescharacteristic, complemented, Sylow
364C2—41no—
365C2—41no—
366C2—122no—
367C2—241no—
368C2#582211yescharacteristic
369C2#582911yescharacteristic
370C2#583711yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#556C2
socleC22×C3C22×D4
prime radical#5550C2
hypercenter#556D3
hypocenterC3#3311
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
1. C3×#556C2semidirect
3. #5089C2semidirect
4. #5084C2semidirect
5. #5052C2semidirect
6. #5048C2semidirect
7. #5047C2semidirect
8. #5046C2semidirect
9. #5028C2semidirect
10. #5026C2semidirect
11. #5011C2semidirect
12. #5003C2semidirect
13. #5002C2semidirect
14. #5001C2semidirect
15. #4996C2semidirect
19. C3×C4⋊Q8C22semidirect
24. C3×C22⋊Q8C22semidirect
26. C3×C4×Q8C22semidirect
29. C3×#117C22semidirect
30. C2×#216C22semidirect
37. C22×C3⋊C8C22semidirect
43. #835C22semidirect
45. #830C22semidirect
46. #810C22semidirect
47. #810C22semidirect
48. #810C22semidirect
55. #808C22semidirect
56. #808C22semidirect
57. #807C22semidirect
58. C4⋊(C3⋊C8)C22semidirect
59. #803C22semidirect
60. #556D3semidirect
134. C4⋊Q8C2×D3semidirect
142. C22⋊Q8C2×D3semidirect
147. C4×Q8C2×D3semidirect
151. #117C2×D3semidirect
363. C3#3311semidirect

Maximal subgroups

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

subgroupindex
C3×#5562
#55502
#50892
#50842
#50522
#50482
#50472
#50462
#50282
#50262
#50112
#50032
#50022
#50012
#49962
#33113