Finite Group Databasebeta

← #81526

Subgroups — #81526

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
1#5630C211yesmaximal, characteristic
2#5533C211yesmaximal, characteristic, complemented
3C3×#485C211yesmaximal, characteristic, complemented
4#5052C211yesmaximal, characteristic, complemented
5#5001C211yesmaximal, characteristic, complemented
6#4889C211yesmaximal, characteristic, complemented
7#4885C211yesmaximal, characteristic, complemented
8#4870C211yesmaximal, characteristic, complemented
9#4864C211yesmaximal, characteristic, complemented
10#4808C211yesmaximal, characteristic, complemented
11#4804C211yesmaximal, characteristic, complemented
12#4798C211yesmaximal, characteristic, complemented
13#4794C211yesmaximal, characteristic, complemented
14#4731C211yesmaximal, characteristic, complemented
15#4728C211yesmaximal, characteristic, complemented
16#3311—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C22×D12C2211yescharacteristic
19C3×C22⋊Q8C2211yescharacteristic
20C3×#117C2211yescharacteristic
21#947C2211yescharacteristic
22C2×#216C2211yescharacteristic, complemented
23C4×#67C2211yescharacteristic
24#924C2211yescharacteristic
25C2×#208C2211yescharacteristic
26C2×#204C2211yescharacteristic
27C2×#204C2211yescharacteristic
28C2×D3×C8C2211yescharacteristic, complemented
29#897C2211yescharacteristic, complemented
30#895—21no—
31#895C2211yescharacteristic
32#894—21no—
33#893C2211yescharacteristic
34#892C2211yescharacteristic, complemented
35#888C2211yescharacteristic, complemented
36#887C2211yescharacteristic, complemented
37#885—21no—
38#882—21no—
39D3×C22⋊C4—21no—
40#879—21no—
41#876—21no—
42C4×Dic6—21no—
43C3×#107C2211yescharacteristic, complemented
44C3×#106C2211yescharacteristic, complemented
45C3×#103C2211yescharacteristic, complemented
46C3×#103C2211yescharacteristic, complemented
47C3×C22⋊C8C2211yescharacteristic
48#835C2211yescharacteristic, complemented
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic, complemented
52#815C2211yescharacteristic, complemented
53Dic3⋊C8C2211yescharacteristic, complemented
54#810C2211yescharacteristic, complemented
55#809C2211yescharacteristic
56#809C2211yescharacteristic
57#809C2211yescharacteristic
58#809C2211yescharacteristic
59#808C2211yescharacteristic, complemented
60#556—31no—
61#546—31no—
62#489—31no—
63#485D311yescharacteristic, complemented
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×Q8C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C2×#67D411yescharacteristic
79#239—22no—
80#239—41no—
81C2×D12—22no—
82C2×D12—21no—
83C2×D12—21no—
84C2×D12—21no—
85C2×D12—41no—
86C2×D12C2311yescharacteristic
87C2×D12C2311yescharacteristic
88C2×C4×D3—21no—
89C2×C4×D3—21no—
90C2×C4×D3C2311yescharacteristic
91C2×Dic6—21no—
92C2×Dic6C2311yescharacteristic
93C2×C3×C8C2311yescharacteristic
94C2×C3×C8C2311yescharacteristic
95C3×C4⋊C4—21no—
96C3×C4⋊C4C2311yescharacteristic
97C3×C4⋊C4C2311yescharacteristic
98C3×C4⋊C4C2311yescharacteristic
99C3×C22⋊C4—21no—
100C3×C22⋊C4—21no—
101C3×C42—21no—
102#217—21no—
103#217D411yescharacteristic
104#216—41no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4D411yescharacteristic
112C4⋊Dic3—21no—
113C4⋊Dic3C2311yescharacteristic
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4D411yescharacteristic
118C4×Dic3—21no—
119C4×Dic3—21no—
120C4×Dic3C2311yescharacteristic
121#208—41no—
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124#204—22no—
125#204—41no—
126D3×C8—41no—
127C2×#41—31no—
128C22×D4—31no—
129C2×Q16—31no—
130C2×SD16—31no—
131C2×SD16—31no—
132C2×M16—31no—
133C22×C8—31no—
134C4⋊Q8—31no—
135#126—61no—
136#126—61no—
137#124—31no—
138#124—61no—
139#123—61no—
140C22⋊Q8—31no—
141C22⋊Q8—31no—
142C22⋊Q8C2×D311yescharacteristic
143#121—31no—
144#121—61no—
145#120—61no—
146C4×Q8—31no—
147C4×Q8—61no—
148C4×D4—31no—
149C4×D4—31no—
150#117—31no—
151#117C2×D311yescharacteristic
152C2×C22⋊C4—61no—
153#107C2×D311yescharacteristic, complemented
154#106—31no—
155#106C2×D311yescharacteristic, complemented
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103C2×D311yescharacteristic, complemented
160#103C2×D311yescharacteristic, complemented
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167C22⋊C8—31no—
168C22⋊C8C2×D311yescharacteristic
169#97—31no—
170C23×C3C2×D411yescharacteristic
171C22×D3—21no—
172C22×D3—21no—
173C22×D3—42no—
174C22×D3—42no—
175C22×D3—81no—
176C22×D3C2×D411yescharacteristic
177C3×Q8—21no—
178C3×Q8—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4C2×D411yescharacteristic
186C2×C3×C4C2411yescharacteristic
187#67—41no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3C2×D411yescharacteristic
191C2×Dic3C2×D411yescharacteristic
192C2×Dic3C2×D411yescharacteristic
193D12—21no—
194D12—21no—
195D12—21no—
196D12—42no—
197D12—42no—
198D12—81no—
199D12#4112yes—
200C4×D3—21no—
201C4×D3—21no—
202C4×D3—42no—
203C4×D3—81no—
204Dic6—21no—
205Dic6—21no—
206Dic6—42no—
207C3×C8—21no—
208C3×C8—21no—
209C3⋊C8—21no—
210C3⋊C8—21no—
211C24—61no—
212#41—62no—
213#41—121no—
214C2×Q8—31no—
215C2×Q8—61no—
216C2×Q8C22×D311yescharacteristic
217C2×D4—31no—
218C2×D4—31no—
219C2×D4—31no—
220C2×D4—62no—
221C2×D4—61no—
222C2×D4—61no—
223C2×D4—61no—
224C2×D4—121no—
225C22×C4—31no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4C22×D311yescharacteristic
229Q16—121no—
230SD16—62no—
231SD16—121no—
232M16—121no—
233C2×C8—31no—
234C2×C8—31no—
235C2×C8—121no—
236C2×C8C22×D311yescharacteristic
237C2×C8C22×D311yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—31no—
240C4⋊C4—31no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4C22×D311yescharacteristic
246C4⋊C4C22×D311yescharacteristic
247C4⋊C4C22×D311yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—21no—
250C22⋊C4—31no—
251C22⋊C4—31no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C42—21no—
260C42—31no—
261C42—61no—
262C42—61no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3C22×D411yescharacteristic
267C2×D3—21no—
268C2×D3—22no—
269C2×D3—21no—
270C2×D3—21no—
271C2×D3—41no—
272C2×D3—41no—
273C2×D3—41no—
274C2×D3—41no—
275C2×D3—41no—
276C2×D3—84no—
277C3×C4—21no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4—41no—
282C3×C4C2×#4111yescharacteristic
283C3×C4H(5,𝔽2)11yescharacteristic
284Dic3—21no—
285Dic3—21no—
286Dic3—21no—
287Dic3—41no—
288Dic3—41no—
289C23—31no—
290C23—61no—
291C23—61no—
292C23—122no—
293C23—122no—
294C23—241no—
295C23D3×D411yescharacteristic
296Q8—21no—
297Q8—21no—
298Q8—61no—
299Q8—61no—
300Q8—122no—
301D4—32no—
302D4—61no—
303D4—61no—
304D4—61no—
305D4—122no—
306D4—122no—
307D4—121no—
308D4—241no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—31no—
316C2×C4—31no—
317C2×C4—31no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—122no—
323C2×C4—241no—
324C2×C4D3×D411yescharacteristic
325C2×C4C23×D311yescharacteristic
326C8—21no—
327C8—21no—
328C8—61no—
329C8—61no—
330C2×C3—41no—
331C2×C3#54611yescharacteristic
332C2×C3#57511yescharacteristic
333C2×C3#57711yescharacteristic
334D3—41no—
335D3—42no—
336D3—81no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—61no—
341C22—62no—
342C22—61no—
343C22—61no—
344C22—121no—
345C22—121no—
346C22—121no—
347C22—121no—
348C22—121no—
349C22—244no—
350C22C2×D3×D411yescharacteristic
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—61no—
357C4—61no—
358C4—61no—
359C4—121no—
360C4—121no—
361C4D3×#4111yescharacteristic
362C4#100911yescharacteristic
363C3#331111yescharacteristic, complemented, Sylow
364C2—41no—
365C2—121no—
366C2—122no—
367C2—241no—
368C2#565411yescharacteristic
369C2#577011yescharacteristic
370C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#485C2
socleC22×C3C22×D4
prime radical#5630C2
hypercenter#485D3
hypocenterC3#3311
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
2. #5533C2semidirect
3. C3×#485C2semidirect
4. #5052C2semidirect
5. #5001C2semidirect
6. #4889C2semidirect
7. #4885C2semidirect
8. #4870C2semidirect
9. #4864C2semidirect
10. #4808C2semidirect
11. #4804C2semidirect
12. #4798C2semidirect
13. #4794C2semidirect
14. #4731C2semidirect
15. #4728C2semidirect
22. C2×#216C22semidirect
28. C2×D3×C8C22semidirect
29. #897C22semidirect
34. #892C22semidirect
35. #888C22semidirect
36. #887C22semidirect
43. C3×#107C22semidirect
44. C3×#106C22semidirect
45. C3×#103C22semidirect
46. C3×#103C22semidirect
48. #835C22semidirect
51. (C2×D3)⋊C8C22semidirect
52. #815C22semidirect
53. Dic3⋊C8C22semidirect
54. #810C22semidirect
59. #808C22semidirect
63. #485D3semidirect
153. #107C2×D3semidirect
155. #106C2×D3semidirect
159. #103C2×D3semidirect
160. #103C2×D3semidirect
363. C3#3311semidirect

Maximal subgroups

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

subgroupindex
#56302
#55332
C3×#4852
#50522
#50012
#48892
#48852
#48702
#48642
#48082
#48042
#47982
#47942
#47312
#47282
#33113