Finite Group Databasebeta

← #81025

Subgroups — #81025

Order 384.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5734C211yesmaximal, characteristic, complemented
2#5576C211yesmaximal, characteristic, complemented
3C3×#475C211yesmaximal, characteristic, complemented
4#5089C211yesmaximal, characteristic, complemented
5#5026C211yesmaximal, characteristic, complemented
6#4868C211yesmaximal, characteristic, complemented
7#4866C211yesmaximal, characteristic, complemented
8#4865C211yesmaximal, characteristic, complemented
9#4864C211yesmaximal, characteristic, complemented
10#4844C211yesmaximal, characteristic, complemented
11#4836C211yesmaximal, characteristic, complemented
12#4810C211yesmaximal, characteristic, complemented
13#4806C211yesmaximal, characteristic, complemented
14#4804C211yesmaximal, characteristic, complemented
15#4802C211yesmaximal, characteristic, complemented
16#3321—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C3×C4⋊Q8C2211yescharacteristic
20C3×C4×Q8C2211yescharacteristic
21#947—21no—
22#947—21no—
23C2×#215C2211yescharacteristic
24C2×D24C2211yescharacteristic
25C2×#204C2211yescharacteristic
26C2×ΓL(1,𝔽25)C2211yescharacteristic
27C2×D3×C8C2211yescharacteristic
28#897—21no—
29#897C2211yescharacteristic
30#895—21no—
31#895—21no—
32#895C2211yescharacteristic
33#895C2211yescharacteristic
34#895C2211yescharacteristic
35#892—21no—
36#892C2211yescharacteristic
37#892C2211yescharacteristic
38#875—21no—
39#875C2211yescharacteristic
40C4×D12—21no—
41C4×D12C2211yescharacteristic
42C4×D12C2211yescharacteristic
43C3×#106C2211yescharacteristic
44C3×#106C2211yescharacteristic
45C3×C4⋊C8C2211yescharacteristic
46C3×#103C2211yescharacteristic
47C3×#103C2211yescharacteristic
48#835C2211yescharacteristic
49#821C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52#816C2211yescharacteristic
53#810C2211yescharacteristic
54#810C2211yescharacteristic
55#809C2211yescharacteristic
56#809C2211yescharacteristic
57#809C2211yescharacteristic
58#807C2211yescharacteristic
59#803C2211yescharacteristic
60#555—31no—
61#548—31no—
62#489—31no—
63#485—31no—
64#482—31no—
65#475D311yescharacteristic, complemented
66#469—31no—
67#465—31no—
68#463—31no—
69#449—31no—
70#448—31no—
71#440—31no—
72#435—31no—
73#419—31no—
74#416—31no—
75C2×C3×Q8—21no—
76C2×C3×Q8C2311yescharacteristic
77#239—22no—
78#239—41no—
79D3×Q8—22no—
80D3×Q8—41no—
81C2×D12—21no—
82C2×D12—21no—
83C2×D12—21no—
84C2×D12—21no—
85C2×D12D411yescharacteristic
86C2×D12C2311yescharacteristic
87C2×D12C2311yescharacteristic
88C2×C4×D3—21no—
89C2×C4×D3—21no—
90C2×C4×D3—21no—
91C2×C4×D3C2311yescharacteristic
92C2×C4×D3C2311yescharacteristic
93C2×Dic6—21no—
94C2×Dic6C2311yescharacteristic
95C2×C3×C8C2311yescharacteristic
96C2×C3×C8C2311yescharacteristic
97C3×C4⋊C4—21no—
98C3×C4⋊C4—21no—
99C3×C4⋊C4C2311yescharacteristic
100C3×C4⋊C4C2311yescharacteristic
101C3×C4⋊C4C2311yescharacteristic
102C3×C42—21no—
103C3×C42C2311yescharacteristic
104#215—22no—
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)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4D411yescharacteristic
115(C2×D3)⋊C4D411yescharacteristic
116C4⋊Dic3—21no—
117C4⋊Dic3D411yescharacteristic
118C4⋊Dic3C2311yescharacteristic
119C4×Dic3—21no—
120C4×Dic3—21no—
121C2×C3⋊C8C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123D24—41no—
124#204—41no—
125ΓL(1,𝔽25)—41no—
126D3×C8—41no—
127C2×#41—31no—
128C22×Q8—31no—
129C2×SD16—31no—
130C2×SD16—31no—
131C2×D8—31no—
132C2×M16—31no—
133C22×C8—31no—
134C4⋊Q8C2×D311yescharacteristic
135#124—31no—
136#124—61no—
137#124—61no—
138#124—61no—
139C22⋊Q8—31no—
140C22⋊Q8—61no—
141#121—31no—
142#121—31no—
143#121—31no—
144#121—61no—
145#121—61no—
146C4×Q8C2×D311yescharacteristic
147C4×D4—31no—
148C4×D4—31no—
149C4×D4—61no—
150#117—31no—
151#117—31no—
152#117—61no—
153#107—31no—
154#106C2×D311yescharacteristic
155#106C2×D311yescharacteristic
156C4⋊C8C2×D311yescharacteristic
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103C2×D311yescharacteristic
162#103C2×D311yescharacteristic
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167C22⋊C8—31no—
168C22⋊C8—31no—
169#97—31no—
170C22×D3—21no—
171C22×D3—21no—
172C22×D3C2×D411yescharacteristic
173C22×D3C2×D411yescharacteristic
174C3×Q8—21no—
175C3×Q8—81no—
176C3×Q8#4112yes—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4C2×D411yescharacteristic
182C2×C3×C4C2×D411yescharacteristic
183C2×C3×C4C2411yescharacteristic
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3C2×D411yescharacteristic
187C2×Dic3C2×D411yescharacteristic
188D12—21no—
189D12—21no—
190D12—21no—
191D12—21no—
192D12—41no—
193D12—42no—
194D12—42no—
195D12—81no—
196D12—81no—
197C4×D3—21no—
198C4×D3—21no—
199C4×D3—21no—
200C4×D3—21no—
201C4×D3—42no—
202C4×D3—42no—
203C4×D3—42no—
204Dic6—21no—
205Dic6—21no—
206Dic6—42no—
207C3×C8—21no—
208C3×C8—21no—
209C3⋊C8—21no—
210C3⋊C8—21no—
211#41—62no—
212#41—121no—
213C2×Q8—21no—
214C2×Q8—31no—
215C2×Q8—61no—
216C2×Q8—62no—
217C2×Q8—121no—
218C2×Q8C22×D311yescharacteristic
219C2×D4—31no—
220C2×D4—31no—
221C2×D4—31no—
222C2×D4—61no—
223C2×D4—61no—
224C2×D4—61no—
225C2×D4—61no—
226C22×C4—31no—
227C22×C4—31no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4—61no—
231SD16—62no—
232SD16—121no—
233D8—121no—
234M16—121no—
235C2×C8—31no—
236C2×C8—31no—
237C2×C8—121no—
238C2×C8C22×D311yescharacteristic
239C2×C8C22×D311yescharacteristic
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—31no—
243C4⋊C4—31no—
244C4⋊C4—61no—
245C4⋊C4C22×D311yescharacteristic
246C4⋊C4C22×D311yescharacteristic
247C4⋊C4C22×D311yescharacteristic
248C22⋊C4—31no—
249C22⋊C4—31no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C42—21no—
260C42—61no—
261C42—61no—
262C42C22×D311yescharacteristic
263C22×C3C22×D411yescharacteristic
264C2×D3—21no—
265C2×D3—21no—
266C2×D3—21no—
267C2×D3—21no—
268C2×D3—21no—
269C2×D3—21no—
270C2×D3—41no—
271C2×D3—41no—
272C2×D3—41no—
273C2×D3—41no—
274C2×D3—41no—
275C2×D3—41no—
276C3×C4—22no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4—41no—
282C3×C4C2×#4111yescharacteristic
283C3×C4H(5,𝔽2)11yescharacteristic
284Dic3—21no—
285Dic3—21no—
286Dic3—41no—
287Dic3—41no—
288C23—31no—
289C23—31no—
290C23—61no—
291C23—61no—
292Q8—21no—
293Q8—61no—
294Q8—61no—
295Q8—81no—
296Q8—122no—
297Q8#23912yes—
298D4—61no—
299D4—61no—
300D4—61no—
301D4—61no—
302D4—121no—
303D4—122no—
304D4—122no—
305D4—241no—
306D4—241no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—122no—
320C2×C4—122no—
321C2×C4—122no—
322C2×C4D3×D411yescharacteristic
323C2×C4D3×D411yescharacteristic
324C2×C4C23×D311yescharacteristic
325C8—21no—
326C8—21no—
327C8—61no—
328C8—61no—
329C2×C3#54611yescharacteristic
330C2×C3#57511yescharacteristic
331C2×C3#57711yescharacteristic
332D3—41no—
333D3—41no—
334D3—81no—
335D3—81no—
336C22—61no—
337C22—61no—
338C22—61no—
339C22—61no—
340C22—61no—
341C22—61no—
342C22—121no—
343C22—121no—
344C22—121no—
345C22—121no—
346C22—121no—
347C22—121no—
348C22C2×D3×D411yescharacteristic
349C4—21no—
350C4—21no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—61no—
356C4—61no—
357C4—121no—
358C4—121no—
359C4C2×#23911yescharacteristic
360C4#100911yescharacteristic
361C3#332111yescharacteristic, complemented, Sylow
362C2—121no—
363C2—121no—
364C2—241no—
365C2—241no—
366C2#563011yescharacteristic
367C2#576211yescharacteristic
368C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#475C2
socleC22×C3C22×D4
hypercenter#475D3
hypocenterC3#3321
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
1. #5734C2semidirect
2. #5576C2semidirect
3. C3×#475C2semidirect
4. #5089C2semidirect
5. #5026C2semidirect
6. #4868C2semidirect
7. #4866C2semidirect
8. #4865C2semidirect
9. #4864C2semidirect
10. #4844C2semidirect
11. #4836C2semidirect
12. #4810C2semidirect
13. #4806C2semidirect
14. #4804C2semidirect
15. #4802C2semidirect
65. #475D3semidirect
361. C3#3321semidirect

Maximal subgroups

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

subgroupindex
#57342
#55762
C3×#4752
#50892
#50262
#48682
#48662
#48652
#48642
#48442
#48362
#48102
#48062
#48042
#48022
#33213