Finite Group Databasebeta

← #81127

Subgroups — #81127

Order 384.

376 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5726C211yesmaximal, characteristic
2#5581C211yesmaximal, characteristic, complemented
3C3×#477C211yesmaximal, characteristic, complemented
4#5081C211yesmaximal, characteristic, complemented
5#5026C211yesmaximal, characteristic, complemented
6#4885C211yesmaximal, characteristic, complemented
7#4882C211yesmaximal, characteristic, complemented
8#4881C211yesmaximal, characteristic, complemented
9D3×#107C211yesmaximal, characteristic, complemented
10#4847C211yesmaximal, characteristic, complemented
11#4836C211yesmaximal, characteristic, complemented
12#4812C211yesmaximal, characteristic, complemented
13#4811C211yesmaximal, characteristic, complemented
14#4807C211yesmaximal, characteristic, complemented
15#4805C211yesmaximal, characteristic, complemented
16#3343—31nomaximal, Sylow
17C2×#239—21no—
18C3×C4⋊Q8C2211yescharacteristic
19C3×C4×Q8C2211yescharacteristic
20#946—21no—
21#946C2211yescharacteristic
22#946C2211yescharacteristic
23C2×#215C2211yescharacteristic
24C2×ΓL(1,𝔽25)C2211yescharacteristic
25C2×D3×C8C2211yescharacteristic
26#895C2211yescharacteristic
27#895C2211yescharacteristic
28#894—21no—
29#894—21no—
30#894—21no—
31#892C2211yescharacteristic
32D3×C4⋊C4C2211yescharacteristic
33D3×C4⋊C4C2211yescharacteristic
34D3×C4⋊C4C2211yescharacteristic
35#889—21no—
36#876—21no—
37C4×D12C2211yescharacteristic
38C4×D12C2211yescharacteristic
39#872—21no—
40#870C2211yescharacteristic
41C3×#107C2211yescharacteristic
42C3×#107C2211yescharacteristic
43C3×C4⋊C8C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×#103C2211yescharacteristic
46#835C2211yescharacteristic
47#821C2211yescharacteristic
48#821C2211yescharacteristic
49(C2×D3)⋊C8C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic
51#818C2211yescharacteristic
52#817C2211yescharacteristic
53#809C2211yescharacteristic
54#808C2211yescharacteristic
55#808C2211yescharacteristic
56#807C2211yescharacteristic
57#807C2211yescharacteristic
58#807C2211yescharacteristic
59#803C2211yescharacteristic
60#552—31no—
61#547—31no—
62#501—31no—
63#484—31no—
64#483—31no—
65#482—31no—
66#480—31no—
67#478—31no—
68#477D311yescharacteristic, complemented
69#468—31no—
70#465—31no—
71#440—31no—
72#435—31no—
73#428—31no—
74C2×#107—31no—
75C2×C3×Q8—21no—
76C2×C3×Q8C2311yescharacteristic
77#239—44no—
78C2×D12—21no—
79C2×D12—21no—
80C2×D12D411yescharacteristic
81C2×D12C2311yescharacteristic
82C2×C4×D3—21no—
83C2×C4×D3—21no—
84C2×C4×D3—21no—
85C2×C4×D3—21no—
86C2×C4×D3C2311yescharacteristic
87C2×C4×D3C2311yescharacteristic
88C2×C3×C8C2311yescharacteristic
89C2×C3×C8C2311yescharacteristic
90C3×C4⋊C4—21no—
91C3×C4⋊C4—21no—
92C3×C4⋊C4C2311yescharacteristic
93C3×C4⋊C4C2311yescharacteristic
94C3×C4⋊C4C2311yescharacteristic
95C3×C42—21no—
96C3×C42C2311yescharacteristic
97#215—22no—
98(C2×D3)⋊C4—21no—
99(C2×D3)⋊C4—21no—
100(C2×D3)⋊C4—21no—
101(C2×D3)⋊C4—21no—
102(C2×D3)⋊C4—21no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4D411yescharacteristic
105(C2×D3)⋊C4D411yescharacteristic
106C4⋊Dic3D411yescharacteristic
107C4⋊Dic3C2311yescharacteristic
108C4⋊Dic3C2311yescharacteristic
109C4⋊Dic3C2311yescharacteristic
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117C4×Dic3—21no—
118C2×C3⋊C8C2311yescharacteristic
119C2×C3⋊C8C2311yescharacteristic
120ΓL(1,𝔽25)—22no—
121D3×C8—22no—
122C2×#41—61no—
123C2×SD16—31no—
124C2×M16—31no—
125C22×C8—31no—
126C4⋊Q8C2×D311yescharacteristic
127#126—61no—
128#125—31no—
129#125—61no—
130#123—61no—
131#123—61no—
132#123—61no—
133C22⋊Q8—31no—
134C22⋊Q8—31no—
135C22⋊Q8—61no—
136#121—31no—
137#121—31no—
138C4×Q8C2×D311yescharacteristic
139C4×D4—31no—
140C4×D4—31no—
141#117—31no—
142#117—61no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—31no—
146#107—31no—
147#107—31no—
148#107—31no—
149#107—31no—
150#107C2×D311yescharacteristic
151#107C2×D311yescharacteristic
152#106—31no—
153#106—31no—
154#106—31no—
155C4⋊C8C2×D311yescharacteristic
156#103—31no—
157#103C2×D311yescharacteristic
158#103C2×D311yescharacteristic
159#102—31no—
160#102—31no—
161#102—31no—
162C22⋊C8—31no—
163C22⋊C8—31no—
164#97—31no—
165C22×D3—21no—
166C22×D3C2×D411yescharacteristic
167C22×D3C2×D411yescharacteristic
168C3×Q8—21no—
169C3×Q8—21no—
170C3×Q8—42no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4C2×D411yescharacteristic
176C2×C3×C4C2×D411yescharacteristic
177C2×C3×C4C2411yescharacteristic
178C2×Dic3—21no—
179C2×Dic3—21no—
180C2×Dic3—21no—
181C2×Dic3C2×D411yescharacteristic
182C2×Dic3C2×D411yescharacteristic
183D12—22no—
184D12—21no—
185D12—21no—
186D12—42no—
187D12—42no—
188C4×D3—21no—
189C4×D3—21no—
190C4×D3—21no—
191C4×D3—21no—
192C4×D3—42no—
193C4×D3—42no—
194C4×D3—42no—
195C4×D3—42no—
196C3×C8—21no—
197C3×C8—21no—
198C3⋊C8—21no—
199C3⋊C8#4112yescomplemented
200#41—124no—
201C2×Q8—21no—
202C2×Q8C22×D311yescharacteristic
203C2×D4—31no—
204C2×D4—31no—
205C2×D4—61no—
206C2×D4—61no—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—61no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4—61no—
213SD16—62no—
214M16—62no—
215C2×C8—31no—
216C2×C8—31no—
217C2×C8—62no—
218C2×C8C22×D311yescharacteristic
219C2×C8C22×D311yescharacteristic
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—31no—
223C4⋊C4—31no—
224C4⋊C4—31no—
225C4⋊C4—31no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4C22×D311yescharacteristic
234C4⋊C4C22×D311yescharacteristic
235C4⋊C4C22×D311yescharacteristic
236C22⋊C4—31no—
237C22⋊C4—31no—
238C22⋊C4—61no—
239C22⋊C4—61no—
240C22⋊C4—61no—
241C22⋊C4—61no—
242C22⋊C4—61no—
243C22⋊C4—61no—
244C42—21no—
245C42—61no—
246C42C22×D311yescharacteristic
247C22×C3C22×D411yescharacteristic
248C2×D3—21no—
249C2×D3—21no—
250C2×D3—21no—
251C2×D3—21no—
252C2×D3—21no—
253C2×D3—41no—
254C2×D3—41no—
255C2×D3—41no—
256C2×D3AGL(1,ℤ/8)12yes—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—41no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—41no—
263C3×C4C2×#4111yescharacteristic
264C3×C42−1+411yescharacteristic
265Dic3—21no—
266Dic3—21no—
267Dic3—41no—
268Dic3—41no—
269Dic3—41no—
270C23—31no—
271C23—31no—
272C23—61no—
273Q8—21no—
274Q8—21no—
275Q8—42no—
276D4—62no—
277D4—61no—
278D4—61no—
279D4—122no—
280D4—122no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—31no—
286C2×C4—31no—
287C2×C4—61no—
288C2×C4—61no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—122no—
295C2×C4—122no—
296C2×C4—122no—
297C2×C4—122no—
298C2×C4D3×D411yescharacteristic
299C2×C4D3×D411yescharacteristic
300C2×C4C23×D311yescharacteristic
301C8—21no—
302C8—21no—
303C8—32no—
304C8—61no—
305C2×C3#54711yescharacteristic
306C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
307C2×C3#57811yescharacteristic
308D3—22no—
309D3—41no—
310D3—81no—
311C22—32no—
312C22—61no—
313C22—61no—
314C22—61no—
315C22—61no—
316C22—61no—
317C22—121no—
318C22—121no—
319C22—121no—
320C22C2×D3×D411yescharacteristic
321C4—21no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—61no—
328C4—61no—
329C4—121no—
330C4—121no—
331C4—121no—
332C4#100711yescharacteristic
333C4D3×#4111yescharacteristic
334C3#334311yescharacteristic, complemented, Sylow
335C2—62no—
336C2—121no—
337C2—241no—
338C2#563111yescharacteristic
339C2#576611yescharacteristic
340C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#477C2
socleC22×C3C22×D4
prime radical#5726C2
hypercenter#477D3
hypocenterC3#3343
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #5581C2semidirect
3. C3×#477C2semidirect
4. #5081C2semidirect
5. #5026C2semidirect
6. #4885C2semidirect
7. #4882C2semidirect
8. #4881C2semidirect
9. D3×#107C2semidirect
10. #4847C2semidirect
11. #4836C2semidirect
12. #4812C2semidirect
13. #4811C2semidirect
14. #4807C2semidirect
15. #4805C2semidirect
68. #477D3semidirect
199. C3⋊C8#41semidirect
334. C3#3343semidirect

Maximal subgroups

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

subgroupindex
#57262
#55812
C3×#4772
#50812
#50262
#48852
#48822
#48812
D3×#1072
#48472
#48362
#48122
#48112
#48072
#48052
#33433