Finite Group Databasebeta

← #81035

Subgroups — #81035

Order 384.

392 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5725C211yesmaximal, characteristic, complemented
2#5580C211yesmaximal, characteristic, complemented
3C3×#475C211yesmaximal, characteristic, complemented
4#5087C211yesmaximal, characteristic, complemented
5#5027C211yesmaximal, characteristic, complemented
6#4871C212yesmaximal, complemented
7#4865C212yesmaximal, complemented
8#4843C211yesmaximal, characteristic, complemented
9#4837C211yesmaximal, characteristic, complemented
10#4819C212yesmaximal, complemented
11#4812C212yesmaximal, complemented
12#3251—31nomaximal, Sylow
13C2×#239—21no—
14C3×C4⋊Q8C2211yescharacteristic
15C3×C4×Q8C2211yescharacteristic
16#947—21no—
17#947C2211yescharacteristic
18#946C2211yescharacteristic
19C2×#215C2212yes—
20C2×D24C2212yes—
21C2×ΓL(1,𝔽25)C2212yes—
22#898—21no—
23#895—21no—
24#895C2211yescharacteristic
25#895C2212yes—
26#894—21no—
27#893—21no—
28#893C2212yes—
29#890C2211yescharacteristic
30#874C2211yescharacteristic
31C4×D12—21no—
32C4×D12C2211yescharacteristic
33#872—21no—
34C42×D3C2211yescharacteristic
35C3×#106C2212yes—
36C3×C4⋊C8C2211yescharacteristic
37C3×#103C2212yes—
38#821C2212yes—
39Dic3⋊C8C2212yes—
40#809C2212yes—
41#809C2212yes—
42#807C2212yes—
43C4⋊(C3⋊C8)C2211yescharacteristic
44#532—31no—
45#529—31no—
46#478—32no—
47#475D311yescharacteristic, complemented
48#474—31no—
49#469—32no—
50#468—32no—
51#463—32no—
52#460—31no—
53#459—31no—
54#423—31no—
55C2×C3×Q8—21no—
56C2×C3×Q8C2311yescharacteristic
57#239—44no—
58C2×D12—21no—
59C2×D12—22no—
60C2×D12—21no—
61C2×D12—21no—
62C2×D12C2311yescharacteristic
63C2×D12C2312yes—
64C2×C4×D3—21no—
65C2×C4×D3—22no—
66C2×C4×D3D411yescharacteristic
67C2×C4×D3D411yescharacteristic
68C2×C4×D3C2311yescharacteristic
69C2×C3×C8C2312yes—
70C3×C4⋊C4—21no—
71C3×C4⋊C4—21no—
72C3×C4⋊C4C2312yes—
73C3×C4⋊C4C2311yescharacteristic
74C3×C42—21no—
75C3×C42C2311yescharacteristic
76#215—42no—
77(C2×D3)⋊C4—21no—
78(C2×D3)⋊C4—21no—
79(C2×D3)⋊C4—21no—
80(C2×D3)⋊C4—21no—
81(C2×D3)⋊C4—21no—
82(C2×D3)⋊C4—22no—
83(C2×D3)⋊C4—22no—
84C4⋊Dic3—21no—
85C4⋊Dic3C2311yescharacteristic
86Dic3⋊C4—21no—
87Dic3⋊C4—21no—
88C4×Dic3D411yescharacteristic
89C4×Dic3D411yescharacteristic
90C4×Dic3C2311yescharacteristic
91C2×C3⋊C8C2312yes—
92D24—42no—
93ΓL(1,𝔽25)—42no—
94C2×#41—61no—
95C2×SD16—32no—
96C2×D8—32no—
97C2×M16—32no—
98C4⋊Q8C2×D311yescharacteristic
99#127—31no—
100#126—61no—
101#125—31no—
102#124—31no—
103#124—61no—
104#123—61no—
105C22⋊Q8—31no—
106#121—31no—
107#121—32no—
108#121—61no—
109C4×Q8C2×D311yescharacteristic
110C4×D4—32no—
111C4×D4—31no—
112C4×D4—61no—
113C4×D4—61no—
114#117—61no—
115C2×C42—31no—
116#107—32no—
117#106C2×D312yes—
118C4⋊C8—32no—
119C4⋊C8—31no—
120C4⋊C8C2×D311yescharacteristic
121#103C2×D312yes—
122#102—32no—
123#102—32no—
124#102—32no—
125C22×D3—21no—
126C22×D3—22no—
127C22×D3C2×D411yescharacteristic
128C3×Q8—21no—
129C3×Q8—21no—
130C3×Q8—42no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4—22no—
134C2×C3×C4C2×D411yescharacteristic
135C2×C3×C4C2×D411yescharacteristic
136C2×C3×C4C2411yescharacteristic
137C2×Dic3—21no—
138C2×Dic3C2×D411yescharacteristic
139C2×Dic3C2×D411yescharacteristic
140C2×Dic3C2×D411yescharacteristic
141D12—21no—
142D12—22no—
143D12—21no—
144D12—22no—
145D12—42no—
146D12—42no—
147D12—44no—
148D12—42no—
149C4×D3—22no—
150C4×D3—21no—
151C4×D3—21no—
152C4×D3—41no—
153C4×D3—44no—
154C4×D3—42no—
155C3×C8—22no—
156C3⋊C8—22no—
157#41—124no—
158C2×Q8—21no—
159C2×Q8C22×D311yescharacteristic
160C2×D4—31no—
161C2×D4—32no—
162C2×D4—61no—
163C2×D4—62no—
164C2×D4—61no—
165C2×D4—61no—
166C22×C4—31no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—62no—
170C22×C4—61no—
171SD16—122no—
172D8—122no—
173M16—122no—
174C2×C8—32no—
175C2×C8C22×D312yes—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—31no—
179C4⋊C4—61no—
180C4⋊C4—61no—
181C4⋊C4—61no—
182C4⋊C4C22×D311yescharacteristic
183C4⋊C4C22×D312yes—
184C22⋊C4—61no—
185C22⋊C4—61no—
186C22⋊C4—62no—
187C22⋊C4—62no—
188C22⋊C4—61no—
189C22⋊C4—61no—
190C22⋊C4—61no—
191C42—21no—
192C42—31no—
193C42—31no—
194C42—31no—
195C42C22×D311yescharacteristic
196C22×C3C22×D411yescharacteristic
197C2×D3—21no—
198C2×D3—21no—
199C2×D3—21no—
200C2×D3—41no—
201C2×D3—42no—
202C2×D3—42no—
203C2×D3—41no—
204C2×D3—42no—
205C2×D3—41no—
206C3×C4—21no—
207C3×C4—41no—
208C3×C4—42no—
209C3×C4—41no—
210C3×C4AGL(1,ℤ/8)12yes—
211C3×C4H(5,𝔽2)11yescharacteristic
212C3×C42−1+411yescharacteristic
213Dic3—21no—
214Dic3—21no—
215Dic3—21no—
216Dic3—41no—
217C23—31no—
218C23—61no—
219C23—62no—
220Q8—21no—
221Q8—21no—
222Q8—42no—
223D4—61no—
224D4—62no—
225D4—61no—
226D4—62no—
227D4—122no—
228D4—124no—
229D4—122no—
230D4—122no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—31no—
235C2×C4—31no—
236C2×C4—31no—
237C2×C4—62no—
238C2×C4—61no—
239C2×C4—61no—
240C2×C4—61no—
241C2×C4—121no—
242C2×C4—124no—
243C2×C4—122no—
244C2×C4D3×D411yescharacteristic
245C2×C4D3×D411yescharacteristic
246C2×C4C23×D311yescharacteristic
247C8—22no—
248C8—62no—
249C2×C3#53711yescharacteristic
250C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
251C2×C3#57511yescharacteristic
252D3—41no—
253D3—82no—
254D3—81no—
255C22—61no—
256C22—61no—
257C22—61no—
258C22—122no—
259C22—121no—
260C22—122no—
261C22—121no—
262C22—121no—
263C22—122no—
264C22C2×D3×D411yescharacteristic
265C4—21no—
266C4—42no—
267C4—41no—
268C4—41no—
269C4—61no—
270C4—61no—
271C4—61no—
272C4—121no—
273C4#91412yes—
274C4#100711yescharacteristic
275C4#100911yescharacteristic
276C3#325111yescharacteristic, complemented, Sylow
277C2—121no—
278C2—242no—
279C2—241no—
280C2#562911yescharacteristic
281C2C2×#91411yescharacteristic
282C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#475C2
socleC22×C3C22×D4
hypercenter#475D3
hypocenterC3#3251
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. #5725C2semidirect
2. #5580C2semidirect
3. C3×#475C2semidirect
4. #5087C2semidirect
5. #5027C2semidirect
6. #4871C2semidirect
7. #4865C2semidirect
8. #4843C2semidirect
9. #4837C2semidirect
10. #4819C2semidirect
11. #4812C2semidirect
47. #475D3semidirect
276. C3#3251semidirect

Maximal subgroups

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

subgroupindex
#57252
#55802
C3×#4752
#50872
#50272
#48712
#48652
#48432
#48372
#48192
#48122
#32513