Finite Group Databasebeta

← #81463

Subgroups — #81463

Order 384.

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

normal subgroups only

This group has 324 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#5622C211yesmaximal, characteristic
2C2×#890C211yesmaximal, characteristic
3C3×#484C211yesmaximal, characteristic
4#5043C211yesmaximal, characteristic
5#4970C211yesmaximal, characteristic
6#4877C212yesmaximal, complemented
7#4876C212yesmaximal, complemented
8#4799C212yesmaximal, complemented
9#4797C212yesmaximal, complemented
10#4721C211yesmaximal, characteristic
11#4719C211yesmaximal, characteristic
12#3236—31nomaximal, Sylow
13C3×C22⋊Q8C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15Q8×Dic3C2211yescharacteristic
16#944C2211yescharacteristic
17C2×C4⋊Dic3—21no—
18C2×C4⋊Dic3C2211yescharacteristic
19#924C2211yescharacteristic
20C2×Dic3⋊C4—21no—
21C2×C4×Dic3C2211yescharacteristic
22#890—21no—
23#890—21no—
24#890C2212yes—
25#890C2212yes—
26#889—21no—
27#888C2211yescharacteristic
28#887—21no—
29#887C2211yescharacteristic
30#878—21no—
31#877—21no—
32C3×#107C2212yes—
33C3×#103C2212yes—
34C3×C22⋊C8C2211yescharacteristic
35#835C2212yes—
36#830C2211yescharacteristic
37#817C2212yes—
38#816C2212yes—
39#815C2212yes—
40Dic3⋊C8C2212yes—
41#810C2212yes—
42#808C2212yes—
43#807C2212yes—
44#533—31no—
45C2×#125—31no—
46#501—32no—
47#491—32no—
48#484—31no—
49#484D311yescharacteristic
50#483—31no—
51#483—31no—
52#479—32no—
53#479—32no—
54#432—31no—
55C2×C3×Q8C2311yescharacteristic
56C22×C3×C4—21no—
57C22×C3×C4C2311yescharacteristic
58C22×Dic3—21no—
59C22×Dic3D411yescharacteristic
60C22×Dic3D411yescharacteristic
61C2×Dic6—21no—
62C2×Dic6C2311yescharacteristic
63C2×C3×C8C2312yes—
64C3×C4⋊C4—21no—
65C3×C4⋊C4—21no—
66C3×C4⋊C4C2312yes—
67C3×C4⋊C4C2311yescharacteristic
68C3×C22⋊C4—21no—
69#217—21no—
70C4⋊Dic3—21no—
71C4⋊Dic3—21no—
72C4⋊Dic3—22no—
73C4⋊Dic3—22no—
74C4⋊Dic3C2311yescharacteristic
75C4⋊Dic3C2312yes—
76Dic3⋊C4—21no—
77Dic3⋊C4—21no—
78Dic3⋊C4—21no—
79Dic3⋊C4—21no—
80Dic3⋊C4—21no—
81Dic3⋊C4—22no—
82Dic3⋊C4—22no—
83C4×Dic3—21no—
84C4×Dic3—21no—
85C4×Dic3D411yescharacteristic
86C4×Dic3D411yescharacteristic
87C4×Dic3C2311yescharacteristic
88C4×Dic3C2311yescharacteristic
89C2×C3⋊C8C2312yes—
90C4⋊Q8—31no—
91C4⋊Q8—31no—
92#125—32no—
93#125—32no—
94#125—61no—
95#125—61no—
96#125—61no—
97C22⋊Q8—31no—
98C22⋊Q8—61no—
99C22⋊Q8C2×D311yescharacteristic
100C4×Q8—31no—
101C4×Q8—31no—
102C4×Q8—61no—
103#117—61no—
104C2×C4⋊C4—31no—
105C2×C4⋊C4—61no—
106C2×C4⋊C4—61no—
107C2×C4⋊C4C2×D311yescharacteristic
108C2×C42—31no—
109#107—32no—
110#107C2×D312yes—
111#106—32no—
112#106—32no—
113C4⋊C8—32no—
114#103—32no—
115#103—32no—
116#103—32no—
117#103C2×D312yes—
118C22⋊C8—31no—
119C22⋊C8C2×D311yescharacteristic
120#97—32no—
121C23×C3C2×D411yescharacteristic
122C3×Q8—22no—
123C2×C3×C4—22no—
124C2×C3×C4—21no—
125C2×C3×C4—21no—
126C2×C3×C4—22no—
127C2×C3×C4—42no—
128C2×C3×C4C2×D411yescharacteristic
129C2×C3×C4C2411yescharacteristic
130C2×Dic3—22no—
131C2×Dic3—21no—
132C2×Dic3—21no—
133C2×Dic3—22no—
134C2×Dic3—41no—
135C2×Dic3—42no—
136C2×Dic3C2×D411yescharacteristic
137C2×Dic3C2×D411yescharacteristic
138C2×Dic3C2×D411yescharacteristic
139C2×Dic3C2×D411yescharacteristic
140Dic6—22no—
141Dic6—42no—
142C3×C8—22no—
143C3⋊C8—22no—
144C2×Q8—31no—
145C2×Q8—61no—
146C2×Q8C22×D311yescharacteristic
147C22×C4—21no—
148C22×C4—31no—
149C22×C4—31no—
150C22×C4—61no—
151C22×C4C22×D311yescharacteristic
152C2×C8—32no—
153C2×C8C22×D312yes—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—32no—
157C4⋊C4—31no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4—62no—
161C4⋊C4—62no—
162C4⋊C4—61no—
163C4⋊C4—61no—
164C4⋊C4—61no—
165C4⋊C4—61no—
166C4⋊C4—61no—
167C4⋊C4—61no—
168C4⋊C4—61no—
169C4⋊C4C22×D312yes—
170C4⋊C4C22×D311yescharacteristic
171C22⋊C4—21no—
172C22⋊C4—61no—
173C42—31no—
174C42—31no—
175C42—31no—
176C42—31no—
177C42—61no—
178C42—61no—
179C22×C3—21no—
180C22×C3—21no—
181C22×C3#13712yes—
182C22×C3C22×D411yescharacteristic
183C3×C4—21no—
184C3×C4—41no—
185C3×C4—41no—
186C3×C4—42no—
187C3×C42−1+412yes—
188Dic3—21no—
189Dic3—21no—
190Dic3—21no—
191Dic3—21no—
192Dic3—41no—
193Dic3—42no—
194Dic3—41no—
195C23D3×D411yescharacteristic
196Q8—22no—
197Q8—62no—
198Q8—122no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—31no—
204C2×C4—31no—
205C2×C4—31no—
206C2×C4—31no—
207C2×C4—42no—
208C2×C4—62no—
209C2×C4—61no—
210C2×C4—61no—
211C2×C4—62no—
212C2×C4—121no—
213C2×C4—122no—
214C2×C4D3×D411yescharacteristic
215C2×C4C23×D311yescharacteristic
216C8—22no—
217C8—62no—
218C2×C3—22no—
219C2×C3#53611yescharacteristic
220C2×C3C2×#13711yescharacteristic
221C2×C3#57511yescharacteristic
222C22—21no—
223C22—21no—
224C22#91812yes—
225C22C2×D3×D411yescharacteristic
226C4—21no—
227C4—41no—
228C4—42no—
229C4—41no—
230C4—61no—
231C4—61no—
232C4—61no—
233C4—61no—
234C4—121no—
235C4—122no—
236C4—121no—
237C4#101012yes—
238C3#323611yescharacteristic, complemented, Sylow
239C2—22no—
240C2#564811yescharacteristic
241C2C2×#91811yescharacteristic
242C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#484C2
socleC22×C3C22×D4
prime radicalC23×C3C2×D4
hypercenter#484D3
hypocenterC3#3236
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
6. #4877C2semidirect
7. #4876C2semidirect
8. #4799C2semidirect
9. #4797C2semidirect
238. C3#3236semidirect

Maximal subgroups

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

subgroupindex
#56222
C2×#8902
C3×#4842
#50432
#49702
#48772
#48762
#47992
#47972
#47212
#47192
#32363