Finite Group Databasebeta

← #81136

Subgroups — #81136

Order 384.

338 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 125 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#5730C211yesmaximal, characteristic
2#5570C211yesmaximal, characteristic, complemented
3C3×#477C211yesmaximal, characteristic, complemented
4#5080C211yesmaximal, characteristic, complemented
5C4×#215C211yesmaximal, characteristic
6#4891C212yesmaximal
7#4878C212yesmaximal, complemented
8#4846C211yesmaximal, characteristic
9#4830C211yesmaximal, characteristic, complemented
10#4796C212yesmaximal, complemented
11#4788C212yesmaximal
12#3381—31nomaximal, Sylow
13C3×C4⋊Q8C2211yescharacteristic
14C3×C4×Q8C2211yescharacteristic
15#946—22no—
16Q8×Dic3C2212yes—
17C2×#215C2211yescharacteristic
18#897—21no—
19#893C2212yes—
20D3×C4⋊C4—21no—
21#890—22no—
22#890C2212yes—
23#890C2212yes—
24C4×D12C2211yescharacteristic
25#869—21no—
26#869C2211yescharacteristic
27C4×Dic6—21no—
28C4×Dic6C2211yescharacteristic
29C3×#107C2212yes—
30C3×C4⋊C8C2211yescharacteristic
31C3×#103C2212yes—
32#835C2211yescharacteristic
33#821C2212yes—
34#818C2212yes—
35Dic3⋊C8C2212yes—
36C8×Dic3C2212yes—
37#809C2211yescharacteristic
38#808C2211yescharacteristic
39#808C2212yes—
40#808C2212yes—
41C4×C3⋊C8C2211yescharacteristic
42#557—31no—
43#554—31no—
44#499—32no—
45#498—31no—
46#479—32no—
47#478—32no—
48#477D311yescharacteristic, complemented
49#474—31no—
50C8×Q8—31no—
51C4×SD16—31no—
52C4×SD16—32no—
53C2×C3×Q8—21no—
54C2×C3×Q8C2311yescharacteristic
55C2×D12C2311yescharacteristic
56C2×C4×D3—21no—
57C2×C4×D3—22no—
58C2×Dic6—21no—
59C2×Dic6D411yescharacteristic
60C2×C3×C8C2312yes—
61C3×C4⋊C4—21no—
62C3×C4⋊C4—21no—
63C3×C4⋊C4C2312yes—
64C3×C4⋊C4C2311yescharacteristic
65C3×C42—21no—
66C3×C42C2311yescharacteristic
67#215Q814yescomplemented
68(C2×D3)⋊C4—21no—
69(C2×D3)⋊C4—22no—
70C4⋊Dic3—21no—
71C4⋊Dic3—22no—
72C4⋊Dic3—22no—
73C4⋊Dic3—22no—
74C4⋊Dic3D411yescharacteristic
75C4⋊Dic3C2312yes—
76C4⋊Dic3C2311yescharacteristic
77Dic3⋊C4—22no—
78Dic3⋊C4—22no—
79Dic3⋊C4D412yes—
80C4×Dic3—22no—
81C4×Dic3C2312yes—
82C2×C3⋊C8C2311yescharacteristic
83C2×C3⋊C8C2311yescharacteristic
84C2×SD16—31no—
85C4⋊Q8—31no—
86C4⋊Q8—61no—
87C4⋊Q8C2×D311yescharacteristic
88#125—32no—
89#125—32no—
90#125—62no—
91C22⋊Q8—61no—
92C22⋊Q8—62no—
93C4×Q8—31no—
94C4×Q8—32no—
95C4×Q8—61no—
96C4×Q8C2×D311yescharacteristic
97C4×D4—32no—
98C4×D4—31no—
99C2×C4⋊C4—61no—
100#107—32no—
101#107C2×D312yes—
102#106—32no—
103#106—31no—
104#106—32no—
105C4⋊C8—32no—
106C4⋊C8C2×D311yescharacteristic
107#103—31no—
108#103C2×D312yes—
109#102—32no—
110#102—31no—
111C4×C8—32no—
112C4×C8—31no—
113C22×D3—21no—
114C3×Q8—21no—
115C3×Q8—81no—
116C3×Q8C2×Q812yes—
117C2×C3×C4—21no—
118C2×C3×C4—21no—
119C2×C3×C4—22no—
120C2×C3×C4C2×D411yescharacteristic
121C2×C3×C4C2×D411yescharacteristic
122C2×C3×C4C2411yescharacteristic
123C2×Dic3—22no—
124C2×Dic3—21no—
125C2×Dic3C2×D412yes—
126C2×Dic3C2×D412yes—
127D12—21no—
128D12C2×Q812yes—
129C4×D3—42no—
130C4×D3—82no—
131Dic6—22no—
132Dic6—42no—
133C3×C8—22no—
134C3⋊C8—21no—
135C3⋊C8C2×Q812yes—
136C2×Q8—21no—
137C2×Q8—31no—
138C2×Q8—61no—
139C2×Q8C22×D311yescharacteristic
140C2×D4—31no—
141C22×C4—62no—
142C22×C4—61no—
143SD16—34no—
144C2×C8—31no—
145C2×C8—31no—
146C2×C8C22×D312yes—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—31no—
150C4⋊C4—32no—
151C4⋊C4—32no—
152C4⋊C4—31no—
153C4⋊C4—62no—
154C4⋊C4—62no—
155C4⋊C4—62no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—61no—
159C4⋊C4C22×D311yescharacteristic
160C4⋊C4C22×D312yes—
161C22⋊C4—62no—
162C22⋊C4—61no—
163C42—21no—
164C42—32no—
165C42—62no—
166C42C22×D311yescharacteristic
167C22×C3C22×D411yescharacteristic
168C2×D3—22no—
169C2×D3—41no—
170C2×D3—41no—
171C3×C4—22no—
172C3×C4—21no—
173C3×C4—42no—
174C3×C4—41no—
175C3×C4#13512yes—
176C3×C4C22×Q811yescharacteristic
177C3×C42−1+411yescharacteristic
178Dic3—22no—
179Dic3—22no—
180Dic3—42no—
181Dic3—41no—
182C23—61no—
183Q8—21no—
184Q8—62no—
185Q8—81no—
186Q8—122no—
187Q8D3×Q812yes—
188D4—32no—
189D4—61no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—32no—
194C2×C4—32no—
195C2×C4—62no—
196C2×C4—61no—
197C2×C4—122no—
198C2×C4—242no—
199C2×C4D3×D411yescharacteristic
200C2×C4D3×D411yescharacteristic
201C2×C4C23×D311yescharacteristic
202C8—22no—
203C8—32no—
204C8—61no—
205C2×C3D4×Q811yescharacteristic
206C2×C3C2×#13511yescharacteristic
207C2×C3#57711yescharacteristic
208D3—42no—
209C22—62no—
210C22—121no—
211C22—121no—
212C22C2×D3×D411yescharacteristic
213C4—22no—
214C4—21no—
215C4—42no—
216C4—41no—
217C4—62no—
218C4—62no—
219C4—121no—
220C4—122no—
221C4#91912yes—
222C4C2×D3×Q811yescharacteristic
223C4#101011yescharacteristic
224C3#338111yescharacteristic, complemented, Sylow
225C2—122no—
226C2#563211yescharacteristic
227C2C2×#91911yescharacteristic
228C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#477C2
socleC22×C3C22×D4
prime radicalC2×D12C23
hypercenter#477D3
hypocenterC3#3381
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. #5570C2semidirect
3. C3×#477C2semidirect
4. #5080C2semidirect
7. #4878C2semidirect
9. #4830C2semidirect
10. #4796C2semidirect
48. #477D3semidirect
67. #215Q8semidirect
224. C3#3381semidirect

Maximal subgroups

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

subgroupindex
#57302
#55702
C3×#4772
#50802
C4×#2152
#48912
#48782
#48462
#48302
#47962
#47882
#33813