Finite Group Databasebeta

← #82811

Subgroups — #82811

Order 384.

644 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 201 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
1C2×C3×C22⋊Q8C211yesmaximal, characteristic
2C2×#944C211yesmaximal, characteristic, complemented
3C2×#924C211yesmaximal, characteristic
4C22×C4×Dic3C211yesmaximal, characteristic
5#5229C211yesmaximal, characteristic, complemented
6#4976C214yesmaximal, complemented
7#4975C211yesmaximal, characteristic, complemented
8#4973C211yesmaximal, characteristic, complemented
9#4942C214yesmaximal
10#2857—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C23×C3×C4C2211yescharacteristic
13C23×Dic3C2212yes—
14C22×Dic6C2211yescharacteristic
15C3×C22⋊Q8—24no—
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2212yes—
18C2×C3×C22⋊C4C2212yes—
19C2×#217C2212yes—
20#944—24no—
21C2×C4⋊Dic3C2211yescharacteristic
22#924—24no—
23C2×Dic3⋊C4C2214yes—
24C2×Dic3⋊C4C2214yes—
25C2×Dic3⋊C4C2212yes—
26C2×C4×Dic3—24no—
27C2×C4×Dic3C2211yescharacteristic
28C2×C4×Dic3C2212yes—
29C2×C4×Dic3C2211yescharacteristic
30C3⋊#95C2214yes—
31C3⋊#95C2214yes—
32C2×C4⋊Q8—31no—
33C2×C22⋊Q8—31no—
34C2×C22⋊Q8D311yescharacteristic
35C22×C42—31no—
36#391—31no—
37#391—31no—
38#389—34no—
39#384—31no—
40#380—34no—
41C24×C3C2311yescharacteristic
42C2×C3×Q8—22no—
43C2×C3×Q8—22no—
44C2×C3×Q8—22no—
45C22×C3×C4—22no—
46C22×C3×C4—22no—
47C22×C3×C4D414yes—
48C22×C3×C4C2312yes—
49C22×C3×C4C2312yes—
50C22×C3×C4C2311yescharacteristic
51C22×C3×C4C2311yescharacteristic
52C22×Dic3—24no—
53C22×Dic3—24no—
54C22×Dic3Q818yescomplemented
55C22×Dic3C2312yes—
56C22×Dic3C2312yes—
57C22×Dic3C2312yes—
58C22×Dic3C2312yes—
59C2×Dic6—22no—
60C2×Dic6—22no—
61C2×Dic6—22no—
62C3×C4⋊C4—22no—
63C3×C4⋊C4—24no—
64C3×C22⋊C4—24no—
65#217—24no—
66C4⋊Dic3—22no—
67Dic3⋊C4—24no—
68Dic3⋊C4—28no—
69Dic3⋊C4—28no—
70C4×Dic3—22no—
71C4×Dic3—24no—
72C4×Dic3D414yes—
73C22×Q8—31no—
74C22×Q8C2×D311yescharacteristic
75C23×C4—32no—
76C23×C4C2×D311yescharacteristic
77C4⋊Q8—64no—
78C22⋊Q8—24no—
79C22⋊Q8—64no—
80C2×C4⋊C4—31no—
81C2×C4⋊C4—32no—
82C2×C4⋊C4—34no—
83C2×C4⋊C4—34no—
84C2×C4⋊C4C2×D311yescharacteristic
85C2×C4⋊C4C2×D312yes—
86C2×C22⋊C4—32no—
87C2×C22⋊C4C2×D312yes—
88C2×C42—31no—
89C2×C42—32no—
90C2×C42—31no—
91C2×C42—64no—
92#95—34no—
93#95—34no—
94C23×C3—21no—
95C23×C3—22no—
96C23×C3—21no—
97C23×C3C2×D412yes—
98C23×C3C2×Q814yes—
99C23×C3C2411yescharacteristic
100C3×Q8—44no—
101C2×C3×C4—21no—
102C2×C3×C4—21no—
103C2×C3×C4—28no—
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4—24no—
108C2×C3×C4—22no—
109C2×C3×C4C2×D412yes—
110C2×C3×C4C2×D412yes—
111C2×C3×C4C2×D412yes—
112C2×C3×C4#4112yes—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—216no—
116C2×Dic3—22no—
117C2×Dic3—22no—
118C2×Dic3—22no—
119C2×Dic3—24no—
120C2×Dic3—22no—
121C2×Dic3C2×D414yes—
122C2×Dic3C2×Q814yes—
123C2×Dic3C2×Q814yes—
124C2×Dic3#4114yes—
125Dic6—44no—
126C24C22×D311yescharacteristic
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—22no—
130C2×Q8—62no—
131C2×Q8—62no—
132C2×Q8—62no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—32no—
136C22×C4—38no—
137C22×C4—32no—
138C22×C4—32no—
139C22×C4—32no—
140C22×C4—64no—
141C22×C4—64no—
142C22×C4#6714yes—
143C22×C4C22×D312yes—
144C22×C4C22×D312yes—
145C22×C4C22×D311yescharacteristic
146C22×C4C22×D311yescharacteristic
147C4⋊C4—22no—
148C4⋊C4—24no—
149C4⋊C4—68no—
150C4⋊C4—68no—
151C4⋊C4—64no—
152C4⋊C4—62no—
153C22⋊C4—24no—
154C22⋊C4—64no—
155C42—34no—
156C42—64no—
157C42—62no—
158C22×C3—22no—
159C22×C3—24no—
160C22×C3—22no—
161C22×C3—24no—
162C22×C3C4⋊Q814yescomplemented
163C22×C3C22×D411yescharacteristic
164C22×C3C22×D411yescharacteristic
165C22×C3C22×Q812yes—
166C22×C3C2×#4111yescharacteristic
167C22×C3C2×#4112yes—
168C3×C4—22no—
169C3×C4—42no—
170C3×C4—42no—
171C3×C4#12114yes—
172Dic3—24no—
173Dic3—42no—
174Dic3—42no—
175Dic3C22⋊Q818yescomplemented
176C23—21no—
177C23—21no—
178C23—22no—
179C23D3×Q814yes—
180C23C2×#6712yes—
181C23C23×D311yescharacteristic
182Q8—44no—
183Q8—124no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—28no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—34no—
193C2×C4—34no—
194C2×C4—34no—
195C2×C4—34no—
196C2×C4—616no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—62no—
201C2×C4—64no—
202C2×C4—62no—
203C2×C4—62no—
204C2×C4D3×D412yes—
205C2×C4#23712yes—
206C2×C4C2×#6712yes—
207C2×C4C2×#6712yes—
208C2×C3—24no—
209C2×C3C2×#12111yescharacteristic
210C2×C3C2×C22⋊Q812yes—
211C2×C3C2×C4⋊Q811yescharacteristic
212C2×C3#53211yescharacteristic
213C2×C3#53312yes—
214C22—22no—
215C22—22no—
216C22—24no—
217C22—24no—
218C22#94414yescomplemented
219C22C2×D3×D411yescharacteristic
220C22C2×#23711yescharacteristic
221C22C2×D3×Q812yes—
222C22D3×#4112yes—
223C22C22×#6711yescharacteristic
224C4—22no—
225C4—38no—
226C4—42no—
227C4—42no—
228C4—64no—
229C4—122no—
230C4—122no—
231C4#93914yes—
232C3#285711yescharacteristic, complemented, Sylow
233C2—24no—
234C2#562212yes—
235C2D3×C22⋊Q812yes—
236C2C2×#93911yescharacteristic
237C2C2×#94411yescharacteristic
238C2#583211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C22⋊Q8C2
prime radicalC24×C3C23
hypercenterC2×C22⋊Q8D3
hypocenterC3#2857
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#944C2semidirect
5. #5229C2semidirect
6. #4976C2semidirect
7. #4975C2semidirect
8. #4973C2semidirect
54. C22×Dic3Q8semidirect
162. C22×C3C4⋊Q8semidirect
175. Dic3C22⋊Q8semidirect
218. C22#944semidirect
232. C3#2857semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊Q82
C2×#9442
C2×#9242
C22×C4×Dic32
#52292
#49762
#49752
#49732
#49422
#28573