Finite Group Databasebeta

← #82117

Subgroups — #82117

Order 384.

668 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C22×C3×C22⋊C4C211yesmaximal, characteristic
2C2×#924C211yesmaximal, characteristic, complemented
3#5247C212yesmaximal
4#5212C211yesmaximal, characteristic, complemented
5#5210C212yesmaximal, complemented
6#4949C212yesmaximal, complemented
7#4948C214yesmaximal, complemented
8#4947C212yesmaximal, complemented
9#2626—31nomaximal, Sylow
10C25×C3C2211yescharacteristic
11C23×C3×C4C2211yescharacteristic
12C23×C3×C4C2211yescharacteristic
13C22×Dic6C2211yescharacteristic, complemented
14C2×C3×C22⋊C4—24no—
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2212yes—
17C2×#217C2212yes—
18C2×#217C2212yes—
19C2×#217C2214yes—
20C2×#217C2214yes—
21C2×C4⋊Dic3C2211yescharacteristic, complemented
22C2×C4⋊Dic3C2212yescomplemented
23#924—24no—
24C2×Dic3⋊C4C2212yescomplemented
25C2×Dic3⋊C4C2214yescomplemented
26C3⋊#95C2214yescomplemented
27C3⋊#95C2212yescomplemented
28C2×C22⋊Q8—31no—
29C22×C22⋊C4D311yescharacteristic
30#399—32no—
31#396—34no—
32#393—32no—
33#385—32no—
34#380—31no—
35#379—32no—
36C24×C3—22no—
37C24×C3—22no—
38C24×C3—42no—
39C24×C3—42no—
40C24×C3Q814yescomplemented
41C24×C3C2312yes—
42C24×C3C2311yescharacteristic
43C22×C3×C4—22no—
44C22×C3×C4—42no—
45C22×C3×C4—42no—
46C22×C3×C4D414yes—
47C22×C3×C4C2312yes—
48C22×C3×C4C2311yescharacteristic
49C22×C3×C4C2311yescharacteristic
50C22×Dic3C2314yes—
51C22×Dic3C2312yes—
52C22×Dic3C2312yes—
53C2×Dic6—22no—
54C2×Dic6—42no—
55C3×C22⋊C4—24no—
56C3×C22⋊C4—24no—
57#217—24no—
58#217—24no—
59#217—44no—
60#217—44no—
61C4⋊Dic3—22no—
62C4⋊Dic3—42no—
63Dic3⋊C4—24no—
64Dic3⋊C4—44no—
65C25C2×D311yescharacteristic
66C22×Q8—31no—
67C23×C4C2×D311yescharacteristic
68C23×C4C2×D311yescharacteristic
69C22⋊Q8—64no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4—31no—
72C2×C4⋊C4—32no—
73C2×C4⋊C4—34no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—32no—
76C2×C22⋊C4—32no—
77C2×C22⋊C4—34no—
78C2×C22⋊C4—34no—
79C2×C22⋊C4C2×D312yes—
80C2×C22⋊C4C2×D312yes—
81#95—32no—
82#95—34no—
83C23×C3—22no—
84C23×C3—22no—
85C23×C3—28no—
86C23×C3—22no—
87C23×C3—22no—
88C23×C3—24no—
89C23×C3—24no—
90C23×C3—48no—
91C23×C3—44no—
92C23×C3—44no—
93C23×C3—48no—
94C23×C3C2×D412yes—
95C23×C3C2×Q814yes—
96C23×C3C2×Q812yes—
97C23×C3#4112yes—
98C23×C3C2411yescharacteristic
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—24no—
103C2×C3×C4—44no—
104C2×C3×C4—44no—
105C2×C3×C4C2×D412yes—
106C2×C3×C4C2×D412yes—
107C2×Dic3—22no—
108C2×Dic3—22no—
109C2×Dic3—24no—
110C2×Dic3—24no—
111C2×Dic3—24no—
112C2×Dic3—24no—
113C2×Dic3—24no—
114Dic6—44no—
115C24—22no—
116C24—22no—
117C24—42no—
118C24—42no—
119C24Dic614yescomplemented
120C24C22×D312yes—
121C24C22×D311yescharacteristic
122C2×Q8—62no—
123C2×Q8—122no—
124C22×C4—22no—
125C22×C4—34no—
126C22×C4—32no—
127C22×C4—32no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4#6714yes—
131C22×C4C22×D312yes—
132C22×C4C22×D311yescharacteristic
133C22×C4C22×D311yescharacteristic
134C4⋊C4—64no—
135C4⋊C4—62no—
136C4⋊C4—124no—
137C4⋊C4—122no—
138C22⋊C4—24no—
139C22⋊C4—24no—
140C22⋊C4—64no—
141C22⋊C4—64no—
142C22⋊C4—124no—
143C22⋊C4—124no—
144C22×C3—22no—
145C22×C3—22no—
146C22×C3—24no—
147C22×C3—24no—
148C22×C3—24no—
149C22×C3—28no—
150C22×C3—416no—
151C22×C3—42no—
152C22×C3—42no—
153C22×C3—44no—
154C22×C3C22⋊Q814yes—
155C22×C3C22×D411yescharacteristic
156C22×C3C22×Q811yescharacteristic
157C22×C3C2×#4111yescharacteristic
158C22×C3H(5,𝔽2)12yes—
159C22×C3H(5,𝔽2)12yes—
160C3×C4—22no—
161C3×C4—22no—
162C3×C4—42no—
163Dic3—42no—
164Dic3—42no—
165Dic3—44no—
166C23—22no—
167C23—22no—
168C23—28no—
169C23—22no—
170C23—24no—
171C23—24no—
172C23—22no—
173C23—48no—
174C23—48no—
175C23—44no—
176C23—44no—
177C23C2×Dic614yes—
178C23C2×Dic612yes—
179C23#23512yes—
180C23C2×#6712yes—
181C23C23×D311yescharacteristic
182Q8—124no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—24no—
186C2×C4—22no—
187C2×C4—44no—
188C2×C4—44no—
189C2×C4—64no—
190C2×C4—64no—
191C2×C4—64no—
192C2×C4—64no—
193C2×C4—64no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4C2×#6712yes—
197C2×C4C2×#6712yes—
198C2×C3—24no—
199C2×C3—44no—
200C2×C3C2×C22⋊Q811yescharacteristic
201C2×C3#53411yescharacteristic
202C2×C3#53511yescharacteristic
203C2×C3#53812yes—
204C2×C3#54312yes—
205C22—24no—
206C22—24no—
207C22—22no—
208C22—22no—
209C22—28no—
210C22—24no—
211C22—416no—
212C22—44no—
213C22—42no—
214C22—42no—
215C22#92414yes—
216C22C22×Dic611yescharacteristic
217C22C2×#23511yescharacteristic
218C22#100412yes—
219C22#100412yes—
220C22C22×#6711yescharacteristic
221C4—22no—
222C4—22no—
223C4—42no—
224C4—124no—
225C4—122no—
226C4—122no—
227C3#262611yescharacteristic, complemented, Sylow
228C2—24no—
229C2—44no—
230C2#548312yes—
231C2#549412yes—
232C2C2×#92411yescharacteristic
233C2#580411yescharacteristic
234C2#580511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC22×C3×C22⋊C4C2
prime radicalC25×C3C22
hypercenterC22×C22⋊C4D3
hypocenterC3#2626
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#924C2semidirect
4. #5212C2semidirect
5. #5210C2semidirect
6. #4949C2semidirect
7. #4948C2semidirect
8. #4947C2semidirect
13. C22×Dic6C22semidirect
21. C2×C4⋊Dic3C22semidirect
22. C2×C4⋊Dic3C22semidirect
24. C2×Dic3⋊C4C22semidirect
25. C2×Dic3⋊C4C22semidirect
26. C3⋊#95C22semidirect
27. C3⋊#95C22semidirect
40. C24×C3Q8semidirect
119. C24Dic6semidirect
227. C3#2626semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊C42
C2×#9242
#52472
#52122
#52102
#49492
#49482
#49472
#26263