Finite Group Databasebeta

← #82116

Subgroups — #82116

Order 384.

672 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 157 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#5211C212yesmaximal, complemented
5C4×#217C211yesmaximal, characteristic, complemented
6#4953C212yesmaximal, complemented
7#4947C212yesmaximal, complemented
8#4945C214yesmaximal, complemented
9#2561—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×#217C2214yes—
18C2×#217C2212yes—
19C2×#217C2212yes—
20C2×#217C2214yes—
21C2×C4⋊Dic3C2211yescharacteristic, complemented
22#924—24no—
23C2×Dic3⋊C4C2212yescomplemented
24C2×C4×Dic3C2212yescomplemented
25C3⋊#95C2214yescomplemented
26C3⋊#95C2214yescomplemented
27C3⋊#95C2212yescomplemented
28C2×C22⋊Q8—31no—
29C22×C22⋊C4D311yescharacteristic
30#397—32no—
31#393—32no—
32#388—34no—
33#381—32no—
34#379—32no—
35C4×C22⋊C4—31no—
36C24×C3—22no—
37C24×C3—24no—
38C24×C3—24no—
39C24×C3—42no—
40C24×C3C2312yes—
41C24×C3C2311yescharacteristic
42C22×C3×C4—22no—
43C22×C3×C4—24no—
44C22×C3×C4—42no—
45C22×C3×C4D414yes—
46C22×C3×C4C2312yes—
47C22×C3×C4C2311yescharacteristic
48C22×C3×C4C2311yescharacteristic
49C22×Dic3C2314yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C2×Dic6—22no—
53C2×Dic6—42no—
54C3×C22⋊C4—24no—
55C3×C22⋊C4—24no—
56#217—24no—
57#217—28no—
58#217—24no—
59#217—44no—
60C4⋊Dic3—22no—
61Dic3⋊C4—24no—
62C4×Dic3—42no—
63C25C2×D311yescharacteristic
64C22×Q8—31no—
65C23×C4C2×D311yescharacteristic
66C23×C4C2×D311yescharacteristic
67C22⋊Q8—64no—
68C2×C4⋊C4—32no—
69C2×C4⋊C4—31no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—32no—
72C2×C22⋊C4—32no—
73C2×C22⋊C4—34no—
74C2×C22⋊C4—34no—
75C2×C22⋊C4C2×D312yes—
76C2×C22⋊C4C2×D312yes—
77C2×C42—32no—
78#95—32no—
79#95—34no—
80#95—34no—
81C23×C3—22no—
82C23×C3—22no—
83C23×C3—28no—
84C23×C3—22no—
85C23×C3—22no—
86C23×C3—24no—
87C23×C3—24no—
88C23×C3—416no—
89C23×C3—44no—
90C23×C3—44no—
91C23×C3C2×D412yes—
92C23×C3#4114yes—
93C23×C3#4114yes—
94C23×C3C2411yescharacteristic
95C2×C3×C4—22no—
96C2×C3×C4—22no—
97C2×C3×C4—22no—
98C2×C3×C4—24no—
99C2×C3×C4—44no—
100C2×C3×C4—44no—
101C2×C3×C4C2×D412yes—
102C2×C3×C4C2×D412yes—
103C2×Dic3—22no—
104C2×Dic3—22no—
105C2×Dic3—24no—
106C2×Dic3—24no—
107C2×Dic3—24no—
108C2×Dic3—24no—
109C2×Dic3—24no—
110Dic6—44no—
111C24—24no—
112C24—22no—
113C24—24no—
114C24—42no—
115C24C22×D312yes—
116C24C22×D311yescharacteristic
117C2×Q8—62no—
118C2×Q8—122no—
119C22×C4—24no—
120C22×C4—22no—
121C22×C4—34no—
122C22×C4—32no—
123C22×C4—32no—
124C22×C4—42no—
125C22×C4#6714yes—
126C22×C4C22×D312yes—
127C22×C4C22×D311yescharacteristic
128C22×C4C22×D311yescharacteristic
129C4⋊C4—64no—
130C4⋊C4—62no—
131C22⋊C4—24no—
132C22⋊C4—24no—
133C22⋊C4—68no—
134C22⋊C4—64no—
135C22⋊C4—64no—
136C22⋊C4—124no—
137C42—122no—
138C22×C3—22no—
139C22×C3—22no—
140C22×C3—28no—
141C22×C3—24no—
142C22×C3—28no—
143C22×C3—416no—
144C22×C3—42no—
145C22×C3—42no—
146C22×C3—44no—
147C22×C3#12414yes—
148C22×C3C22×D411yescharacteristic
149C22×C3C2×#4112yes—
150C22×C3C2×#4112yes—
151C22×C3H(5,𝔽2)12yes—
152C3×C4—22no—
153C3×C4—22no—
154C3×C4—42no—
155Dic3—42no—
156Dic3—42no—
157Dic3—44no—
158C23—22no—
159C23—22no—
160C23—28no—
161C23—22no—
162C23—24no—
163C23—24no—
164C23—22no—
165C23—416no—
166C23—44no—
167C23—44no—
168C23#23514yes—
169C23#23714yes—
170C23C2×#6712yes—
171C23C23×D311yescharacteristic
172Q8—124no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—22no—
177C2×C4—44no—
178C2×C4—44no—
179C2×C4—64no—
180C2×C4—64no—
181C2×C4—64no—
182C2×C4—64no—
183C2×C4—64no—
184C2×C4—62no—
185C2×C4—62no—
186C2×C4C2×#6712yes—
187C2×C4C2×#6712yes—
188C2×C3—24no—
189C2×C3—44no—
190C2×C3#52511yescharacteristic
191C2×C3C2×#12411yescharacteristic
192C2×C3#53411yescharacteristic
193C2×C3#55114yes—
194C22—28no—
195C22—22no—
196C22—22no—
197C22—28no—
198C22—24no—
199C22—416no—
200C22—44no—
201C22—42no—
202C22—42no—
203C22#93614yes—
204C22C2×#23512yes—
205C22C2×#23712yes—
206C22#100412yes—
207C22C22×#6711yescharacteristic
208C4—22no—
209C4—22no—
210C4—42no—
211C4—124no—
212C4—122no—
213C4—122no—
214C3#256111yescharacteristic, complemented, Sylow
215C2—24no—
216C2—44no—
217C2#549514yes—
218C2#579111yescharacteristic
219C2C2×#93611yescharacteristic
220C2#580411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC22×C3×C22⋊C4C2
prime radicalC25×C3C22
hypercenterC22×C22⋊C4D3
hypocenterC3#2561
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. #5211C2semidirect
5. C4×#217C2semidirect
6. #4953C2semidirect
7. #4947C2semidirect
8. #4945C2semidirect
13. C22×Dic6C22semidirect
21. C2×C4⋊Dic3C22semidirect
23. C2×Dic3⋊C4C22semidirect
24. C2×C4×Dic3C22semidirect
25. C3⋊#95C22semidirect
26. C3⋊#95C22semidirect
27. C3⋊#95C22semidirect
214. C3#2561semidirect

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
#52112
C4×#2172
#49532
#49472
#49452
#25613