Finite Group Databasebeta

← #82113

Subgroups — #82113

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×#926C211yesmaximal, characteristic, complemented
3#5247C212yesmaximal
4#5211C211yesmaximal, characteristic, complemented
5#5210C212yesmaximal, complemented
6#4953C212yesmaximal, complemented
7#4949C212yesmaximal, complemented
8#4944C214yesmaximal, complemented
9#2482—31nomaximal, Sylow
10C25×C3C2211yescharacteristic
11C23×C3×C4C2211yescharacteristic
12C23×C3×C4C2211yescharacteristic
13C2×C3×C22⋊C4—24no—
14C2×C3×C22⋊C4C2212yes—
15C2×C3×C22⋊C4C2212yes—
16C2×#217C2214yes—
17C2×#217C2212yes—
18C2×#217C2212yes—
19C2×#217C2214yes—
20#926—24no—
21C2×C4⋊Dic3C2211yescharacteristic, complemented
22C2×C4⋊Dic3C2211yescharacteristic, complemented
23C2×Dic3⋊C4C2214yescomplemented
24C2×C4×Dic3C2212yescomplemented
25C3⋊#95C2212yescomplemented
26C3⋊#95C2214yescomplemented
27C3⋊#95C2212yescomplemented
28C2×#117—31no—
29C22×C22⋊C4D311yescharacteristic
30#399—32no—
31#397—32no—
32#388—34no—
33#385—32no—
34#381—31no—
35#379—32no—
36C24×C3—22no—
37C24×C3—24no—
38C24×C3—24no—
39C24×C3—42no—
40C24×C3C2312yes—
41C24×C3C2311yescharacteristic
42C22×C3×C4—24no—
43C22×C3×C4—22no—
44C22×C3×C4—42no—
45C22×C3×C4D414yes—
46C22×C3×C4C2312yes—
47C22×C3×C4C2311yescharacteristic
48C22×C3×C4C2311yescharacteristic
49C22×Dic3C2314yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C3×C22⋊C4—24no—
53C3×C22⋊C4—24no—
54#217—24no—
55#217—28no—
56#217—24no—
57#217—44no—
58C4⋊Dic3—22no—
59C4⋊Dic3—22no—
60Dic3⋊C4—44no—
61C4×Dic3—24no—
62C25C2×D311yescharacteristic
63C23×C4C2×D311yescharacteristic
64C23×C4C2×D311yescharacteristic
65#117—64no—
66C2×C4⋊C4—31no—
67C2×C4⋊C4—31no—
68C2×C4⋊C4—34no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4—32no—
71C2×C22⋊C4—32no—
72C2×C22⋊C4—34no—
73C2×C22⋊C4—34no—
74C2×C22⋊C4C2×D312yes—
75C2×C22⋊C4C2×D312yes—
76C2×C42—32no—
77#95—32no—
78#95—32no—
79#95—34no—
80C23×C3—22no—
81C23×C3—22no—
82C23×C3—28no—
83C23×C3—22no—
84C23×C3—22no—
85C23×C3—24no—
86C23×C3—24no—
87C23×C3—416no—
88C23×C3—44no—
89C23×C3—44no—
90C23×C3C2×D412yes—
91C23×C3#4114yes—
92C23×C3#4114yes—
93C23×C3C2411yescharacteristic
94C2×C3×C4—28no—
95C2×C3×C4—22no—
96C2×C3×C4—22no—
97C2×C3×C4—22no—
98C2×C3×C4—24no—
99C2×C3×C4—44no—
100C2×C3×C4C2×D412yes—
101C2×C3×C4C2×D412yes—
102C2×Dic3—22no—
103C2×Dic3—22no—
104C2×Dic3—24no—
105C2×Dic3—24no—
106C2×Dic3—24no—
107C2×Dic3—24no—
108C2×Dic3—24no—
109C24—24no—
110C24—22no—
111C24—24no—
112C24—42no—
113C24C22×D312yes—
114C24C22×D311yescharacteristic
115C22×C4—24no—
116C22×C4—22no—
117C22×C4—34no—
118C22×C4—32no—
119C22×C4—32no—
120C22×C4—42no—
121C22×C4#6714yes—
122C22×C4C22×D312yes—
123C22×C4C22×D311yescharacteristic
124C22×C4C22×D311yescharacteristic
125C4⋊C4—62no—
126C4⋊C4—62no—
127C4⋊C4—124no—
128C22⋊C4—24no—
129C22⋊C4—24no—
130C22⋊C4—68no—
131C22⋊C4—64no—
132C22⋊C4—64no—
133C22⋊C4—124no—
134C42—64no—
135C22×C3—22no—
136C22×C3—22no—
137C22×C3—28no—
138C22×C3—24no—
139C22×C3—28no—
140C22×C3—416no—
141C22×C3—42no—
142C22×C3—42no—
143C22×C3—44no—
144C22×C3#12314yes—
145C22×C3C22×D411yescharacteristic
146C22×C3C2×#4112yes—
147C22×C3C2×#4112yes—
148C22×C3H(5,𝔽2)12yes—
149C3×C4—22no—
150C3×C4—22no—
151C3×C4—42no—
152Dic3—42no—
153Dic3—42no—
154Dic3—44no—
155C23—22no—
156C23—22no—
157C23—28no—
158C23—22no—
159C23—24no—
160C23—24no—
161C23—22no—
162C23—416no—
163C23—44no—
164C23—44no—
165C23#23514yes—
166C23#23714yes—
167C23C2×#6712yes—
168C23C23×D311yescharacteristic
169C2×C4—28no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—44no—
175C2×C4—64no—
176C2×C4—64no—
177C2×C4—64no—
178C2×C4—64no—
179C2×C4—64no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4C2×#6712yes—
183C2×C4C2×#6712yes—
184C2×C3—24no—
185C2×C3—44no—
186C2×C3C2×#12311yescharacteristic
187C2×C3#52511yescharacteristic
188C2×C3#53511yescharacteristic
189C2×C3#55114yes—
190C22—28no—
191C22—22no—
192C22—22no—
193C22—28no—
194C22—24no—
195C22—416no—
196C22—44no—
197C22—42no—
198C22—42no—
199C22#93514yes—
200C22C2×#23512yes—
201C22C2×#23712yes—
202C22#100412yes—
203C22C22×#6711yescharacteristic
204C4—22no—
205C4—22no—
206C4—42no—
207C4—124no—
208C4—122no—
209C4—122no—
210C3#248211yescharacteristic, complemented, Sylow
211C2—24no—
212C2—44no—
213C2#549514yes—
214C2#579111yescharacteristic
215C2C2×#93511yescharacteristic
216C2#580511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC22×C3×C22⋊C4C2
prime radicalC25×C3C22
hypercenterC22×C22⋊C4D3
hypocenterC3#2482
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×#926C2semidirect
4. #5211C2semidirect
5. #5210C2semidirect
6. #4953C2semidirect
7. #4949C2semidirect
8. #4944C2semidirect
21. C2×C4⋊Dic3C22semidirect
22. C2×C4⋊Dic3C22semidirect
23. C2×Dic3⋊C4C22semidirect
24. C2×C4×Dic3C22semidirect
25. C3⋊#95C22semidirect
26. C3⋊#95C22semidirect
27. C3⋊#95C22semidirect
210. C3#2482semidirect

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×#9262
#52472
#52112
#52102
#49532
#49492
#49442
#24823