Finite Group Databasebeta

← #83262

Subgroups — #83262

Order 384.

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

normal subgroups only

This group has 488 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
1C2×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#872C211yesmaximal, characteristic
3#5231C212yesmaximal
4#5230C212yesmaximal, complemented
5#4993C212yesmaximal
6#4987C212yesmaximal
7#4985C212yesmaximal, complemented
8#4972C212yesmaximal, complemented
9#4936C211yesmaximal, characteristic, complemented
10#2628—31nomaximal, Sylow
11C22×C3×Q8C2212yes—
12C22×C4×D3C2211yescharacteristic
13C3×C4⋊Q8—24no—
14C2×C3×C4⋊C4C2212yes—
15C2×C3×C4⋊C4C2212yes—
16C2×C3×C42C2211yescharacteristic
17C2×(C2×D3)⋊C4C2211yescharacteristic
18C2×(C2×D3)⋊C4C2211yescharacteristic
19C2×(C2×D3)⋊C4C2214yes—
20C2×C4⋊Dic3C2212yes—
21C2×Dic3⋊C4C2211yescharacteristic
22C2×Dic3⋊C4C2211yescharacteristic
23C2×Dic3⋊C4C2214yes—
24C2×C4×Dic3C2211yescharacteristic
25#872—24no—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2214yes—
29C3⋊#95C2214yes—
30C2×C4⋊Q8D311yescharacteristic, complemented
31C2×#117—31no—
32#400—32no—
33#399—32no—
34#395—32no—
35#393—32no—
36#389—32no—
37#382—31no—
38#380—32no—
39C23×D3C2311yescharacteristic
40C2×C3×Q8—24no—
41C2×C3×Q8—44no—
42C22×C3×C4C2314yes—
43C22×C3×C4C2311yescharacteristic
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2311yescharacteristic
46C22×Dic3C2311yescharacteristic
47C22×Dic3C2314yes—
48C22×Dic3C2311yescharacteristic
49C22×Dic3C2311yescharacteristic
50C2×C4×D3—22no—
51C2×C4×D3—22no—
52C2×C4×D3Q814yescomplemented
53C3×C4⋊C4—24no—
54C3×C4⋊C4—24no—
55C3×C42D414yes—
56(C2×D3)⋊C4—22no—
57(C2×D3)⋊C4—22no—
58(C2×D3)⋊C4—44no—
59C4⋊Dic3—24no—
60Dic3⋊C4—22no—
61Dic3⋊C4—22no—
62Dic3⋊C4—44no—
63C4×Dic3—22no—
64C22×Q8C2×D312yes—
65C23×C4—31no—
66C4⋊Q8—24no—
67#117—64no—
68C2×C4⋊C4—31no—
69C2×C4⋊C4—31no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4—34no—
72C2×C4⋊C4C2×D312yes—
73C2×C4⋊C4C2×D312yes—
74C2×C22⋊C4—31no—
75C2×C22⋊C4—31no—
76C2×C22⋊C4—34no—
77C2×C42—31no—
78C2×C42C2×D311yescharacteristic
79#95—34no—
80#95—32no—
81#95—32no—
82#95—34no—
83C23×C3C2411yescharacteristic
84C22×D3—21no—
85C22×D3—21no—
86C22×D3—22no—
87C22×D3—22no—
88C22×D3C2×Q812yes—
89C3×Q8—48no—
90C2×C3×C4—22no—
91C2×C3×C4—22no—
92C2×C3×C4—24no—
93C2×C3×C4—24no—
94C2×C3×C4—24no—
95C2×C3×C4C2×D412yes—
96C2×C3×C4C2×D412yes—
97C2×C3×C4C2×D412yes—
98C2×C3×C4C2×Q812yes—
99C2×C3×C4#4112yes—
100C2×Dic3—21no—
101C2×Dic3—21no—
102C2×Dic3—21no—
103C2×Dic3—21no—
104C2×Dic3—24no—
105C2×Dic3—22no—
106C2×Dic3—24no—
107C2×Dic3—22no—
108C2×Dic3—24no—
109C2×Dic3C2×Q812yes—
110C4×D3—28no—
111C24—31no—
112C2×Q8—24no—
113C2×Q8—44no—
114C22×C4—31no—
115C22×C4—34no—
116C22×C4—31no—
117C22×C4—34no—
118C22×C4—31no—
119C22×C4—62no—
120C22×C4—62no—
121C22×C4C22×D314yes—
122C22×C4C22×D311yescharacteristic
123C22×C4C22×D311yescharacteristic
124C22×C4C22×D311yescharacteristic
125C4⋊C4—24no—
126C4⋊C4—24no—
127C4⋊C4—64no—
128C4⋊C4—62no—
129C4⋊C4—62no—
130C4⋊C4—124no—
131C22⋊C4—62no—
132C22⋊C4—62no—
133C22⋊C4—124no—
134C42—62no—
135C42#6714yes—
136C22×C3C22×D411yescharacteristic
137C22×C3C22×Q811yescharacteristic
138C22×C3C2×#4111yescharacteristic
139C22×C3H(5,𝔽2)12yes—
140C22×C32−1+412yes—
141C2×D3—22no—
142C2×D3—24no—
143C2×D3—41no—
144C2×D3—41no—
145C2×D3—42no—
146C3×C4—22no—
147C3×C4—22no—
148C3×C4—44no—
149C3×C4C22⋊Q814yes—
150Dic3—22no—
151Dic3—41no—
152Dic3—41no—
153Dic3—44no—
154C23—32no—
155C23—62no—
156C23—61no—
157C23—62no—
158C23—61no—
159C23C23×D311yescharacteristic
160Q8—48no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—24no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—32no—
167C2×C4—61no—
168C2×C4—61no—
169C2×C4—68no—
170C2×C4—64no—
171C2×C4—62no—
172C2×C4—64no—
173C2×C4—64no—
174C2×C4—62no—
175C2×C4—61no—
176C2×C4—61no—
177C2×C4D3×Q812yes—
178C2×C4#23912yes—
179C2×C4C2×#6712yes—
180C2×C4C2×#6712yes—
181C2×C4C2×#6712yes—
182C2×C3C2×C22⋊Q811yescharacteristic
183C2×C3#53511yescharacteristic
184C2×C3#53611yescharacteristic
185C2×C3#54212yes—
186C2×C3#54412yes—
187D3—42no—
188C22—64no—
189C22—62no—
190C22—122no—
191C22—121no—
192C22—121no—
193C22#100412yes—
194C22C2×D3×Q811yescharacteristic
195C22C2×#23911yescharacteristic
196C22#100712yes—
197C22C22×#6711yescharacteristic
198C4—22no—
199C4—22no—
200C4—44no—
201C4—62no—
202C4—124no—
203C4—121no—
204C4—121no—
205C4#94614yes—
206C3#262811yescharacteristic, complemented, Sylow
207C2—122no—
208C2#572912yes—
209C2#573312yes—
210C2#580511yescharacteristic
211C2C2×#94611yescharacteristic
212C2#581611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C4⋊Q8C2
prime radicalC23×D3C23
hypercenterC2×C4⋊Q8D3
hypocenterC3#2628
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4⋊Q8C2semidirect
4. #5230C2semidirect
7. #4985C2semidirect
8. #4972C2semidirect
9. #4936C2semidirect
30. C2×C4⋊Q8D3semidirect
52. C2×C4×D3Q8semidirect
206. C3#2628semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊Q82
C2×#8722
#52312
#52302
#49932
#49872
#49852
#49722
#49362
#26283