Finite Group Databasebeta

← #81802

Subgroups — #81802

Order 384.

384 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#5728C211yesmaximal, characteristic
2D3×C4⋊Q8C211yesmaximal, characteristic
3C3×#494C211yesmaximal, characteristic, complemented
4#5186C214yesmaximal
5#5182C214yesmaximal, complemented
6#5092C211yesmaximal, characteristic, complemented
7#5079C211yesmaximal, characteristic, complemented
8#4685C211yesmaximal, characteristic
9#4681C211yesmaximal, characteristic, complemented
10#3209—31nomaximal, Sylow
11C2×D3×Q8C2212yes—
12C2×C3×Q16C2214yes—
13C3×C4⋊Q8C2211yescharacteristic
14C3×C4⋊Q8C2211yescharacteristic
15#946C2212yes—
16Q8×Dic3C2212yes—
17#944C2212yes—
18C2×#216C2214yes—
19#897—21no—
20#892—21no—
21D3×C4⋊C4—21no—
22D3×C4⋊C4—21no—
23#890—21no—
24#888—21no—
25#888—21no—
26#887—21no—
27C42×D3C2211yescharacteristic
28#869C2211yescharacteristic
29#869C2211yescharacteristic
30C3×C4×C8C2211yescharacteristic
31#835C2214yes—
32(C2×D3)⋊C8C2212yes—
33#816C2214yes—
34Dic3⋊C8C2212yes—
35C4×C3⋊C8C2211yescharacteristic
36#533—31no—
37C2×C4⋊Q8—31no—
38#494—31no—
39#494D311yescharacteristic, complemented
40#487—31no—
41#487—31no—
42#462—34no—
43#451—34no—
44#431—31no—
45C2×C3×Q8C2312yes—
46C2×C3×Q8C2312yes—
47D3×Q8—44no—
48C2×C4×D3—22no—
49C2×C4×D3D411yescharacteristic
50C2×C4×D3D411yescharacteristic
51C2×C4×D3C2311yescharacteristic
52C2×Dic6—21no—
53C2×Dic6—22no—
54C2×Dic6C2312yes—
55C3×Q16—44no—
56C2×C3×C8C2312yes—
57C3×C4⋊C4—21no—
58C3×C4⋊C4—21no—
59C3×C4⋊C4—21no—
60C3×C4⋊C4—21no—
61C3×C42C2311yescharacteristic
62#216—44no—
63(C2×D3)⋊C4—22no—
64C4⋊Dic3—21no—
65C4⋊Dic3—21no—
66C4⋊Dic3—21no—
67C4⋊Dic3—22no—
68C4⋊Dic3C2312yes—
69Dic3⋊C4—22no—
70Dic3⋊C4—22no—
71Dic3⋊C4—22no—
72C4×Dic3—22no—
73C4×Dic3D411yescharacteristic
74C4×Dic3D411yescharacteristic
75C4×Dic3C2311yescharacteristic
76C2×C3⋊C8C2312yes—
77C22×Q8—32no—
78C2×Q16—34no—
79C2×Q16C2×D314yes—
80C4⋊Q8—32no—
81C4⋊Q8—31no—
82C4⋊Q8—31no—
83C4⋊Q8—61no—
84C4⋊Q8—61no—
85C4⋊Q8C2×D311yescharacteristic
86C4⋊Q8C2×D311yescharacteristic
87#125—61no—
88C22⋊Q8—32no—
89C22⋊Q8—61no—
90C4×Q8—32no—
91C4×Q8—61no—
92#117—61no—
93C2×C4⋊C4—61no—
94C2×C4⋊C4—61no—
95C2×C42—31no—
96C4⋊C8—32no—
97#103—34no—
98#103—34no—
99C22⋊C8—32no—
100C4×C8—31no—
101C4×C8C2×D311yescharacteristic
102C22×D3C2×D411yescharacteristic
103C3×Q8—24no—
104C3×Q8—24no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4C2×D411yescharacteristic
109C2×C3×C4C2411yescharacteristic
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3C2×D411yescharacteristic
113C2×Dic3C2×D411yescharacteristic
114C2×Dic3C2×D411yescharacteristic
115C4×D3—22no—
116C4×D3—22no—
117C4×D3—44no—
118C4×D3Q1614yescomplemented
119Dic6—24no—
120Dic6—44no—
121Dic6—42no—
122C3×C8—22no—
123C3⋊C8—22no—
124C2×Q8—32no—
125C2×Q8—62no—
126C2×Q8—61no—
127C2×Q8—124no—
128C2×Q8C22×D312yes—
129C2×Q8C22×D312yes—
130C22×C4—31no—
131C22×C4—31no—
132C22×C4—31no—
133C22×C4—62no—
134Q16—44no—
135Q16—124no—
136C2×C8—32no—
137C2×C8C22×D312yes—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—32no—
143C4⋊C4—62no—
144C4⋊C4—62no—
145C4⋊C4—62no—
146C4⋊C4—62no—
147C4⋊C4—61no—
148C4⋊C4—61no—
149C4⋊C4—61no—
150C22⋊C4—62no—
151C42—31no—
152C42—31no—
153C42—31no—
154C42—62no—
155C42C22×D311yescharacteristic
156C22×C3C22×D411yescharacteristic
157C2×D3—21no—
158C2×D3—21no—
159C2×D3C2×Q1612yes—
160C3×C4—42no—
161C3×C4—42no—
162C3×C4C2×Q1612yes—
163C3×C4#13712yes—
164C3×C4H(5,𝔽2)12yes—
165Dic3—21no—
166Dic3—21no—
167Dic3—42no—
168Dic3—42no—
169Dic3C2×Q1612yes—
170C23—31no—
171Q8—24no—
172Q8—24no—
173Q8—64no—
174Q8—124no—
175Q8—122no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—31no—
179C2×C4—34no—
180C2×C4—31no—
181C2×C4—31no—
182C2×C4—62no—
183C2×C4—62no—
184C2×C4—62no—
185C2×C4—62no—
186C2×C4—124no—
187C2×C4D3×D411yescharacteristic
188C2×C4D3×D411yescharacteristic
189C2×C4C23×D311yescharacteristic
190C8—22no—
191C8—62no—
192C2×C3#53511yescharacteristic
193C2×C3C22×Q1611yescharacteristic
194C2×C3C2×#13711yescharacteristic
195D3—22no—
196C22—32no—
197C22—61no—
198C22—61no—
199C22C2×D3×D411yescharacteristic
200C4—32no—
201C4—42no—
202C4—42no—
203C4—61no—
204C4—61no—
205C4—122no—
206C4—122no—
207C4D3×Q1612yes—
208C4#91812yes—
209C4#100412yes—
210C3#320911yescharacteristic, complemented, Sylow
211C2—62no—
212C2#571511yescharacteristic
213C2C2×D3×Q1611yescharacteristic
214C2C2×#91811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#494C2
socleC22×C3C22×D4
prime radicalC22×D3C2×D4
hypercenter#494D3
hypocenterC3#3209
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
3. C3×#494C2semidirect
5. #5182C2semidirect
6. #5092C2semidirect
7. #5079C2semidirect
9. #4681C2semidirect
39. #494D3semidirect
118. C4×D3Q16semidirect
210. C3#3209semidirect

Maximal subgroups

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

subgroupindex
#57282
D3×C4⋊Q82
C3×#4942
#51862
#51822
#50922
#50792
#46852
#46812
#32093