Finite Group Databasebeta

← #69442

Subgroups — #69442

Order 384.

306 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 89 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×#926C211yesmaximal, characteristic
2C3×C4×C22⋊C4C211yesmaximal, characteristic
3#5207C211yesmaximal, characteristic
4#4523C214yesmaximal, complemented
5#1826—31nomaximal, Sylow
6C23×C3×C4C2211yescharacteristic
7C2×C3×C22⋊C4—21no—
8C2×C3×C42C2212yes—
9C2×#217—21no—
10#926—21no—
11#926—21no—
12#926C412yescomplemented
13#926C2212yes—
14C2×C4⋊Dic3C411yescharacteristic, complemented
15C2×C4⋊Dic3C411yescharacteristic, complemented
16C2×C4×Dic3—21no—
17C2×#208C2212yes—
18C3×#95—21no—
19#830—21no—
20#830—21no—
21C2×#117—31no—
22#407—31no—
23C4×C22⋊C4D311yescharacteristic
24#339—34no—
25C24×C3D411yescharacteristic
26C22×C3×C4—24no—
27C22×C3×C4—22no—
28C22×C3×C4C2×C411yescharacteristic
29C22×C3×C4C2×C411yescharacteristic
30C22×C3×C4D411yescharacteristic
31C22×C3×C4D411yescharacteristic
32C22×C3×C4D411yescharacteristic
33C22×C3×C4C2311yescharacteristic
34C22×Dic3—21no—
35C22×Dic3—21no—
36C3×C22⋊C4—24no—
37C3×C42—24no—
38#217—22no—
39#217—22no—
40C4⋊Dic3—21no—
41C4⋊Dic3—21no—
42C4⋊Dic3C2×C412yes—
43C4⋊Dic3C2×C412yes—
44C4×Dic3—22no—
45C4×Dic3—22no—
46#208—24no—
47C2×C3⋊C8—22no—
48C23×C4C2×D311yescharacteristic
49C2×M16—32no—
50#117—32no—
51#117—32no—
52#117—61no—
53#117—61no—
54C2×C4⋊C4—31no—
55C2×C4⋊C4—31no—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—61no—
58C2×C42—61no—
59C2×C42C2×D312yes—
60C22⋊C8—61no—
61C22⋊C8—61no—
62#95—21no—
63C23×C3—22no—
64C23×C3—22no—
65C23×C3—41no—
66C23×C3C22⋊C411yescharacteristic
67C23×C3C22⋊C411yescharacteristic
68C23×C3C2×D411yescharacteristic
69C2×C3×C4—22no—
70C2×C3×C4—22no—
71C2×C3×C4—22no—
72C2×C3×C4—22no—
73C2×C3×C4—22no—
74C2×C3×C4—24no—
75C2×C3×C4—42no—
76C2×C3×C4—42no—
77C2×C3×C4C22⋊C411yescharacteristic
78C2×C3×C4C22⋊C411yescharacteristic
79C2×C3×C4C22×C411yescharacteristic
80C2×C3×C4C2×D411yescharacteristic
81C2×C3×C4#4114yes—
82C2×Dic3—22no—
83C2×Dic3—22no—
84C2×Dic3—42no—
85C2×Dic3—42no—
86C3⋊C8—42no—
87C24#6711yescharacteristic
88C22×C4—24no—
89C22×C4—22no—
90C22×C4—61no—
91C22×C4—61no—
92C22×C4C4×D311yescharacteristic
93C22×C4C4×D311yescharacteristic
94C22×C4D1211yescharacteristic
95C22×C4D1211yescharacteristic
96C22×C4#6711yescharacteristic
97C22×C4C22×D311yescharacteristic
98M16—64no—
99C2×C8—62no—
100C4⋊C4—32no—
101C4⋊C4—32no—
102C4⋊C4—61no—
103C4⋊C4—61no—
104C22⋊C4—24no—
105C22⋊C4—62no—
106C22⋊C4—62no—
107C42—24no—
108C42—62no—
109C42—62no—
110C22×C3—21no—
111C22×C3—21no—
112C22×C3—21no—
113C22×C3—21no—
114C22×C3—21no—
115C22×C3—21no—
116C22×C3—21no—
117C22×C3—44no—
118C22×C3#10412yes—
119C22×C3C2×C22⋊C411yescharacteristic
120C22×C3#12312yes—
121C3×C4—21no—
122C3×C4—21no—
123C3×C4—44no—
124C3×C4#11712yes—
125C3×C4#12312yes—
126Dic3—42no—
127Dic3—42no—
128C23—22no—
129C23—22no—
130C23—41no—
131C23(C2×D3)⋊C411yescharacteristic
132C23(C2×D3)⋊C411yescharacteristic
133C23C2×#6711yescharacteristic
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—24no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—62no—
143C2×C4—62no—
144C2×C4—122no—
145C2×C4—122no—
146C2×C4(C2×D3)⋊C411yescharacteristic
147C2×C4(C2×D3)⋊C411yescharacteristic
148C2×C4C2×C4×D311yescharacteristic
149C2×C4C2×D1211yescharacteristic
150C2×C4#23714yes—
151C8—122no—
152C2×C3—22no—
153C2×C3—22no—
154C2×C3—41no—
155C2×C3#38111yescharacteristic
156C2×C3C2×#10411yescharacteristic
157C2×C3#42111yescharacteristic
158C22—21no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—21no—
164C22—21no—
165C22—44no—
166C22#80512yes—
167C22C2×(C2×D3)⋊C411yescharacteristic
168C22#93512yes—
169C4—21no—
170C4—21no—
171C4—44no—
172C4—122no—
173C4—122no—
174C4#87712yes—
175C4#88612yes—
176C3#182611yescharacteristic, complemented, Sylow
177C2—22no—
178C2—22no—
179C2—41no—
180C2C2×#80511yescharacteristic
181C2#494311yescharacteristic
182C2#500511yescharacteristic
subgroupG/H
centerC2×C4C2×D12
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×C4×C22⋊C4C2
socleC22×C3C2×C22⋊C4
prime radicalC24×C3D4
cosocleC22×C3×C4C23
hypercenterC4×C22⋊C4D3
hypocenterC3#1826
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. #4523C2semidirect
12. #926C4semidirect
14. C2×C4⋊Dic3C4semidirect
15. C2×C4⋊Dic3C4semidirect
176. C3#1826semidirect

Maximal subgroups

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

subgroupindex
C2×#9262
C3×C4×C22⋊C42
#52072
#45232
#18263