Finite Group Databasebeta

← #70147

Subgroups — #70147

Order 384.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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#5825C211yesmaximal, characteristic
2C3×#413C211yesmaximal, characteristic
3#5240C211yesmaximal, characteristic
4#4476C214yesmaximal, complemented
5#2123—31nomaximal, Sylow
6C2×C3×#41C2211yescharacteristic
7C22×Dic6C411yescharacteristic, complemented
8C2×C3×M16—21no—
9#944—21no—
10#936—21no—
11#935C412yescomplemented
12#935C2212yes—
13#926—21no—
14#926C411yescharacteristic, complemented
15#924—21no—
16#924—21no—
17C3×#101—21no—
18C3×#100C2212yes—
19#834—21no—
20#834C2212yes—
21#536—31no—
22#413D311yescharacteristic
23#410—31no—
24#352—34no—
25C3×#41—41no—
26C3×#41—41no—
27C2×C3×Q8D411yescharacteristic
28C2×C3×D4C2×C411yescharacteristic
29C2×C3×D4D411yescharacteristic
30C2×C3×D4C2311yescharacteristic
31C22×C3×C4C2×C411yescharacteristic
32C22×C3×C4D411yescharacteristic
33C22×C3×C4D411yescharacteristic
34C22×Dic3C2×C412yes—
35C2×Dic6—21no—
36C2×Dic6—21no—
37C2×Dic6—41no—
38C2×Dic6—41no—
39C3×M16—22no—
40C3×M16—41no—
41C2×C3×C8—41no—
42#217—22no—
43#217—22no—
44#217—22no—
45#217C2×C412yes—
46C4⋊Dic3—21no—
47C4⋊Dic3—41no—
48Dic3⋊C4—22no—
49Dic3⋊C4—22no—
50C4×Dic3—21no—
51C4×Dic3—41no—
52C2×#41C2×D311yescharacteristic
53C22×Q8—31no—
54C2×M16—21no—
55C4⋊Q8—61no—
56#124—61no—
57#123—32no—
58#123—32no—
59C22⋊Q8—61no—
60C22⋊Q8—61no—
61#117—31no—
62#117—61no—
63#101—21no—
64#100C2×D312yes—
65#99—32no—
66#99—61no—
67C23×C3C22⋊C411yescharacteristic
68C23×C3C22×C411yescharacteristic
69C23×C3C2×D411yescharacteristic
70C3×Q8—41no—
71C3×D4—41no—
72C3×D4—41no—
73C3×D4—41no—
74C2×C3×C4—21no—
75C2×C3×C4—21no—
76C2×C3×C4—21no—
77C2×C3×C4—21no—
78C2×C3×C4—21no—
79C2×C3×C4—21no—
80C2×C3×C4C22⋊C411yescharacteristic
81C2×C3×C4C22⋊C411yescharacteristic
82C2×C3×C4C22⋊C411yescharacteristic
83C2×C3×C4C2×D411yescharacteristic
84C2×Dic3—22no—
85C2×Dic3—22no—
86C2×Dic3—42no—
87C2×Dic3—42no—
88Dic6—42no—
89Dic6—41no—
90Dic6—41no—
91C3×C8—42no—
92#41—41no—
93#41—41no—
94C2×Q8—61no—
95C2×Q8—61no—
96C2×Q8—121no—
97C2×Q8—121no—
98C2×Q8#6711yescharacteristic
99C2×D4C4×D311yescharacteristic
100C2×D4D1211yescharacteristic
101C2×D4C22×D311yescharacteristic
102C22×C4—32no—
103C22×C4C4×D311yescharacteristic
104C22×C4D1211yescharacteristic
105C22×C4#6711yescharacteristic
106M16—22no—
107M16—41no—
108C2×C8—41no—
109C4⋊C4—62no—
110C4⋊C4—61no—
111C4⋊C4—62no—
112C4⋊C4—121no—
113C22⋊C4—32no—
114C22⋊C4—62no—
115C22⋊C4—62no—
116C22⋊C4—62no—
117C42—61no—
118C42—121no—
119C22×C3—21no—
120C22×C3—21no—
121C22×C3—21no—
122C22×C3—41no—
123C22×C3—41no—
124C22×C3—41no—
125C22×C3C2×C22⋊C411yescharacteristic
126C3×C4—21no—
127C3×C4—21no—
128C3×C4—21no—
129C3×C4#9911yescharacteristic
130C3×C4#9911yescharacteristic
131Dic3—42no—
132Dic3—42no—
133Dic3—82no—
134C23(C2×D3)⋊C411yescharacteristic
135C23C2×C4×D311yescharacteristic
136C23C2×D1211yescharacteristic
137Q8—41no—
138Q8—122no—
139Q8—121no—
140Q8—121no—
141D4—41no—
142D4—41no—
143D4—41no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—62no—
151C2×C4—62no—
152C2×C4—122no—
153C2×C4—122no—
154C2×C4(C2×D3)⋊C411yescharacteristic
155C2×C4(C2×D3)⋊C411yescharacteristic
156C2×C4(C2×D3)⋊C411yescharacteristic
157C2×C4C2×#6711yescharacteristic
158C8—42no—
159C2×C3—21no—
160C2×C3—41no—
161C2×C3—41no—
162C2×C3—41no—
163C2×C3C2×#9911yescharacteristic
164C22—21no—
165C22—21no—
166C22—21no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22C2×(C2×D3)⋊C411yescharacteristic
171C4—21no—
172C4—21no—
173C4—21no—
174C4—122no—
175C4—122no—
176C4—242no—
177C4#80611yescharacteristic
178C4#80611yescharacteristic
179C3#212311yescharacteristic, complemented, Sylow
180C2—21no—
181C2—41no—
182C2—41no—
183C2—41no—
184C2C2×#80611yescharacteristic
subgroupG/H
centerC2C2×#806
commutatorC23×C3C22×C4
FrattiniC2×D4C22×D3
FittingC3×#413C2
socleC2×C3C2×#99
prime radicalC2×C3×#41C22
cosocleC2×C3×D4C23
hypercenter#413D3
hypocenterC3#2123
LCSG ▷ C23×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
4. #4476C2semidirect
7. C22×Dic6C4semidirect
11. #935C4semidirect
14. #926C4semidirect
179. C3#2123semidirect

Maximal subgroups

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

subgroupindex
#58252
C3×#4132
#52402
#44762
#21233