Finite Group Databasebeta

← #70046

Subgroups — #70046

Order 384.

312 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 85 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
2#5819C211yesmaximal, characteristic
3C3×#410C211yesmaximal, characteristic
4#5004C212yesmaximal, complemented
5#4531C212yesmaximal, complemented
6#1909—31nomaximal, Sylow
7C2×C3×#41C2211yescharacteristic
8C22×Dic6C2211yescharacteristic
9C3×#117C2212yes—
10#948—24no—
11#944C412yescomplemented
12#936C412yescomplemented
13#935—22no—
14#926C2211yescharacteristic
15#924—22no—
16C2×#208—22no—
17C2×#208C2212yes—
18C3×#99—22no—
19#810—24no—
20#568—31no—
21#536—31no—
22#419—32no—
23#410D311yescharacteristic
24H(3,ℤ/4)—32no—
25C3×#41—22no—
26C3×#41—22no—
27C2×C3×Q8C2×C411yescharacteristic
28C2×C3×D4C2×C411yescharacteristic
29C2×C3×D4D412yes—
30C22×C3×C4D412yes—
31C22×C3×C4C2311yescharacteristic
32C22×Dic3—21no—
33C2×Dic6—22no—
34C2×Dic6—22no—
35C2×Dic6—21no—
36C2×Dic6C2×C412yes—
37C3×C4⋊C4—22no—
38C3×C22⋊C4—22no—
39C3×C42—22no—
40#217—24no—
41#217—21no—
42C4⋊Dic3—21no—
43Dic3⋊C4—24no—
44C4×Dic3C2×C412yes—
45#208—24no—
46#208—22no—
47#208—22no—
48C2×C3⋊C8—24no—
49C2×C3⋊C8D414yes—
50C2×#41C2×D311yescharacteristic
51C22×Q8—31no—
52#131—64no—
53C2×M16—32no—
54C2×M16—62no—
55C4⋊Q8—32no—
56#124—32no—
57#123—62no—
58C22⋊Q8—62no—
59#117—31no—
60#117C2×D312yes—
61#103—64no—
62#99—22no—
63C23×C3C22⋊C412yes—
64C23×C3C2×D411yescharacteristic
65C3×Q8—22no—
66C3×D4—24no—
67C3×D4—22no—
68C2×C3×C4—22no—
69C2×C3×C4—22no—
70C2×C3×C4—42no—
71C2×C3×C4C22⋊C412yes—
72C2×C3×C4C22×C411yescharacteristic
73C2×C3×C4C2×D411yescharacteristic
74C2×C3×C4C2×D412yes—
75C2×C3×C4#4112yes—
76C2×Dic3—22no—
77C2×Dic3—22no—
78C2×Dic3—42no—
79Dic6—24no—
80Dic6—41no—
81Dic6—41no—
82C3⋊C8—24no—
83#41—22no—
84#41—22no—
85C2×Q8—32no—
86C2×Q8—61no—
87C2×Q8—62no—
88C2×Q8—62no—
89C2×Q8C4×D311yescharacteristic
90C2×D4C4×D311yescharacteristic
91C2×D4#6712yes—
92C22×C4—61no—
93C22×C4D1212yes—
94C22×C4C22×D311yescharacteristic
95M16—64no—
96M16—62no—
97M16—62no—
98C2×C8—34no—
99C2×C8—64no—
100C4⋊C4—22no—
101C4⋊C4—64no—
102C4⋊C4—61no—
103C22⋊C4—22no—
104C22⋊C4—64no—
105C22⋊C4—61no—
106C42—22no—
107C42—32no—
108C22×C3—22no—
109C22×C3—41no—
110C22×C3—42no—
111C22×C3C2×C22⋊C411yescharacteristic
112C22×C3#12112yes—
113C3×C4—22no—
114C3×C4—82no—
115C3×C4C4×D412yes—
116C3×C4#12011yescharacteristic
117C3×C4#12311yescharacteristic
118Dic3—42no—
119Dic3—42no—
120C23(C2×D3)⋊C412yes—
121C23C2×#6711yescharacteristic
122Q8—22no—
123Q8—64no—
124Q8—121no—
125Q8—121no—
126D4—24no—
127D4—22no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—42no—
131C2×C4—62no—
132C2×C4—62no—
133C2×C4—122no—
134C2×C4(C2×D3)⋊C412yes—
135C2×C4C2×C4×D311yescharacteristic
136C2×C4C2×D1211yescharacteristic
137C2×C4D3×D412yes—
138C2×C4#23712yes—
139C8—64no—
140C2×C3—21no—
141C2×C3—22no—
142C2×C3—42no—
143C2×C3#38611yescharacteristic
144C22—22no—
145C22—41no—
146C22—42no—
147C22C2×(C2×D3)⋊C411yescharacteristic
148C22#93912yes—
149C4—22no—
150C4—82no—
151C4—122no—
152C4—122no—
153C4#67⋊C412yes—
154C4#88211yescharacteristic
155C4#88611yescharacteristic
156C3#190911yescharacteristic, complemented, Sylow
157C2—22no—
158C2—21no—
159C2—42no—
160C2#495811yescharacteristic
subgroupG/H
centerC2#4958
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#410C2
socleC2×C3#386
prime radicalC2×C3×#41C22
cosocleC22×C3×C4C23
hypercenter#410D3
hypocenterC3#1909
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
4. #5004C2semidirect
5. #4531C2semidirect
11. #944C4semidirect
12. #936C4semidirect
156. C3#1909semidirect

Maximal subgroups

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

subgroupindex
#58252
#58192
C3×#4102
#50042
#45312
#19093