Finite Group Databasebeta

← #69159

Subgroups — #69159

Order 384.

238 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 55 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#5675C211yesmaximal, characteristic
2#4740C212yesmaximal
3C3×#354C211yesmaximal, characteristic, complemented
4#4541C211yesmaximal, characteristic, complemented
5#4473C212yesmaximal, complemented
6#2129—31nomaximal, Sylow
7C2×D3×Q8C411yescharacteristic, complemented
8C2×#237C2211yescharacteristic
9C3×#124C2211yescharacteristic
10#944C411yescharacteristic, complemented
11#936C2211yescharacteristic
12#883—22no—
13#878—22no—
14#877—22no—
15#872C411yescharacteristic, complemented
16#869C411yescharacteristic, complemented
17C3×#99C2212yes—
18#834C2212yes—
19#806—22no—
20#536—31no—
21#410—32no—
22#354—31no—
23#354—32no—
24#354D311yescharacteristic, complemented
25C2×C3×Q8C2×C411yescharacteristic
26C2×C3×D4C2311yescharacteristic
27C2×#67D412yes—
28C22×Dic3D412yes—
29D3×Q8—42no—
30#237—42no—
31C2×C4×D3—21no—
32C2×C4×D3C2×C411yescharacteristic
33C2×Dic6—21no—
34C2×Dic6C2×C411yescharacteristic
35C2×Dic6C2×C411yescharacteristic
36C3×C22⋊C4—22no—
37C3×C22⋊C4—22no—
38C3×C42C2×C411yescharacteristic
39#217—22no—
40#217—22no—
41(C2×D3)⋊C4—21no—
42C4⋊Dic3—22no—
43Dic3⋊C4—21no—
44Dic3⋊C4—22no—
45Dic3⋊C4—42no—
46C4×Dic3—42no—
47C4×Dic3C2×C411yescharacteristic
48C2×#41—31no—
49C22×Q8—31no—
50C4⋊Q8—31no—
51C4⋊Q8—31no—
52#124—31no—
53#124C2×D311yescharacteristic
54#123—62no—
55C22⋊Q8—62no—
56#117—31no—
57#117—62no—
58#99—32no—
59#99—62no—
60#99C2×D312yes—
61C23×C3C2×D412yes—
62C22×D3C22⋊C411yescharacteristic
63C3×Q8—41no—
64C3×D4—41no—
65C2×C3×C4—21no—
66C2×C3×C4—21no—
67C2×C3×C4—42no—
68C2×C3×C4C22×C411yescharacteristic
69#67—42no—
70C2×Dic3—22no—
71C2×Dic3—22no—
72C2×Dic3—21no—
73C2×Dic3—21no—
74C2×Dic3—42no—
75C2×Dic3C22⋊C412yes—
76C2×Dic3C22⋊C411yescharacteristic
77C4×D3—22no—
78C4×D3—42no—
79Dic6—41no—
80Dic6—42no—
81Dic6—41no—
82#41—122no—
83C2×Q8—31no—
84C2×Q8—31no—
85C2×Q8—61no—
86C2×Q8—122no—
87C2×Q8C4×D311yescharacteristic
88C2×D4—32no—
89C2×D4C22×D311yescharacteristic
90C22×C4—32no—
91C22×C4—31no—
92C22×C4—61no—
93C4⋊C4—61no—
94C4⋊C4—62no—
95C4⋊C4—62no—
96C4⋊C4—122no—
97C22⋊C4—22no—
98C22⋊C4—22no—
99C22⋊C4—61no—
100C22⋊C4—62no—
101C22⋊C4—62no—
102C42—31no—
103C42—122no—
104C42C4×D311yescharacteristic
105C22×C3—22no—
106C22×C3—42no—
107C22×C3C2×C22⋊C411yescharacteristic
108C2×D3—21no—
109C2×D3—41no—
110C3×C4—21no—
111C3×C4—41no—
112C3×C4—41no—
113C3×C4—82no—
114Dic3—22no—
115Dic3—41no—
116Dic3—41no—
117Dic3—82no—
118Dic3#9912yes—
119C23—31no—
120C23D3×D412yes—
121Q8—41no—
122Q8—121no—
123Q8—121no—
124Q8—122no—
125D4—41no—
126D4—122no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—31no—
130C2×C4—32no—
131C2×C4—42no—
132C2×C4—62no—
133C2×C4—62no—
134C2×C4—62no—
135C2×C4—61no—
136C2×C4—61no—
137C2×C4—122no—
138C2×C4—122no—
139C2×C4C2×C4×D311yescharacteristic
140C2×C3—21no—
141C2×C3—42no—
142C2×C3C2×#9911yescharacteristic
143D3—41no—
144C22—22no—
145C22—42no—
146C22—61no—
147C22—121no—
148C22D3×C22⋊C411yescharacteristic
149C4—21no—
150C4—32no—
151C4—41no—
152C4—41no—
153C4—62no—
154C4—82no—
155C4—121no—
156C4—121no—
157C4—242no—
158C3#212911yescharacteristic, complemented, Sylow
159C2—21no—
160C2—42no—
161C2—121no—
162C2D3×#9911yescharacteristic
subgroupG/H
centerC2D3×#99
commutatorC2×C3×C4C22×C4
FrattiniC2×D4C22×D3
FittingC3×#354C2
socleC2×C3C2×#99
prime radicalC2×#237C22
cosocleC2×C3×D4C23
hypercenter#354D3
hypocenterC3#2129
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. C3×#354C2semidirect
4. #4541C2semidirect
5. #4473C2semidirect
7. C2×D3×Q8C4semidirect
10. #944C4semidirect
15. #872C4semidirect
16. #869C4semidirect
24. #354D3semidirect
158. C3#2129semidirect

Maximal subgroups

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

subgroupindex
#56752
#47402
C3×#3542
#45412
#44732
#21293