Finite Group Databasebeta

← #69155

Subgroups — #69155

Order 384.

226 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 41 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#5672C211yesmaximal, characteristic, complemented
2#5061C211yesmaximal, characteristic
3#4742C212yesmaximal
4C3×#354C211yesmaximal, characteristic, complemented
5#4472C212yesmaximal, complemented
6#2201—31nomaximal, Sylow
7#1004C2211yescharacteristic
8C3×#124C2211yescharacteristic
9#944—21no—
10#942—21no—
11#935C2212yes—
12#886—22no—
13#879—22no—
14#878—22no—
15#870—21no—
16C3×#99C2212yes—
17#833C2211yescharacteristic
18#806—22no—
19#805—21no—
20#563—31no—
21#458—32no—
22#455—31no—
23#354D311yescharacteristic, complemented
24#352—32no—
25C2×C3×Q8D411yescharacteristic
26C2×C3×D4C2311yescharacteristic
27C2×#67—22no—
28C22×Dic3D412yes—
29#237—41no—
30D3×D4—41no—
31#235—21no—
32C2×Dic6—21no—
33C3×C22⋊C4—22no—
34C3×C22⋊C4—22no—
35C3×C42D411yescharacteristic
36#217—22no—
37#217D412yes—
38#216—41no—
39#215—41no—
40(C2×D3)⋊C4—42no—
41C4⋊Dic3—21no—
42C4⋊Dic3—21no—
43C4⋊Dic3—42no—
44Dic3⋊C4—22no—
45Dic3⋊C4—22no—
46Dic3⋊C4—22no—
47C4×Dic3—21no—
48#208—21no—
49H(5,𝔽2)—31no—
50#137—61no—
51C4⋊Q8—61no—
52#126—62no—
53#125—61no—
54#124C2×D311yescharacteristic
55#123—32no—
56#123—62no—
57C22⋊Q8—62no—
58#104—61no—
59#100—31no—
60#99—62no—
61#99C2×D312yes—
62C23×C3C2×D412yes—
63C22×D3—22no—
64C3×Q8—41no—
65C3×D4—41no—
66C2×C3×C4—21no—
67C2×C3×C4—21no—
68C2×C3×C4—42no—
69C2×C3×C4C2×D411yescharacteristic
70#67—42no—
71#67—41no—
72C2×Dic3—22no—
73C2×Dic3—22no—
74C2×Dic3—22no—
75C2×Dic3—42no—
76D12—21no—
77C4×D3—41no—
78Dic6—21no—
79Dic6—81no—
80C3⋊C8—41no—
81#41—61no—
82#41—121no—
83C2×Q8—61no—
84C2×Q8D1211yescharacteristic
85C2×D4—62no—
86C2×D4—121no—
87C2×D4C22×D311yescharacteristic
88C22×C4—32no—
89Q16—121no—
90SD16—121no—
91M16—61no—
92C4⋊C4—61no—
93C4⋊C4—62no—
94C4⋊C4—62no—
95C4⋊C4—61no—
96C4⋊C4—62no—
97C4⋊C4—122no—
98C22⋊C4—22no—
99C22⋊C4—22no—
100C22⋊C4—32no—
101C22⋊C4—62no—
102C22⋊C4—122no—
103C42—61no—
104C42D1211yescharacteristic
105C22×C3—22no—
106C22×C3—42no—
107C22×C3#12011yescharacteristic
108C2×D3—41no—
109C2×D3—82no—
110C3×C4—21no—
111C3×C4—41no—
112C3×C4—41no—
113C3×C4—82no—
114Dic3—42no—
115Dic3—41no—
116Dic3—42no—
117C23—62no—
118C23D3×D412yes—
119Q8—41no—
120Q8—61no—
121Q8—241no—
122D4—41no—
123D4—61no—
124D4—122no—
125D4—121no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—42no—
129C2×C4—62no—
130C2×C4—62no—
131C2×C4—62no—
132C2×C4—121no—
133C2×C4—122no—
134C2×C4C2×D1211yescharacteristic
135C8—121no—
136C2×C3—21no—
137C2×C3—42no—
138C2×C3B(4,𝔽2)11yescharacteristic
139D3—81no—
140C22—22no—
141C22—42no—
142C22—121no—
143C22—242no—
144C22#88211yescharacteristic
145C4—21no—
146C4—41no—
147C4—41no—
148C4—82no—
149C4—122no—
150C4—121no—
151C4—122no—
152C3#220111yescharacteristic, complemented, Sylow
153C2—21no—
154C2—42no—
155C2—241no—
156C2#474111yescharacteristic
subgroupG/H
centerC2#4741
commutator, cosocleC2×C3×D4C23
FrattiniC2×D4C22×D3
FittingC3×#354C2
socleC2×C3B(4,𝔽2)
prime radical#1004C22
hypercenter#354D3
hypocenterC3#2201
LCSG ▷ C2×C3×D4 ▷ 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
1. #5672C2semidirect
4. C3×#354C2semidirect
5. #4472C2semidirect
23. #354D3semidirect
152. C3#2201semidirect

Maximal subgroups

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

subgroupindex
#56722
#50612
#47422
C3×#3542
#44722
#22013