Finite Group Databasebeta

← #69162

Subgroups — #69162

Order 384.

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

normal subgroups only

This group has 216 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#5062C211yesmaximal, characteristic
3#4752C212yesmaximal, complemented
4C3×#355C211yesmaximal, characteristic, complemented
5#4478C212yesmaximal, complemented
6#2203—31nomaximal, Sylow
7#1007C2211yescharacteristic
8C3×#124C2211yescharacteristic
9#944C2212yes—
10#935—21no—
11#932—21no—
12#909—22no—
13#879—22no—
14#878—22no—
15#870—21no—
16C3×#101C2212yes—
17#836C2211yescharacteristic
18#825—22no—
19#805—21no—
20#563—31no—
21#456—32no—
22#454—31no—
23#356—32no—
24#355D311yescharacteristic, complemented
25C2×C3×Q8C2311yescharacteristic
26C2×C3×D4D411yescharacteristic
27C22×Dic3—21no—
28#239—41no—
29D3×Q8—41no—
30#235—21no—
31#235—22no—
32C2×Dic6D412yes—
33C3×M16—22no—
34C3×C22⋊C4—22no—
35C3×C42D411yescharacteristic
36#217—22no—
37#217—21no—
38#214—41no—
39#213—41no—
40C4⋊Dic3—21no—
41C4⋊Dic3—21no—
42Dic3⋊C4—22no—
43Dic3⋊C4—22no—
44Dic3⋊C4—22no—
45C4×Dic3D412yes—
46#208—21no—
47Dic12—42no—
48#204—42no—
492−1+4—31no—
50#137—62no—
51AGL(1,ℤ/8)—61no—
52C4⋊Q8—32no—
53#126—62no—
54#125—61no—
55#124C2×D311yescharacteristic
56#123—61no—
57C22⋊Q8—62no—
58#104—62no—
59#104—61no—
60#101—31no—
61#101C2×D312yes—
62C23×C3—21no—
63C3×Q8—41no—
64C3×D4—41no—
65C2×C3×C4—21no—
66C2×C3×C4C2×D411yescharacteristic
67C2×C3×C4C2×D412yes—
68#67—41no—
69C2×Dic3—22no—
70C2×Dic3—22no—
71C2×Dic3—81no—
72D12—22no—
73D12—21no—
74C4×D3—41no—
75C4×D3—42no—
76Dic6—22no—
77Dic6—21no—
78Dic6—42no—
79C3×C8—42no—
80C3⋊C8—41no—
81#41—61no—
82#41—62no—
83#41—121no—
84C2×Q8—32no—
85C2×Q8—121no—
86C2×Q8C22×D311yescharacteristic
87C2×D4D1211yescharacteristic
88C22×C4—61no—
89Q16—122no—
90SD16—122no—
91SD16—121no—
92D8—121no—
93M16—22no—
94M16—61no—
95C4⋊C4—61no—
96C4⋊C4—62no—
97C4⋊C4—61no—
98C4⋊C4—62no—
99C4⋊C4—62no—
100C22⋊C4—22no—
101C22⋊C4—62no—
102C22⋊C4—61no—
103C42—32no—
104C42D1211yescharacteristic
105C22×C3—42no—
106C22×C3—41no—
107C22×C3#12011yescharacteristic
108C2×D3—41no—
109C3×C4—21no—
110C3×C4—22no—
111C3×C4—41no—
112Dic3—42no—
113Dic3—41no—
114Dic3—42no—
115C23—21no—
116Q8—41no—
117Q8—61no—
118Q8—62no—
119Q8—122no—
120D4—41no—
121D4—61no—
122D4—62no—
123D4—121no—
124C2×C4—21no—
125C2×C4—62no—
126C2×C4—62no—
127C2×C4—122no—
128C2×C4—121no—
129C2×C4—241no—
130C2×C4C2×D1211yescharacteristic
131C2×C4D3×D412yes—
132C8—42no—
133C8—121no—
134C2×C3—21no—
135C2×C3—81no—
136C2×C3B(4,𝔽2)11yescharacteristic
137D3—81no—
138C22—42no—
139C22—41no—
140C22—121no—
141C22#88211yescharacteristic
142C4—22no—
143C4—21no—
144C4—41no—
145C4—122no—
146C4—122no—
147C4—121no—
148C3#220311yescharacteristic, complemented, Sylow
149C2—21no—
150C2—81no—
151C2—241no—
152C2#474111yescharacteristic
subgroupG/H
centerC2#4741
commutator, cosocleC2×C3×Q8C23
FrattiniC2×Q8C22×D3
FittingC3×#355C2
socleC2×C3B(4,𝔽2)
prime radical#5062C2
hypercenter#355D3
hypocenterC3#2203
LCSG ▷ C2×C3×Q8 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
1. #5672C2semidirect
3. #4752C2semidirect
4. C3×#355C2semidirect
5. #4478C2semidirect
24. #355D3semidirect
148. C3#2203semidirect

Maximal subgroups

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

subgroupindex
#56722
#50622
#47522
C3×#3552
#44782
#22033