Finite Group Databasebeta

← #69168

Subgroups — #69168

Order 384.

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

normal subgroups only

This group has 236 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#5734C211yesmaximal, characteristic, complemented
2#5095C211yesmaximal, characteristic, complemented
3#4753C212yesmaximal, complemented
4C3×#356C211yesmaximal, characteristic, complemented
5#4477C212yesmaximal, complemented
6#2204—31nomaximal, Sylow
7#1007C2211yescharacteristic
8C3×C4⋊Q8C2211yescharacteristic
9#947—21no—
10#947C2212yes—
11#942—21no—
12#908—22no—
13#895—22no—
14#892—22no—
15C4×D12—21no—
16C3×#101C2212yes—
17#836C2211yescharacteristic
18#825—22no—
19#805—21no—
20#555—31no—
21#456—31no—
22#454—32no—
23#356D311yescharacteristic, complemented
24#355—32no—
25C2×C3×Q8D411yescharacteristic
26C2×C3×Q8C2311yescharacteristic
27#239—41no—
28D3×Q8—41no—
29#235—21no—
30#235—22no—
31C2×D12—21no—
32C2×D12—21no—
33C2×D12D412yes—
34C2×C4×D3—22no—
35C3×M16—22no—
36C3×C4⋊C4—22no—
37C3×C42D411yescharacteristic
38#216—41no—
39#215—41no—
40(C2×D3)⋊C4—22no—
41(C2×D3)⋊C4—22no—
42(C2×D3)⋊C4—22no—
43C4⋊Dic3—21no—
44C4×Dic3—21no—
45C4×Dic3D412yes—
46#208—21no—
47D24—42no—
48#204—42no—
492−1+4—31no—
50#137—61no—
51AGL(1,ℤ/8)—62no—
52C4⋊Q8C2×D311yescharacteristic
53#124—32no—
54#124—61no—
55#121—62no—
56C4×D4—61no—
57#117—62no—
58#104—62no—
59#104—61no—
60#101—31no—
61#101C2×D312yes—
62C22×D3—22no—
63C3×Q8—41no—
64C3×Q8—41no—
65C2×C3×C4—21no—
66C2×C3×C4—21no—
67C2×C3×C4C2×D411yescharacteristic
68C2×C3×C4C2×D412yes—
69#67—41no—
70C2×Dic3—22no—
71D12—22no—
72D12—21no—
73D12—42no—
74D12—42no—
75D12—81no—
76C4×D3—44no—
77C4×D3—41no—
78C4×D3—42no—
79Dic6—22no—
80Dic6—21no—
81C3×C8—42no—
82C3⋊C8—41no—
83#41—61no—
84#41—62no—
85#41—121no—
86C2×Q8—121no—
87C2×Q8D1211yescharacteristic
88C2×Q8C22×D311yescharacteristic
89C2×D4—32no—
90C2×D4—61no—
91C2×D4—61no—
92C22×C4—62no—
93Q16—121no—
94SD16—122no—
95SD16—121no—
96D8—122no—
97M16—22no—
98M16—61no—
99C4⋊C4—22no—
100C4⋊C4—61no—
101C22⋊C4—62no—
102C22⋊C4—62no—
103C22⋊C4—62no—
104C42—32no—
105C42—61no—
106C42D1211yescharacteristic
107C22×C3#12011yescharacteristic
108C2×D3—42no—
109C2×D3—42no—
110C2×D3—41no—
111C2×D3—42no—
112C3×C4—22no—
113C3×C4—21no—
114C3×C4—22no—
115C3×C4—41no—
116Dic3—42no—
117Dic3—41no—
118C23—62no—
119Q8—41no—
120Q8—41no—
121Q8—62no—
122Q8—61no—
123D4—62no—
124D4—61no—
125D4—122no—
126D4—122no—
127D4—121no—
128D4—241no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—62no—
132C2×C4—124no—
133C2×C4—122no—
134C2×C4—121no—
135C2×C4C2×D1211yescharacteristic
136C2×C4D3×D412yes—
137C8—42no—
138C8—121no—
139C2×C3—21no—
140C2×C3B(4,𝔽2)11yescharacteristic
141D3—81no—
142D3—82no—
143C22—122no—
144C22—122no—
145C22—121no—
146C22—122no—
147C22#88211yescharacteristic
148C4—22no—
149C4—22no—
150C4—21no—
151C4—41no—
152C4—122no—
153C4—121no—
154C3#220411yescharacteristic, complemented, Sylow
155C2—21no—
156C2—242no—
157C2—241no—
158C2#474111yescharacteristic
subgroupG/H
centerC2#4741
commutator, cosocleC2×C3×Q8C23
FrattiniC2×Q8C22×D3
FittingC3×#356C2
socleC2×C3B(4,𝔽2)
hypercenter#356D3
hypocenterC3#2204
LCSG ▷ C2×C3×Q8 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
1. #5734C2semidirect
2. #5095C2semidirect
3. #4753C2semidirect
4. C3×#356C2semidirect
5. #4477C2semidirect
23. #356D3semidirect
154. C3#2204semidirect

Maximal subgroups

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

subgroupindex
#57342
#50952
#47532
C3×#3562
#44772
#22043