Finite Group Databasebeta

← #68976

Subgroups — #68976

Order 384.

306 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 67 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#5510C211yesmaximal, characteristic
2C2×#825C212yesmaximal, complemented
3#5133C212yesmaximal, complemented
4#4975C211yesmaximal, characteristic, complemented
5C3×#343C211yesmaximal, characteristic, complemented
6#2039—31nomaximal, Sylow
7C2×#235C2212yes—
8C2×C3×M16C2212yes—
9C2×C3×C4⋊C4C2211yescharacteristic
10#930—22no—
11C2×C4⋊Dic3C2211yescharacteristic
12C2×Dic3⋊C4—22no—
13C2×C4×Dic3C2211yescharacteristic
14#897—22no—
15#895—22no—
16#825—24no—
17#821—22no—
18#816—22no—
19#537—31no—
20C2×#104—32no—
21#417—32no—
22#384—31no—
23#343D311yescharacteristic, complemented
24C22×C3×C4D412yes—
25C22×C3×C4C2311yescharacteristic
26C2×#67—22no—
27C22×Dic3—21no—
28C22×Dic3—21no—
29#235—22no—
30#235—22no—
31#235—22no—
32#235—22no—
33C2×D12—22no—
34C2×D12D412yes—
35C2×C4×D3—22no—
36C2×Dic6D412yes—
37C3×M16—22no—
38C3×M16—22no—
39C2×C3×C8—22no—
40C3×C4⋊C4—22no—
41(C2×D3)⋊C4—24no—
42C4⋊Dic3—22no—
43C4⋊Dic3—22no—
44Dic3⋊C4—44no—
45C4×Dic3—21no—
46C4×Dic3D412yescomplemented
47C2×#41—32no—
48C2×M16C2×D312yes—
49C22⋊Q8—62no—
50#121—62no—
51#121—62no—
52C2×C4⋊C4—31no—
53C2×C4⋊C4—62no—
54C2×C4⋊C4C2×D311yescharacteristic
55C2×C42—31no—
56#104—64no—
57#103—62no—
58#102—62no—
59C23×C3C2×D411yescharacteristic
60C22×D3—22no—
61C2×C3×C4—22no—
62C2×C3×C4—42no—
63C2×C3×C4C2×D411yescharacteristic
64C2×C3×C4C2×D412yes—
65C2×C3×C4#4112yes—
66C2×C3×C4#4111yescharacteristic
67#67—42no—
68#67—42no—
69C2×Dic3—22no—
70C2×Dic3—22no—
71C2×Dic3—22no—
72C2×Dic3—22no—
73C2×Dic3—42no—
74C2×Dic3—41no—
75D12—22no—
76D12—22no—
77D12—82no—
78C4×D3—42no—
79C4×D3—42no—
80Dic6—22no—
81Dic6—22no—
82C3×C8—42no—
83#41—62no—
84#41—62no—
85#41—62no—
86#41—62no—
87C2×Q8—32no—
88C2×D4—32no—
89C2×D4—62no—
90C2×D4—62no—
91C22×C4—61no—
92C22×C4—62no—
93C22×C4—61no—
94C22×C4D1212yes—
95C22×C4C22×D311yescharacteristic
96M16—22no—
97M16—22no—
98C2×C8—22no—
99C4⋊C4—22no—
100C4⋊C4—62no—
101C4⋊C4—62no—
102C4⋊C4—124no—
103C22⋊C4—64no—
104C42—32no—
105C42—61no—
106C22×C3—21no—
107C22×C3—21no—
108C22×C3#12011yescharacteristic
109C22×C3#12111yescharacteristic
110C22×C3#12411yescharacteristic
111C2×D3—42no—
112C2×D3—42no—
113C2×D3—42no—
114C3×C4—42no—
115C3×C4#12112yes—
116C3×C4#12312yes—
117Dic3—24no—
118Dic3—42no—
119Dic3—81no—
120C23—62no—
121C23C2×D1211yescharacteristic
122Q8—62no—
123Q8—62no—
124D4—62no—
125D4—62no—
126D4—122no—
127D4—122no—
128D4—242no—
129C2×C4—22no—
130C2×C4—42no—
131C2×C4—62no—
132C2×C4—62no—
133C2×C4—62no—
134C2×C4—62no—
135C2×C4—122no—
136C2×C4—122no—
137C2×C4—122no—
138C2×C4—121no—
139C2×C4#23512yes—
140C2×C4D3×D411yescharacteristic
141C2×C4D3×D412yes—
142C2×C4#23911yescharacteristic
143C8—42no—
144C2×C3—21no—
145C2×C3—21no—
146C2×C3#39411yescharacteristic
147C2×C3#45311yescharacteristic
148C2×C3#45611yescharacteristic
149D3—82no—
150C22—21no—
151C22—21no—
152C22—122no—
153C22—122no—
154C22—122no—
155C22#87511yescharacteristic
156C22#88211yescharacteristic
157C22#89511yescharacteristic
158C4—42no—
159C4—64no—
160C4—122no—
161C4—241no—
162C4#88412yes—
163C4#89412yes—
164C3#203911yescharacteristic, complemented, Sylow
165C2—21no—
166C2—21no—
167C2—242no—
168C2#467211yescharacteristic
169C2#474711yescharacteristic
170C2#475211yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#343C2
socleC22×C3#120
prime radical#5510C2
hypercenter#343D3
hypocenterC3#2039
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
2. C2×#825C2semidirect
3. #5133C2semidirect
4. #4975C2semidirect
5. C3×#343C2semidirect
23. #343D3semidirect
46. C4×Dic3D4semidirect
164. C3#2039semidirect

Maximal subgroups

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

subgroupindex
#55102
C2×#8252
#51332
#49752
C3×#3432
#20393