Finite Group Databasebeta

← #68133

Subgroups — #68133

Order 384.

282 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 53 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
2#4979C211yesmaximal, characteristic, complemented
3#4738C212yesmaximal, complemented
4C3×#324C211yesmaximal, characteristic, complemented
5#4470C212yesmaximal, complemented
6#1604—31nomaximal, Sylow
7C2×#235—21no—
8C22×Dic6C2211yescharacteristic
9C2×C3×C4⋊C4C2211yescharacteristic
10#930—21no—
11#930C2212yes—
12C2×Dic3⋊C4—21no—
13C2×#204—22no—
14#897—22no—
15#895—22no—
16C3×C22⋊C8C2212yes—
17C3⋊#95—21no—
18C3⋊#95C2211yescharacteristic
19#816—22no—
20#537—31no—
21#448—32no—
22#395—31no—
23#327—32no—
24#324D311yescharacteristic, complemented
25C22×C3×C4D411yescharacteristic
26C22×C3×C4D411yescharacteristic
27C22×C3×C4C2311yescharacteristic
28C2×#67—22no—
29C22×Dic3—21no—
30C22×Dic3—21no—
31#235—42no—
32#235—42no—
33C2×D12—21no—
34C2×D12—21no—
35C2×D12D412yes—
36C2×C4×D3—22no—
37C2×Dic6—22no—
38C2×Dic6—21no—
39C2×Dic6—42no—
40C2×C3×C8—22no—
41C3×C4⋊C4—22no—
42(C2×D3)⋊C4—22no—
43(C2×D3)⋊C4—22no—
44C4⋊Dic3—21no—
45C4⋊Dic3D412yes—
46Dic3⋊C4—42no—
47#204—44no—
48C2×#41—61no—
49C22×Q8—31no—
50C2×SD16—62no—
51C22⋊Q8—62no—
52#121—32no—
53#121—62no—
54#121—61no—
55C2×C4⋊C4—61no—
56C2×C4⋊C4C2×D311yescharacteristic
57#103—62no—
58C22⋊C8C2×D312yes—
59#95—31no—
60#95—61no—
61C23×C3C2×D411yescharacteristic
62C22×D3—22no—
63C2×C3×C4—21no—
64C2×C3×C4—21no—
65C2×C3×C4—21no—
66C2×C3×C4—41no—
67C2×C3×C4—41no—
68C2×C3×C4SD1612yescomplemented
69C2×C3×C4C2×D412yes—
70#67—42no—
71#67—42no—
72C2×Dic3—22no—
73C2×Dic3—22no—
74C2×Dic3—22no—
75C2×Dic3—41no—
76C2×Dic3—41no—
77C2×Dic3—41no—
78C2×Dic3—41no—
79D12—42no—
80D12—42no—
81D12—81no—
82C4×D3—44no—
83Dic6—24no—
84Dic6—42no—
85Dic6—42no—
86C3×C8—42no—
87#41—122no—
88#41—122no—
89C2×Q8—61no—
90C2×Q8—62no—
91C2×Q8—122no—
92C2×D4—32no—
93C2×D4—61no—
94C2×D4—61no—
95C2×D4—62no—
96C22×C4—62no—
97C22×C4—61no—
98C22×C4—61no—
99C22×C4D1211yescharacteristic
100C22×C4D1211yescharacteristic
101C22×C4C22×D311yescharacteristic
102SD16—124no—
103C2×C8—22no—
104C4⋊C4—22no—
105C4⋊C4—32no—
106C4⋊C4—61no—
107C4⋊C4—122no—
108C22⋊C4—62no—
109C22⋊C4—62no—
110C22×C3—21no—
111C22×C3—21no—
112C22×C3#12011yescharacteristic
113C22×C3C2×SD1611yescharacteristic
114C22×C3AGL(1,ℤ/8)11yescharacteristic
115C2×D3—42no—
116C2×D3—42no—
117C2×D3—42no—
118C3×C4—22no—
119C3×C4—22no—
120C3×C4—41no—
121Dic3—42no—
122Dic3—42no—
123Dic3—42no—
124C23—62no—
125C23C2×D1211yescharacteristic
126Q8—64no—
127Q8—122no—
128Q8—122no—
129D4—122no—
130D4—122no—
131D4—122no—
132D4—122no—
133D4—241no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—62no—
140C2×C4—62no—
141C2×C4—62no—
142C2×C4—124no—
143C2×C4—121no—
144C2×C4—121no—
145C2×C4—121no—
146C2×C4—121no—
147C2×C4#20412yescomplemented
148C2×C4D3×D412yes—
149C8—42no—
150C2×C3—21no—
151C2×C3—21no—
152C2×C3#45011yescharacteristic
153C2×C3#45611yescharacteristic
154C2×C3B(4,𝔽2)11yescharacteristic
155D3—82no—
156C22—21no—
157C22—21no—
158C22—122no—
159C22—122no—
160C22—122no—
161C22#88211yescharacteristic
162C22C2×#20411yescharacteristic
163C22#90811yescharacteristic
164C4—22no—
165C4—22no—
166C4—41no—
167C4—122no—
168C4—122no—
169C4—122no—
170C3#160411yescharacteristic, complemented, Sylow
171C2—21no—
172C2—21no—
173C2—242no—
174C2#473111yescharacteristic
175C2#474111yescharacteristic
176C2#475211yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#324C2
socleC22×C3#120
prime radical#5510C2
hypercenter#324D3
hypocenterC3#1604
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #4979C2semidirect
3. #4738C2semidirect
4. C3×#324C2semidirect
5. #4470C2semidirect
24. #324D3semidirect
68. C2×C3×C4SD16semidirect
147. C2×C4#204semidirect
170. C3#1604semidirect

Maximal subgroups

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

subgroupindex
#55102
#49792
#47382
C3×#3242
#44702
#16043