Finite Group Databasebeta

← #68492

Subgroups — #68492

Order 384.

214 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 55 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#5730C211yesmaximal, characteristic
2#5092C211yesmaximal, characteristic, complemented
3#4850C211yesmaximal, characteristic, complemented
4#4842C211yesmaximal, characteristic
5C3×#333C211yesmaximal, characteristic, complemented
6#4484C211yesmaximal, characteristic, complemented
7#4483C211yesmaximal, characteristic
8#1632—31nomaximal, Sylow
9C3×C4⋊Q8C2211yescharacteristic
10#946—22no—
11C2×#216—22no—
12#897—21no—
13#893—21no—
14D3×C4⋊C4—21no—
15#890—21no—
16C4×D12C2211yescharacteristic
17#869C2211yescharacteristic
18C4×Dic6C2211yescharacteristic
19C3×C4⋊C8C2211yescharacteristic
20C3×C4⋊C8C2211yescharacteristic
21#821—21no—
22#818—21no—
23#817—21no—
24#816—21no—
25C4×C3⋊C8C2211yescharacteristic
26#554—31no—
27#494—31no—
28#477—31no—
29#476—31no—
30#333D311yescharacteristic, complemented
31Q8⋊C8—31no—
32D4⋊C8—31no—
33C2×C3×Q8D412yes—
34C2×D12D411yescharacteristic
35C2×C4×D3—21no—
36C2×C4×D3—22no—
37C2×Dic6—21no—
38C2×Dic6D411yescharacteristic
39C2×C3×C8—21no—
40C2×C3×C8—21no—
41C3×C4⋊C4—21no—
42C3×C4⋊C4—21no—
43C3×C42C2311yescharacteristic
44#216—44no—
45(C2×D3)⋊C4—22no—
46(C2×D3)⋊C4—21no—
47C4⋊Dic3—22no—
48C4⋊Dic3—21no—
49C4⋊Dic3—21no—
50C4⋊Dic3D411yescharacteristic
51C4⋊Dic3D411yescharacteristic
52Dic3⋊C4—22no—
53Dic3⋊C4—21no—
54C4×Dic3—21no—
55C2×C3⋊C8—21no—
56C2×Q16—62no—
57C4⋊Q8—31no—
58C4⋊Q8C2×D311yescharacteristic
59#125—61no—
60C22⋊Q8—61no—
61C22⋊Q8—62no—
62C4×Q8—31no—
63C4×D4—31no—
64C4×D4—61no—
65C2×C4⋊C4—61no—
66#107—61no—
67#106—61no—
68C4⋊C8C2×D311yescharacteristic
69C4⋊C8C2×D311yescharacteristic
70#103—61no—
71#102—61no—
72C4×C8—31no—
73C22×D3—21no—
74C3×Q8—42no—
75C2×C3×C4—22no—
76C2×C3×C4C2×D411yescharacteristic
77C2×C3×C4C2×D411yescharacteristic
78C2×C3×C4C2×D411yescharacteristic
79C2×Dic3—21no—
80C2×Dic3—21no—
81C2×Dic3—21no—
82C2×Dic3—41no—
83D12—21no—
84D12Q1612yescomplemented
85C4×D3—41no—
86C4×D3—41no—
87C4×D3—82no—
88Dic6—21no—
89Dic6—21no—
90Dic6—42no—
91C3×C8—41no—
92C3×C8—41no—
93C3⋊C8—22no—
94C2×Q8—31no—
95C2×Q8—61no—
96C2×Q8D1212yes—
97C2×D4—31no—
98C22×C4—62no—
99C22×C4—61no—
100Q16—124no—
101C2×C8—21no—
102C2×C8—21no—
103C2×C8—61no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—31no—
107C4⋊C4—31no—
108C4⋊C4—62no—
109C4⋊C4—62no—
110C4⋊C4—61no—
111C4⋊C4—61no—
112C4⋊C4—61no—
113C22⋊C4—62no—
114C22⋊C4—61no—
115C42—61no—
116C42C22×D311yescharacteristic
117C22×C3#12011yescharacteristic
118C2×D3—22no—
119C2×D3—41no—
120C2×D3—41no—
121C3×C4—21no—
122C3×C4—42no—
123C3×C4C2×Q1611yescharacteristic
124C3×C4#13511yescharacteristic
125C3×C4#13711yescharacteristic
126C3×C4#13711yescharacteristic
127Dic3—41no—
128Dic3—41no—
129Dic3—41no—
130Dic3—81no—
131C23—61no—
132Q8—42no—
133Q8—61no—
134Q8—61no—
135Q8—122no—
136D4—32no—
137D4—61no—
138C2×C4—22no—
139C2×C4—61no—
140C2×C4—61no—
141C2×C4—61no—
142C2×C4—121no—
143C2×C4—121no—
144C2×C4—121no—
145C2×C4—242no—
146C2×C4C2×D1211yescharacteristic
147C2×C4D3×D411yescharacteristic
148C2×C4D3×D411yescharacteristic
149C8—41no—
150C8—41no—
151C8—62no—
152C2×C3#45111yescharacteristic
153C2×C3#45211yescharacteristic
154C2×C3#45311yescharacteristic
155D3—42no—
156C22—62no—
157C22—121no—
158C22—121no—
159C22#88211yescharacteristic
160C4—21no—
161C4—42no—
162C4—121no—
163C4—121no—
164C4—121no—
165C4—241no—
166C4#91511yescharacteristic
167C4#91611yescharacteristic
168C4D3×Q1611yescharacteristic
169C4#91811yescharacteristic
170C3#163211yescharacteristic, complemented, Sylow
171C2—122no—
172C2#474711yescharacteristic
173C2#480811yescharacteristic
174C2#480911yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#333C2
socleC22×C3#120
prime radicalC2×D12D4
hypercenter#333D3
hypocenterC3#1632
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #5092C2semidirect
3. #4850C2semidirect
5. C3×#333C2semidirect
6. #4484C2semidirect
30. #333D3semidirect
84. D12Q16semidirect
170. C3#1632semidirect

Maximal subgroups

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

subgroupindex
#57302
#50922
#48502
#48422
C3×#3332
#44842
#44832
#16323