Finite Group Databasebeta

← #68461

Subgroups — #68461

Order 384.

234 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 57 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#5727C211yesmaximal, characteristic
2#5078C211yesmaximal, characteristic, complemented
3#4849C211yesmaximal, characteristic, complemented
4#4846C211yesmaximal, characteristic
5C3×#332C211yesmaximal, characteristic, complemented
6#4484C211yesmaximal, characteristic, complemented
7#4483C211yesmaximal, characteristic
8#1649—31nomaximal, Sylow
9C2×D3×Q8—21no—
10C3×C4⋊Q8C2211yescharacteristic
11#946—21no—
12C2×Dic12—21no—
13#896—22no—
14#893—21no—
15#888—21no—
16C4×D12C2211yescharacteristic
17#869C2211yescharacteristic
18C4×Dic6C2211yescharacteristic
19C3×C4⋊C8C2211yescharacteristic
20C3×C4⋊C8C2211yescharacteristic
21#821—21no—
22#818—21no—
23#816—21no—
24#807—22no—
25C4×C3⋊C8C2211yescharacteristic
26D4×Q8—31no—
27#500—31no—
28#474—31no—
29#462—31no—
30#332D311yescharacteristic, complemented
31Q8⋊C8—31no—
32D4⋊C8—31no—
33C2×C3×Q8—21no—
34D3×Q8—42no—
35D3×Q8—42no—
36C2×D12D411yescharacteristic
37C2×C4×D3—22no—
38C2×C4×D3—21no—
39C2×Dic6—22no—
40C2×Dic6—21no—
41C2×Dic6D411yescharacteristic
42C2×C3×C8—21no—
43C2×C3×C8—21no—
44C3×C4⋊C4—21no—
45C3×C4⋊C4D412yes—
46C3×C42C2311yescharacteristic
47(C2×D3)⋊C4—22no—
48(C2×D3)⋊C4—21no—
49C4⋊Dic3—21no—
50C4⋊Dic3—21no—
51C4⋊Dic3D411yescharacteristic
52C4⋊Dic3D411yescharacteristic
53Dic3⋊C4—22no—
54Dic3⋊C4—21no—
55C4×Dic3—21no—
56C2×C3⋊C8—21no—
57Dic12—42no—
58C22×Q8—61no—
59C2×Q16—61no—
60C4⋊Q8—31no—
61C4⋊Q8—61no—
62C4⋊Q8C2×D311yescharacteristic
63C22⋊Q8—61no—
64C22⋊Q8—62no—
65C4×Q8—31no—
66C4×D4—31no—
67C4×D4—61no—
68#107—61no—
69#107—62no—
70C4⋊C8C2×D311yescharacteristic
71C4⋊C8C2×D311yescharacteristic
72#103—61no—
73#102—61no—
74C4×C8—31no—
75C22×D3—21no—
76C3×Q8—41no—
77C3×Q8—41no—
78C2×C3×C4—22no—
79C2×C3×C4C2×D411yescharacteristic
80C2×C3×C4C2×D411yescharacteristic
81C2×C3×C4C2×D411yescharacteristic
82C2×Dic3—21no—
83C2×Dic3—21no—
84C2×Dic3—21no—
85C2×Dic3—41no—
86D12—21no—
87D12Q1612yescomplemented
88C4×D3—44no—
89C4×D3—41no—
90C4×D3—41no—
91Dic6—21no—
92Dic6—44no—
93Dic6—41no—
94Dic6—41no—
95Dic6D812yes—
96C3×C8—41no—
97C3×C8—41no—
98C3⋊C8—22no—
99C2×Q8—21no—
100C2×Q8—31no—
101C2×Q8—62no—
102C2×Q8—61no—
103C2×Q8—122no—
104C2×Q8—122no—
105C2×D4—31no—
106C22×C4—62no—
107C22×C4—61no—
108Q16—122no—
109C2×C8—21no—
110C2×C8—21no—
111C2×C8—61no—
112C4⋊C4—21no—
113C4⋊C4—31no—
114C4⋊C4—31no—
115C4⋊C4—62no—
116C4⋊C4—61no—
117C4⋊C4—61no—
118C4⋊C4—61no—
119C4⋊C4D1212yes—
120C22⋊C4—62no—
121C22⋊C4—61no—
122C42—61no—
123C42C22×D311yescharacteristic
124C22×C3#12011yescharacteristic
125C2×D3—22no—
126C2×D3—41no—
127C2×D3—41no—
128C3×C4—21no—
129C3×C4—42no—
130C3×C4C2×D811yescharacteristic
131C3×C4C2×Q1611yescharacteristic
132C3×C4AGL(1,ℤ/8)11yescharacteristic
133C3×C4#13711yescharacteristic
134Dic3—22no—
135Dic3—41no—
136Dic3—41no—
137Dic3—81no—
138C23—61no—
139Q8—32no—
140Q8—41no—
141Q8—41no—
142Q8—61no—
143Q8—124no—
144Q8—121no—
145Q8—121no—
146D4—32no—
147D4—61no—
148C2×C4—22no—
149C2×C4—61no—
150C2×C4—61no—
151C2×C4—61no—
152C2×C4—124no—
153C2×C4—121no—
154C2×C4—121no—
155C2×C4—121no—
156C2×C4C2×D1211yescharacteristic
157C2×C4D3×D411yescharacteristic
158C2×C4D3×D411yescharacteristic
159C8—41no—
160C8—41no—
161C8—62no—
162C2×C3#44711yescharacteristic
163C2×C3#45111yescharacteristic
164C2×C3#45611yescharacteristic
165D3—42no—
166C22—62no—
167C22—121no—
168C22—121no—
169C22#88211yescharacteristic
170C4—21no—
171C4—42no—
172C4—62no—
173C4—121no—
174C4—121no—
175C4—241no—
176C4D3×D811yescharacteristic
177C4#91411yescharacteristic
178C4#91511yescharacteristic
179C4D3×Q1611yescharacteristic
180C3#164911yescharacteristic, complemented, Sylow
181C2—122no—
182C2#475211yescharacteristic
183C2#477311yescharacteristic
184C2#480911yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#332C2
socleC22×C3#120
prime radicalC2×D12D4
hypercenter#332D3
hypocenterC3#1649
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #5078C2semidirect
3. #4849C2semidirect
5. C3×#332C2semidirect
6. #4484C2semidirect
30. #332D3semidirect
87. D12Q16semidirect
180. C3#1649semidirect

Maximal subgroups

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

subgroupindex
#57272
#50782
#48492
#48462
C3×#3322
#44842
#44832
#16493