Finite Group Databasebeta

← #68457

Subgroups — #68457

Order 384.

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

normal subgroups only

This group has 250 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, complemented
2#5088C211yesmaximal, characteristic, complemented
3#4848C211yesmaximal, characteristic, complemented
4#4843C211yesmaximal, characteristic
5C3×#332C211yesmaximal, characteristic, complemented
6#4484C211yesmaximal, characteristic, complemented
7#4483C211yesmaximal, characteristic, complemented
8#1627—31nomaximal, Sylow
9C2×D3×Q8—21no—
10C3×C4⋊Q8C2211yescharacteristic
11#946—21no—
12C2×D24—21no—
13C2×#204—21no—
14#896—22no—
15#893—21no—
16#888—21no—
17#874C2211yescharacteristic
18C4×D12C2211yescharacteristic
19C4×Dic6C2211yescharacteristic
20C3×C4⋊C8C2211yescharacteristic
21C3×C4⋊C8C2211yescharacteristic
22#821—21no—
23#821—21no—
24#809—22no—
25C4×C3⋊C8C2211yescharacteristic
26D4×Q8—31no—
27#486—31no—
28#460—31no—
29#459—31no—
30#332D311yescharacteristic, complemented
31Q8⋊C8—31no—
32D4⋊C8—31no—
33C2×C3×Q8—21no—
34D3×Q8—42no—
35D3×Q8—42no—
36C2×D12—21no—
37C2×D12—21no—
38C2×D12—21no—
39C2×D12D411yescharacteristic
40C2×C4×D3—22no—
41C2×C4×D3—21no—
42C2×Dic6—22no—
43C2×Dic6D411yescharacteristic
44C2×C3×C8—21no—
45C2×C3×C8—21no—
46C3×C4⋊C4—21no—
47C3×C4⋊C4D412yes—
48C3×C42C2311yescharacteristic
49(C2×D3)⋊C4—22no—
50(C2×D3)⋊C4—21no—
51C4⋊Dic3D411yescharacteristic
52C4⋊Dic3D411yescharacteristic
53Dic3⋊C4—22no—
54Dic3⋊C4—21no—
55C4×Dic3—21no—
56C2×C3⋊C8—21no—
57D24—42no—
58#204—42no—
59C22×Q8—61no—
60C2×SD16—61no—
61C2×D8—61no—
62C4⋊Q8—61no—
63C4⋊Q8C2×D311yescharacteristic
64#127—31no—
65C22⋊Q8—61no—
66C22⋊Q8—62no—
67C4×Q8—31no—
68C4×D4—31no—
69C4×D4—61no—
70C4⋊C8C2×D311yescharacteristic
71C4⋊C8C2×D311yescharacteristic
72#102—61no—
73#102—61no—
74#102—62no—
75C4×C8—31no—
76C22×D3—21no—
77C22×D3—41no—
78C3×Q8—41no—
79C3×Q8—41no—
80C2×C3×C4—22no—
81C2×C3×C4C2×D411yescharacteristic
82C2×C3×C4C2×D411yescharacteristic
83C2×C3×C4C2×D411yescharacteristic
84C2×Dic3—21no—
85C2×Dic3—21no—
86C2×Dic3—21no—
87D12—21no—
88D12—42no—
89D12—41no—
90D12—41no—
91D12—41no—
92D12—41no—
93D12SD1612yescomplemented
94C4×D3—44no—
95C4×D3—41no—
96C4×D3—41no—
97Dic6—21no—
98Dic6—44no—
99Dic6D812yescomplemented
100C3×C8—41no—
101C3×C8—41no—
102C3⋊C8—41no—
103C2×Q8—21no—
104C2×Q8—31no—
105C2×Q8—62no—
106C2×Q8—122no—
107C2×Q8—122no—
108C2×D4—31no—
109C2×D4—61no—
110C2×D4—61no—
111C2×D4—61no—
112C22×C4—62no—
113C22×C4—61no—
114SD16—122no—
115D8—122no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—61no—
119C4⋊C4—21no—
120C4⋊C4—31no—
121C4⋊C4—31no—
122C4⋊C4—62no—
123C4⋊C4—61no—
124C4⋊C4D1212yes—
125C22⋊C4—62no—
126C22⋊C4—61no—
127C42—61no—
128C42C22×D311yescharacteristic
129C22×C3#12011yescharacteristic
130C2×D3—22no—
131C2×D3—41no—
132C2×D3—41no—
133C2×D3—81no—
134C2×D3—81no—
135C2×D3—81no—
136C3×C4—21no—
137C3×C4—42no—
138C3×C4C2×D811yescharacteristic
139C3×C4C2×SD1611yescharacteristic
140C3×C4AGL(1,ℤ/8)11yescharacteristic
141C3×C4#13711yescharacteristic
142Dic3—22no—
143Dic3—41no—
144Dic3—41no—
145C23—61no—
146C23—121no—
147Q8—32no—
148Q8—41no—
149Q8—41no—
150Q8—61no—
151Q8—124no—
152D4—32no—
153D4—61no—
154D4—122no—
155D4—121no—
156D4—121no—
157D4—121no—
158D4—121no—
159C2×C4—22no—
160C2×C4—61no—
161C2×C4—61no—
162C2×C4—61no—
163C2×C4—124no—
164C2×C4—121no—
165C2×C4—121no—
166C2×C4C2×D1211yescharacteristic
167C2×C4D3×D411yescharacteristic
168C2×C4D3×D411yescharacteristic
169C8—41no—
170C8—41no—
171C8—121no—
172C2×C3#44711yescharacteristic
173C2×C3#44811yescharacteristic
174C2×C3#45411yescharacteristic
175D3—42no—
176D3—161no—
177C22—62no—
178C22—121no—
179C22—121no—
180C22—241no—
181C22—241no—
182C22—241no—
183C22#88211yescharacteristic
184C4—21no—
185C4—42no—
186C4—62no—
187C4—121no—
188C4—121no—
189C4D3×D811yescharacteristic
190C4D3×SD1611yescharacteristic
191C4#91411yescharacteristic
192C4#91811yescharacteristic
193C3#162711yescharacteristic, complemented, Sylow
194C2—122no—
195C2—481no—
196C2#475311yescharacteristic
197C2#477311yescharacteristic
198C2#480611yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#332C2
socleC22×C3#120
prime radical#4843C2
hypercenter#332D3
hypocenterC3#1627
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
1. #5727C2semidirect
2. #5088C2semidirect
3. #4848C2semidirect
5. C3×#332C2semidirect
6. #4484C2semidirect
7. #4483C2semidirect
30. #332D3semidirect
93. D12SD16semidirect
99. Dic6D8semidirect
193. C3#1627semidirect

Maximal subgroups

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

subgroupindex
#57272
#50882
#48482
#48432
C3×#3322
#44842
#44832
#16273