Finite Group Databasebeta

← #11723708

Subgroups — #11723708

Order 864.

362 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C3×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
2C4⋊#1113C211yesmaximal, characteristic
3C3×#3755C212yesmaximal, complemented
4#88811C212yesmaximal, complemented
5#88808C211yesmaximal, characteristic, complemented
6#63940—31nomaximal
7D3×#215—32nomaximal
8C2×#1113—21no—
9C33×Q8C2211yescharacteristic, complemented
10C3×C4⋊(C3⋊D3)C2212yes—
11C3×C4×C3⋊D3—21no—
12C3×C4×C3⋊D3C2211yescharacteristic
13C3×#634—21no—
14C3×#634C2211yescharacteristic, complemented
15#6295—21no—
16C3×C3⋊(C3⋊C8)C2212yescomplemented
17Q8×C3⋊D3D311yescharacteristic, complemented
18C3×D3×Q8—32no—
19C3×D3×Q8—61no—
20#3802—31no—
21D3×D12—32no—
22#3755D312yescomplemented
23C3×#215—32no—
24C3×#215—32no—
25C3×#215—62no—
26#3717—32no—
27#3716—32no—
28#3715—32no—
29#3710—31no—
30D3×C3⋊C8—32no—
31C2×C3×C3⋊D3—22no—
32C2×C3×C3⋊D3D411yescharacteristic
33#1113—42no—
34C33×C4—21no—
35C33×C4C2311yescharacteristic
36C3×C3⋊Dic3—21no—
37C3×C3⋊Dic3D411yescharacteristic
38C2×#215—91no—
39D3×SD16—92no—
40C2×D32—62no—
41C2×D32—61no—
42C32×Q8—21no—
43C32×Q8—22no—
44C32×Q8—41no—
45C32×Q8C2×D311yescharacteristic, complemented
46C32×Q8C2×D312yescomplemented
47C4⋊(C3⋊D3)C2×D312yes—
48C4×C3⋊D3—21no—
49C4×C3⋊D3C2×D311yescharacteristic
50#634—21no—
51#634C2×D311yescharacteristic, complemented
52C3×D12—32no—
53C3×D12—32no—
54C3×D12—62no—
55C3×C4×D3—32no—
56C3×C4×D3—61no—
57C3×C4×D3—62no—
58C3×C4×D3—62no—
59C3×Dic6—32no—
60C3×Dic6—61no—
61C3×Dic6—62no—
62C3×Dic6—62no—
63#626—62no—
64#625—61no—
65C3⋊(C3⋊C8)C2×D312yescomplemented
66C3×C3⋊C8—32no—
67C3×C3⋊C8—32no—
68C3×C3⋊C8—62no—
69C2×C33C2×D411yescharacteristic
70C3×C3⋊D3—42no—
71C3×C3⋊D3SD1612yescomplemented
72C2×C3×Q8—91no—
73D3×Q8—32no—
74D3×Q8—61no—
75D3×D4—92no—
76C2×D12—91no—
77C3×SD16—92no—
78#215—32no—
79#215—32no—
80#215—62no—
81#215—92no—
82#214—92no—
83C2×C3⋊C8—91no—
84#204—92no—
85D3×C8—92no—
86C2×C3⋊D3—22no—
87C2×C3⋊D3#6711yescharacteristic
88C2×C3×D3—32no—
89C2×C3×D3—61no—
90C2×C3×D3—62no—
91C2×C3×D3—62no—
92C2×C3×D3—122no—
93D32—124no—
94D32—122no—
95C32×C4—21no—
96C32×C4—22no—
97C32×C4—22no—
98C32×C4—22no—
99C32×C4—21no—
100C32×C4—41no—
101C32×C4—42no—
102C32×C4—42no—
103C32×C4C22×D312yes—
104C32×C4C22×D311yescharacteristic
105C3⋊Dic3—21no—
106C3⋊Dic3#6711yescharacteristic
107C3×Dic3—32no—
108C3×Dic3—61no—
109C3×Dic3—62no—
110C3×Dic3—62no—
111C2×SD16—271noSylow
112C33C2×SD1611yescharacteristic, complemented, Sylow
113C22×D3—182no—
114C22×D3—181no—
115C3×Q8—21no—
116C3×Q8—22no—
117C3×Q8—41no—
118C3×Q8—91no—
119C3×Q8—181no—
120C3×Q8D3211yescharacteristic, complemented
121C3×Q8D3212yescomplemented
122C3×D4—92no—
123C2×C3×C4—91no—
124C2×C3×C4—181no—
125#67—182no—
126D12—32no—
127D12—32no—
128D12—62no—
129D12—181no—
130C4×D3—32no—
131C4×D3—61no—
132C4×D3—62no—
133C4×D3—62no—
134Dic6—32no—
135Dic6—61no—
136Dic6—62no—
137Dic6—62no—
138C3×C8—92no—
139C3⋊C8—32no—
140C3⋊C8—32no—
141C3⋊C8—62no—
142C2×C32—22no—
143C2×C32—21no—
144C2×C32—41no—
145C2×C32D3×D412yes—
146C2×C32C2×#6711yescharacteristic
147C3⋊D3—42no—
148C3⋊D3#21512yescomplemented
149C3×D3—34no—
150C3×D3—62no—
151C3×D3—122no—
152C3×D3—122no—
153C3×D3—242no—
154C2×Q8—91no—
155C2×D4—271no—
156SD16—92no—
157SD16—272no—
158C2×C8—271no—
159C22×C3—91no—
160C22×C3—182no—
161C2×D3—32no—
162C2×D3—61no—
163C2×D3—62no—
164C2×D3—62no—
165C2×D3—122no—
166C2×D3—364no—
167C2×D3—362no—
168C3×C4—22no—
169C3×C4—22no—
170C3×C4—21no—
171C3×C4—22no—
172C3×C4—21no—
173C3×C4—42no—
174C3×C4—42no—
175C3×C4—41no—
176C3×C4—91no—
177C3×C4—181no—
178C3×C4C2×D3212yes—
179C3×C4C2×D3211yescharacteristic
180Dic3—32no—
181Dic3—61no—
182Dic3—62no—
183Dic3—62no—
184C32—22no—
185C32—21no—
186C32—41no—
187C32D3×SD1612yescomplemented
188C32C2×#21511yescharacteristic, complemented
189C23—541no—
190Q8—91no—
191Q8—181no—
192Q8#111311yescharacteristic, complemented
193D4—92no—
194D4—541no—
195C2×C4—91no—
196C2×C4—181no—
197C8—92no—
198C2×C3—21no—
199C2×C3—22no—
200C2×C3—41no—
201C2×C3—92no—
202C2×C3—362no—
203C2×C3D3×#6712yes—
204C2×C3#381111yescharacteristic
205D3—34no—
206D3—62no—
207D3—122no—
208D3—122no—
209D3—242no—
210C22—91no—
211C22—182no—
212C22—1082no—
213C4—21no—
214C4—91no—
215C4—181no—
216C4C2×#111311yescharacteristic
217C3—21no—
218C3—22no—
219C3—41no—
220C3D3×#21512yescomplemented
221C3#6394011yescharacteristic, complemented
222C2—92no—
223C2—362no—
224C2#8904911yescharacteristic
subgroupG/H
centerC2#89049
commutator, cosocleC33×C4C23
FrattiniC4C2×#1113
FittingC33×Q8C22
socleC2×C33C2×D4
prime radicalC4⋊#1113C2
hypercenterQ8#1113
hypocenterC33C2×SD16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×C3⋊D3C2semidirect
3. C3×#3755C2semidirect
4. #88811C2semidirect
5. #88808C2semidirect
9. C33×Q8C22semidirect
14. C3×#634C22semidirect
16. C3×C3⋊(C3⋊C8)C22semidirect
17. Q8×C3⋊D3D3semidirect
22. #3755D3semidirect
45. C32×Q8C2×D3semidirect
46. C32×Q8C2×D3semidirect
51. #634C2×D3semidirect
65. C3⋊(C3⋊C8)C2×D3semidirect
71. C3×C3⋊D3SD16semidirect
112. C33C2×SD16semidirect
120. C3×Q8D32semidirect
121. C3×Q8D32semidirect
148. C3⋊D3#215semidirect
187. C32D3×SD16semidirect
188. C32C2×#215semidirect
192. Q8#1113semidirect
220. C3D3×#215semidirect
221. C3#63940semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C3⋊D32
C4⋊#11132
C3×#37552
#888112
#888082
#639403
D3×#2153