Finite Group Databasebeta

← #11724145

Subgroups — #11724145

Order 864.

230 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 32 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#88965C211yesmaximal, characteristic
2#88958C211yesmaximal, characteristic, complemented
3#88942C211yesmaximal, characteristic, complemented
4#88938C211yesmaximal, characteristic, complemented
5#88936C211yesmaximal, characteristic, complemented
6C3×#3776C211yesmaximal, characteristic, complemented
7#88926C211yesmaximal, characteristic, complemented
8#64226—31nomaximal
9#6296C2211yescharacteristic
10#6295C2211yescharacteristic
11#6292—21no—
12#6289—21no—
13#6289C2211yescharacteristic
14C3×#627—21no—
15C3×#627C2211yescharacteristic
16C3×#627C2211yescharacteristic
17C3×#624—21no—
18C3⋊#622C2211yescharacteristic
19C3×#622C2211yescharacteristic
20#3799—61no—
21#3798—31no—
22#3797—31no—
23#3797—61no—
24#3788—31no—
25#3776—31no—
26#3776D311yescharacteristic, complemented
27#3775—31no—
28#3775—31no—
29C2×C3⋊(C3⋊D3)D411yescharacteristic
30C2×C3×C3⋊D3—21no—
31C3×C3⋊Dic3—21no—
32C3×C3⋊Dic3C2311yescharacteristic
33C32×Dic3—21no—
34C32×Dic3—21no—
35C32×Dic3D411yescharacteristic
36#918—91nomaximal
37C4⋊(C3⋊D3)—61no—
38C4×C3⋊D3—31no—
39C4×C3⋊D3—61no—
40C3×C4×D3—61no—
41C3×C4×D3—61no—
42C3×Dic6—61no—
43C3×Dic6—61no—
44C3×Dic6—61no—
45C3×Dic6—61no—
46#627—21no—
47#627—31no—
48#627—61no—
49#627C2×D311yescharacteristic
50#627C2×D311yescharacteristic
51#626—61no—
52#626—61no—
53#626—61no—
54#626—121no—
55#625—31no—
56#624—21no—
57#624—61no—
58#624—61no—
59#624—61no—
60#624—61no—
61#624—121no—
62#622—31no—
63#622C2×D311yescharacteristic
64C2×C33C2×D411yescharacteristic
65C3⋊(C3⋊D3)—21no—
66C3×C3⋊D3—41no—
67#239—91no—
68D3×Q8—91no—
69D3×Q8—181no—
70#235—181no—
71C3×Q16—91no—
72#216—91no—
73#215—91no—
74#204—91no—
75ΓL(1,𝔽25)—91no—
76C2×C3⋊D3—21no—
77C2×C3⋊D3—31no—
78C2×C3⋊D3—61no—
79C2×C3⋊D3—61no—
80C2×C3⋊D3—121no—
81C2×C3⋊D3—121no—
82C2×C3×D3—61no—
83C2×C3×D3—61no—
84C2×C3×D3—121no—
85C32×C4—31no—
86C32×C4—61no—
87C32×C4—61no—
88C3⋊Dic3—21no—
89C3⋊Dic3C22×D311yescharacteristic
90C3×Dic3—21no—
91C3×Dic3—21no—
92C3×Dic3—21no—
93C3×Dic3—21no—
94C3×Dic3—21no—
95C3×Dic3—21no—
96C3×Dic3—41no—
97C3×Dic3—41no—
98C3×Dic3—61no—
99C3×Dic3—61no—
100C3×Dic3—61no—
101C3×Dic3—61no—
102C3×Dic3—121no—
103#137—271noSylow
104C33#13711yescharacteristic, complemented, Sylow
105C3×Q8—91no—
106C3×Q8—91no—
107C3×Q8—181no—
108C2×C3×C4—181no—
109#67—181no—
110#67—181no—
111D12—91no—
112D12—181no—
113D12—181no—
114D12—361no—
115C4×D3—61no—
116C4×D3—61no—
117C4×D3—91no—
118C4×D3—181no—
119C4×D3—181no—
120C4×D3—181no—
121C4×D3—181no—
122C4×D3—181no—
123C4×D3—361no—
124Dic6—61no—
125Dic6—61no—
126Dic6—61no—
127Dic6—61no—
128Dic6—91no—
129Dic6—181no—
130C3×C8—91no—
131C3⋊C8—91no—
132C2×C32—21no—
133C2×C32—21no—
134C2×C32—41no—
135C2×C32—41no—
136C2×C32D3×D411yescharacteristic
137C3⋊D3—41no—
138C3⋊D3—62no—
139C3⋊D3—61no—
140C3⋊D3—121no—
141C3⋊D3—122no—
142C3⋊D3—241no—
143C3×D3—121no—
144C3×D3—121no—
145C3×D3—241no—
146#41—271no—
147C2×Q8—271no—
148Q16—91no—
149Q16—271no—
150SD16—271no—
151SD16—271no—
152M16—271no—
153C22×C3—181no—
154C2×D3—61no—
155C2×D3—61no—
156C2×D3—91no—
157C2×D3—121no—
158C2×D3—181no—
159C2×D3—181no—
160C2×D3—361no—
161C2×D3—361no—
162C3×C4—61no—
163C3×C4—61no—
164C3×C4—61no—
165C3×C4—61no—
166C3×C4—61no—
167C3×C4—61no—
168C3×C4—91no—
169C3×C4—121no—
170C3×C4—121no—
171C3×C4—181no—
172Dic3—21no—
173Dic3—21no—
174Dic3—61no—
175Dic3—61no—
176Dic3—61no—
177Dic3—61no—
178Dic3—121no—
179Dic3GO+(4,𝔽2)11yescharacteristic
180C32—21no—
181C32—21no—
182C32—41no—
183C32—41no—
184C32#91811yescharacteristic, complemented
185Q8—91no—
186Q8—91no—
187Q8—181no—
188Q8—271no—
189D4—271no—
190D4—541no—
191C2×C4—181no—
192C2×C4—271no—
193C2×C4—541no—
194C8—91no—
195C8—271no—
196C2×C3—21no—
197C2×C3—21no—
198C2×C3—41no—
199C2×C3—41no—
200C2×C3—361no—
201C2×C3C2×GO+(4,𝔽2)11yescharacteristic
202D3—121no—
203D3—121no—
204D3—182no—
205D3—181no—
206D3—241no—
207D3—361no—
208D3—362no—
209D3—721no—
210C22—181no—
211C22—271no—
212C4—31no—
213C4—61no—
214C4—61no—
215C4—91no—
216C4—181no—
217C3—21no—
218C3—21no—
219C3—41no—
220C3—41no—
221C3#6422611yescharacteristic, complemented
222C2—361no—
223C2—541no—
224C2D3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×GO+(4,𝔽2)
commutator, cosocleC3×C3⋊Dic3C23
Fitting, socleC2×C33C2×D4
prime radical#88965C2
hypocenterC33#137
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. #88958C2semidirect
3. #88942C2semidirect
4. #88938C2semidirect
5. #88936C2semidirect
6. C3×#3776C2semidirect
7. #88926C2semidirect
26. #3776D3semidirect
104. C33#137semidirect
184. C32#918semidirect
221. C3#64226semidirect

Maximal subgroups

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

subgroupindex
#889652
#889582
#889422
#889382
#889362
C3×#37762
#889262
#642263
#9189