Finite Group Databasebeta

← #11724143

Subgroups — #11724143

Order 864.

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

normal subgroups only

This group has 226 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#88964C211yesmaximal, characteristic
2D3×#627C211yesmaximal, characteristic, complemented
3#88942C211yesmaximal, characteristic, complemented
4#88938C211yesmaximal, characteristic, complemented
5#88936C211yesmaximal, characteristic, complemented
6C3×#3776C211yesmaximal, characteristic, complemented
7#88925C211yesmaximal, characteristic, complemented
8#64239—31nomaximal
9#6296C2211yescharacteristic
10#6295C2211yescharacteristic
11#6293—21no—
12Dic3×C3⋊D3—21no—
13D3×C3⋊Dic3C2211yescharacteristic
14C3×#627C2211yescharacteristic
15C3×#627C2211yescharacteristic
16C3×#626—21no—
17C3×D3×Dic3—21no—
18C3⋊#622C2211yescharacteristic
19C3×#622C2211yescharacteristic
20C2×#627—31no—
21#3805—31no—
22#3796—61no—
23D3×Dic6—61no—
24#3792—31no—
25#3776—31no—
26#3776D311yescharacteristic, complemented
27#3775—31no—
28#3775—31no—
29C2×C3×C3⋊D3—21no—
30C2×C32×D3D411yescharacteristic
31C3⋊(C3⋊Dic3)D411yescharacteristic
32C3×C3⋊Dic3—21no—
33C3×C3⋊Dic3C2311yescharacteristic
34C32×Dic3—21no—
35C32×Dic3—21no—
36#918—91nomaximal
37C2×C3⋊Dic3—31no—
38C4×C3⋊D3—61no—
39#634—61no—
40C3×#67—61no—
41C2×C3×Dic3—61no—
42C3×D12—61no—
43C3×C4×D3—61no—
44C3×Dic6—61no—
45C3×Dic6—61no—
46#627—31no—
47#627—61no—
48#627—61no—
49#627—61no—
50#627—121no—
51#627C2×D311yescharacteristic
52#627C2×D311yescharacteristic
53#626—21no—
54#626—61no—
55#625—31no—
56D3×Dic3—21no—
57D3×Dic3—61no—
58D3×Dic3—61no—
59D3×Dic3—61no—
60D3×Dic3—61no—
61D3×Dic3—121no—
62#622—31no—
63#622C2×D311yescharacteristic
64C2×C33C2×D411yescharacteristic
65C3×C3⋊D3—41no—
66C32×D3—21no—
67#239—91no—
68D3×Q8—91no—
69#237—181no—
70C2×Dic6—181no—
71C3×Q16—91no—
72#216—91no—
73#215—91no—
74#204—91no—
75ΓL(1,𝔽25)—91no—
76C22×C32—31no—
77C2×C3⋊D3—21no—
78C2×C3×D3—21no—
79C2×C3×D3—21no—
80C2×C3×D3—61no—
81C2×C3×D3—61no—
82C2×C3×D3—121no—
83C32×C4—61no—
84C32×C4—61no—
85C3⋊Dic3—21no—
86C3⋊Dic3—31no—
87C3⋊Dic3—61no—
88C3⋊Dic3—61no—
89C3⋊Dic3—121no—
90C3⋊Dic3—121no—
91C3⋊Dic3C22×D311yescharacteristic
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×D4—181no—
108C2×C3×C4—181no—
109#67—61no—
110#67—181no—
111C2×Dic3—61no—
112C2×Dic3—181no—
113C2×Dic3—181no—
114C2×Dic3—181no—
115D12—61no—
116D12—91no—
117C4×D3—61no—
118C4×D3—91no—
119C4×D3—181no—
120C4×D3—181no—
121C4×D3—361no—
122Dic6—61no—
123Dic6—61no—
124Dic6—91no—
125Dic6—181no—
126Dic6—181no—
127Dic6—181no—
128Dic6—181no—
129Dic6—361no—
130C3×C8—91no—
131C3⋊C8—91no—
132C2×C32—21no—
133C2×C32—21no—
134C2×C32—41no—
135C2×C32—41no—
136C2×C32—61no—
137C2×C32D3×D411yescharacteristic
138C3⋊D3—41no—
139C3×D3—22no—
140C3×D3—41no—
141C3×D3—121no—
142C3×D3—122no—
143C3×D3—121no—
144#41—271no—
145C2×Q8—271no—
146Q16—91no—
147Q16—271no—
148SD16—271no—
149SD16—271no—
150M16—271no—
151C22×C3—61no—
152C22×C3—61no—
153C22×C3—181no—
154C2×D3—61no—
155C2×D3—61no—
156C2×D3—121no—
157C2×D3GO+(4,𝔽2)11yescharacteristic
158C3×C4—61no—
159C3×C4—61no—
160C3×C4—61no—
161C3×C4—61no—
162C3×C4—91no—
163C3×C4—121no—
164C3×C4—121no—
165C3×C4—181no—
166Dic3—21no—
167Dic3—21no—
168Dic3—61no—
169Dic3—61no—
170Dic3—61no—
171Dic3—61no—
172Dic3—91no—
173Dic3—121no—
174Dic3—181no—
175Dic3—181no—
176Dic3—361no—
177Dic3—361no—
178C32—21no—
179C32—21no—
180C32—41no—
181C32—41no—
182C32#91811yescharacteristic, complemented
183Q8—91no—
184Q8—91no—
185Q8—271no—
186Q8—541no—
187D4—181no—
188D4—271no—
189C2×C4—181no—
190C2×C4—271no—
191C2×C4—541no—
192C8—91no—
193C8—271no—
194C2×C3—21no—
195C2×C3—21no—
196C2×C3—41no—
197C2×C3—41no—
198C2×C3—62no—
199C2×C3—121no—
200C2×C3—361no—
201C2×C3C2×GO+(4,𝔽2)11yescharacteristic
202D3—21no—
203D3—121no—
204D3—122no—
205D3—121no—
206C22—31no—
207C22—181no—
208C4—61no—
209C4—61no—
210C4—91no—
211C4—181no—
212C4—271no—
213C3—21no—
214C3—21no—
215C3—41no—
216C3—41no—
217C3#6423911yescharacteristic, complemented
218C2—61no—
219C2—361no—
220C2D3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×GO+(4,𝔽2)
commutator, cosocleC3×C3⋊Dic3C23
Fitting, socleC2×C33C2×D4
prime radical#88964C2
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. D3×#627C2semidirect
3. #88942C2semidirect
4. #88938C2semidirect
5. #88936C2semidirect
6. C3×#3776C2semidirect
7. #88925C2semidirect
26. #3776D3semidirect
104. C33#137semidirect
182. C32#918semidirect
217. C3#64239semidirect

Maximal subgroups

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

subgroupindex
#889642
D3×#6272
#889422
#889382
#889362
C3×#37762
#889252
#642393
#9189