Finite Group Databasebeta

← #11721267

Subgroups — #11721267

Order 864.

276 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C22×C3⋊Dic9C211yesmaximal, characteristic
2C2×#6189C211yesmaximal, characteristic, complemented
3C9×#217C211yesmaximal, characteristic
4C3×#3676C211yesmaximal, characteristic
5Dic3⋊Dic9C211yesmaximal, characteristic, complemented
6#88442C211yesmaximal, characteristic, complemented
7Dic9⋊Dic3C211yesmaximal, characteristic, complemented
8#63982—31nomaximal
9#63440—31nomaximal
10C23×C3×C9C2211yescharacteristic
11C2×C3⋊Dic9—22no—
12C2×C3⋊Dic9C2211yescharacteristic
13C2×C3⋊Dic9C2211yescharacteristic
14C2×C9×Dic3C2211yescharacteristic
15C2×C9×Dic3C2211yescharacteristic
16C2×C3×Dic9C2211yescharacteristic
17C2×C3×Dic9C2211yescharacteristic
18#6189—22no—
19C22×C3⋊Dic3—31no—
20C2×#627—31no—
21C3×#217—31no—
22C3×#217D311yescharacteristic
23#3724—31no—
24Dic3⋊Dic3—31no—
25Dic3⋊Dic3—31no—
26C22×Dic9—31no—
27C22×Dic9—61no—
28C2×Dic18—31no—
29C9×C22⋊C4—31no—
30#3676D311yescharacteristic
31C4⋊Dic9—31no—
32Dic9⋊C4—31no—
33Dic9⋊C4—31no—
34C22×C3×C9—21no—
35C22×C3×C9—21no—
36C22×C3×C9Q812yescomplemented
37C22×C3×C9C2311yescharacteristic
38C3⋊Dic9—21no—
39C3⋊Dic9D412yes—
40C9×Dic3—21no—
41C9×Dic3—21no—
42C3×Dic9—21no—
43C3×Dic9—21no—
44#878—91no—
45#878—91no—
46C23×C32C2×D311yescharacteristic
47C2×C3⋊Dic3—31no—
48C2×C3⋊Dic3—31no—
49C2×C3⋊Dic3—62no—
50C2×C3×Dic3—31no—
51C2×C3×Dic3—31no—
52C2×C3×Dic3C2×D311yescharacteristic
53C2×C3×Dic3C2×D311yescharacteristic
54#627—62no—
55C23×C9—21no—
56C23×C9C2×D311yescharacteristic
57C2×C4×C9—31no—
58C2×C4×C9—31no—
59C2×Dic9—31no—
60C2×Dic9—31no—
61C2×Dic9—61no—
62C2×Dic9—61no—
63C2×Dic9—62no—
64C2×Dic9—64no—
65C2×Dic9C2×D311yescharacteristic
66C2×Dic9C2×D311yescharacteristic
67Dic18—62no—
68C2×C3×C9—22no—
69C2×C3×C9C2×D411yescharacteristic
70C2×C3×C9C2×Q811yescharacteristic
71C2×C3×C9#4111yescharacteristic
72C22×Dic3—91no—
73C22×Dic3—91no—
74C22×Dic3—181no—
75C2×Dic6—91no—
76C2×Dic6—91no—
77C3×C22⋊C4—31no—
78C3×C22⋊C4—91no—
79#217—31no—
80#217D911yescharacteristic
81C4⋊Dic3—91no—
82C4⋊Dic3—91no—
83Dic3⋊C4—91no—
84Dic3⋊C4—91no—
85Dic3⋊C4—91no—
86Dic3⋊C4—91no—
87C22×C32—21no—
88C22×C32—21no—
89C22×C32Dic612yes—
90C22×C32C22×D311yescharacteristic
91C3⋊Dic3—32no—
92C3⋊Dic3—61no—
93C3×Dic3—21no—
94C3×Dic3—21no—
95C3×Dic3—61no—
96C3×Dic3—61no—
97C22×C9—22no—
98C22×C9—22no—
99C22×C9—21no—
100C22×C9—21no—
101C22×C9—22no—
102C22×C9—21no—
103C22×C9Dic612yescomplemented
104C22×C9C22×D311yescharacteristic
105C4×C9—61no—
106C4×C9—61no—
107Dic9—21no—
108Dic9—21no—
109Dic9—32no—
110Dic9—62no—
111Dic9—62no—
112Dic9—61no—
113C22⋊Q8—271noSylow
114C3×C9C22⋊Q811yescharacteristic, complemented, Sylow
115C23×C3—21no—
116C23×C3C2×D911yescharacteristic
117C23×C3D3211yescharacteristic
118C2×C3×C4—31no—
119C2×C3×C4—31no—
120C2×C3×C4—91no—
121C2×C3×C4—91no—
122C2×Dic3—31no—
123C2×Dic3—31no—
124C2×Dic3—91no—
125C2×Dic3—91no—
126C2×Dic3—91no—
127C2×Dic3—91no—
128C2×Dic3—181no—
129C2×Dic3—181no—
130C2×Dic3—182no—
131C2×Dic3—182no—
132C2×Dic3—184no—
133C2×Dic3C2×D911yescharacteristic
134C2×Dic3C2×D911yescharacteristic
135Dic6—182no—
136Dic6—182no—
137C2×C32—22no—
138C2×C32C2×Dic611yescharacteristic
139C2×C32D3×D411yescharacteristic
140C2×C32#23711yescharacteristic
141C2×C9—24no—
142C2×C9—22no—
143C2×C9—21no—
144C2×C9—21no—
145C2×C9—21no—
146C2×C9C2×Dic611yescharacteristic
147C2×C9D3×D411yescharacteristic
148C2×C9#23711yescharacteristic
149C2×Q8—271no—
150C22×C4—271no—
151C4⋊C4—271no—
152C4⋊C4—271no—
153C4⋊C4—271no—
154C22⋊C4—31no—
155C22⋊C4—91no—
156C22×C3—22no—
157C22×C3—22no—
158C22×C3—21no—
159C22×C3—21no—
160C22×C3—21no—
161C22×C3—21no—
162C22×C3—22no—
163C22×C3—21no—
164C22×C3Dic1812yescomplemented
165C22×C3C22×D911yescharacteristic
166C22×C3#62712yes—
167C22×C3C2×D3211yescharacteristic
168C3×C4—61no—
169C3×C4—61no—
170C3×C4—181no—
171C3×C4—181no—
172Dic3—21no—
173Dic3—21no—
174Dic3—61no—
175Dic3—61no—
176Dic3—92no—
177Dic3—92no—
178Dic3—182no—
179Dic3—182no—
180Dic3—181no—
181Dic3—181no—
182C32#87811yescharacteristic
183C9—21no—
184C9#87811yescharacteristic, complemented
185C23D3×D911yescharacteristic
186Q8—542no—
187C2×C4—31no—
188C2×C4—31no—
189C2×C4—91no—
190C2×C4—91no—
191C2×C4—271no—
192C2×C4—271no—
193C2×C4—542no—
194C2×C3—24no—
195C2×C3—22no—
196C2×C3—22no—
197C2×C3—21no—
198C2×C3—21no—
199C2×C3—21no—
200C2×C3C2×Dic1811yescharacteristic
201C2×C3D4×D911yescharacteristic
202C2×C3#369911yescharacteristic
203C2×C3#380511yescharacteristic
204C2×C3C2×#62711yescharacteristic
205C2×C3#381111yescharacteristic
206C22—21no—
207C22—21no—
208C22#618912yescomplemented
209C22C2×D3×D911yescharacteristic
210C4—61no—
211C4—61no—
212C4—181no—
213C4—181no—
214C4—272no—
215C4—541no—
216C3—21no—
217C3#6344011yescharacteristic, complemented
218C3#6398211yescharacteristic
219C2—22no—
220C2C2×#618911yescharacteristic
221C2#8866311yescharacteristic
222C2#8866811yescharacteristic
subgroupG/H
centerC22C2×D3×D9
commutator, cosocleC22×C3×C9C23
FrattiniC22×C3C2×D32
FittingC23×C3×C9C22
socleC22×C32C22×D3
prime radicalC23×C32C2×D3
hypercenterC23D3×D9
hypocenterC3×C9C22⋊Q8
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#6189C2semidirect
5. Dic3⋊Dic9C2semidirect
6. #88442C2semidirect
7. Dic9⋊Dic3C2semidirect
36. C22×C3×C9Q8semidirect
103. C22×C9Dic6semidirect
114. C3×C9C22⋊Q8semidirect
164. C22×C3Dic18semidirect
184. C9#878semidirect
208. C22#6189semidirect
217. C3#63440semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊Dic92
C2×#61892
C9×#2172
C3×#36762
Dic3⋊Dic92
#884422
Dic9⋊Dic32
#639823
#634403