Finite Group Databasebeta

← #11720996

Subgroups — #11720996

Order 864.

260 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 65 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
1C2×#6226C211yesmaximal, characteristic, complemented
2C2×Dic3×D9C211yesmaximal, characteristic
3C3×(C2×D9)⋊C4C211yesmaximal, characteristic
4C9×Dic3⋊C4C211yesmaximal, characteristic, complemented
5(C2×D9)⋊Dic3C211yesmaximal, characteristic
6#88442C211yesmaximal, characteristic, complemented
7Dic9⋊Dic3C211yesmaximal, characteristic, complemented
8#63896—31nomaximal
9#63458—31nomaximal
10C22×C3×D9C2211yescharacteristic
11C2×C3×C4×C9C2211yescharacteristic
12C2×C3⋊Dic9C2211yescharacteristic
13C2×C3⋊Dic9C2211yescharacteristic
14#6226—22no—
15C2×C9×Dic3C2211yescharacteristic
16C2×C9×Dic3C2211yescharacteristic
17C2×C3×Dic9C2211yescharacteristic
18Dic3×D9—22no—
19C2×#634—31no—
20C2×D3×Dic3—31no—
21C3×(C2×D3)⋊C4—31no—
22C3×Dic3⋊C4D311yescharacteristic
23#3724—31no—
24Dic3⋊Dic3—31no—
25(C2×D3)⋊Dic3—31no—
26C2×C4×D9—31no—
27C2×Dic18—31no—
28C2×Dic18—61no—
29C9×C4⋊C4—31no—
30(C2×D9)⋊C4—31no—
31(C2×D9)⋊C4D311yescharacteristic
32Dic9⋊C4—31no—
33Dic9⋊C4—31no—
34C22×C3×C9C2311yescharacteristic
35C2×C3×D9—21no—
36C2×C3×D9—21no—
37C2×C3×D9Q812yescomplemented
38C3×C4×C9—21no—
39C3⋊Dic9—21no—
40C3⋊Dic9D412yes—
41C9×Dic3—21no—
42C9×Dic3—21no—
43C3×Dic9—21no—
44#896—91no—
45#878—91no—
46C22×C3×D3—31no—
47C2×C32×C4C2×D311yescharacteristic
48C2×C3⋊Dic3—31no—
49C2×C3⋊Dic3—31no—
50#634—62no—
51C2×C3×Dic3—31no—
52C2×C3×Dic3C2×D311yescharacteristic
53C2×C3×Dic3C2×D311yescharacteristic
54D3×Dic3—62no—
55C22×D9C2×D311yescharacteristic
56C2×C4×C9—21no—
57C2×C4×C9—31no—
58C2×C4×C9—31no—
59C2×C4×C9C2×D311yescharacteristic
60C2×Dic9—31no—
61C2×Dic9—31no—
62C2×Dic9—61no—
63C2×Dic9—61no—
64C2×Dic9C2×D311yescharacteristic
65C4×D9—62no—
66Dic18—62no—
67Dic18—64no—
68C2×C3×C9C2×D411yescharacteristic
69C2×C3×C9C2×Q811yescharacteristic
70C2×C3×C9#4111yescharacteristic
71C3×D9—22no—
72C22×Dic3—91no—
73C2×C4×D3—91no—
74C2×Dic6—91no—
75C2×Dic6—91no—
76C2×Dic6—181no—
77C3×C4⋊C4—31no—
78C3×C22⋊C4—91no—
79#217—91no—
80(C2×D3)⋊C4—31no—
81(C2×D3)⋊C4—91no—
82C4⋊Dic3—91no—
83Dic3⋊C4—91no—
84Dic3⋊C4—91no—
85Dic3⋊C4—91no—
86Dic3⋊C4D911yescharacteristic, complemented
87C22×C32C22×D311yescharacteristic
88C2×C3×D3—32no—
89C2×C3×D3—61no—
90C2×C3×D3—61no—
91C32×C4—21no—
92C3⋊Dic3—32no—
93C3⋊Dic3—61no—
94C3×Dic3—21no—
95C3×Dic3—21no—
96C3×Dic3—61no—
97C22×C9—21no—
98C22×C9C22×D311yescharacteristic
99C2×D9—21no—
100C2×D9—21no—
101C2×D9Dic612yescomplemented
102C4×C9—22no—
103C4×C9—21no—
104C4×C9—61no—
105C4×C9—61no—
106Dic9—21no—
107Dic9—32no—
108Dic9—62no—
109Dic9—62no—
110Dic9—61no—
111C22⋊Q8—271noSylow
112C3×C9C22⋊Q811yescharacteristic, complemented, Sylow
113C23×C3—91no—
114C22×D3—31no—
115C2×C3×C4—21no—
116C2×C3×C4—31no—
117C2×C3×C4—31no—
118C2×C3×C4—91no—
119C2×C3×C4C2×D911yescharacteristic
120C2×C3×C4D3211yescharacteristic
121C2×Dic3—31no—
122C2×Dic3—91no—
123C2×Dic3—91no—
124C2×Dic3—91no—
125C2×Dic3—91no—
126C2×Dic3—181no—
127C2×Dic3—181no—
128C2×Dic3—182no—
129C2×Dic3C2×D911yescharacteristic
130C2×Dic3C2×D911yescharacteristic
131C4×D3—182no—
132Dic6—182no—
133Dic6—182no—
134Dic6—184no—
135C2×C32#23511yescharacteristic
136C2×C32D3×D411yescharacteristic
137C2×C32D3×Q811yescharacteristic
138C3×D3—62no—
139C2×C9—21no—
140C2×C9—21no—
141C2×C9—21no—
142C2×C9C2×Dic611yescharacteristic
143C2×C9D3×D411yescharacteristic
144C2×C9#23711yescharacteristic
145D9—22no—
146C2×Q8—271no—
147C22×C4—271no—
148C4⋊C4—31no—
149C4⋊C4—271no—
150C4⋊C4—271no—
151C22⋊C4—91no—
152C22⋊C4—271no—
153C22×C3—21no—
154C22×C3—92no—
155C22×C3—181no—
156C22×C3—181no—
157C22×C3C22×D911yescharacteristic
158C22×C3C2×D3211yescharacteristic
159C2×D3—32no—
160C2×D3—61no—
161C2×D3—61no—
162C3×C4—22no—
163C3×C4—21no—
164C3×C4—21no—
165C3×C4—61no—
166C3×C4—61no—
167C3×C4—181no—
168Dic3—21no—
169Dic3—21no—
170Dic3—61no—
171Dic3—92no—
172Dic3—92no—
173Dic3—182no—
174Dic3—182no—
175Dic3—181no—
176Dic3—181no—
177C32#89611yescharacteristic
178C9—21no—
179C9#87811yescharacteristic, complemented
180C23—91no—
181Q8—542no—
182C2×C4—31no—
183C2×C4—31no—
184C2×C4—91no—
185C2×C4—271no—
186C2×C4—271no—
187C2×C4—542no—
188C2×C4D3×D911yescharacteristic
189C2×C3—21no—
190C2×C3—21no—
191C2×C3—21no—
192C2×C3—182no—
193C2×C3#369711yescharacteristic
194C2×C3D4×D911yescharacteristic
195C2×C3Q8×D911yescharacteristic
196C2×C3D3×Dic611yescharacteristic
197C2×C3#379511yescharacteristic
198C2×C3#381111yescharacteristic
199D3—62no—
200C22—92no—
201C22—181no—
202C22—181no—
203C22C2×D3×D911yescharacteristic
204C4—21no—
205C4—61no—
206C4—61no—
207C4—181no—
208C4—272no—
209C4—541no—
210C3—21no—
211C3#6345811yescharacteristic, complemented
212C3#6389611yescharacteristic
213C2—182no—
214C2D9×Dic611yescharacteristic
215C2#8864111yescharacteristic
216C2#8866811yescharacteristic
subgroupG/H
centerC22C2×D3×D9
commutator, cosocleC22×C3×C9C23
FrattiniC22×C3C2×D32
FittingC2×C3×C4×C9C22
socleC22×C32C22×D3
prime radicalC22×C3×D9C22
hypercenterC2×C4D3×D9
hypocenterC3×C9C22⋊Q8
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#6226C2semidirect
4. C9×Dic3⋊C4C2semidirect
6. #88442C2semidirect
7. Dic9⋊Dic3C2semidirect
37. C2×C3×D9Q8semidirect
86. Dic3⋊C4D9semidirect
101. C2×D9Dic6semidirect
112. C3×C9C22⋊Q8semidirect
179. C9#878semidirect
211. C3#63458semidirect

Maximal subgroups

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

subgroupindex
C2×#62262
C2×Dic3×D92
C3×(C2×D9)⋊C42
C9×Dic3⋊C42
(C2×D9)⋊Dic32
#884422
Dic9⋊Dic32
#638963
#634583