Finite Group Databasebeta

← #11722823

Subgroups — #11722823

Order 864.

322 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 82 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×#3798C211yesmaximal, characteristic
2#88809C211yesmaximal, characteristic, complemented
3C3⋊D3×C3⋊C8C211yesmaximal, characteristic, complemented
4D3×C3⋊(C3⋊C8)C212yesmaximal, complemented
5C3×#3711C212yesmaximal, complemented
6#63815—32nomaximal
7#63805—31nomaximal
8C3×C4×C3⋊D3C2211yescharacteristic
9C32×C4×D3C2212yes—
10C3×#627C411yescharacteristic
11C3×#626C412yes—
12C3×#625C411yescharacteristic
13C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C3⋊(C3⋊C8)C2212yes—
15C32×C3⋊C8C2211yescharacteristic
16C3×#235—32no—
17#3798D311yescharacteristic
18C2×C3⋊(C3⋊C8)—32no—
19C8×C3⋊D3—31no—
20C3×#208—32no—
21C3×ΓL(1,𝔽25)—32no—
22#3712—31no—
23#3711—32no—
24#3711D312yescomplemented
25D3×C3⋊C8—32no—
26D3×C3⋊C8—32no—
27D3×C3⋊C8—32no—
28D3×C3⋊C8—61no—
29D3×C3⋊C8—62no—
30C2×C3×C3⋊D3C2×C411yescharacteristic
31C2×C32×D3C2×C412yes—
32C33×C4C2311yescharacteristic
33C3×C3⋊Dic3C2×C411yescharacteristic
34C32×Dic3C2×C412yes—
35#948—91no—
36#907—92no—
37C2×C32×C4—32no—
38C4×C3⋊D3C2×D311yescharacteristic
39C3×#67—32no—
40C3×#67—32no—
41C3×D12—32no—
42C3×C4×D3—22no—
43C3×C4×D3—32no—
44C3×C4×D3—61no—
45C3×C4×D3C2×D312yes—
46C3×C4×D3C2×D312yes—
47C3×Dic6—32no—
48#627Dic311yescharacteristic
49#626Dic312yes—
50#625Dic311yescharacteristic
51C32×C8—31no—
52C3⋊(C3⋊C8)—31no—
53C3⋊(C3⋊C8)—32no—
54C3⋊(C3⋊C8)—62no—
55C3⋊(C3⋊C8)—61no—
56C3⋊(C3⋊C8)—121no—
57C3⋊(C3⋊C8)C2×D312yes—
58C3×C3⋊C8—21no—
59C3×C3⋊C8—32no—
60C3×C3⋊C8—32no—
61C3×C3⋊C8—62no—
62C3×C3⋊C8C2×D312yes—
63C2×C33C22×C411yescharacteristic
64C3×C3⋊D3—21no—
65C32×D3—22no—
66C3×#41—91no—
67#235—32no—
68C3×M16—92no—
69#208—32no—
70#208—91no—
71C2×C3⋊C8—92no—
72C2×C3⋊C8—92no—
73C2×C3⋊C8—91no—
74C2×C3⋊C8—182no—
75ΓL(1,𝔽25)—32no—
76ΓL(1,𝔽25)—92no—
77D3×C8—92no—
78D3×C8—92no—
79D3×C8—181no—
80C22×C32—32no—
81C2×C3⋊D3C2×Dic311yescharacteristic
82C2×C3×D3—22no—
83C2×C3×D3—32no—
84C2×C3×D3—61no—
85C2×C3×D3C4×D312yes—
86C2×C3×D3C2×Dic312yes—
87C32×C4—21no—
88C32×C4—22no—
89C32×C4—32no—
90C32×C4—41no—
91C32×C4C22×D312yes—
92C32×C4C22×D311yescharacteristic
93C3⋊Dic3C2×Dic311yescharacteristic
94C3×Dic3—22no—
95C3×Dic3—32no—
96C3×Dic3—61no—
97C3×Dic3C4×D312yes—
98C3×Dic3C2×Dic312yes—
99#131—271noSylow
100C33#13111yescharacteristic, complemented, Sylow
101C3×Q8—91no—
102C3×D4—92no—
103C3×D4—91no—
104C2×C3×C4—32no—
105C2×C3×C4—32no—
106C2×C3×C4—62no—
107C2×C3×C4—91no—
108#67—32no—
109#67—32no—
110D12—32no—
111C4×D3—32no—
112C4×D3—61no—
113C4×D3D3212yes—
114Dic6—32no—
115C3×C8—32no—
116C3×C8—61no—
117C3×C8—92no—
118C3⋊C8—32no—
119C3⋊C8—32no—
120C3⋊C8—62no—
121C3⋊C8—92no—
122C3⋊C8—91no—
123C3⋊C8—181no—
124C3⋊C8—182no—
125C3⋊C8—361no—
126C3⋊C8D3211yescharacteristic
127C2×C32—21no—
128C2×C32—22no—
129C2×C32—41no—
130C2×C32—62no—
131C2×C32C2×C4×D312yes—
132C2×C32C22×Dic311yescharacteristic
133C3⋊D3—21no—
134C3×D3—24no—
135C3×D3—22no—
136C3×D3—22no—
137C3×D3—62no—
138C3×D3—62no—
139#41—91no—
140M16—92no—
141M16—271no—
142C2×C8—271no—
143C2×C8—272no—
144C22×C3—32no—
145C22×C3—32no—
146C22×C3—62no—
147C22×C3—91no—
148C2×D3—32no—
149C2×D3—61no—
150C2×D3D3×Dic312yes—
151C3×C4—22no—
152C3×C4—21no—
153C3×C4—32no—
154C3×C4—32no—
155C3×C4—41no—
156C3×C4—62no—
157C3×C4—91no—
158C3×C4C2×D3211yescharacteristic
159C3×C4C2×D3212yes—
160Dic3—32no—
161Dic3—61no—
162Dic3D3×Dic312yes—
163C32—21no—
164C32—22no—
165C32—41no—
166C32#90712yescomplemented
167C32#94811yescharacteristic, complemented
168Q8—91no—
169D4—92no—
170D4—91no—
171C2×C4—32no—
172C2×C4—91no—
173C8—31no—
174C8—92no—
175C8—271no—
176C2×C3—21no—
177C2×C3—22no—
178C2×C3—41no—
179C2×C3—64no—
180C2×C3—62no—
181C2×C3—62no—
182C2×C3—181no—
183C2×C3C4×D3211yescharacteristic
184C2×C3C2×D3×Dic312yes—
185D3—22no—
186D3—62no—
187D3—62no—
188C22—32no—
189C22—91no—
190C4—32no—
191C4—91no—
192C4D3311yescharacteristic
193C3—21no—
194C3—22no—
195C3—41no—
196C3#6380511yescharacteristic, complemented
197C3#6381512yescomplemented
198C2—62no—
199C2—181no—
200C2D32×Dic311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC4D33
commutator, socleC2×C33C22×C4
Fitting, cosocleC33×C4C23
prime radicalC3×#3798C2
hypocenterC33#131
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #88809C2semidirect
3. C3⋊D3×C3⋊C8C2semidirect
4. D3×C3⋊(C3⋊C8)C2semidirect
5. C3×#3711C2semidirect
24. #3711D3semidirect
100. C33#131semidirect
166. C32#907semidirect
167. C32#948semidirect
196. C3#63805semidirect
197. C3#63815semidirect

Maximal subgroups

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

subgroupindex
C3×#37982
#888092
C3⋊D3×C3⋊C82
D3×C3⋊(C3⋊C8)2
C3×#37112
#638153
#638053