Finite Group Databasebeta

← #11721317

Subgroups — #11721317

Order 864.

432 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 60 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×#6235C211yesmaximal, characteristic, complemented
2C2×#638⋊C3C212yesmaximal, complemented
3C2×#6200C212yesmaximal, complemented
4C2×#6198C211yesmaximal, characteristic, complemented
5#88447C211yesmaximal, characteristic, complemented
6#63979—32nomaximal
7C23×H(3,𝔽3)C2211yescharacteristic
8C22×#272C2211yescharacteristic
9C22×(C3⋊D3)⋊C3C2212yes—
10#6235—22no—
11C2×#1084C2211yescharacteristic
12#638⋊C3—24no—
13C2×(C3⋊Dic3)⋊C3C2212yes—
14#6200—24no—
15#6198—22no—
16C2×#638D312yescomplemented
17C2×C3×#67—32no—
18C2×C3×#67—32no—
19C2×C3×#67—61no—
20C2×#626—32no—
21C2×#626—32no—
22C2×#625—32no—
23Dic32—32no—
24C22×H(3,𝔽3)—21no—
25C22×H(3,𝔽3)—22no—
26C22×H(3,𝔽3)C2311yescharacteristic
27C2×#272—21no—
28C2×#272—22no—
29C2×(C3⋊D3)⋊C3—22no—
30C2×(C3⋊D3)⋊C3—22no—
31C2×(C3⋊D3)⋊C3—22no—
32#1084D412yescomplemented
33(C3⋊Dic3)⋊C3D414yescomplemented
34#940—91no—
35#940—92no—
36C23×C32—21no—
37C23×C32C2×D312yes—
38C22×C3⋊D3C2×D312yes—
39C22×C3×D3—32no—
40C22×C3×D3—32no—
41C22×C3×D3—61no—
42#638—24no—
43C2×C3⋊Dic3C2×D312yes—
44C3×#67—64no—
45C3×#67—64no—
46C3×#67—64no—
47C2×C3×Dic3—32no—
48C2×C3×Dic3—32no—
49C2×C3×Dic3—61no—
50#626—64no—
51#626—64no—
52#625—64no—
53C2×H(3,𝔽3)—41no—
54C2×H(3,𝔽3)C2×D411yescharacteristic
55C2×H(3,𝔽3)C2×D412yes—
56#272—41no—
57(C3⋊D3)⋊C3—42no—
58C2×C3×D4—92no—
59C2×C3×D4—91no—
60C2×#67—32no—
61C2×#67—32no—
62C2×#67—61no—
63C2×#67—92no—
64C2×#67—92no—
65C2×#67—92no—
66C2×#67—92no—
67C2×D12—91no—
68C2×D12—92no—
69C4×Dic3—91no—
70C4×Dic3—92no—
71C22×C32—22no—
72C22×C32—22no—
73C22×C32—22no—
74C22×C32—22no—
75C22×C32—21no—
76C22×C32—42no—
77C22×C32C22×D312yes—
78C2×C3⋊D3—22no—
79C2×C3⋊D3—22no—
80C2×C3⋊D3—22no—
81C2×C3×D3—62no—
82C2×C3×D3—62no—
83C2×C3×D3—62no—
84C2×C3×D3—62no—
85C2×C3×D3—62no—
86C2×C3×D3—62no—
87C2×C3×D3—62no—
88C2×C3×D3—122no—
89C3⋊Dic3#6714yescomplemented
90C3×Dic3—34no—
91C3×Dic3—34no—
92C3×Dic3—62no—
93#127—271noSylow
94H(3,𝔽3)#12711yescharacteristic, complemented, Sylow
95C23×C3—32no—
96C23×C3—61no—
97C23×C3—92no—
98C23×C3—91no—
99C23×C3D3211yescharacteristic
100C22×D3—32no—
101C22×D3—32no—
102C22×D3—61no—
103C22×D3—92no—
104C3×D4—182no—
105C3×D4—184no—
106C2×C3×C4—92no—
107C2×C3×C4—91no—
108#67—64no—
109#67—64no—
110#67—64no—
111#67—184no—
112#67—184no—
113#67—184no—
114#67—184no—
115C2×Dic3—32no—
116C2×Dic3—32no—
117C2×Dic3—61no—
118C2×Dic3—92no—
119D12—182no—
120D12—184no—
121C2×C32—22no—
122C2×C32—21no—
123C2×C32—42no—
124C2×C32—42no—
125C2×C32D3×D412yes—
126C2×C32D3×D412yes—
127C2×C32C2×#6712yes—
128C3⋊D3—42no—
129C3×D3—122no—
130C3×D3—122no—
131C3×D3—122no—
132C2×D4—91no—
133C2×D4—92no—
134C2×D4—271no—
135C2×D4—272no—
136C42—271no—
137C22×C3—22no—
138C22×C3—21no—
139C22×C3—32no—
140C22×C3—62no—
141C22×C3—62no—
142C22×C3—62no—
143C22×C3—62no—
144C22×C3—61no—
145C22×C3—122no—
146C22×C3—182no—
147C22×C3—182no—
148C22×C3—182no—
149C22×C3—182no—
150C22×C3—181no—
151C22×C3C2×D3211yescharacteristic
152C2×D3—62no—
153C2×D3—62no—
154C2×D3—62no—
155C2×D3—62no—
156C2×D3—62no—
157C2×D3—62no—
158C2×D3—62no—
159C2×D3—122no—
160C2×D3—182no—
161C2×D3—182no—
162C2×D3—182no—
163C3×C4—94no—
164C3×C4—92no—
165Dic3—34no—
166Dic3—34no—
167Dic3—62no—
168Dic3—94no—
169C32—21no—
170C32#94012yescomplemented
171C23—91no—
172C23—92no—
173C23((C3⋊D3)⋊C3)⋊C211yescharacteristic
174D4—182no—
175D4—184no—
176D4—544no—
177D4—542no—
178C2×C4—91no—
179C2×C4—92no—
180C2×C3—32no—
181C2×C3—32no—
182C2×C3—32no—
183C2×C3—41no—
184C2×C3—62no—
185C2×C3—61no—
186C2×C3—122no—
187C2×C3—122no—
188C2×C3—362no—
189C2×C3—361no—
190C2×C3D3×#6712yes—
191C2×C3#381111yescharacteristic
192D3—122no—
193D3—122no—
194D3—122no—
195D3—362no—
196C22—22no—
197C22—21no—
198C22—182no—
199C22—182no—
200C22—182no—
201C22—181no—
202C22—182no—
203C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
204C4—94no—
205C4—92no—
206C3—32no—
207C3—61no—
208C3#6397911yescharacteristic
209C2—41no—
210C2—361no—
211C2—362no—
212C2#8867612yes—
213C2#8867711yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C23
Frattini, socleC22×C3C2×D32
FittingC23×H(3,𝔽3)C22
hypercenterC23((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#127
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×#6235C2semidirect
2. C2×#638⋊C3C2semidirect
3. C2×#6200C2semidirect
4. C2×#6198C2semidirect
5. #88447C2semidirect
16. C2×#638D3semidirect
32. #1084D4semidirect
33. (C3⋊Dic3)⋊C3D4semidirect
89. C3⋊Dic3#67semidirect
94. H(3,𝔽3)#127semidirect
170. C32#940semidirect

Maximal subgroups

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

subgroupindex
C2×#62352
C2×#638⋊C32
C2×#62002
C2×#61982
#884472
#639793