Finite Group Databasebeta

← #11721318

Subgroups — #11721318

Order 864.

383 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 62 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×#1084C211yesmaximal, characteristic, complemented
2C2×#638⋊C3C212yesmaximal, complemented
3C2×#6198C211yesmaximal, characteristic, complemented
4#88450C212yesmaximal, complemented
5#88449C211yesmaximal, characteristic, complemented
6#63978—32nomaximal
7C23×H(3,𝔽3)C2211yescharacteristic
8C22×(C3⋊D3)⋊C3C2212yes—
9C2×#1084—22no—
10C2×#1084C2211yescharacteristic
11C2×#1084C2211yescharacteristic
12#638⋊C3—24no—
13C2×(C3⋊Dic3)⋊C3C2212yes—
14#6198—22no—
15C2×#638D312yescomplemented
16C2×C3×#67—32no—
17C22×C3×Dic3—32no—
18C22×C3×Dic3—61no—
19C2×#626—32no—
20Dic3⋊Dic3—32no—
21#3722—32no—
22(C2×D3)⋊Dic3—32no—
23C22×H(3,𝔽3)—21no—
24C22×H(3,𝔽3)—21no—
25C22×H(3,𝔽3)D412yescomplemented
26C22×H(3,𝔽3)C2311yescharacteristic
27C2×(C3⋊D3)⋊C3—22no—
28C2×(C3⋊D3)⋊C3—22no—
29C2×(C3⋊D3)⋊C3—22no—
30#1084—21no—
31#1084D412yescomplemented
32(C3⋊Dic3)⋊C3—22no—
33#939—92no—
34#930—91no—
35C23×C32—21no—
36C23×C32C2×D312yes—
37C22×C3⋊D3C2×D312yes—
38C22×C3×D3—32no—
39#638—24no—
40C2×C3⋊Dic3C2×D312yes—
41C3×#67—64no—
42C2×C3×Dic3—32no—
43C2×C3×Dic3—32no—
44C2×C3×Dic3—32no—
45C2×C3×Dic3—61no—
46C2×C3×Dic3—61no—
47C2×C3×Dic3—64no—
48C2×C3×Dic3—64no—
49#626—64no—
50C2×H(3,𝔽3)—22no—
51C2×H(3,𝔽3)C2×D411yescharacteristic
52C2×H(3,𝔽3)C2×D411yescharacteristic
53C2×H(3,𝔽3)#4111yescharacteristic
54(C3⋊D3)⋊C3—42no—
55C2×C3×D4—92no—
56C22×C3×C4—91no—
57C2×#67—32no—
58C2×#67—92no—
59C2×#67—92no—
60C22×Dic3—32no—
61C22×Dic3—61no—
62C2×D12—91no—
63#217—92no—
64(C2×D3)⋊C4—92no—
65(C2×D3)⋊C4—92no—
66C4⋊Dic3—91no—
67Dic3⋊C4—92no—
68C22×C32—22no—
69C22×C32—22no—
70C22×C32—22no—
71C22×C32—22no—
72C22×C32—22no—
73C22×C32—21no—
74C22×C32#6714yescomplemented
75C22×C32C22×D312yes—
76C2×C3⋊D3—22no—
77C2×C3⋊D3—22no—
78C2×C3⋊D3—22no—
79C2×C3×D3—62no—
80C2×C3×D3—62no—
81C2×C3×D3—62no—
82C3⋊Dic3—22no—
83C3×Dic3—34no—
84C3×Dic3—62no—
85C3×Dic3—62no—
86C3×Dic3—62no—
87C3×Dic3—62no—
88#121—271noSylow
89H(3,𝔽3)#12111yescharacteristic, complemented, Sylow
90C23×C3—32no—
91C23×C3—61no—
92C23×C3—92no—
93C23×C3D3211yescharacteristic
94C22×D3—32no—
95C22×D3—92no—
96C3×D4—184no—
97C2×C3×C4—91no—
98C2×C3×C4—92no—
99C2×C3×C4—91no—
100C2×C3×C4—182no—
101#67—64no—
102#67—184no—
103#67—184no—
104C2×Dic3—32no—
105C2×Dic3—32no—
106C2×Dic3—32no—
107C2×Dic3—61no—
108C2×Dic3—61no—
109C2×Dic3—64no—
110C2×Dic3—64no—
111C2×Dic3—92no—
112D12—182no—
113C2×C32—24no—
114C2×C32—24no—
115C2×C32—21no—
116C2×C32—21no—
117C2×C32—21no—
118C2×C32D3×D412yes—
119C2×C32#23712yes—
120C2×C32C2×#6712yes—
121C3⋊D3—42no—
122C3×D3—122no—
123C2×D4—92no—
124C2×D4—271no—
125C22×C4—91no—
126C4⋊C4—271no—
127C22⋊C4—272no—
128C22×C3—21no—
129C22×C3—21no—
130C22×C3—34no—
131C22×C3—32no—
132C22×C3—62no—
133C22×C3—62no—
134C22×C3—62no—
135C22×C3—62no—
136C22×C3—62no—
137C22×C3—61no—
138C22×C3—182no—
139C22×C3—182no—
140C22×C3—182no—
141C22×C3#62512yes—
142C22×C3C2×D3211yescharacteristic
143C2×D3—62no—
144C2×D3—62no—
145C2×D3—62no—
146C2×D3—182no—
147C2×D3—182no—
148C2×D3—182no—
149C3×C4—92no—
150C3×C4—182no—
151C3×C4—181no—
152Dic3—34no—
153Dic3—62no—
154Dic3—62no—
155Dic3—62no—
156Dic3—62no—
157Dic3—182no—
158C32—21no—
159C32#93912yescomplemented
160C23—92no—
161C23((C3⋊D3)⋊C3)⋊C211yescharacteristic
162D4—184no—
163D4—542no—
164C2×C4—91no—
165C2×C4—91no—
166C2×C4—92no—
167C2×C4—182no—
168C2×C3—22no—
169C2×C3—32no—
170C2×C3—32no—
171C2×C3—32no—
172C2×C3—64no—
173C2×C3—64no—
174C2×C3—61no—
175C2×C3—61no—
176C2×C3—61no—
177C2×C3—362no—
178C2×C3#380511yescharacteristic
179C2×C3C2×#62511yescharacteristic
180C2×C3#381111yescharacteristic
181D3—122no—
182D3—362no—
183C22—21no—
184C22—21no—
185C22—182no—
186C22—182no—
187C22—182no—
188C22#619812yescomplemented
189C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
190C4—92no—
191C4—182no—
192C4—181no—
193C3—32no—
194C3—61no—
195C3#6398011yescharacteristic
196C2—22no—
197C2—362no—
198C2#8867211yescharacteristic
199C2C2×#619811yescharacteristic
200C2#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)#121
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. C22×#1084C2semidirect
2. C2×#638⋊C3C2semidirect
3. C2×#6198C2semidirect
4. #88450C2semidirect
5. #88449C2semidirect
15. C2×#638D3semidirect
25. C22×H(3,𝔽3)D4semidirect
31. #1084D4semidirect
74. C22×C32#67semidirect
89. H(3,𝔽3)#121semidirect
159. C32#939semidirect
188. C22#6198semidirect

Maximal subgroups

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

subgroupindex
C22×#10842
C2×#638⋊C32
C2×#61982
#884502
#884492
#639783