Finite Group Databasebeta

← #11721740

Subgroups — #11721740

Order 864.

517 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 78 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×#6235C211yesmaximal, characteristic, complemented
3C2×C4⋊#272C211yesmaximal, characteristic, complemented
4C2×C4×#272C211yesmaximal, characteristic, complemented
5C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
6#88542C211yesmaximal, characteristic, complemented
7#88540C211yesmaximal, characteristic, complemented
8C3×#884—34nomaximal
9C23×H(3,𝔽3)C2211yescharacteristic
10C22×#272C2211yescharacteristic
11C22×#272C2211yescharacteristic
12C2×C4×H(3,𝔽3)C2211yescharacteristic
13C2×C4×H(3,𝔽3)C2211yescharacteristic
14#6235—22no—
15#6235—22no—
16C2×#1084C2211yescharacteristic
17C2×#1084C2211yescharacteristic
18C4⋊#272—22no—
19C4×#272—22no—
20C2×C3×#67—34no—
21C2×C3×#67—34no—
22C2×C3×D12—34no—
23C2×C3×C4×D3—34no—
24C32×C22⋊C4D314yescomplemented
25C3×(C2×D3)⋊C4—34no—
26C3×Dic3⋊C4—34no—
27C22×H(3,𝔽3)—21no—
28C22×H(3,𝔽3)—21no—
29C22×H(3,𝔽3)—21no—
30C22×H(3,𝔽3)C2311yescharacteristic
31C2×#272—21no—
32C2×#272—21no—
33C2×#272—21no—
34C2×#272—21no—
35C2×#272—21no—
36C2×#272D412yescomplemented
37C4×H(3,𝔽3)—21no—
38C4×H(3,𝔽3)—21no—
39#1084—21no—
40#1084D412yescomplemented
41C3×#121—91no—
42#884—34no—
43C23×C32C2×D314yes—
44C22×C3×D3—34no—
45C22×C3×D3—34no—
46C2×C32×C4C2×D314yes—
47C2×C32×C4C2×D314yes—
48C3×#67—68no—
49C3×#67—68no—
50C2×C3×Dic3—34no—
51C2×C3×Dic3—34no—
52C3×D12—68no—
53C3×C4×D3—68no—
54C2×H(3,𝔽3)—41no—
55C2×H(3,𝔽3)C2×D411yescharacteristic
56C2×H(3,𝔽3)C2×D411yescharacteristic
57C2×H(3,𝔽3)#4111yescharacteristic
58#272—22no—
59#272—41no—
60C2×C3×D4—91no—
61C2×C3×D4—91no—
62C2×C3×D4—91no—
63C22×C3×C4—91no—
64C2×#67—34no—
65C2×#67—34no—
66C2×D12—34no—
67C2×C4×D3—34no—
68C3×C4⋊C4—91no—
69C3×C22⋊C4—34no—
70C3×C22⋊C4—91no—
71C3×C22⋊C4C3⋊D311yescharacteristic
72(C2×D3)⋊C4—34no—
73Dic3⋊C4—34no—
74C22×C32—24no—
75C22×C32—24no—
76C22×C32—24no—
77C22×C32C22×D314yes—
78C2×C3×D3—38no—
79C2×C3×D3—64no—
80C2×C3×D3—64no—
81C2×C3×D3—64no—
82C2×C3×D3—64no—
83C2×C3×D3—64no—
84C32×C4—24no—
85C32×C4—24no—
86C3×Dic3—38no—
87C3×Dic3—64no—
88#121—91noSylow
89H(3,𝔽3)#12111yescharacteristic, complemented, Sylow
90C23×C3—34no—
91C23×C3—91no—
92C23×C3—91no—
93C23×C3C2×C3⋊D311yescharacteristic
94C22×D3—34no—
95C22×D3—34no—
96C3×D4—182no—
97C3×D4—182no—
98C3×D4—182no—
99C2×C3×C4—34no—
100C2×C3×C4—34no—
101C2×C3×C4—91no—
102C2×C3×C4—91no—
103C2×C3×C4—182no—
104C2×C3×C4C2×C3⋊D311yescharacteristic
105C2×C3×C4C2×C3⋊D311yescharacteristic
106#67—68no—
107#67—68no—
108C2×Dic3—34no—
109C2×Dic3—34no—
110D12—68no—
111C4×D3—68no—
112C2×C32—44no—
113C2×C32#23514yes—
114C2×C32D3×D414yes—
115C2×C32D3×D414yes—
116C3×D3—68no—
117C3×D3—124no—
118C2×D4—91no—
119C2×D4—91no—
120C2×D4—91no—
121C22×C4—91no—
122C4⋊C4—91no—
123C22⋊C4—91no—
124C22⋊C4#27211yescharacteristic, complemented
125C22×C3—21no—
126C22×C3—21no—
127C22×C3—21no—
128C22×C3—34no—
129C22×C3—64no—
130C22×C3—64no—
131C22×C3—64no—
132C22×C3—92no—
133C22×C3—181no—
134C22×C3—181no—
135C22×C3—181no—
136C22×C3—181no—
137C22×C3—181no—
138C22×C3C22×C3⋊D311yescharacteristic
139C2×D3—38no—
140C2×D3—64no—
141C2×D3—64no—
142C2×D3—64no—
143C2×D3—64no—
144C2×D3—64no—
145C3×C4—21no—
146C3×C4—21no—
147C3×C4—64no—
148C3×C4—64no—
149C3×C4—92no—
150C3×C4—181no—
151Dic3—38no—
152Dic3—64no—
153C32#88414yescomplemented
154C23—91no—
155C23—91no—
156C23C2×#27211yescharacteristic
157D4—182no—
158D4—182no—
159D4—182no—
160C2×C4—91no—
161C2×C4—91no—
162C2×C4—182no—
163C2×C4C2×#27211yescharacteristic
164C2×C4C2×#27211yescharacteristic
165C2×C3—34no—
166C2×C3—34no—
167C2×C3—34no—
168C2×C3—41no—
169C2×C3—124no—
170C2×C3—182no—
171C2×C3—361no—
172C2×C3#382811yescharacteristic
173C2×C3D4×C3⋊D311yescharacteristic
174C2×C3D4×C3⋊D311yescharacteristic
175D3—68no—
176D3—124no—
177C22—21no—
178C22—21no—
179C22—21no—
180C22—92no—
181C22—181no—
182C22—181no—
183C22—181no—
184C22—181no—
185C22—181no—
186C22C22×#27211yescharacteristic
187C4—21no—
188C4—21no—
189C4—92no—
190C4—181no—
191C3—34no—
192C3#6409311yescharacteristic
193C2—41no—
194C2—182no—
195C2—361no—
196C2#8874011yescharacteristic
197C2D4×#27211yescharacteristic
198C2D4×#27211yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C23
FittingC22⋊C4×H(3,𝔽3)C2
hypercenterC3×C22⋊C4C3⋊D3
hypocenterH(3,𝔽3)#121
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#6235C2semidirect
2. C2×#6235C2semidirect
3. C2×C4⋊#272C2semidirect
4. C2×C4×#272C2semidirect
5. C22⋊C4×H(3,𝔽3)C2semidirect
6. #88542C2semidirect
7. #88540C2semidirect
24. C32×C22⋊C4D3semidirect
36. C2×#272D4semidirect
40. #1084D4semidirect
89. H(3,𝔽3)#121semidirect
124. C22⋊C4#272semidirect
153. C32#884semidirect

Maximal subgroups

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

subgroupindex
C2×#62352
C2×C4⋊#2722
C2×C4×#2722
C22⋊C4×H(3,𝔽3)2
#885422
#885402
C3×#8843