Finite Group Databasebeta

← #11721739

Subgroups — #11721739

Order 864.

429 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 76 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
2C2×C4×#272C211yesmaximal, characteristic, complemented
3C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#88552C211yesmaximal, characteristic, complemented
5#88542C211yesmaximal, characteristic, complemented
6C4⋊#1084C211yesmaximal, characteristic, complemented
7#88540C211yesmaximal, characteristic, complemented
8C3×#883—34nomaximal
9C23×H(3,𝔽3)C2211yescharacteristic
10C22×#272C2211yescharacteristic
11C2×C4×H(3,𝔽3)C2211yescharacteristic
12C2×C4×H(3,𝔽3)C2211yescharacteristic
13#6235—22no—
14C2×#1084C2211yescharacteristic
15C2×#1084C2211yescharacteristic
16C2×#1084C2211yescharacteristic
17C4×#272—22no—
18C2×C3×#67—34no—
19C2×C3×C4×D3—34no—
20C32×C22⋊C4D314yescomplemented
21C3×#217—34no—
22C3×(C2×D3)⋊C4—34no—
23C3×C4⋊Dic3—34no—
24C3×Dic3⋊C4—34no—
25C22×H(3,𝔽3)—21no—
26C22×H(3,𝔽3)—21no—
27C22×H(3,𝔽3)—21no—
28C22×H(3,𝔽3)C2311yescharacteristic
29C2×#272—21no—
30C2×#272—21no—
31C2×#272D412yes—
32C4×H(3,𝔽3)—21no—
33C4×H(3,𝔽3)—21no—
34#1084—21no—
35#1084—21no—
36#1084—21no—
37C3×#123—91no—
38#883—34no—
39C23×C32C2×D314yes—
40C22×C3×D3—34no—
41C2×C32×C4C2×D314yes—
42C2×C32×C4C2×D314yes—
43C3×#67—68no—
44C2×C3×Dic3—34no—
45C2×C3×Dic3—34no—
46C2×C3×Dic3—34no—
47C3×C4×D3—68no—
48C2×H(3,𝔽3)—41no—
49C2×H(3,𝔽3)C2×D411yescharacteristic
50C2×H(3,𝔽3)#4111yescharacteristic
51C2×H(3,𝔽3)#4111yescharacteristic
52#272—22no—
53C2×C3×D4—91no—
54C22×C3×C4—91no—
55C2×#67—34no—
56C2×C4×D3—34no—
57C3×C4⋊C4—91no—
58C3×C4⋊C4—91no—
59C3×C22⋊C4—34no—
60C3×C22⋊C4—91no—
61C3×C22⋊C4—91no—
62C3×C22⋊C4C3⋊D311yescharacteristic
63#217—34no—
64(C2×D3)⋊C4—34no—
65C4⋊Dic3—34no—
66Dic3⋊C4—34no—
67C22×C32—24no—
68C22×C32—24no—
69C22×C32—24no—
70C22×C32C22×D314yes—
71C2×C3×D3—38no—
72C2×C3×D3—64no—
73C2×C3×D3—64no—
74C32×C4—24no—
75C32×C4—24no—
76C3×Dic3—64no—
77C3×Dic3—64no—
78C3×Dic3—64no—
79#123—91noSylow
80H(3,𝔽3)#12311yescharacteristic, complemented, Sylow
81C23×C3—34no—
82C23×C3—91no—
83C23×C3C2×C3⋊D311yescharacteristic
84C22×D3—34no—
85C3×D4—182no—
86C2×C3×C4—34no—
87C2×C3×C4—34no—
88C2×C3×C4—91no—
89C2×C3×C4—91no—
90C2×C3×C4—91no—
91C2×C3×C4—182no—
92C2×C3×C4C2×C3⋊D311yescharacteristic
93C2×C3×C4C2×C3⋊D311yescharacteristic
94#67—68no—
95C2×Dic3—34no—
96C2×Dic3—34no—
97C2×Dic3—34no—
98C4×D3—68no—
99C2×C32—44no—
100C2×C32#23514yes—
101C2×C32D3×D414yes—
102C2×C32#23714yes—
103C3×D3—68no—
104C2×D4—91no—
105C22×C4—91no—
106C4⋊C4—91no—
107C4⋊C4—91no—
108C22⋊C4—91no—
109C22⋊C4—91no—
110C22⋊C4#27211yescharacteristic, complemented
111C22×C3—21no—
112C22×C3—21no—
113C22×C3—21no—
114C22×C3—34no—
115C22×C3—64no—
116C22×C3—64no—
117C22×C3—64no—
118C22×C3—92no—
119C22×C3—181no—
120C22×C3—181no—
121C22×C3C22×C3⋊D311yescharacteristic
122C2×D3—38no—
123C2×D3—64no—
124C2×D3—64no—
125C3×C4—21no—
126C3×C4—21no—
127C3×C4—64no—
128C3×C4—64no—
129C3×C4—181no—
130C3×C4—181no—
131C3×C4—181no—
132Dic3—64no—
133Dic3—64no—
134Dic3—64no—
135C32#88314yescomplemented
136C23—91no—
137C23C2×#27211yescharacteristic
138D4—182no—
139C2×C4—91no—
140C2×C4—91no—
141C2×C4—91no—
142C2×C4—182no—
143C2×C4C2×#27211yescharacteristic
144C2×C4C2×#27211yescharacteristic
145C2×C3—34no—
146C2×C3—34no—
147C2×C3—34no—
148C2×C3—41no—
149C2×C3—124no—
150C2×C3—182no—
151C2×C3#382811yescharacteristic
152C2×C3D4×C3⋊D311yescharacteristic
153C2×C3#383011yescharacteristic
154D3—68no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—92no—
159C22—181no—
160C22—181no—
161C22C22×#27211yescharacteristic
162C4—21no—
163C4—21no—
164C4—181no—
165C4—181no—
166C4—181no—
167C3—34no—
168C3#6409211yescharacteristic
169C2—41no—
170C2—182no—
171C2#8874011yescharacteristic
172C2D4×#27211yescharacteristic
173C2#8874411yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C23
FittingC22⋊C4×H(3,𝔽3)C2
prime radicalC2×#6235C2
hypercenterC3×C22⋊C4C3⋊D3
hypocenterH(3,𝔽3)#123
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
2. C2×C4×#272C2semidirect
3. C22⋊C4×H(3,𝔽3)C2semidirect
4. #88552C2semidirect
5. #88542C2semidirect
6. C4⋊#1084C2semidirect
7. #88540C2semidirect
20. C32×C22⋊C4D3semidirect
80. H(3,𝔽3)#123semidirect
110. C22⋊C4#272semidirect
135. C32#883semidirect

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
C22⋊C4×H(3,𝔽3)2
#885522
#885422
C4⋊#10842
#885402
C3×#8833