Finite Group Databasebeta

← #11721764

Subgroups — #11721764

Order 864.

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

normal subgroups only

This group has 407 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×C4×#272C211yesmaximal, characteristic
2C2×#6229C211yesmaximal, characteristic, complemented
3C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#88542C211yesmaximal, characteristic
5#88542C211yesmaximal, characteristic
6#88540C211yesmaximal, characteristic, complemented
7#88540C211yesmaximal, characteristic, complemented
8C3×#896—34nomaximal
9C22×#272C2211yescharacteristic
10C2×C4×H(3,𝔽3)C2211yescharacteristic
11C2×C4×H(3,𝔽3)C2211yescharacteristic
12C2×C4×H(3,𝔽3)C2211yescharacteristic
13C2×#1084C2211yescharacteristic
14C2×#1084C2211yescharacteristic
15C2×#1084C2211yescharacteristic
16C4×#272—22no—
17#6229—22no—
18C2×C3×C4×D3—34no—
19C2×C3×Dic6—34no—
20C32×C4⋊C4D314yescomplemented
21C3×(C2×D3)⋊C4—34no—
22C3×(C2×D3)⋊C4—34no—
23C3×Dic3⋊C4—34no—
24C3×Dic3⋊C4—34no—
25C22×H(3,𝔽3)C2311yescharacteristic
26C2×#272—21no—
27C2×#272—21no—
28C2×#272Q812yescomplemented
29C4×H(3,𝔽3)—21no—
30C4×H(3,𝔽3)—21no—
31C4×H(3,𝔽3)—21no—
32#1084—21no—
33#1084—21no—
34#1084D412yes—
35C3×C22⋊Q8—91no—
36#896—34no—
37C22×C3×D3—34no—
38C2×C32×C4C2×D314yes—
39C2×C32×C4C2×D314yes—
40C2×C32×C4C2×D314yes—
41C2×C3×Dic3—34no—
42C2×C3×Dic3—34no—
43C2×C3×Dic3—34no—
44C3×C4×D3—68no—
45C3×Dic6—68no—
46C2×H(3,𝔽3)C2×D411yescharacteristic
47C2×H(3,𝔽3)C2×Q811yescharacteristic
48C2×H(3,𝔽3)#4111yescharacteristic
49#272—22no—
50C2×C3×Q8—91no—
51C22×C3×C4—91no—
52C2×C4×D3—34no—
53C2×Dic6—34no—
54C3×C4⋊C4—64no—
55C3×C4⋊C4—91no—
56C3×C4⋊C4—91no—
57C3×C4⋊C4C3⋊D311yescharacteristic
58C3×C22⋊C4—91no—
59C3×C22⋊C4—91no—
60(C2×D3)⋊C4—34no—
61(C2×D3)⋊C4—34no—
62Dic3⋊C4—34no—
63Dic3⋊C4—34no—
64C22×C32C22×D314yes—
65C2×C3×D3—38no—
66C2×C3×D3—64no—
67C2×C3×D3—64no—
68C32×C4—24no—
69C32×C4—24no—
70C32×C4—24no—
71C3×Dic3—38no—
72C3×Dic3—64no—
73C3×Dic3—64no—
74C22⋊Q8—91noSylow
75H(3,𝔽3)C22⋊Q811yescharacteristic, complemented, Sylow
76C23×C3—91no—
77C22×D3—34no—
78C3×Q8—182no—
79C2×C3×C4—64no—
80C2×C3×C4—64no—
81C2×C3×C4—64no—
82C2×C3×C4—91no—
83C2×C3×C4—91no—
84C2×C3×C4—91no—
85C2×C3×C4—182no—
86C2×C3×C4C2×C3⋊D311yescharacteristic
87C2×C3×C4C2×C3⋊D311yescharacteristic
88C2×C3×C4C2×C3⋊D311yescharacteristic
89C2×Dic3—34no—
90C2×Dic3—34no—
91C2×Dic3—34no—
92C4×D3—68no—
93Dic6—68no—
94C2×C32#23514yes—
95C2×C32D3×D414yes—
96C2×C32D3×Q814yes—
97C3×D3—68no—
98C2×Q8—91no—
99C22×C4—91no—
100C4⋊C4—91no—
101C4⋊C4—91no—
102C4⋊C4#27211yescharacteristic, complemented
103C22⋊C4—91no—
104C22⋊C4—91no—
105C22×C3—64no—
106C22×C3—92no—
107C22×C3—181no—
108C22×C3—181no—
109C22×C3C22×C3⋊D311yescharacteristic
110C2×D3—38no—
111C2×D3—64no—
112C2×D3—64no—
113C3×C4—21no—
114C3×C4—21no—
115C3×C4—21no—
116C3×C4—64no—
117C3×C4—92no—
118C3×C4—124no—
119C3×C4—124no—
120C3×C4—181no—
121C3×C4—181no—
122Dic3—38no—
123Dic3—64no—
124Dic3—64no—
125C32#89614yescomplemented
126C23—91no—
127Q8—182no—
128C2×C4—91no—
129C2×C4—91no—
130C2×C4—91no—
131C2×C4—182no—
132C2×C4C2×#27211yescharacteristic
133C2×C4C2×#27211yescharacteristic
134C2×C4C2×#27211yescharacteristic
135C2×C3—64no—
136C2×C3—64no—
137C2×C3—64no—
138C2×C3—182no—
139C2×C3#382811yescharacteristic
140C2×C3D4×C3⋊D311yescharacteristic
141C2×C3Q8×C3⋊D311yescharacteristic
142D3—68no—
143C22—92no—
144C22—181no—
145C22—181no—
146C22C22×#27211yescharacteristic
147C4—21no—
148C4—21no—
149C4—21no—
150C4—92no—
151C4—181no—
152C4—181no—
153C3—64no—
154C3#6410511yescharacteristic
155C2—182no—
156C2#8874011yescharacteristic
157C2D4×#27211yescharacteristic
158C2Q8×#27211yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C23
FittingC4⋊C4×H(3,𝔽3)C2
prime radicalC22×#272C22
hypercenterC3×C4⋊C4C3⋊D3
hypocenterH(3,𝔽3)C22⋊Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#6229C2semidirect
3. C4⋊C4×H(3,𝔽3)C2semidirect
6. #88540C2semidirect
7. #88540C2semidirect
20. C32×C4⋊C4D3semidirect
28. C2×#272Q8semidirect
75. H(3,𝔽3)C22⋊Q8semidirect
102. C4⋊C4#272semidirect
125. C32#896semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2722
C2×#62292
C4⋊C4×H(3,𝔽3)2
#885422
#885402
C3×#8963