Finite Group Databasebeta

← #11721116

Subgroups — #11721116

Order 864.

264 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 68 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×#6196C212yesmaximal
2C4⋊#1084C211yesmaximal, characteristic
3C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic
4C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
5#88448C212yesmaximal
6#63919—31nomaximal
7#63918—31nomaximal
8C2×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#1084C2212yes—
10C2×(C3⋊Dic3)⋊C3C2212yes—
11C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
12C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
13#6196—24no—
14C2×#627—32no—
15C2×#627—32no—
16C4⋊(C3⋊Dic3)D311yescharacteristic
17C4×C3⋊Dic3D311yescharacteristic
18C3×C4⋊Dic3—31no—
19C3×C4⋊Dic3—31no—
20C3×C4⋊Dic3—31no—
21C3×C4⋊Dic3—61no—
22C3×C4×Dic3—31no—
23#3724—32no—
24Dic3⋊Dic3—32no—
25C22×H(3,𝔽3)C2311yescharacteristic
26C4×H(3,𝔽3)Q812yescomplemented
27#1084—22no—
28(C3⋊Dic3)⋊C3—22no—
29(C3⋊Dic3)⋊C3D412yes—
30(C3⋊Dic3)⋊C3Q812yescomplemented
31#888—91no—
32#888—91no—
33#869—91no—
34C2×C32×C4—21no—
35C2×C32×C4C2×D311yescharacteristic
36C2×C32×C4C2×D311yescharacteristic
37C2×C3⋊Dic3C2×D312yes—
38C2×C3⋊Dic3C2×D311yescharacteristic
39C2×C3⋊Dic3C2×D311yescharacteristic
40C2×C3×Dic3—31no—
41C2×C3×Dic3—31no—
42C2×C3×Dic3—32no—
43C2×C3×Dic3—32no—
44C2×C3×Dic3—32no—
45C2×C3×Dic3—62no—
46#627—64no—
47#627—64no—
48C2×H(3,𝔽3)C2×D411yescharacteristic
49C2×H(3,𝔽3)C2×Q811yescharacteristic
50C2×H(3,𝔽3)C2×Q811yescharacteristic
51C2×Dic6—92no—
52C2×Dic6—92no—
53C2×Dic6—92no—
54C3×C4⋊C4—91no—
55C3×C4⋊C4—91no—
56C3×C42—91no—
57C4⋊Dic3—31no—
58C4⋊Dic3—31no—
59C4⋊Dic3—31no—
60C4⋊Dic3—61no—
61C4⋊Dic3—91no—
62C4⋊Dic3—92no—
63Dic3⋊C4—92no—
64Dic3⋊C4—92no—
65C4×Dic3—31no—
66C4×Dic3—181no—
67C22×C32—21no—
68C22×C32C22×D311yescharacteristic
69C22×C32C22×D311yescharacteristic
70C32×C4—22no—
71C32×C4Dic612yescomplemented
72C32×C4Dic612yescomplemented
73C3⋊Dic3—22no—
74C3⋊Dic3Dic612yescomplemented
75C3⋊Dic3D1212yes—
76C3×Dic3—32no—
77C3×Dic3—32no—
78C3×Dic3—62no—
79C3×Dic3—62no—
80C3×Dic3—62no—
81C3×Dic3—122no—
82C4⋊Q8—271noSylow
83H(3,𝔽3)C4⋊Q811yescharacteristic, complemented, Sylow
84C2×C3×C4—31no—
85C2×C3×C4—61no—
86C2×C3×C4—91no—
87C2×C3×C4—91no—
88C2×C3×C4—92no—
89C2×C3×C4—92no—
90C2×C3×C4—121no—
91C2×C3×C4D3211yescharacteristic
92C2×Dic3—31no—
93C2×Dic3—31no—
94C2×Dic3—32no—
95C2×Dic3—32no—
96C2×Dic3—32no—
97C2×Dic3—62no—
98C2×Dic3—92no—
99C2×Dic3—181no—
100C2×Dic3—181no—
101Dic6—184no—
102Dic6—184no—
103Dic6—184no—
104C2×C32—21no—
105C2×C32—21no—
106C2×C32—21no—
107C2×C32C2×Dic611yescharacteristic
108C2×C32C2×Dic611yescharacteristic
109C2×C32C2×Dic611yescharacteristic
110C2×C32C2×D1211yescharacteristic
111C2×C32D3×D411yescharacteristic
112C2×C32D3×Q811yescharacteristic
113C2×Q8—272no—
114C4⋊C4—91no—
115C4⋊C4—91no—
116C4⋊C4—272no—
117C42—91no—
118C22×C3—31no—
119C22×C3—61no—
120C22×C3—121no—
121C22×C3C2×D3211yescharacteristic
122C3×C4—32no—
123C3×C4—62no—
124C3×C4—92no—
125C3×C4—92no—
126C3×C4—122no—
127C3×C4—182no—
128C3×C4—182no—
129C3×C4#62712yes—
130Dic3—32no—
131Dic3—32no—
132Dic3—62no—
133Dic3—62no—
134Dic3—62no—
135Dic3—122no—
136Dic3—182no—
137Dic3—182no—
138Dic3—182no—
139C32—21no—
140C32#86911yescharacteristic, complemented
141C32#88811yescharacteristic, complemented
142Q8—544no—
143C2×C4—91no—
144C2×C4—91no—
145C2×C4—92no—
146C2×C4—92no—
147C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
148C2×C3—31no—
149C2×C3—31no—
150C2×C3—31no—
151C2×C3—61no—
152C2×C3—61no—
153C2×C3—61no—
154C2×C3—121no—
155C2×C3—121no—
156C2×C3—121no—
157C2×C3D3×Dic611yescharacteristic
158C2×C3D3×D1211yescharacteristic
159C2×C3C2×#62711yescharacteristic
160C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
161C4—92no—
162C4—92no—
163C4—182no—
164C4—182no—
165C4#619612yescomplemented
166C3—31no—
167C3—61no—
168C3—121no—
169C3#6391811yescharacteristic
170C2((C3⋊D3)⋊C3)⋊Q811yescharacteristic
171C2#8865411yescharacteristic
172C2C2×#619611yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, prime radical, cosocleC22×H(3,𝔽3)C23
Frattini, socleC22×C3C2×D32
FittingC2×C4×H(3,𝔽3)C22
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4⋊Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
26. C4×H(3,𝔽3)Q8semidirect
30. (C3⋊Dic3)⋊C3Q8semidirect
71. C32×C4Dic6semidirect
72. C32×C4Dic6semidirect
74. C3⋊Dic3Dic6semidirect
83. H(3,𝔽3)C4⋊Q8semidirect
140. C32#869semidirect
141. C32#888semidirect
165. C4#6196semidirect

Maximal subgroups

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

subgroupindex
C2×#61962
C4⋊#10842
C4⋊((C3⋊Dic3)⋊C3)2
C4×(C3⋊Dic3)⋊C32
#884482
#639193
#639183