Finite Group Databasebeta

← #11721105

Subgroups — #11721105

Order 864.

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

normal subgroups only

This group has 296 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×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C4×#1084C211yesmaximal, characteristic, complemented
3C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4#88450C212yesmaximal
5#88448C212yesmaximal, complemented
6#63874—31nomaximal
7#63855—31nomaximal
8C22×(C3⋊D3)⋊C3C2211yescharacteristic
9C2×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#1084C2212yes—
11C2×(C3⋊Dic3)⋊C3C2212yes—
12C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C4×(C3⋊D3)⋊C3C414yescomplemented
14C2×C4×C3⋊D3D311yescharacteristic
15C2×C3×C4×D3—31no—
16C4×C3⋊Dic3D311yescharacteristic, complemented
17C3×C4×Dic3—31no—
18C3×C4×Dic3—31no—
19C3×C4×Dic3—31no—
20C3×C4×Dic3—61no—
21#3724—32no—
22Dic3⋊Dic3—32no—
23#3722—32no—
24(C2×D3)⋊Dic3—32no—
25C22×H(3,𝔽3)C2311yescharacteristic
26C2×(C3⋊D3)⋊C3—22no—
27C2×(C3⋊D3)⋊C3C2×C412yes—
28C4×H(3,𝔽3)C2×C412yes—
29#1084—22no—
30(C3⋊Dic3)⋊C3—22no—
31(C3⋊Dic3)⋊C3C2×C412yes—
32#926—91no—
33#872—91no—
34#872—91no—
35C22×C3⋊D3C2×D311yescharacteristic
36C22×C3×D3—31no—
37C2×C32×C4—21no—
38C2×C32×C4C2×D311yescharacteristic
39C2×C32×C4C2×D311yescharacteristic
40C2×C3⋊Dic3C2×D312yes—
41C2×C3⋊Dic3C2×D311yescharacteristic
42C4×C3⋊D3Dic314yescomplemented
43C2×C3×Dic3—31no—
44C2×C3×Dic3—32no—
45C2×C3×Dic3—32no—
46C2×C3×Dic3—32no—
47C2×C3×Dic3—62no—
48C3×C4×D3—34no—
49C2×H(3,𝔽3)C22×C411yescharacteristic
50C2×H(3,𝔽3)#4112yes—
51(C3⋊D3)⋊C3—22no—
52C22×C3×C4—91no—
53C2×C4×D3—31no—
54C2×C4×D3—181no—
55C3×C42—91no—
56C3×C42—91no—
57#217—92no—
58(C2×D3)⋊C4—92no—
59(C2×D3)⋊C4—92no—
60C4⋊Dic3—92no—
61Dic3⋊C4—92no—
62Dic3⋊C4—92no—
63C4×Dic3—31no—
64C4×Dic3—31no—
65C4×Dic3—31no—
66C4×Dic3—61no—
67C4×Dic3—91no—
68C22×C32—21no—
69C22×C32C22×D311yescharacteristic
70C22×C32C22×D311yescharacteristic
71C2×C3⋊D3—22no—
72C2×C3⋊D3C2×Dic312yes—
73C2×C3×D3—32no—
74C2×C3×D3—62no—
75C32×C4—22no—
76C32×C4C4×D312yes—
77C32×C4C2×Dic312yes—
78C3⋊Dic3—22no—
79C3⋊Dic3C2×Dic312yes—
80C3×Dic3—32no—
81C3×Dic3—64no—
82C3×Dic3—62no—
83C3×Dic3—62no—
84C3×Dic3—62no—
85#117—271noSylow
86H(3,𝔽3)#11711yescharacteristic, complemented, Sylow
87C23×C3—91no—
88C22×D3—31no—
89C22×D3—181no—
90C2×C3×C4—31no—
91C2×C3×C4—61no—
92C2×C3×C4—61no—
93C2×C3×C4—94no—
94C2×C3×C4—91no—
95C2×C3×C4—92no—
96C2×C3×C4—92no—
97C2×C3×C4D3211yescharacteristic
98C2×Dic3—31no—
99C2×Dic3—32no—
100C2×Dic3—32no—
101C2×Dic3—32no—
102C2×Dic3—62no—
103C2×Dic3—92no—
104C2×Dic3—181no—
105C4×D3—34no—
106C4×D3—184no—
107C2×C32—22no—
108C2×C32—21no—
109C2×C32C2×C4×D311yescharacteristic
110C2×C32#23512yes—
111C2×C32#23512yes—
112C2×C32C22×Dic311yescharacteristic
113C3⋊D3—22no—
114C3×D3—62no—
115C22×C4—91no—
116C4⋊C4—272no—
117C22⋊C4—272no—
118C42—91no—
119C42—91no—
120C22×C3—31no—
121C22×C3—61no—
122C22×C3—61no—
123C22×C3—92no—
124C22×C3—182no—
125C22×C3C2×D3211yescharacteristic
126C2×D3—32no—
127C2×D3—62no—
128C2×D3—182no—
129C2×D3—182no—
130C3×C4—32no—
131C3×C4—62no—
132C3×C4—62no—
133C3×C4—92no—
134C3×C4—182no—
135C3×C4—182no—
136C3×C4D3×Dic312yes—
137Dic3—32no—
138Dic3—62no—
139Dic3—62no—
140Dic3—64no—
141Dic3—62no—
142Dic3—182no—
143Dic3—182no—
144C32—21no—
145C32#87211yescharacteristic, complemented
146C32#92611yescharacteristic, complemented
147C23—91no—
148C2×C4—91no—
149C2×C4—94no—
150C2×C4—92no—
151C2×C4—92no—
152C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
153C2×C3—32no—
154C2×C3—31no—
155C2×C3—62no—
156C2×C3—62no—
157C2×C3—61no—
158C2×C3—61no—
159C2×C3—182no—
160C2×C3#379812yes—
161C2×C3C2×D3×Dic311yescharacteristic
162D3—62no—
163D3—182no—
164C22—92no—
165C22—182no—
166C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
167C4—92no—
168C4—182no—
169C4—182no—
170C4((C3⋊D3)⋊C3)⋊C412yes—
171C3—31no—
172C3—61no—
173C3—61no—
174C3#6385511yescharacteristic
175C2—182no—
176C2#8865012yes—
177C2C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
subgroupG/H
center, hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C22×C4
Frattini, socleC22×C3C2×D32
FittingC2×C4×H(3,𝔽3)C22
prime radicalC22×(C3⋊D3)⋊C3C22
cosocleC22×H(3,𝔽3)C23
hypocenterH(3,𝔽3)#117
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C4×#1084C2semidirect
3. C4×(C3⋊Dic3)⋊C3C2semidirect
5. #88448C2semidirect
13. C4×(C3⋊D3)⋊C3C4semidirect
16. C4×C3⋊Dic3D3semidirect
42. C4×C3⋊D3Dic3semidirect
86. H(3,𝔽3)#117semidirect
145. C32#872semidirect
146. C32#926semidirect

Maximal subgroups

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

subgroupindex
C2×C4×(C3⋊D3)⋊C32
C4×#10842
C4×(C3⋊Dic3)⋊C32
#884502
#884482
#638743
#638553