Finite Group Databasebeta

← #11721089

Subgroups — #11721089

Order 864.

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

normal subgroups only

This group has 320 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×C4×#272C211yesmaximal, characteristic
2C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
3C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
4#88451C212yesmaximal
5#88447C212yesmaximal, complemented
6#63841—31nomaximal
7#63841—31nomaximal
8C22×#272C2211yescharacteristic
9C2×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#1084C2211yescharacteristic
11C4×#272C414yescomplemented
12C2×(C3⋊Dic3)⋊C3C2212yes—
13C2×(C3⋊Dic3)⋊C3C2212yes—
14C2×C3×C4×D3—31no—
15C2×C3×C4×D3—31no—
16C2×C3×C4×D3—61no—
17C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
18C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
19C3×C4⋊Dic3—31no—
20C3×C4⋊Dic3—31no—
21(C2×D3)⋊Dic3—32no—
22(C2×D3)⋊Dic3—32no—
23Dic32—32no—
24Dic32—32no—
25C22×H(3,𝔽3)C2311yescharacteristic
26C2×#272—21no—
27C2×#272—21no—
28C2×#272C2×C412yes—
29C4×H(3,𝔽3)C2×C412yes—
30#1084C2×C412yes—
31(C3⋊Dic3)⋊C3—22no—
32(C3⋊Dic3)⋊C3—22no—
33#926—91no—
34#892—91no—
35#892—91no—
36C22×C3×D3—31no—
37C22×C3×D3—31no—
38C22×C3×D3—61no—
39C2×C32×C4—21no—
40C2×C32×C4C2×D311yescharacteristic
41C2×C32×C4C2×D311yescharacteristic
42C2×C3⋊Dic3C2×D312yes—
43C2×C3⋊Dic3C2×D312yes—
44C2×C3×Dic3—31no—
45C2×C3×Dic3—31no—
46C2×C3×Dic3—32no—
47C2×C3×Dic3—32no—
48C2×C3×Dic3—61no—
49C3×C4×D3—34no—
50C3×C4×D3—34no—
51C3×C4×D3—64no—
52C2×H(3,𝔽3)C22×C411yescharacteristic
53C2×H(3,𝔽3)#4111yescharacteristic
54C2×H(3,𝔽3)#4111yescharacteristic
55#272—22no—
56C22×C3×C4—91no—
57C2×C4×D3—31no—
58C2×C4×D3—31no—
59C2×C4×D3—61no—
60C3×C4⋊C4—91no—
61C3×C4⋊C4—91no—
62#217—92no—
63(C2×D3)⋊C4—92no—
64(C2×D3)⋊C4—92no—
65C4⋊Dic3—31no—
66C4⋊Dic3—31no—
67C4⋊Dic3—91no—
68C4⋊Dic3—181no—
69C4×Dic3—92no—
70C4×Dic3—92no—
71C4×Dic3—92no—
72C22×C32—21no—
73C22×C32C22×D311yescharacteristic
74C22×C32C22×D311yescharacteristic
75C2×C3×D3—32no—
76C2×C3×D3—32no—
77C2×C3×D3—62no—
78C2×C3×D3—62no—
79C2×C3×D3—62no—
80C2×C3×D3—61no—
81C2×C3×D3—61no—
82C2×C3×D3—61no—
83C2×C3×D3—61no—
84C32×C4—22no—
85C32×C4C4×D312yes—
86C32×C4C4×D312yes—
87C3⋊Dic3—22no—
88C3⋊Dic3—22no—
89C3×Dic3—32no—
90C3×Dic3—32no—
91C3×Dic3—62no—
92C3×Dic3—62no—
93C3×Dic3—62no—
94#117—271noSylow
95H(3,𝔽3)#11711yescharacteristic, complemented, Sylow
96C23×C3—91no—
97C22×D3—31no—
98C22×D3—31no—
99C22×D3—61no—
100C2×C3×C4—31no—
101C2×C3×C4—61no—
102C2×C3×C4—61no—
103C2×C3×C4—91no—
104C2×C3×C4—94no—
105C2×C3×C4—92no—
106C2×C3×C4—92no—
107C2×C3×C4D3211yescharacteristic
108C2×Dic3—31no—
109C2×Dic3—31no—
110C2×Dic3—32no—
111C2×Dic3—32no—
112C2×Dic3—61no—
113C2×Dic3—92no—
114C2×Dic3—182no—
115C4×D3—34no—
116C4×D3—34no—
117C4×D3—64no—
118C2×C32—21no—
119C2×C32—21no—
120C2×C32—21no—
121C2×C32C2×C4×D311yescharacteristic
122C2×C32C2×C4×D311yescharacteristic
123C2×C32#23711yescharacteristic
124C2×C32#23711yescharacteristic
125C2×C32#23911yescharacteristic
126C2×C32#23911yescharacteristic
127C3×D3—64no—
128C3×D3—62no—
129C3×D3—62no—
130C22×C4—91no—
131C4⋊C4—91no—
132C4⋊C4—91no—
133C22⋊C4—272no—
134C42—272no—
135C22×C3—31no—
136C22×C3—61no—
137C22×C3—61no—
138C22×C3—92no—
139C22×C3—181no—
140C22×C3—181no—
141C22×C3C2×D3211yescharacteristic
142C2×D3—32no—
143C2×D3—32no—
144C2×D3—62no—
145C2×D3—61no—
146C2×D3—61no—
147C2×D3—61no—
148C2×D3—61no—
149C2×D3—62no—
150C2×D3—62no—
151C3×C4—32no—
152C3×C4—62no—
153C3×C4—62no—
154C3×C4—92no—
155C3×C4—182no—
156C3×C4—182no—
157C3×C4#62412yes—
158Dic3—32no—
159Dic3—32no—
160Dic3—62no—
161Dic3—62no—
162Dic3—62no—
163Dic3—182no—
164Dic3—362no—
165C32—21no—
166C32#89211yescharacteristic, complemented
167C32#89211yescharacteristic, complemented
168C23—91no—
169C2×C4—91no—
170C2×C4—94no—
171C2×C4—92no—
172C2×C4—92no—
173C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
174C2×C3—31no—
175C2×C3—31no—
176C2×C3—31no—
177C2×C3—61no—
178C2×C3—61no—
179C2×C3—61no—
180C2×C3—61no—
181C2×C3—61no—
182C2×C3—61no—
183C2×C3—182no—
184C2×C3#379611yescharacteristic
185C2×C3#379611yescharacteristic
186C2×C3C2×#62411yescharacteristic
187D3—62no—
188D3—62no—
189D3—64no—
190C22—92no—
191C22—181no—
192C22—181no—
193C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
194C4—92no—
195C4—182no—
196C4—182no—
197C4#619912yes—
198C3—31no—
199C3—61no—
200C3—61no—
201C3#6384911yescharacteristic
202C2—182no—
203C2#8864911yescharacteristic
204C2#8864911yescharacteristic
205C2C2×#619911yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C22×C4
Frattini, socleC22×C3C2×D32
FittingC2×C4×H(3,𝔽3)C22
prime radicalC22×#272C22
cosocleC22×H(3,𝔽3)C23
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#117
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C4⋊((C3⋊Dic3)⋊C3)C2semidirect
3. C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. #88447C2semidirect
11. C4×#272C4semidirect
17. C4⋊(C3⋊Dic3)D3semidirect
18. C4⋊(C3⋊Dic3)D3semidirect
95. H(3,𝔽3)#117semidirect
166. C32#892semidirect
167. C32#892semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2722
C4⋊((C3⋊Dic3)⋊C3)2
#884512
#884472
#638413