Finite Group Databasebeta

← #11721307

Subgroups — #11721307

Order 864.

327 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 327 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C22×#1084C211yesmaximal, characteristic, complemented
2C2×#638⋊C3C211yesmaximal, characteristic
3#3757⋊C3C211yesmaximal, characteristic, complemented
4#88450C212yesmaximal, complemented
5#88448C212yesmaximal, complemented
6#63966—31nomaximal
7#63961—31nomaximal
8C23×H(3,𝔽3)C2211yescharacteristic
9C22×(C3⋊D3)⋊C3C2211yescharacteristic
10C2×#1084—22no—
11C2×#1084C2212yes—
12#638⋊C3—22no—
13C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C2×(C3⋊Dic3)⋊C3C2212yes—
15C2×#638D311yescharacteristic
16C2×C3×#67—31no—
17C22×C3×Dic3—31no—
18C22×C3×Dic3—31no—
19C22×C3×Dic3—61no—
20#3757D311yescharacteristic, complemented
21C3×#217—31no—
22#3724—32no—
23Dic3⋊Dic3—32no—
24#3722—32no—
25(C2×D3)⋊Dic3—32no—
26C22×H(3,𝔽3)—22no—
27C22×H(3,𝔽3)D412yes—
28C22×H(3,𝔽3)C2311yescharacteristic
29C2×(C3⋊D3)⋊C3—21no—
30C2×(C3⋊D3)⋊C3—22no—
31#1084—22no—
32(C3⋊Dic3)⋊C3—21no—
33(C3⋊Dic3)⋊C3—22no—
34#935—91no—
35#929—91no—
36#886—91no—
37C23×C32—21no—
38C23×C32C2×D311yescharacteristic
39C23×C32C2×D311yescharacteristic
40C22×C3⋊D3C2×D311yescharacteristic
41C22×C3×D3—31no—
42#638—22no—
43C2×C3⋊Dic3C2×D312yes—
44C2×C3⋊Dic3C2×D311yescharacteristic
45C3×#67—62no—
46C2×C3×Dic3—32no—
47C2×C3×Dic3—31no—
48C2×C3×Dic3—32no—
49C2×C3×Dic3—32no—
50C2×C3×Dic3—62no—
51C2×C3×Dic3—64no—
52C2×C3×Dic3—62no—
53C2×C3×Dic3—62no—
54C2×H(3,𝔽3)—22no—
55C2×H(3,𝔽3)C2×D411yescharacteristic
56C2×H(3,𝔽3)#4112yes—
57(C3⋊D3)⋊C3—41no—
58C2×C3×D4—91no—
59C22×C3×C4—91no—
60C2×#67—31no—
61C2×#67—91no—
62C22×Dic3—31no—
63C22×Dic3—31no—
64C22×Dic3—61no—
65C3×C22⋊C4—91no—
66#217—31no—
67#217—91no—
68#217—92no—
69(C2×D3)⋊C4—92no—
70(C2×D3)⋊C4—92no—
71C4⋊Dic3—92no—
72Dic3⋊C4—92no—
73Dic3⋊C4—92no—
74C22×C32—24no—
75C22×C32—22no—
76C22×C32—22no—
77C22×C32—22no—
78C22×C32—21no—
79C22×C32D1212yes—
80C22×C32#6712yes—
81C22×C32C22×D311yescharacteristic
82C22×C32C22×D311yescharacteristic
83C2×C3⋊D3—22no—
84C2×C3⋊D3—21no—
85C2×C3×D3—61no—
86C2×C3×D3—62no—
87C3⋊Dic3—22no—
88C3⋊Dic3—21no—
89C3×Dic3—64no—
90C3×Dic3—61no—
91C3×Dic3—62no—
92C3×Dic3—62no—
93C3×Dic3—62no—
94#123—271noSylow
95H(3,𝔽3)#12311yescharacteristic, complemented, Sylow
96C23×C3—31no—
97C23×C3—31no—
98C23×C3—61no—
99C23×C3—91no—
100C23×C3D3211yescharacteristic
101C22×D3—31no—
102C22×D3—91no—
103C3×D4—182no—
104C2×C3×C4—91no—
105C2×C3×C4—92no—
106C2×C3×C4—92no—
107C2×C3×C4—182no—
108#67—62no—
109#67—182no—
110C2×Dic3—32no—
111C2×Dic3—32no—
112C2×Dic3—31no—
113C2×Dic3—32no—
114C2×Dic3—62no—
115C2×Dic3—62no—
116C2×Dic3—62no—
117C2×Dic3—64no—
118C2×Dic3—92no—
119C2×Dic3—91no—
120C2×C32—24no—
121C2×C32—22no—
122C2×C32—22no—
123C2×C32—22no—
124C2×C32—21no—
125C2×C32C2×D1211yescharacteristic
126C2×C32#23712yes—
127C2×C32#23712yes—
128C2×C32C2×#6711yescharacteristic
129C3⋊D3—41no—
130C3×D3—121no—
131C2×D4—91no—
132C22×C4—91no—
133C4⋊C4—272no—
134C22⋊C4—91no—
135C22⋊C4—272no—
136C22×C3—22no—
137C22×C3—32no—
138C22×C3—32no—
139C22×C3—31no—
140C22×C3—31no—
141C22×C3—62no—
142C22×C3—62no—
143C22×C3—64no—
144C22×C3—62no—
145C22×C3—61no—
146C22×C3—181no—
147C22×C3—182no—
148C22×C3#62612yes—
149C22×C3C2×D3211yescharacteristic
150C2×D3—62no—
151C2×D3—61no—
152C2×D3—181no—
153C2×D3—182no—
154C3×C4—181no—
155C3×C4—182no—
156C3×C4—182no—
157Dic3—62no—
158Dic3—62no—
159Dic3—64no—
160Dic3—62no—
161Dic3—61no—
162Dic3—181no—
163Dic3—182no—
164C32—21no—
165C32#88611yescharacteristic, complemented
166C32#93511yescharacteristic, complemented
167C23—91no—
168C23((C3⋊D3)⋊C3)⋊C211yescharacteristic
169D4—182no—
170C2×C4—92no—
171C2×C4—92no—
172C2×C4—91no—
173C2×C4—182no—
174C2×C3—22no—
175C2×C3—32no—
176C2×C3—32no—
177C2×C3—31no—
178C2×C3—31no—
179C2×C3—62no—
180C2×C3—62no—
181C2×C3—64no—
182C2×C3—62no—
183C2×C3—61no—
184C2×C3—361no—
185C2×C3#380512yes—
186C2×C3C2×#62611yescharacteristic
187D3—121no—
188D3—361no—
189C22—22no—
190C22—182no—
191C22—181no—
192C22#620012yes—
193C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
194C4—182no—
195C4—182no—
196C4—181no—
197C3—31no—
198C3—31no—
199C3—61no—
200C3#6396211yescharacteristic
201C2—22no—
202C2—361no—
203C2#8867212yes—
204C2C2×#620011yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C23
Frattini, socleC22×C3C2×D32
FittingC23×H(3,𝔽3)C22
prime radicalC2×#638⋊C3C2
hypercenterC23((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#123
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×#1084C2semidirect
3. #3757⋊C3C2semidirect
4. #88450C2semidirect
5. #88448C2semidirect
20. #3757D3semidirect
95. H(3,𝔽3)#123semidirect
165. C32#886semidirect
166. C32#935semidirect

Maximal subgroups

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

subgroupindex
C22×#10842
C2×#638⋊C32
#3757⋊C32
#884502
#884482
#639663
#639613