Finite Group Databasebeta

← #11723502

Subgroups — #11723502

Order 864.

548 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 110 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×#6293C211yesmaximal, characteristic, complemented
2C2×Dic3×C3⋊D3C211yesmaximal, characteristic
3#88857C211yesmaximal, characteristic, complemented
4C3×#3752C211yesmaximal, characteristic
5C32×C4⋊Dic3C211yesmaximal, characteristic, complemented
6#88806C211yesmaximal, characteristic, complemented
7#88803C211yesmaximal, characteristic
8#64105—31nomaximal
9#63899—34nomaximal
10C22×C3×C3⋊D3C2211yescharacteristic
11C2×C33×C4C2211yescharacteristic
12C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊Dic3C2211yescharacteristic
15C2×C32×Dic3C2211yescharacteristic
16C2×C32×Dic3C2211yescharacteristic
17#6293—22no—
18Dic3×C3⋊D3—22no—
19C2×C4×C3⋊D3—31no—
20C2×#634—31no—
21C2×#627—34no—
22C2×D3×Dic3—34no—
23C32×C4⋊C4—31no—
24#3752—31no—
25#3752D311yescharacteristic
26#3750—31no—
27#3750—34no—
28#3750—31no—
29#3750—64no—
30C3×(C2×D3)⋊C4—34no—
31C3×C4⋊Dic3D314yescomplemented
32#3724—34no—
33(C2×D3)⋊Dic3—34no—
34C22×C33C2311yescharacteristic
35C2×C3×C3⋊D3—21no—
36C2×C3×C3⋊D3—21no—
37C2×C3×C3⋊D3Q812yescomplemented
38C33×C4—21no—
39C3⋊(C3⋊Dic3)—21no—
40C3⋊(C3⋊Dic3)D412yes—
41C3×C3⋊Dic3—21no—
42C32×Dic3—21no—
43C32×Dic3—21no—
44#896—94no—
45#878—91no—
46C22×C3⋊D3C2×D311yescharacteristic
47C22×C3×D3—34no—
48C2×C32×C4—24no—
49C2×C32×C4—31no—
50C2×C32×C4—31no—
51C2×C32×C4C2×D314yes—
52C2×C32×C4C2×D311yescharacteristic
53C2×C3⋊Dic3—31no—
54C2×C3⋊Dic3—31no—
55C2×C3⋊Dic3—34no—
56C2×C3⋊Dic3—34no—
57C2×C3⋊Dic3—64no—
58C2×C3⋊Dic3—64no—
59C2×C3⋊Dic3C2×D311yescharacteristic
60C4×C3⋊D3—62no—
61#634—62no—
62C2×C3×Dic3—34no—
63C2×C3×Dic3C2×D314yes—
64C2×C3×Dic3C2×D314yes—
65#627—68no—
66D3×Dic3—68no—
67C2×C33C2×D411yescharacteristic
68C2×C33C2×Q811yescharacteristic
69C2×C33#4111yescharacteristic
70C3×C3⋊D3—22no—
71C22×Dic3—91no—
72C2×C4×D3—94no—
73C2×Dic6—94no—
74C2×Dic6—91no—
75C3×C4⋊C4—34no—
76C3×C22⋊C4—91no—
77#217—91no—
78(C2×D3)⋊C4—34no—
79(C2×D3)⋊C4—94no—
80C4⋊Dic3C3⋊D311yescharacteristic, complemented
81Dic3⋊C4—94no—
82Dic3⋊C4—94no—
83Dic3⋊C4—91no—
84Dic3⋊C4—91no—
85Dic3⋊C4—184no—
86C22×C32—24no—
87C22×C32C22×D314yes—
88C22×C32C22×D311yescharacteristic
89C2×C3⋊D3—21no—
90C2×C3⋊D3—21no—
91C2×C3⋊D3Dic612yescomplemented
92C2×C3×D3—38no—
93C2×C3×D3—64no—
94C2×C3×D3—64no—
95C32×C4—21no—
96C32×C4—24no—
97C32×C4—44no—
98C32×C4—61no—
99C32×C4—61no—
100C3⋊Dic3—21no—
101C3⋊Dic3—32no—
102C3⋊Dic3—38no—
103C3⋊Dic3—68no—
104C3⋊Dic3—61no—
105C3⋊Dic3—64no—
106C3⋊Dic3—124no—
107C3×Dic3—24no—
108C3×Dic3—24no—
109C3×Dic3—64no—
110C22⋊Q8—271noSylow
111C33C22⋊Q811yescharacteristic, complemented, Sylow
112C23×C3—91no—
113C22×D3—34no—
114C2×C3×C4—24no—
115C2×C3×C4—34no—
116C2×C3×C4—34no—
117C2×C3×C4—91no—
118C2×C3×C4D3214yes—
119C2×C3×C4C2×C3⋊D311yescharacteristic
120C2×Dic3—34no—
121C2×Dic3—94no—
122C2×Dic3—91no—
123C2×Dic3—94no—
124C2×Dic3—91no—
125C2×Dic3—184no—
126C2×Dic3—184no—
127C2×Dic3—182no—
128C2×Dic3C2×C3⋊D311yescharacteristic
129C2×Dic3C2×C3⋊D311yescharacteristic
130C4×D3—188no—
131Dic6—182no—
132Dic6—188no—
133C2×C32—24no—
134C2×C32—24no—
135C2×C32—24no—
136C2×C32C2×Dic611yescharacteristic
137C2×C32#23514yes—
138C2×C32D3×D414yes—
139C2×C32D3×D411yescharacteristic
140C2×C32#23711yescharacteristic
141C2×C32D3×Q814yes—
142C3⋊D3—22no—
143C3×D3—68no—
144C2×Q8—271no—
145C22×C4—271no—
146C4⋊C4—31no—
147C4⋊C4—271no—
148C4⋊C4—271no—
149C22⋊C4—91no—
150C22⋊C4—271no—
151C22×C3—24no—
152C22×C3—92no—
153C22×C3—181no—
154C22×C3—181no—
155C22×C3C2×D3214yes—
156C22×C3C22×C3⋊D311yescharacteristic
157C2×D3—38no—
158C2×D3—64no—
159C2×D3—64no—
160C3×C4—24no—
161C3×C4—21no—
162C3×C4—44no—
163C3×C4—64no—
164C3×C4—64no—
165C3×C4—181no—
166Dic3—21no—
167Dic3—21no—
168Dic3—64no—
169Dic3—98no—
170Dic3—92no—
171Dic3—188no—
172Dic3—184no—
173Dic3—181no—
174Dic3—364no—
175C32—24no—
176C32#87811yescharacteristic, complemented
177C32#89614yescomplemented
178C23—91no—
179Q8—542no—
180C2×C4—31no—
181C2×C4—31no—
182C2×C4—91no—
183C2×C4—271no—
184C2×C4—271no—
185C2×C4—542no—
186C2×C4D3×C3⋊D311yescharacteristic
187C2×C3—24no—
188C2×C3—24no—
189C2×C3—24no—
190C2×C3—182no—
191C2×C3D3×Dic614yes—
192C2×C3#380214yes—
193C2×C3#380414yes—
194C2×C3#382811yescharacteristic
195C2×C3D4×C3⋊D311yescharacteristic
196C2×C3Q8×C3⋊D311yescharacteristic
197D3—68no—
198C22—92no—
199C22—181no—
200C22—181no—
201C22C2×D3×C3⋊D311yescharacteristic
202C4—21no—
203C4—61no—
204C4—61no—
205C4—181no—
206C4—272no—
207C4—541no—
208C3—24no—
209C3#6389914yescomplemented
210C3#6410511yescharacteristic, complemented
211C2—182no—
212C2C3⋊D3×Dic611yescharacteristic
213C2#8902611yescharacteristic
214C2#8903111yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×C3⋊D3
commutator, socle, cosocleC22×C33C23
FittingC2×C33×C4C22
prime radicalC22×C3×C3⋊D3C22
hypercenterC2×C4D3×C3⋊D3
hypocenterC33C22⋊Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#6293C2semidirect
3. #88857C2semidirect
5. C32×C4⋊Dic3C2semidirect
6. #88806C2semidirect
31. C3×C4⋊Dic3D3semidirect
37. C2×C3×C3⋊D3Q8semidirect
80. C4⋊Dic3C3⋊D3semidirect
91. C2×C3⋊D3Dic6semidirect
111. C33C22⋊Q8semidirect
176. C32#878semidirect
177. C32#896semidirect
209. C3#63899semidirect
210. C3#64105semidirect

Maximal subgroups

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

subgroupindex
C2×#62932
C2×Dic3×C3⋊D32
#888572
C3×#37522
C32×C4⋊Dic32
#888062
#888032
#641053
#638993