Finite Group Databasebeta

← #11723430

Subgroups — #11723430

Order 864.

464 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 104 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
1C4⋊(C3⋊(C3⋊Dic3))C211yesmaximal, characteristic, complemented
2C3×#3750C211yesmaximal, characteristic, complemented
3C32×(C2×D3)⋊C4C211yesmaximal, characteristic
4#88806C211yesmaximal, characteristic, complemented
5(C2×D3)⋊(C3⋊Dic3)C211yesmaximal, characteristic
6(C2×D3)⋊(C3⋊Dic3)C211yesmaximal, characteristic
7Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#64088—31nomaximal
9#63861—34nomaximal
10C22×C32×D3C2211yescharacteristic
11C2×C33×C4C2211yescharacteristic
12C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊Dic3C2211yescharacteristic
16C2×C32×Dic3C2211yescharacteristic
17C32×C22⋊C4—31no—
18#3757—31no—
19#3757—31no—
20C4⋊(C3⋊Dic3)—31no—
21C4⋊(C3⋊Dic3)—34no—
22C4⋊(C3⋊Dic3)—64no—
23#3750—31no—
24#3750D311yescharacteristic, complemented
25C4×C3⋊Dic3—31no—
26C3×(C2×D3)⋊C4D314yes—
27C3×Dic3⋊C4—34no—
28#3724—34no—
29(C2×D3)⋊Dic3—34no—
30(C2×D3)⋊Dic3—34no—
31Dic32—34no—
32C22×C33C2311yescharacteristic
33C2×C32×D3—21no—
34C2×C32×D3—21no—
35C2×C32×D3—21no—
36C33×C4—21no—
37C3⋊(C3⋊Dic3)—21no—
38C3⋊(C3⋊Dic3)—21no—
39C3×C3⋊Dic3—21no—
40C3×C3⋊Dic3—21no—
41C32×Dic3—21no—
42#898—91no—
43#879—94no—
44C23×C32—31no—
45C22×C3×D3C2×D314yes—
46C2×C32×C4—24no—
47C2×C32×C4—31no—
48C2×C32×C4C2×D314yes—
49C2×C32×C4C2×D311yescharacteristic
50C2×C3⋊Dic3—31no—
51C2×C3⋊Dic3—31no—
52C2×C3⋊Dic3—34no—
53C2×C3⋊Dic3—34no—
54C2×C3⋊Dic3—64no—
55C2×C3⋊Dic3—64no—
56C2×C3⋊Dic3C2×D311yescharacteristic
57C2×C3⋊Dic3C2×D311yescharacteristic
58C2×C3×Dic3—34no—
59C2×C3×Dic3—34no—
60C2×C3×Dic3C2×D314yes—
61C2×C33#4111yescharacteristic
62C2×C33#4111yescharacteristic
63C2×C33#4111yescharacteristic
64C32×D3—41no—
65C3×C4⋊C4—91no—
66C3×C22⋊C4—34no—
67#217—94no—
68#217—94no—
69(C2×D3)⋊C4—91no—
70(C2×D3)⋊C4—91no—
71(C2×D3)⋊C4C3⋊D311yescharacteristic
72C4⋊Dic3—94no—
73C4⋊Dic3—91no—
74C4⋊Dic3—184no—
75Dic3⋊C4—34no—
76Dic3⋊C4—94no—
77Dic3⋊C4—91no—
78C4×Dic3—94no—
79C4×Dic3—91no—
80C22×C32—24no—
81C22×C32—61no—
82C22×C32—61no—
83C22×C32—61no—
84C22×C32C22×D314yes—
85C22×C32C22×D311yescharacteristic
86C2×C3×D3—24no—
87C2×C3×D3—24no—
88C2×C3×D3—24no—
89C32×C4—28no—
90C32×C4—24no—
91C32×C4—21no—
92C32×C4—61no—
93C3⋊Dic3—21no—
94C3⋊Dic3—21no—
95C3⋊Dic3—61no—
96C3⋊Dic3—61no—
97C3⋊Dic3—64no—
98C3⋊Dic3—64no—
99C3⋊Dic3—124no—
100C3⋊Dic3—124no—
101C3×Dic3—24no—
102C3×Dic3—64no—
103C3×Dic3—64no—
104#126—271noSylow
105C33#12611yescharacteristic, complemented, Sylow
106C23×C3—34no—
107C22×D3C2×C3⋊D311yescharacteristic
108C2×C3×C4—24no—
109C2×C3×C4—34no—
110C2×C3×C4—91no—
111C2×C3×C4—91no—
112C2×C3×C4D3214yes—
113C2×C3×C4C2×C3⋊D311yescharacteristic
114C2×Dic3—34no—
115C2×Dic3—34no—
116C2×Dic3—94no—
117C2×Dic3—94no—
118C2×Dic3—91no—
119C2×Dic3—91no—
120C2×Dic3—184no—
121C2×Dic3—184no—
122C2×Dic3C2×C3⋊D311yescharacteristic
123C2×C32—24no—
124C2×C32—24no—
125C2×C32—24no—
126C2×C32—121no—
127C2×C32#23514yes—
128C2×C32#23511yescharacteristic
129C2×C32#23714yes—
130C2×C32#23714yes—
131C2×C32#23711yescharacteristic
132C2×C32#23911yescharacteristic
133C3×D3—44no—
134C4⋊C4—91no—
135C4⋊C4—271no—
136C4⋊C4—271no—
137C22⋊C4—31no—
138C22⋊C4—271no—
139C22⋊C4—271no—
140C42—271no—
141C22×C3—24no—
142C22×C3—64no—
143C22×C3—64no—
144C22×C3—64no—
145C22×C3C2×D3214yes—
146C22×C3C22×C3⋊D311yescharacteristic
147C2×D3—21no—
148C2×D3—21no—
149C2×D3—21no—
150C3×C4—28no—
151C3×C4—24no—
152C3×C4—21no—
153C3×C4—64no—
154C3×C4—181no—
155C3×C4—181no—
156Dic3—21no—
157Dic3—64no—
158Dic3—64no—
159Dic3—184no—
160Dic3—184no—
161Dic3—181no—
162Dic3—181no—
163Dic3—364no—
164Dic3—364no—
165C32—24no—
166C32#87914yescomplemented
167C32#89811yescharacteristic, complemented
168C23—31no—
169C2×C4—31no—
170C2×C4—91no—
171C2×C4—91no—
172C2×C4—271no—
173C2×C4—271no—
174C2×C4D3×C3⋊D311yescharacteristic
175C2×C3—24no—
176C2×C3—24no—
177C2×C3—24no—
178C2×C3—124no—
179C2×C3#379514yes—
180C2×C3#379914yes—
181C2×C3#380514yes—
182C2×C3#382811yescharacteristic
183C2×C3#383011yescharacteristic
184C2×C3#383011yescharacteristic
185D3—41no—
186C22—61no—
187C22—61no—
188C22—61no—
189C22C2×D3×C3⋊D311yescharacteristic
190C4—21no—
191C4—61no—
192C4—181no—
193C4—181no—
194C4—541no—
195C4—541no—
196C3—24no—
197C3#6386114yescomplemented
198C3#6408811yescharacteristic, complemented
199C2—121no—
200C2#8901411yescharacteristic
201C2#8902211yescharacteristic
202C2#8902911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×C3⋊D3
commutator, socle, cosocleC22×C33C23
FittingC2×C33×C4C22
prime radicalC22×C32×D3C22
hypercenterC2×C4D3×C3⋊D3
hypocenterC33#126
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C4⋊(C3⋊(C3⋊Dic3))C2semidirect
2. C3×#3750C2semidirect
4. #88806C2semidirect
7. Dic3×C3⋊Dic3C2semidirect
24. #3750D3semidirect
105. C33#126semidirect
166. C32#879semidirect
167. C32#898semidirect
197. C3#63861semidirect
198. C3#64088semidirect

Maximal subgroups

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

subgroupindex
C4⋊(C3⋊(C3⋊Dic3))2
C3×#37502
C32×(C2×D3)⋊C42
#888062
(C2×D3)⋊(C3⋊Dic3)2
Dic3×C3⋊Dic32
#640883
#638613