Finite Group Databasebeta

← #11723596

Subgroups — #11723596

Order 864.

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

normal subgroups only

This group has 496 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
1#88864C211yesmaximal, characteristic
2C3×#3757C211yesmaximal, characteristic
3C32×#217C211yesmaximal, characteristic
4#88807C211yesmaximal, characteristic, complemented
5#88806C211yesmaximal, characteristic, complemented
6Dic3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
7Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#64088—31nomaximal
9#63956—34nomaximal
10C23×C33C2211yescharacteristic
11C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
12C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C2×C3×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊Dic3C2211yescharacteristic
15C2×C32×Dic3C2211yescharacteristic
16C2×C32×Dic3C2211yescharacteristic
17C32×C22⋊C4—31no—
18#3757—31no—
19#3757—34no—
20#3757—64no—
21#3757D311yescharacteristic
22C4⋊(C3⋊Dic3)—31no—
23#3750—31no—
24#3750—31no—
25C4×C3⋊Dic3—31no—
26C3×#217—34no—
27C3×#217D314yes—
28#3724—34no—
29Dic3⋊Dic3—34no—
30Dic3⋊Dic3—34no—
31Dic32—34no—
32C22×C33—21no—
33C22×C33—21no—
34C22×C33—21no—
35C22×C33C2311yescharacteristic
36C3⋊(C3⋊Dic3)—21no—
37C3⋊(C3⋊Dic3)—21no—
38C3×C3⋊Dic3—21no—
39C3×C3⋊Dic3—21no—
40C32×Dic3—21no—
41C32×Dic3—21no—
42#879—94no—
43#879—91no—
44C23×C32—24no—
45C23×C32C2×D314yes—
46C23×C32C2×D311yescharacteristic
47C2×C32×C4—31no—
48C2×C32×C4—31no—
49C2×C3⋊Dic3—31no—
50C2×C3⋊Dic3—31no—
51C2×C3⋊Dic3—34no—
52C2×C3⋊Dic3—34no—
53C2×C3⋊Dic3—64no—
54C2×C3⋊Dic3—64no—
55C2×C3⋊Dic3C2×D311yescharacteristic
56C2×C3⋊Dic3C2×D311yescharacteristic
57C2×C3×Dic3—34no—
58C2×C3×Dic3—34no—
59C2×C3×Dic3C2×D314yes—
60C2×C3×Dic3C2×D314yes—
61C2×C33—41no—
62C2×C33#4111yescharacteristic
63C2×C33#4111yescharacteristic
64C2×C33#4111yescharacteristic
65C3×C22⋊C4—34no—
66C3×C22⋊C4—91no—
67#217—34no—
68#217—94no—
69#217—91no—
70#217—184no—
71#217C3⋊D311yescharacteristic
72C4⋊Dic3—94no—
73C4⋊Dic3—91no—
74Dic3⋊C4—94no—
75Dic3⋊C4—94no—
76Dic3⋊C4—91no—
77Dic3⋊C4—91no—
78C4×Dic3—94no—
79C4×Dic3—91no—
80C22×C32—28no—
81C22×C32—21no—
82C22×C32—24no—
83C22×C32—21no—
84C22×C32—21no—
85C22×C32—24no—
86C22×C32—24no—
87C22×C32—24no—
88C22×C32—44no—
89C22×C32—44no—
90C22×C32C22×D314yes—
91C22×C32C22×D311yescharacteristic
92C32×C4—61no—
93C32×C4—61no—
94C3⋊Dic3—21no—
95C3⋊Dic3—21no—
96C3⋊Dic3—61no—
97C3⋊Dic3—61no—
98C3⋊Dic3—64no—
99C3⋊Dic3—64no—
100C3⋊Dic3—124no—
101C3⋊Dic3—124no—
102C3×Dic3—24no—
103C3×Dic3—24no—
104C3×Dic3—64no—
105C3×Dic3—64no—
106#126—271noSylow
107C33#12611yescharacteristic, complemented, Sylow
108C23×C3—24no—
109C23×C3D3214yes—
110C23×C3C2×C3⋊D311yescharacteristic
111C2×C3×C4—34no—
112C2×C3×C4—34no—
113C2×C3×C4—91no—
114C2×C3×C4—91no—
115C2×Dic3—34no—
116C2×Dic3—34no—
117C2×Dic3—94no—
118C2×Dic3—94no—
119C2×Dic3—91no—
120C2×Dic3—91no—
121C2×Dic3—184no—
122C2×Dic3—184no—
123C2×Dic3C2×C3⋊D311yescharacteristic
124C2×Dic3C2×C3⋊D311yescharacteristic
125C2×C32—24no—
126C2×C32—24no—
127C2×C32—24no—
128C2×C32—48no—
129C2×C32—44no—
130C2×C32—41no—
131C2×C32#23514yes—
132C2×C32#23511yescharacteristic
133C2×C32#23714yes—
134C2×C32#23714yes—
135C2×C32#23711yescharacteristic
136C2×C32#23711yescharacteristic
137C4⋊C4—271no—
138C4⋊C4—271no—
139C4⋊C4—271no—
140C22⋊C4—31no—
141C22⋊C4—91no—
142C22⋊C4—271no—
143C42—271no—
144C22×C3—28no—
145C22×C3—24no—
146C22×C3—24no—
147C22×C3—24no—
148C22×C3—21no—
149C22×C3—21no—
150C22×C3—21no—
151C22×C3—24no—
152C22×C3—44no—
153C22×C3—44no—
154C22×C3C2×D3214yes—
155C22×C3C22×C3⋊D311yescharacteristic
156C3×C4—64no—
157C3×C4—64no—
158C3×C4—181no—
159C3×C4—181no—
160Dic3—21no—
161Dic3—21no—
162Dic3—64no—
163Dic3—64no—
164Dic3—184no—
165Dic3—184no—
166Dic3—181no—
167Dic3—181no—
168Dic3—364no—
169Dic3—364no—
170C32—24no—
171C32#87914yescomplemented
172C32#87911yescharacteristic, complemented
173C23D3×C3⋊D311yescharacteristic
174C2×C4—31no—
175C2×C4—31no—
176C2×C4—91no—
177C2×C4—91no—
178C2×C4—271no—
179C2×C4—271no—
180C2×C3—24no—
181C2×C3—24no—
182C2×C3—24no—
183C2×C3—48no—
184C2×C3—41no—
185C2×C3—44no—
186C2×C3#380414yes—
187C2×C3#380414yes—
188C2×C3#380514yes—
189C2×C3#382811yescharacteristic
190C2×C3#383011yescharacteristic
191C2×C3#383011yescharacteristic
192C22—21no—
193C22—21no—
194C22—21no—
195C22C2×D3×C3⋊D311yescharacteristic
196C4—61no—
197C4—61no—
198C4—181no—
199C4—181no—
200C4—541no—
201C4—541no—
202C3—24no—
203C3#6395614yescomplemented
204C3#6408811yescharacteristic, complemented
205C2—41no—
206C2#8902811yescharacteristic
207C2#8902911yescharacteristic
208C2#8903111yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×C3⋊D3
commutator, socle, cosocleC22×C33C23
Fitting, prime radicalC23×C33C22
hypercenterC23D3×C3⋊D3
hypocenterC33#126
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #88807C2semidirect
5. #88806C2semidirect
6. Dic3⋊(C3⋊Dic3)C2semidirect
7. Dic3×C3⋊Dic3C2semidirect
107. C33#126semidirect
171. C32#879semidirect
172. C32#879semidirect
203. C3#63956semidirect
204. C3#64088semidirect

Maximal subgroups

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

subgroupindex
#888642
C3×#37572
C32×#2172
#888072
#888062
Dic3⋊(C3⋊Dic3)2
Dic3×C3⋊Dic32
#640883
#639563