Finite Group Databasebeta

← #11723433

Subgroups — #11723433

Order 864.

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

normal subgroups only

This group has 480 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×#3752C211yesmaximal, characteristic
3C32×Dic3⋊C4C211yesmaximal, characteristic, complemented
4#88807C211yesmaximal, characteristic, complemented
5Dic3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
6#88803C211yesmaximal, characteristic
7#88803C211yesmaximal, characteristic
8#64107—31nomaximal
9#63862—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
17C32×C4⋊C4—31no—
18#3752—31no—
19#3752—31no—
20#3752D311yescharacteristic
21C4⋊(C3⋊Dic3)—31no—
22#3750—31no—
23C4×C3⋊Dic3—31no—
24C4×C3⋊Dic3—34no—
25C4×C3⋊Dic3—64no—
26C3×(C2×D3)⋊C4—34no—
27C3×Dic3⋊C4D314yescomplemented
28Dic3⋊Dic3—34no—
29Dic3⋊Dic3—34no—
30(C2×D3)⋊Dic3—34no—
31(C2×D3)⋊Dic3—34no—
32C22×C33C2311yescharacteristic
33C2×C3×C3⋊D3—21no—
34C2×C3×C3⋊D3—21no—
35C2×C3×C3⋊D3—21no—
36C33×C4—21no—
37C3⋊(C3⋊Dic3)—21no—
38C3⋊(C3⋊Dic3)—21no—
39C3×C3⋊Dic3—21no—
40C32×Dic3—21no—
41C32×Dic3—21no—
42#898—94no—
43#879—91no—
44C22×C3⋊D3C2×D311yescharacteristic
45C22×C3×D3—34no—
46C2×C32×C4—24no—
47C2×C32×C4—31no—
48C2×C32×C4—31no—
49C2×C32×C4C2×D314yes—
50C2×C32×C4C2×D311yescharacteristic
51C2×C3⋊Dic3—31no—
52C2×C3⋊Dic3—31no—
53C2×C3⋊Dic3—34no—
54C2×C3⋊Dic3—34no—
55C2×C3⋊Dic3—64no—
56C2×C3⋊Dic3—64no—
57C2×C3⋊Dic3C2×D311yescharacteristic
58C2×C3×Dic3—34no—
59C2×C3×Dic3C2×D314yes—
60C2×C3×Dic3C2×D314yes—
61C2×C33#4111yescharacteristic
62C2×C33#4111yescharacteristic
63C2×C33#4111yescharacteristic
64C3×C3⋊D3—41no—
65C3×C4⋊C4—34no—
66C3×C22⋊C4—91no—
67#217—91no—
68#217—91no—
69(C2×D3)⋊C4—34no—
70(C2×D3)⋊C4—94no—
71(C2×D3)⋊C4—94no—
72C4⋊Dic3—94no—
73C4⋊Dic3—91no—
74Dic3⋊C4—94no—
75Dic3⋊C4—91no—
76Dic3⋊C4C3⋊D311yescharacteristic, complemented
77C4×Dic3—94no—
78C4×Dic3—91no—
79C4×Dic3—184no—
80C22×C32—24no—
81C22×C32C22×D314yes—
82C22×C32C22×D311yescharacteristic
83C2×C3⋊D3—21no—
84C2×C3⋊D3—21no—
85C2×C3⋊D3—21no—
86C2×C3×D3—64no—
87C2×C3×D3—64no—
88C2×C3×D3—64no—
89C32×C4—28no—
90C32×C4—24no—
91C32×C4—21no—
92C32×C4—61no—
93C32×C4—61no—
94C3⋊Dic3—21no—
95C3⋊Dic3—68no—
96C3⋊Dic3—68no—
97C3⋊Dic3—64no—
98C3⋊Dic3—61no—
99C3⋊Dic3—61no—
100C3⋊Dic3—64no—
101C3×Dic3—24no—
102C3×Dic3—24no—
103C3×Dic3—64no—
104#126—271noSylow
105C33#12611yescharacteristic, complemented, Sylow
106C23×C3—91no—
107C22×D3—34no—
108C2×C3×C4—24no—
109C2×C3×C4—34no—
110C2×C3×C4—34no—
111C2×C3×C4—91no—
112C2×C3×C4D3214yes—
113C2×C3×C4C2×C3⋊D311yescharacteristic
114C2×Dic3—34no—
115C2×Dic3—94no—
116C2×Dic3—94no—
117C2×Dic3—91no—
118C2×Dic3—91no—
119C2×Dic3—184no—
120C2×Dic3—184no—
121C2×Dic3C2×C3⋊D311yescharacteristic
122C2×Dic3C2×C3⋊D311yescharacteristic
123C2×C32—24no—
124C2×C32—24no—
125C2×C32—24no—
126C2×C32#23514yes—
127C2×C32#23511yescharacteristic
128C2×C32#23714yes—
129C2×C32#23711yescharacteristic
130C2×C32#23711yescharacteristic
131C2×C32#23914yes—
132C3⋊D3—41no—
133C3×D3—124no—
134C4⋊C4—31no—
135C4⋊C4—271no—
136C4⋊C4—271no—
137C22⋊C4—91no—
138C22⋊C4—271no—
139C22⋊C4—271no—
140C42—271no—
141C22×C3—24no—
142C22×C3—181no—
143C22×C3—181no—
144C22×C3—181no—
145C22×C3C2×D3214yes—
146C22×C3C22×C3⋊D311yescharacteristic
147C2×D3—64no—
148C2×D3—64no—
149C2×D3—64no—
150C3×C4—28no—
151C3×C4—24no—
152C3×C4—21no—
153C3×C4—64no—
154C3×C4—64no—
155C3×C4—181no—
156Dic3—21no—
157Dic3—21no—
158Dic3—64no—
159Dic3—188no—
160Dic3—188no—
161Dic3—184no—
162Dic3—184no—
163Dic3—181no—
164Dic3—181no—
165C32—24no—
166C32#87911yescharacteristic, complemented
167C32#89814yescomplemented
168C23—91no—
169C2×C4—31no—
170C2×C4—31no—
171C2×C4—91no—
172C2×C4—271no—
173C2×C4—271no—
174C2×C4D3×C3⋊D311yescharacteristic
175C2×C3—24no—
176C2×C3—24no—
177C2×C3—24no—
178C2×C3—361no—
179C2×C3#379614yes—
180C2×C3#379814yes—
181C2×C3#380514yes—
182C2×C3#382811yescharacteristic
183C2×C3#383011yescharacteristic
184C2×C3#383211yescharacteristic
185D3—124no—
186C22—181no—
187C22—181no—
188C22—181no—
189C22C2×D3×C3⋊D311yescharacteristic
190C4—21no—
191C4—61no—
192C4—61no—
193C4—181no—
194C4—541no—
195C4—541no—
196C3—24no—
197C3#6386214yescomplemented
198C3#6410711yescharacteristic, complemented
199C2—361no—
200C2#8901711yescharacteristic
201C2#8902011yescharacteristic
202C2#8902911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×C3⋊D3
commutator, socle, cosocleC22×C33C23
FittingC2×C33×C4C22
prime radicalC22×C3×C3⋊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
3. C32×Dic3⋊C4C2semidirect
4. #88807C2semidirect
5. Dic3⋊(C3⋊Dic3)C2semidirect
27. C3×Dic3⋊C4D3semidirect
76. Dic3⋊C4C3⋊D3semidirect
105. C33#126semidirect
166. C32#879semidirect
167. C32#898semidirect
197. C3#63862semidirect
198. C3#64107semidirect

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×#37522
C32×Dic3⋊C42
#888072
Dic3⋊(C3⋊Dic3)2
#888032
#641073
#638623