Finite Group Databasebeta

← #11723591

Subgroups — #11723591

Order 864.

592 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 108 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×C3×#638C211yesmaximal, characteristic
2C2×#6293C211yesmaximal, characteristic, complemented
3#88864C211yesmaximal, characteristic, complemented
4C32×#217C211yesmaximal, characteristic, complemented
5#88803C211yesmaximal, characteristic, complemented
6#88803C211yesmaximal, characteristic, complemented
7Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#64094—31nomaximal
9#63959—34nomaximal
10C23×C33C2211yescharacteristic
11C22×C3×C3⋊D3C2211yescharacteristic
12C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C3×#638—22no—
15C2×C3×C3⋊Dic3C2211yescharacteristic
16C2×C32×Dic3C2211yescharacteristic
17C2×C32×Dic3C2211yescharacteristic
18#6293—22no—
19C2×#638D311yescharacteristic
20C2×#634—31no—
21C2×C3×#67—34no—
22C2×#627—34no—
23C32×C22⋊C4—31no—
24#3757—31no—
25#3757—34no—
26#3757—64no—
27#3752—31no—
28#3752—31no—
29C4×C3⋊Dic3—31no—
30C3×#217D314yescomplemented
31(C2×D3)⋊Dic3—34no—
32(C2×D3)⋊Dic3—34no—
33Dic32—34no—
34C22×C33—21no—
35C22×C33—21no—
36C22×C33—21no—
37C22×C33C2311yescharacteristic
38C2×C3×C3⋊D3—21no—
39C2×C3×C3⋊D3—21no—
40C2×C3×C3⋊D3—21no—
41C3⋊(C3⋊Dic3)—21no—
42C3⋊(C3⋊Dic3)—21no—
43C3×C3⋊Dic3D412yes—
44C32×Dic3—21no—
45C32×Dic3—21no—
46#936—91no—
47#885—94no—
48C23×C32—24no—
49C23×C32C2×D314yes—
50C23×C32C2×D311yescharacteristic
51C22×C3⋊D3C2×D311yescharacteristic
52C22×C3×D3—34no—
53C2×C32×C4—31no—
54C2×C32×C4—31no—
55#638—22no—
56C2×C3⋊Dic3—31no—
57C2×C3⋊Dic3—31no—
58C2×C3⋊Dic3—34no—
59C2×C3⋊Dic3—34no—
60C2×C3⋊Dic3—64no—
61C2×C3⋊Dic3—64no—
62C2×C3⋊Dic3C2×D311yescharacteristic
63#634—62no—
64C3×#67—68no—
65C2×C3×Dic3—34no—
66C2×C3×Dic3C2×D314yes—
67C2×C3×Dic3C2×D314yes—
68#627—68no—
69C2×C33—41no—
70C2×C33C2×D411yescharacteristic
71C2×C33#4111yescharacteristic
72C2×C33#4111yescharacteristic
73C3×C3⋊D3—41no—
74C2×C3×D4—91no—
75C2×#67—34no—
76C2×Dic6—94no—
77C2×Dic6—91no—
78C3×C22⋊C4—34no—
79#217—94no—
80#217—91no—
81#217—91no—
82#217—91no—
83#217—184no—
84#217C3⋊D311yescharacteristic, complemented
85(C2×D3)⋊C4—94no—
86(C2×D3)⋊C4—94no—
87C4×Dic3—94no—
88C4×Dic3—91no—
89C22×C32—28no—
90C22×C32—24no—
91C22×C32—21no—
92C22×C32—21no—
93C22×C32—21no—
94C22×C32—24no—
95C22×C32—24no—
96C22×C32—24no—
97C22×C32—44no—
98C22×C32—44no—
99C22×C32C22×D314yes—
100C22×C32C22×D311yescharacteristic
101C2×C3⋊D3—21no—
102C2×C3⋊D3—21no—
103C2×C3⋊D3—21no—
104C2×C3×D3—64no—
105C2×C3×D3—64no—
106C2×C3×D3—64no—
107C32×C4—61no—
108C32×C4—61no—
109C3⋊Dic3—61no—
110C3⋊Dic3—61no—
111C3⋊Dic3—64no—
112C3⋊Dic3—64no—
113C3⋊Dic3—124no—
114C3⋊Dic3—124no—
115C3⋊Dic3#6712yes—
116C3×Dic3—24no—
117C3×Dic3—24no—
118C3×Dic3—38no—
119#124—271noSylow
120C33#12411yescharacteristic, complemented, Sylow
121C23×C3—24no—
122C23×C3—91no—
123C23×C3D3214yes—
124C23×C3C2×C3⋊D311yescharacteristic
125C22×D3—34no—
126C3×D4—182no—
127C2×C3×C4—34no—
128C2×C3×C4—34no—
129C2×C3×C4—91no—
130#67—68no—
131C2×Dic3—34no—
132C2×Dic3—94no—
133C2×Dic3—94no—
134C2×Dic3—91no—
135C2×Dic3—91no—
136C2×Dic3—184no—
137C2×Dic3—184no—
138C2×Dic3C2×C3⋊D311yescharacteristic
139C2×Dic3C2×C3⋊D311yescharacteristic
140Dic6—182no—
141Dic6—188no—
142C2×C32—24no—
143C2×C32—24no—
144C2×C32—24no—
145C2×C32—48no—
146C2×C32—44no—
147C2×C32—41no—
148C2×C32#23514yes—
149C2×C32D3×D414yes—
150C2×C32#23714yes—
151C2×C32#23711yescharacteristic
152C2×C32#23711yescharacteristic
153C2×C32C2×#6711yescharacteristic
154C3⋊D3—41no—
155C3×D3—124no—
156C2×Q8—271no—
157C2×D4—91no—
158C22⋊C4—31no—
159C22⋊C4—271no—
160C22⋊C4—271no—
161C22⋊C4—271no—
162C42—271no—
163C22×C3—28no—
164C22×C3—24no—
165C22×C3—24no—
166C22×C3—24no—
167C22×C3—21no—
168C22×C3—21no—
169C22×C3—21no—
170C22×C3—24no—
171C22×C3—44no—
172C22×C3—44no—
173C22×C3—181no—
174C22×C3—181no—
175C22×C3—181no—
176C22×C3C2×D3214yes—
177C22×C3C22×C3⋊D311yescharacteristic
178C2×D3—64no—
179C2×D3—64no—
180C2×D3—64no—
181C3×C4—64no—
182C3×C4—64no—
183C3×C4—92no—
184Dic3—21no—
185Dic3—21no—
186Dic3—38no—
187Dic3—184no—
188Dic3—184no—
189Dic3—181no—
190Dic3—181no—
191Dic3—364no—
192Dic3—364no—
193C32—24no—
194C32#88514yescomplemented
195C32#93611yescharacteristic, complemented
196C23—91no—
197C23D3×C3⋊D311yescharacteristic
198Q8—542no—
199D4—182no—
200C2×C4—31no—
201C2×C4—31no—
202C2×C4—91no—
203C2×C4—271no—
204C2×C4—271no—
205C2×C3—24no—
206C2×C3—24no—
207C2×C3—24no—
208C2×C3—48no—
209C2×C3—41no—
210C2×C3—44no—
211C2×C3—361no—
212C2×C3#380414yes—
213C2×C3#380514yes—
214C2×C3D3×#6714yes—
215C2×C3#382811yescharacteristic
216C2×C3D4×C3⋊D311yescharacteristic
217C2×C3#383011yescharacteristic
218D3—124no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—181no—
223C22—181no—
224C22—181no—
225C22C2×D3×C3⋊D311yescharacteristic
226C4—61no—
227C4—61no—
228C4—92no—
229C4—541no—
230C4—541no—
231C3—24no—
232C3#6395914yescomplemented
233C3#6409411yescharacteristic, complemented
234C2—41no—
235C2—361no—
236C2#8902911yescharacteristic
237C2#8903111yescharacteristic
238C2C3⋊D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×C3⋊D3
commutator, socle, cosocleC22×C33C23
FittingC23×C33C22
prime radicalC2×C3×#638C2
hypercenterC23D3×C3⋊D3
hypocenterC33#124
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#6293C2semidirect
3. #88864C2semidirect
4. C32×#217C2semidirect
5. #88803C2semidirect
6. #88803C2semidirect
7. Dic3×C3⋊Dic3C2semidirect
30. C3×#217D3semidirect
84. #217C3⋊D3semidirect
120. C33#124semidirect
194. C32#885semidirect
195. C32#936semidirect
232. C3#63959semidirect
233. C3#64094semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#6382
C2×#62932
#888642
C32×#2172
#888032
Dic3×C3⋊Dic32
#640943
#639593