Finite Group Databasebeta

← #11722992

Subgroups — #11722992

Order 864.

434 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 74 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×#6290C211yesmaximal, characteristic
2C2×C3×#624C211yesmaximal, characteristic, complemented
3#88814C211yesmaximal, characteristic, complemented
4#88807C212yesmaximal, complemented
5C3×(C2×D3)⋊Dic3C212yesmaximal, complemented
6#63975—31nomaximal
7#63905—32nomaximal
8C22×C3×C3⋊D3C2211yescharacteristic
9C22×C32×D3C2211yescharacteristic
10C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
11C2×C3×C3⋊Dic3C2212yes—
12C2×C32×Dic3C2212yes—
13#6290—22no—
14C3×#624—22no—
15C2×#638—31no—
16C2×C3×C4×D3—32no—
17C2×#625—32no—
18C2×#625—61no—
19C2×#624D311yescharacteristic, complemented
20#3750—32no—
21C3×#217—32no—
22C3×(C2×D3)⋊C4—32no—
23Dic3⋊Dic3—32no—
24Dic3⋊Dic3—32no—
25Dic3⋊Dic3—62no—
26#3722—31no—
27(C2×D3)⋊Dic3—32no—
28(C2×D3)⋊Dic3D312yescomplemented
29C22×C33C2311yescharacteristic
30C2×C3×C3⋊D3—22no—
31C2×C3×C3⋊D3D412yes—
32C2×C32×D3—21no—
33C2×C32×D3—22no—
34C3⋊(C3⋊Dic3)—21no—
35C3×C3⋊Dic3—22no—
36C32×Dic3—22no—
37#929—91no—
38#883—92no—
39C23×C32—31no—
40C22×C3⋊D3C2×D311yescharacteristic
41C22×C3×D3—21no—
42C22×C3×D3—32no—
43C22×C3×D3—61no—
44C22×C3×D3C2×D312yes—
45C2×C32×C4—32no—
46#638—62no—
47C2×C3⋊Dic3—31no—
48C2×C3⋊Dic3—32no—
49C2×C3⋊Dic3—61no—
50C2×C3⋊Dic3—62no—
51C2×C3⋊Dic3—121no—
52C2×C3⋊Dic3C2×D312yes—
53C2×C3×Dic3—22no—
54C2×C3×Dic3—32no—
55C2×C3×Dic3—32no—
56C2×C3×Dic3—62no—
57C2×C3×Dic3C2×D312yes—
58C2×C3×Dic3C2×D312yes—
59C3×C4×D3—64no—
60#625—64no—
61#625—64no—
62#624—22no—
63C2×C33C2×D411yescharacteristic
64C2×C33#4112yes—
65C3×C3⋊D3—22no—
66C32×D3—41no—
67C22×C3×C4—91no—
68C2×#67—92no—
69C2×#67—91no—
70C2×#67—181no—
71C2×C4×D3—32no—
72C3×C22⋊C4—92no—
73#217—32no—
74#217—91no—
75(C2×D3)⋊C4—32no—
76(C2×D3)⋊C4—92no—
77C4⋊Dic3—92no—
78Dic3⋊C4—92no—
79Dic3⋊C4—92no—
80Dic3⋊C4—182no—
81C22×C32—22no—
82C22×C32—21no—
83C22×C32—41no—
84C22×C32—61no—
85C22×C32—62no—
86C22×C32C22×D311yescharacteristic
87C22×C32C22×D312yes—
88C2×C3⋊D3—22no—
89C2×C3⋊D3#6712yes—
90C2×C3×D3—22no—
91C2×C3×D3—22no—
92C2×C3×D3—22no—
93C2×C3×D3—22no—
94C2×C3×D3—34no—
95C2×C3×D3—42no—
96C2×C3×D3—62no—
97C2×C3×D3—62no—
98C2×C3×D3—62no—
99C2×C3×D3—122no—
100C32×C4—62no—
101C3⋊Dic3—22no—
102C3⋊Dic3—61no—
103C3⋊Dic3—62no—
104C3⋊Dic3—62no—
105C3⋊Dic3—122no—
106C3⋊Dic3—122no—
107C3×Dic3—22no—
108C3×Dic3—22no—
109C3×Dic3—42no—
110C3×Dic3—64no—
111C3×Dic3—62no—
112C3×Dic3—62no—
113#123—271noSylow
114C33#12311yescharacteristic, complemented, Sylow
115C23×C3—32no—
116C23×C3—61no—
117C23×C3—91no—
118C22×D3—32no—
119C22×D3—61no—
120C22×D3D3211yescharacteristic
121C2×C3×C4—32no—
122C2×C3×C4—32no—
123C2×C3×C4—62no—
124C2×C3×C4—92no—
125C2×C3×C4—182no—
126#67—182no—
127#67—184no—
128#67—184no—
129C2×Dic3—32no—
130C2×Dic3—32no—
131C2×Dic3—62no—
132C2×Dic3—91no—
133C2×Dic3—92no—
134C2×Dic3—181no—
135C2×Dic3—182no—
136C2×Dic3—361no—
137C2×Dic3D3212yes—
138C4×D3—64no—
139C2×C32—22no—
140C2×C32—22no—
141C2×C32—22no—
142C2×C32—22no—
143C2×C32—21no—
144C2×C32—41no—
145C2×C32—42no—
146C2×C32—121no—
147C2×C32#23512yes—
148C2×C32#23512yes—
149C2×C32D3×D412yes—
150C2×C32#23712yes—
151C2×C32C2×#6711yescharacteristic
152C3⋊D3—22no—
153C3×D3—42no—
154C3×D3—42no—
155C3×D3—64no—
156C3×D3—122no—
157C2×D4—271no—
158C22×C4—91no—
159C4⋊C4—272no—
160C22⋊C4—92no—
161C22⋊C4—271no—
162C22×C3—22no—
163C22×C3—21no—
164C22×C3—41no—
165C22×C3—62no—
166C22×C3—62no—
167C22×C3—62no—
168C22×C3—62no—
169C22×C3—92no—
170C22×C3—122no—
171C22×C3—182no—
172C22×C3C2×D3212yes—
173C22×C3C2×D3211yescharacteristic
174C2×D3—22no—
175C2×D3—21no—
176C2×D3—34no—
177C2×D3—62no—
178C2×D3—62no—
179C2×D3—62no—
180C2×D3—122no—
181C3×C4—62no—
182C3×C4—62no—
183C3×C4—122no—
184C3×C4—182no—
185Dic3—22no—
186Dic3—64no—
187Dic3—62no—
188Dic3—62no—
189Dic3—182no—
190Dic3—182no—
191Dic3—181no—
192Dic3—362no—
193Dic3—362no—
194C32—22no—
195C32—21no—
196C32—41no—
197C32#88312yescomplemented
198C32#92911yescharacteristic, complemented
199C23—31no—
200C23—91no—
201D4—542no—
202C2×C4—32no—
203C2×C4—92no—
204C2×C4—182no—
205C2×C4—271no—
206C2×C3—21no—
207C2×C3—22no—
208C2×C3—22no—
209C2×C3—22no—
210C2×C3—22no—
211C2×C3—41no—
212C2×C3—42no—
213C2×C3—122no—
214C2×C3—122no—
215C2×C3—182no—
216C2×C3#379512yes—
217C2×C3#379812yes—
218C2×C3#380412yes—
219C2×C3D3×#6712yes—
220C2×C3#381111yescharacteristic
221D3—41no—
222D3—64no—
223D3—122no—
224C22—61no—
225C22—62no—
226C22—92no—
227C22—182no—
228C22D3311yescharacteristic
229C4—62no—
230C4—182no—
231C4—541no—
232C3—21no—
233C3—22no—
234C3—41no—
235C3#6390512yescomplemented
236C3#6397511yescharacteristic, complemented
237C2—121no—
238C2—182no—
239C2#8895211yescharacteristic
240C2#8896012yes—
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, cosocleC22×C33C23
prime radicalC2×#6290C2
hypocenterC33#123
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C3×#624C2semidirect
3. #88814C2semidirect
4. #88807C2semidirect
5. C3×(C2×D3)⋊Dic3C2semidirect
19. C2×#624D3semidirect
28. (C2×D3)⋊Dic3D3semidirect
114. C33#123semidirect
197. C32#883semidirect
198. C32#929semidirect
235. C3#63905semidirect
236. C3#63975semidirect

Maximal subgroups

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

subgroupindex
C2×#62902
C2×C3×#6242
#888142
#888072
C3×(C2×D3)⋊Dic32
#639753
#639053