Finite Group Databasebeta

← #11722981

Subgroups — #11722981

Order 864.

418 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 82 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×#6296C211yesmaximal, characteristic, complemented
2C2×D3×C3⋊Dic3C211yesmaximal, characteristic
3Dic3⋊(C3⋊Dic3)C212yesmaximal, complemented
4C3×#3724C211yesmaximal, characteristic, complemented
5C3×(C2×D3)⋊Dic3C212yesmaximal
6#63982—31nomaximal
7#63893—32nomaximal
8C22×C32×D3C2211yescharacteristic
9C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
10C2×C3×C3⋊Dic3C2212yes—
11C2×C3×C3⋊Dic3C2211yescharacteristic
12C2×C32×Dic3C2212yes—
13#6296—22no—
14D3×C3⋊Dic3—22no—
15C22×C3⋊Dic3—31no—
16C2×#627—32no—
17C2×#627—31no—
18C2×D3×Dic3—32no—
19C2×D3×Dic3—61no—
20C4⋊(C3⋊Dic3)—32no—
21C3×#217—32no—
22C3×(C2×D3)⋊C4—32no—
23C3×Dic3⋊C4—32no—
24#3724D311yescharacteristic, complemented
25Dic3⋊Dic3—32no—
26Dic3⋊Dic3—32no—
27Dic3⋊Dic3—62no—
28(C2×D3)⋊Dic3D312yes—
29C22×C33C2311yescharacteristic
30C2×C32×D3—21no—
31C2×C32×D3—21no—
32C2×C32×D3Q812yescomplemented
33C3⋊(C3⋊Dic3)—21no—
34C3×C3⋊Dic3—22no—
35C3×C3⋊Dic3D412yes—
36C32×Dic3—22no—
37#897—91no—
38#878—92no—
39C23×C32—31no—
40C22×C3×D3—21no—
41C22×C3×D3C2×D312yes—
42C2×C32×C4—32no—
43C2×C3⋊Dic3—31no—
44C2×C3⋊Dic3—32no—
45C2×C3⋊Dic3—61no—
46C2×C3⋊Dic3—62no—
47C2×C3⋊Dic3—62no—
48C2×C3⋊Dic3—121no—
49C2×C3⋊Dic3C2×D312yes—
50C2×C3⋊Dic3C2×D311yescharacteristic
51C2×C3×Dic3—22no—
52C2×C3×Dic3—32no—
53C2×C3×Dic3—32no—
54C2×C3×Dic3—32no—
55C2×C3×Dic3—61no—
56C2×C3×Dic3—62no—
57C2×C3×Dic3C2×D312yes—
58C2×C3×Dic3C2×D312yes—
59#627—64no—
60#627—62no—
61D3×Dic3—64no—
62D3×Dic3—64no—
63C2×C33C2×D411yescharacteristic
64C2×C33C2×Q811yescharacteristic
65C2×C33#4111yescharacteristic
66C32×D3—22no—
67C22×Dic3—92no—
68C22×Dic3—181no—
69C2×C4×D3—91no—
70C2×Dic6—92no—
71C2×Dic6—91no—
72C3×C4⋊C4—91no—
73C3×C22⋊C4—92no—
74#217—32no—
75(C2×D3)⋊C4—32no—
76C4⋊Dic3—92no—
77C4⋊Dic3—92no—
78C4⋊Dic3—182no—
79Dic3⋊C4—32no—
80Dic3⋊C4—92no—
81C22×C32—22no—
82C22×C32—21no—
83C22×C32—32no—
84C22×C32—41no—
85C22×C32—61no—
86C22×C32—61no—
87C22×C32C22×D312yes—
88C22×C32C22×D311yescharacteristic
89C2×C3×D3—22no—
90C2×C3×D3—22no—
91C2×C3×D3—22no—
92C2×C3×D3—22no—
93C2×C3×D3—22no—
94C2×C3×D3Dic614yescomplemented
95C32×C4—62no—
96C3⋊Dic3—22no—
97C3⋊Dic3—61no—
98C3⋊Dic3—62no—
99C3⋊Dic3—62no—
100C3⋊Dic3—122no—
101C3⋊Dic3—122no—
102C3⋊Dic3D1212yes—
103C3×Dic3—24no—
104C3×Dic3—22no—
105C3×Dic3—22no—
106C3×Dic3—34no—
107C3×Dic3—62no—
108C3×Dic3—62no—
109C3×Dic3—62no—
110C3×Dic3—122no—
111C22⋊Q8—271noSylow
112C33C22⋊Q811yescharacteristic, complemented, Sylow
113C23×C3—32no—
114C23×C3—61no—
115C22×D3D3211yescharacteristic
116C2×C3×C4—32no—
117C2×C3×C4—32no—
118C2×C3×C4—62no—
119C2×C3×C4—91no—
120C2×C3×C4—92no—
121C2×Dic3—32no—
122C2×Dic3—32no—
123C2×Dic3—32no—
124C2×Dic3—61no—
125C2×Dic3—62no—
126C2×Dic3—92no—
127C2×Dic3—91no—
128C2×Dic3—181no—
129C2×Dic3—182no—
130C2×Dic3—184no—
131C2×Dic3—184no—
132C2×Dic3—361no—
133C2×Dic3D3212yes—
134C4×D3—182no—
135Dic6—184no—
136Dic6—182no—
137C2×C32—22no—
138C2×C32—22no—
139C2×C32—22no—
140C2×C32—21no—
141C2×C32—21no—
142C2×C32—21no—
143C2×C32—41no—
144C2×C32—41no—
145C2×C32—41no—
146C2×C32—62no—
147C2×C32C2×Dic612yes—
148C2×C32C2×D1211yescharacteristic
149C2×C32D3×D412yes—
150C2×C32#23712yes—
151C2×C32#23711yescharacteristic
152C2×C32D3×Q811yescharacteristic
153C3×D3—24no—
154C3×D3—24no—
155C2×Q8—271no—
156C22×C4—271no—
157C4⋊C4—91no—
158C4⋊C4—272no—
159C22⋊C4—92no—
160C22×C3—22no—
161C22×C3—21no—
162C22×C3—34no—
163C22×C3—41no—
164C22×C3—62no—
165C22×C3—62no—
166C22×C3—62no—
167C22×C3—62no—
168C22×C3—62no—
169C22×C3C2×D3212yes—
170C22×C3C2×D3211yescharacteristic
171C2×D3—21no—
172C2×D3—21no—
173C2×D3#62712yescomplemented
174C3×C4—64no—
175C3×C4—62no—
176C3×C4—62no—
177C3×C4—92no—
178C3×C4—182no—
179Dic3—22no—
180Dic3—34no—
181Dic3—62no—
182Dic3—62no—
183Dic3—62no—
184Dic3—122no—
185Dic3—182no—
186Dic3—182no—
187Dic3—181no—
188Dic3—362no—
189Dic3—362no—
190C32—22no—
191C32—21no—
192C32—41no—
193C32#87812yescomplemented
194C32#89711yescharacteristic, complemented
195C23—31no—
196Q8—542no—
197C2×C4—32no—
198C2×C4—91no—
199C2×C4—92no—
200C2×C4—271no—
201C2×C4—542no—
202C2×C3—21no—
203C2×C3—22no—
204C2×C3—21no—
205C2×C3—22no—
206C2×C3—22no—
207C2×C3—21no—
208C2×C3—41no—
209C2×C3—41no—
210C2×C3—41no—
211C2×C3—64no—
212C2×C3—64no—
213C2×C3D3×Dic612yes—
214C2×C3D3×D1212yes—
215C2×C3#380512yes—
216C2×C3#380511yescharacteristic
217C2×C3C2×#62711yescharacteristic
218C2×C3#381111yescharacteristic
219D3—22no—
220C22—32no—
221C22—61no—
222C22—61no—
223C22D3311yescharacteristic
224C4—62no—
225C4—92no—
226C4—182no—
227C4—541no—
228C3—21no—
229C3—22no—
230C3—41no—
231C3#6389312yescomplemented
232C3#6398211yescharacteristic, complemented
233C2—62no—
234C2#8895511yescharacteristic
235C2D3×#62711yescharacteristic
236C2#8896311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, cosocleC22×C33C23
prime radicalC22×C32×D3C22
hypocenterC33C22⋊Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#6296C2semidirect
3. Dic3⋊(C3⋊Dic3)C2semidirect
4. C3×#3724C2semidirect
24. #3724D3semidirect
32. C2×C32×D3Q8semidirect
94. C2×C3×D3Dic6semidirect
112. C33C22⋊Q8semidirect
173. C2×D3#627semidirect
193. C32#878semidirect
194. C32#897semidirect
231. C3#63893semidirect
232. C3#63982semidirect

Maximal subgroups

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

subgroupindex
C2×#62962
C2×D3×C3⋊Dic32
Dic3⋊(C3⋊Dic3)2
C3×#37242
C3×(C2×D3)⋊Dic32
#639823
#638933