Finite Group Databasebeta

← #11722982

Subgroups — #11722982

Order 864.

450 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 76 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×#6289C211yesmaximal, characteristic
3#88804C212yesmaximal
4C3×#3724C211yesmaximal, characteristic, complemented
5C3×Dic3⋊Dic3C212yesmaximal, complemented
6#63895—32nomaximal
7#63865—31nomaximal
8C22×C3⋊(C3⋊D3)C2211yescharacteristic
9C2×C3×C3⋊Dic3C2212yes—
10C2×C3×C3⋊Dic3C2211yescharacteristic
11C2×C32×Dic3C2211yescharacteristic
12C2×C32×Dic3C2212yes—
13#6296—22no—
14#6289—22no—
15C2×C4×C3⋊D3—31no—
16C2×#627—31no—
17C2×#627—32no—
18C2×#624—32no—
19C2×#624—61no—
20#3752—32no—
21C3×C4⋊Dic3—32no—
22C3×Dic3⋊C4—32no—
23C3×Dic3⋊C4—32no—
24#3724D311yescharacteristic, complemented
25Dic3⋊Dic3D312yescomplemented
26#3722—32no—
27#3722—32no—
28#3722—62no—
29C22×C33C2311yescharacteristic
30C2×C3⋊(C3⋊D3)—21no—
31C2×C3⋊(C3⋊D3)—21no—
32C2×C3⋊(C3⋊D3)Q812yescomplemented
33C3×C3⋊Dic3—22no—
34C3×C3⋊Dic3D412yes—
35C32×Dic3—21no—
36C32×Dic3—22no—
37#897—91no—
38#896—92no—
39C22×C3⋊D3—31no—
40C22×C3⋊D3—32no—
41C22×C3⋊D3—61no—
42C22×C3⋊D3—62no—
43C22×C3⋊D3—121no—
44C2×C32×C4—31no—
45C2×C32×C4—32no—
46C2×C3⋊Dic3C2×D312yes—
47C2×C3⋊Dic3C2×D311yescharacteristic
48C4×C3⋊D3—62no—
49C2×C3×Dic3—21no—
50C2×C3×Dic3—22no—
51C2×C3×Dic3—32no—
52C2×C3×Dic3—32no—
53C2×C3×Dic3—32no—
54C2×C3×Dic3—61no—
55C2×C3×Dic3—62no—
56C2×C3×Dic3C2×D312yes—
57C2×C3×Dic3C2×D312yes—
58C2×C3×Dic3C2×D312yes—
59#627—64no—
60#627—62no—
61#624—64no—
62#624—64no—
63C2×C33C2×D411yescharacteristic
64C2×C33C2×Q811yescharacteristic
65C2×C33#4111yescharacteristic
66C3⋊(C3⋊D3)—22no—
67C2×C4×D3—92no—
68C2×C4×D3—91no—
69C2×C4×D3—181no—
70C2×Dic6—92no—
71C2×Dic6—91no—
72C3×C4⋊C4—92no—
73C3×C4⋊C4—91no—
74(C2×D3)⋊C4—92no—
75(C2×D3)⋊C4—92no—
76(C2×D3)⋊C4—92no—
77(C2×D3)⋊C4—182no—
78C4⋊Dic3—32no—
79Dic3⋊C4—32no—
80Dic3⋊C4—32no—
81C22×C32—21no—
82C22×C32—22no—
83C22×C32—41no—
84C22×C32C22×D311yescharacteristic
85C22×C32C22×D312yes—
86C2×C3⋊D3—32no—
87C2×C3⋊D3—34no—
88C2×C3⋊D3—62no—
89C2×C3⋊D3—64no—
90C2×C3⋊D3—61no—
91C2×C3⋊D3—61no—
92C2×C3⋊D3—62no—
93C2×C3⋊D3—62no—
94C2×C3⋊D3—62no—
95C2×C3⋊D3—62no—
96C2×C3⋊D3—122no—
97C2×C3⋊D3—122no—
98C2×C3⋊D3—122no—
99C2×C3⋊D3—122no—
100C2×C3⋊D3—122no—
101C32×C4—61no—
102C32×C4—62no—
103C3⋊Dic3—22no—
104C3⋊Dic3D1212yes—
105C3×Dic3—22no—
106C3×Dic3—22no—
107C3×Dic3—22no—
108C3×Dic3—22no—
109C3×Dic3—34no—
110C3×Dic3—42no—
111C3×Dic3—62no—
112C3×Dic3—62no—
113C3×Dic3—62no—
114C3×Dic3—122no—
115C22⋊Q8—271noSylow
116C33C22⋊Q811yescharacteristic, complemented, Sylow
117C22×D3—91no—
118C22×D3—92no—
119C22×D3—181no—
120C22×D3—182no—
121C22×D3—361no—
122C2×C3×C4—32no—
123C2×C3×C4—32no—
124C2×C3×C4—32no—
125C2×C3×C4—61no—
126C2×C3×C4—62no—
127C2×C3×C4—91no—
128C2×C3×C4—92no—
129C2×Dic3—32no—
130C2×Dic3—32no—
131C2×Dic3—32no—
132C2×Dic3—61no—
133C2×Dic3—62no—
134C2×Dic3D3212yes—
135C2×Dic3D3211yescharacteristic
136C4×D3—182no—
137C4×D3—184no—
138C4×D3—184no—
139Dic6—184no—
140Dic6—182no—
141C2×C32—21no—
142C2×C32—22no—
143C2×C32—21no—
144C2×C32—22no—
145C2×C32—22no—
146C2×C32—21no—
147C2×C32—41no—
148C2×C32—41no—
149C2×C32—41no—
150C2×C32C2×D1211yescharacteristic
151C2×C32#23512yes—
152C2×C32D3×D412yes—
153C2×C32#23711yescharacteristic
154C2×C32D3×Q812yes—
155C2×C32D3×Q811yescharacteristic
156C3⋊D3—62no—
157C3⋊D3—64no—
158C3⋊D3—64no—
159C3⋊D3—124no—
160C3⋊D3—124no—
161C2×Q8—271no—
162C22×C4—271no—
163C4⋊C4—91no—
164C4⋊C4—92no—
165C22⋊C4—272no—
166C22×C3—21no—
167C22×C3—22no—
168C22×C3—41no—
169C22×C3C2×D3211yescharacteristic
170C22×C3C2×D3212yes—
171C2×D3—92no—
172C2×D3—94no—
173C2×D3—182no—
174C2×D3—184no—
175C2×D3—182no—
176C2×D3—182no—
177C2×D3—181no—
178C2×D3—182no—
179C2×D3—182no—
180C2×D3—181no—
181C2×D3—362no—
182C2×D3—362no—
183C2×D3—362no—
184C2×D3—362no—
185C2×D3—362no—
186C3×C4—62no—
187C3×C4—62no—
188C3×C4—62no—
189C3×C4—62no—
190C3×C4—92no—
191C3×C4—122no—
192C3×C4—182no—
193Dic3—22no—
194Dic3—21no—
195Dic3—34no—
196Dic3—62no—
197Dic3—62no—
198Dic3—62no—
199Dic3—122no—
200C32—21no—
201C32—22no—
202C32—41no—
203C32#89612yescomplemented
204C32#89711yescharacteristic, complemented
205C23—271no—
206Q8—542no—
207C2×C4—32no—
208C2×C4—31no—
209C2×C4—91no—
210C2×C4—92no—
211C2×C4—542no—
212C2×C3—21no—
213C2×C3—22no—
214C2×C3—21no—
215C2×C3—22no—
216C2×C3—22no—
217C2×C3—21no—
218C2×C3—41no—
219C2×C3—41no—
220C2×C3—41no—
221C2×C3#379711yescharacteristic
222C2×C3#379712yes—
223C2×C3#379811yescharacteristic
224C2×C3D3×D1212yes—
225C2×C3#380412yes—
226C2×C3#381111yescharacteristic
227D3—184no—
228D3—182no—
229D3—184no—
230D3—364no—
231D3—364no—
232C22—272no—
233C22—541no—
234C22—541no—
235C22D3311yescharacteristic
236C4—61no—
237C4—62no—
238C4—92no—
239C4—182no—
240C3—21no—
241C3—22no—
242C3—41no—
243C3#6386511yescharacteristic, complemented
244C3#6389512yescomplemented
245C2—542no—
246C2#8895511yescharacteristic
247C2#8895811yescharacteristic
248C2#8895911yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, cosocleC22×C33C23
prime radicalC22×C3⋊(C3⋊D3)C22
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
4. C3×#3724C2semidirect
5. C3×Dic3⋊Dic3C2semidirect
24. #3724D3semidirect
25. Dic3⋊Dic3D3semidirect
32. C2×C3⋊(C3⋊D3)Q8semidirect
116. C33C22⋊Q8semidirect
203. C32#896semidirect
204. C32#897semidirect
243. C3#63865semidirect
244. C3#63895semidirect

Maximal subgroups

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

subgroupindex
C2×#62962
C2×#62892
#888042
C3×#37242
C3×Dic3⋊Dic32
#638953
#638653