Finite Group Databasebeta

← #11722938

Subgroups — #11722938

Order 864.

530 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 88 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×#6295C211yesmaximal, characteristic
2C2×Dic3×C3⋊D3C212yesmaximal, complemented
3Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C3×#3724C211yesmaximal, characteristic, complemented
5C3×#3722C212yesmaximal, complemented
6#63902—31nomaximal
7#63881—32nomaximal
8C22×C3×C3⋊D3C2212yes—
9C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
10C2×C3×C3⋊Dic3C2211yescharacteristic
11C2×C32×Dic3C2212yes—
12C2×C32×Dic3C2211yescharacteristic
13#6295C414yescomplemented
14Dic3×C3⋊D3—24no—
15C2×C4×C3⋊D3—32no—
16C2×#626—32no—
17C2×#625—31no—
18C2×D3×Dic3—32no—
19C2×D3×Dic3—32no—
20C2×D3×Dic3—62no—
21C4×C3⋊Dic3—31no—
22C3×(C2×D3)⋊C4—32no—
23C3×(C2×D3)⋊C4—32no—
24C3×Dic3⋊C4—32no—
25#3724D311yescharacteristic, complemented
26#3722D312yescomplemented
27Dic32—32no—
28Dic32—61no—
29C22×C33C2311yescharacteristic
30C2×C3×C3⋊D3—22no—
31C2×C3×C3⋊D3—22no—
32C2×C3×C3⋊D3C2×C414yescomplemented
33C3⋊(C3⋊Dic3)D412yescomplemented
34C3×C3⋊Dic3C2×C412yescomplemented
35C32×Dic3—22no—
36C32×Dic3—21no—
37#893—91no—
38#67⋊C4—92no—
39C22×C3⋊D3C2×D312yes—
40C22×C3×D3—32no—
41C22×C3×D3—32no—
42C22×C3×D3—62no—
43C2×C32×C4—32no—
44C2×C32×C4—31no—
45C2×C3⋊Dic3—32no—
46C2×C3⋊Dic3—31no—
47C2×C3⋊Dic3—61no—
48C2×C3⋊Dic3—62no—
49C2×C3⋊Dic3—121no—
50C2×C3⋊Dic3C2×D311yescharacteristic
51C4×C3⋊D3—64no—
52C2×C3×Dic3—21no—
53C2×C3×Dic3—22no—
54C2×C3×Dic3—32no—
55C2×C3×Dic3—61no—
56C2×C3×Dic3C2×D312yes—
57C2×C3×Dic3C2×D312yes—
58C2×C3×Dic3C2×D312yes—
59#626—38no—
60#625—34no—
61D3×Dic3—64no—
62D3×Dic3—64no—
63D3×Dic3—68no—
64C2×C33C22×C411yescharacteristic
65C2×C33C2×D411yescharacteristic
66C2×C33#4111yescharacteristic
67C3×C3⋊D3—24no—
68C2×#67—92no—
69C22×Dic3—92no—
70C2×D12—91no—
71C2×C4×D3—92no—
72C2×C4×D3—92no—
73C2×C4×D3—182no—
74C3×C4⋊C4—91no—
75C3×C22⋊C4—92no—
76(C2×D3)⋊C4—32no—
77(C2×D3)⋊C4—32no—
78Dic3⋊C4—32no—
79C4×Dic3—92no—
80C4×Dic3—91no—
81C4×Dic3—181no—
82C22×C32—21no—
83C22×C32—22no—
84C22×C32—41no—
85C22×C32C22×D312yes—
86C22×C32C22×D311yescharacteristic
87C2×C3⋊D3—22no—
88C2×C3⋊D3—22no—
89C2×C3⋊D3C4×D314yescomplemented
90C2×C3×D3—34no—
91C2×C3×D3—34no—
92C2×C3×D3—64no—
93C2×C3×D3—64no—
94C2×C3×D3—64no—
95C2×C3×D3—62no—
96C2×C3×D3—62no—
97C2×C3×D3—62no—
98C2×C3×D3—62no—
99C32×C4—62no—
100C32×C4—61no—
101C3⋊Dic3—32no—
102C3⋊Dic3—34no—
103C3⋊Dic3—64no—
104C3⋊Dic3—62no—
105C3⋊Dic3—122no—
106C3⋊Dic3C4×D312yescomplemented
107C3×Dic3—24no—
108C3×Dic3—22no—
109C3×Dic3—22no—
110C3×Dic3—22no—
111C3×Dic3—22no—
112C3×Dic3—34no—
113C3×Dic3—62no—
114C4×D4—271noSylow
115C33C4×D411yescharacteristic, complemented, Sylow
116C23×C3—92no—
117C22×D3—32no—
118C22×D3—32no—
119C22×D3—62no—
120C2×C3×C4—32no—
121C2×C3×C4—32no—
122C2×C3×C4—32no—
123C2×C3×C4—61no—
124C2×C3×C4—62no—
125C2×C3×C4—91no—
126#67—98no—
127C2×Dic3—32no—
128C2×Dic3—61no—
129C2×Dic3—91no—
130C2×Dic3—92no—
131C2×Dic3—181no—
132C2×Dic3—182no—
133C2×Dic3—184no—
134C2×Dic3—361no—
135C2×Dic3D3211yescharacteristic
136C2×Dic3D3212yes—
137D12—94no—
138C4×D3—184no—
139C4×D3—184no—
140C4×D3—188no—
141C2×C32—22no—
142C2×C32—21no—
143C2×C32—22no—
144C2×C32—21no—
145C2×C32—21no—
146C2×C32—22no—
147C2×C32—41no—
148C2×C32—41no—
149C2×C32—41no—
150C2×C32C2×C4×D312yes—
151C2×C32C2×C4×D311yescharacteristic
152C2×C32D3×D412yes—
153C2×C32D3×D411yescharacteristic
154C2×C32#23712yes—
155C2×C32#23911yescharacteristic
156C3⋊D3—24no—
157C3×D3—68no—
158C3×D3—64no—
159C3×D3—64no—
160C2×D4—271no—
161C22×C4—272no—
162C4⋊C4—91no—
163C22⋊C4—92no—
164C42—271no—
165C22×C3—21no—
166C22×C3—22no—
167C22×C3—41no—
168C22×C3—94no—
169C22×C3—182no—
170C22×C3—182no—
171C22×C3C2×D3212yes—
172C22×C3C2×D3211yescharacteristic
173C2×D3—34no—
174C2×D3—34no—
175C2×D3—64no—
176C2×D3—64no—
177C2×D3—64no—
178C2×D3—62no—
179C2×D3—62no—
180C2×D3—62no—
181C2×D3—62no—
182C3×C4—64no—
183C3×C4—62no—
184C3×C4—62no—
185C3×C4—62no—
186C3×C4—62no—
187C3×C4—92no—
188Dic3—21no—
189Dic3—22no—
190Dic3—34no—
191Dic3—62no—
192Dic3—92no—
193Dic3—94no—
194Dic3—184no—
195Dic3—182no—
196Dic3—362no—
197C32—22no—
198C32—21no—
199C32—41no—
200C32#67⋊C412yescomplemented
201C32#89311yescharacteristic, complemented
202C23—92no—
203D4—274no—
204C2×C4—31no—
205C2×C4—32no—
206C2×C4—91no—
207C2×C4—271no—
208C2×C4—544no—
209C2×C3—21no—
210C2×C3—22no—
211C2×C3—22no—
212C2×C3—21no—
213C2×C3—21no—
214C2×C3—22no—
215C2×C3—41no—
216C2×C3—41no—
217C2×C3—41no—
218C2×C3—184no—
219C2×C3#379612yes—
220C2×C3C4×D3212yes—
221C2×C3C4×D3211yescharacteristic
222C2×C3#380211yescharacteristic
223C2×C3#380511yescharacteristic
224C2×C3#381112yes—
225D3—68no—
226D3—64no—
227D3—64no—
228C22—94no—
229C22—182no—
230C22—182no—
231C22D3311yescharacteristic
232C4—62no—
233C4—61no—
234C4—92no—
235C4—272no—
236C3—22no—
237C3—21no—
238C3—41no—
239C3#6388112yescomplemented
240C3#6390211yescharacteristic, complemented
241C2—184no—
242C2#8894911yescharacteristic
243C2#8895411yescharacteristic
244C2#8896411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D33
commutatorC2×C33C22×C4
Fitting, socle, cosocleC22×C33C23
prime radicalC2×#6295C2
hypocenterC33C4×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×C3⋊D3C2semidirect
3. Dic3×C3⋊Dic3C2semidirect
4. C3×#3724C2semidirect
5. C3×#3722C2semidirect
13. #6295C4semidirect
25. #3724D3semidirect
26. #3722D3semidirect
32. C2×C3×C3⋊D3C2×C4semidirect
33. C3⋊(C3⋊Dic3)D4semidirect
34. C3×C3⋊Dic3C2×C4semidirect
89. C2×C3⋊D3C4×D3semidirect
106. C3⋊Dic3C4×D3semidirect
115. C33C4×D4semidirect
200. C32#67⋊C4semidirect
201. C32#893semidirect
239. C3#63881semidirect
240. C3#63902semidirect

Maximal subgroups

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

subgroupindex
C2×#62952
C2×Dic3×C3⋊D32
Dic3×C3⋊Dic32
C3×#37242
C3×#37222
#639023
#638813