Finite Group Databasebeta

← #11723029

Subgroups — #11723029

Order 864.

586 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 84 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, complemented
2C2×#6290C212yesmaximal, complemented
3C2×C3×#626C212yesmaximal, complemented
4C2×C3×#625C211yesmaximal, characteristic, complemented
5Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
6#63979—32nomaximal
7#63914—31nomaximal
8C22×C3×C3⋊D3C2212yes—
9C22×C32×D3C2212yes—
10C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
11C2×C3×C3⋊Dic3C2211yescharacteristic
12C2×C32×Dic3C2211yescharacteristic
13#6295—22no—
14#6290—24no—
15C3×#626—24no—
16C3×#625—22no—
17C2×#638—32no—
18C2×C3×#67—32no—
19C2×C3×#67—32no—
20C2×C3×D12—32no—
21C2×#626—32no—
22C2×#626D312yescomplemented
23C2×#625—32no—
24C2×#625—32no—
25C2×#625—31no—
26C2×#625—62no—
27C2×#625D311yescharacteristic, complemented
28C4×C3⋊Dic3—31no—
29Dic32—32no—
30Dic32—61no—
31C22×C33C2311yescharacteristic
32C2×C3×C3⋊D3—22no—
33C2×C3×C3⋊D3—22no—
34C2×C3×C3⋊D3—22no—
35C2×C32×D3—22no—
36C2×C32×D3—22no—
37C2×C32×D3—22no—
38C3⋊(C3⋊Dic3)D412yescomplemented
39C3×C3⋊Dic3D412yescomplemented
40C32×Dic3D412yescomplemented
41#940—92no—
42#940—91no—
43C23×C32—32no—
44C22×C3⋊D3C2×D312yes—
45C22×C3×D3—22no—
46C22×C3×D3—32no—
47C22×C3×D3—32no—
48C22×C3×D3—62no—
49C22×C3×D3C2×D312yes—
50C22×C3×D3C2×D312yes—
51C2×C32×C4—31no—
52#638—64no—
53C2×C3⋊Dic3—32no—
54C2×C3⋊Dic3—31no—
55C2×C3⋊Dic3—61no—
56C2×C3⋊Dic3—62no—
57C2×C3⋊Dic3—121no—
58C2×C3⋊Dic3C2×D311yescharacteristic
59C3×#67—64no—
60C3×#67—64no—
61C2×C3×Dic3—21no—
62C2×C3×Dic3—32no—
63C2×C3×Dic3—61no—
64C2×C3×Dic3C2×D312yes—
65C3×D12—64no—
66#626—24no—
67#626—64no—
68#625—22no—
69#625—64no—
70#625—64no—
71#625—68no—
72#625—62no—
73C2×C33C2×D411yescharacteristic
74C2×C33C2×D411yescharacteristic
75C2×C33C2×D411yescharacteristic
76C3×C3⋊D3—42no—
77C32×D3—42no—
78C2×C3×D4—92no—
79C2×C3×D4—91no—
80C2×#67—32no—
81C2×#67—32no—
82C2×#67—92no—
83C2×#67—92no—
84C2×#67—92no—
85C2×#67—92no—
86C2×#67—182no—
87C2×D12—32no—
88C2×D12—91no—
89C4×Dic3—92no—
90C4×Dic3—91no—
91C4×Dic3—181no—
92C22×C32—21no—
93C22×C32—22no—
94C22×C32—41no—
95C22×C32—62no—
96C22×C32—62no—
97C22×C32—62no—
98C22×C32C22×D312yes—
99C22×C32C22×D311yescharacteristic
100C2×C3⋊D3—22no—
101C2×C3⋊D3—22no—
102C2×C3⋊D3—22no—
103C2×C3×D3—24no—
104C2×C3×D3—22no—
105C2×C3×D3—22no—
106C2×C3×D3—22no—
107C2×C3×D3—22no—
108C2×C3×D3—22no—
109C2×C3×D3—22no—
110C2×C3×D3—42no—
111C2×C3×D3—42no—
112C2×C3×D3—64no—
113C2×C3×D3—62no—
114C2×C3×D3—62no—
115C2×C3×D3—62no—
116C2×C3×D3—62no—
117C2×C3×D3—62no—
118C2×C3×D3—62no—
119C2×C3×D3—122no—
120C2×C3×D3—122no—
121C32×C4—32no—
122C3⋊Dic3—32no—
123C3⋊Dic3—34no—
124C3⋊Dic3—64no—
125C3⋊Dic3—62no—
126C3⋊Dic3—122no—
127C3⋊Dic3#6712yescomplemented
128C3×Dic3—22no—
129C3×Dic3—34no—
130C3×Dic3—62no—
131C3×Dic3#6714yescomplemented
132#127—271noSylow
133C33#12711yescharacteristic, complemented, Sylow
134C23×C3—32no—
135C23×C3—32no—
136C23×C3—62no—
137C23×C3—92no—
138C22×D3—32no—
139C22×D3—32no—
140C22×D3—62no—
141C22×D3D3212yes—
142C3×D4—184no—
143C3×D4—182no—
144C2×C3×C4—32no—
145C2×C3×C4—61no—
146C2×C3×C4—91no—
147#67—64no—
148#67—64no—
149#67—184no—
150#67—184no—
151#67—184no—
152#67—184no—
153#67—188no—
154C2×Dic3—32no—
155C2×Dic3—61no—
156C2×Dic3—91no—
157C2×Dic3—92no—
158C2×Dic3—181no—
159C2×Dic3—182no—
160C2×Dic3—361no—
161C2×Dic3D3211yescharacteristic
162D12—64no—
163D12—182no—
164C2×C32—21no—
165C2×C32—21no—
166C2×C32—21no—
167C2×C32—22no—
168C2×C32—22no—
169C2×C32—22no—
170C2×C32—41no—
171C2×C32—41no—
172C2×C32—41no—
173C2×C32—122no—
174C2×C32D3×D412yes—
175C2×C32D3×D412yes—
176C2×C32D3×D411yescharacteristic
177C2×C32D3×D411yescharacteristic
178C2×C32C2×#6712yes—
179C2×C32C2×#6711yescharacteristic
180C3⋊D3—42no—
181C3×D3—44no—
182C3×D3—42no—
183C3×D3—42no—
184C3×D3—124no—
185C3×D3—122no—
186C3×D3—122no—
187C2×D4—92no—
188C2×D4—91no—
189C2×D4—271no—
190C2×D4—272no—
191C42—271no—
192C22×C3—21no—
193C22×C3—22no—
194C22×C3—41no—
195C22×C3—64no—
196C22×C3—62no—
197C22×C3—62no—
198C22×C3—62no—
199C22×C3—62no—
200C22×C3—62no—
201C22×C3—62no—
202C22×C3—122no—
203C22×C3—122no—
204C22×C3—182no—
205C22×C3—182no—
206C22×C3—182no—
207C22×C3C2×D3211yescharacteristic
208C22×C3C2×D3212yes—
209C2×D3—22no—
210C2×D3—22no—
211C2×D3—22no—
212C2×D3—64no—
213C2×D3—62no—
214C2×D3—62no—
215C2×D3—62no—
216C2×D3—62no—
217C2×D3—62no—
218C2×D3—62no—
219C2×D3—122no—
220C2×D3—122no—
221C3×C4—34no—
222C3×C4—62no—
223C3×C4—92no—
224Dic3—34no—
225Dic3—62no—
226Dic3—92no—
227Dic3—94no—
228Dic3—184no—
229Dic3—182no—
230Dic3—362no—
231Dic3#62512yescomplemented
232C32—21no—
233C32—22no—
234C32—41no—
235C32#94012yescomplemented
236C32#94011yescharacteristic, complemented
237C23—32no—
238C23—92no—
239D4—184no—
240D4—182no—
241D4—542no—
242D4—544no—
243C2×C4—31no—
244C2×C4—91no—
245C2×C4—271no—
246C2×C3—21no—
247C2×C3—22no—
248C2×C3—21no—
249C2×C3—22no—
250C2×C3—22no—
251C2×C3—21no—
252C2×C3—41no—
253C2×C3—41no—
254C2×C3—41no—
255C2×C3—124no—
256C2×C3—122no—
257C2×C3—122no—
258C2×C3—362no—
259C2×C3#380211yescharacteristic
260C2×C3#380211yescharacteristic
261C2×C3C2×#62511yescharacteristic
262C2×C3D3×#6712yes—
263C2×C3D3×#6712yes—
264C2×C3#381112yes—
265D3—42no—
266D3—124no—
267D3—122no—
268D3—122no—
269C22—62no—
270C22—62no—
271C22—62no—
272C22—182no—
273C22—182no—
274C22—182no—
275C22D3311yescharacteristic
276C4—32no—
277C4—92no—
278C4—272no—
279C3—22no—
280C3—21no—
281C3—41no—
282C3#6391411yescharacteristic, complemented
283C3#6397912yescomplemented
284C2—122no—
285C2—362no—
286C2D3×#62511yescharacteristic
287C2#8895311yescharacteristic
288C2#8895411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, cosocleC22×C33C23
hypocenterC33#127
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#6295C2semidirect
2. C2×#6290C2semidirect
3. C2×C3×#626C2semidirect
4. C2×C3×#625C2semidirect
5. Dic3×C3⋊Dic3C2semidirect
22. C2×#626D3semidirect
27. C2×#625D3semidirect
38. C3⋊(C3⋊Dic3)D4semidirect
39. C3×C3⋊Dic3D4semidirect
40. C32×Dic3D4semidirect
127. C3⋊Dic3#67semidirect
131. C3×Dic3#67semidirect
133. C33#127semidirect
231. Dic3#625semidirect
235. C32#940semidirect
236. C32#940semidirect
282. C3#63914semidirect
283. C3#63979semidirect

Maximal subgroups

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

subgroupindex
C2×#62952
C2×#62902
C2×C3×#6262
C2×C3×#6252
Dic3×C3⋊Dic32
#639793
#639143