Finite Group Databasebeta

← #11722964

Subgroups — #11722964

Order 864.

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

normal subgroups only

This group has 570 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×#6291C212yesmaximal, complemented
2C2×#6289C211yesmaximal, characteristic, complemented
3C2×C3×#625C211yesmaximal, characteristic, complemented
4#88815C211yesmaximal, characteristic, complemented
5C3×(C2×D3)⋊Dic3C212yesmaximal, complemented
6#63958—32nomaximal
7#63912—31nomaximal
8C22×C3⋊(C3⋊D3)C2211yescharacteristic
9C22×C32×D3C2212yes—
10C2×C3×C3⋊Dic3C2212yes—
11C2×C3×C3⋊Dic3C2211yescharacteristic
12C2×C32×Dic3C2211yescharacteristic
13#6291—24no—
14#6289—22no—
15C3×#625—22no—
16C2×#638—32no—
17C2×C4×C3⋊D3—31no—
18C2×C3×#67—32no—
19C2×#626—32no—
20C2×#626—32no—
21C2×#626—62no—
22C2×#625D311yescharacteristic, complemented
23C2×#624—32no—
24C2×#624—61no—
25C3×#217—32no—
26C3×(C2×D3)⋊C4—32no—
27#3724—31no—
28Dic3⋊Dic3—32no—
29(C2×D3)⋊Dic3D312yescomplemented
30C22×C33C2311yescharacteristic
31C2×C3⋊(C3⋊D3)—21no—
32C2×C3⋊(C3⋊D3)—21no—
33C2×C3⋊(C3⋊D3)D412yescomplemented
34C2×C32×D3—22no—
35C2×C32×D3—22no—
36C2×C32×D3—22no—
37C3×C3⋊Dic3—22no—
38C3×C3⋊Dic3D412yescomplemented
39C32×Dic3—21no—
40#938—91no—
41#884—92no—
42C23×C32—32no—
43C22×C3⋊D3—32no—
44C22×C3⋊D3—31no—
45C22×C3⋊D3—61no—
46C22×C3⋊D3—62no—
47C22×C3⋊D3—121no—
48C22×C3×D3—22no—
49C22×C3×D3C2×D312yes—
50C22×C3×D3C2×D312yes—
51C2×C32×C4—31no—
52#638—64no—
53C2×C3⋊Dic3C2×D312yes—
54C2×C3⋊Dic3C2×D311yescharacteristic
55C4×C3⋊D3—62no—
56C3×#67—64no—
57C2×C3×Dic3—21no—
58C2×C3×Dic3—32no—
59C2×C3×Dic3—32no—
60C2×C3×Dic3—32no—
61C2×C3×Dic3—61no—
62C2×C3×Dic3—62no—
63C2×C3×Dic3C2×D312yes—
64#626—64no—
65#626—64no—
66#626—68no—
67#625—22no—
68#624—64no—
69#624—64no—
70C2×C33C2×D411yescharacteristic
71C2×C33C2×D411yescharacteristic
72C2×C33#4111yescharacteristic
73C3⋊(C3⋊D3)—22no—
74C32×D3—42no—
75C2×C3×D4—91no—
76C2×#67—32no—
77C2×#67—92no—
78C2×#67—92no—
79C2×#67—182no—
80C2×D12—92no—
81C2×C4×D3—92no—
82C2×C4×D3—91no—
83C2×C4×D3—181no—
84C3×C22⋊C4—92no—
85#217—32no—
86(C2×D3)⋊C4—32no—
87C4⋊Dic3—91no—
88Dic3⋊C4—92no—
89C22×C32—21no—
90C22×C32—22no—
91C22×C32—41no—
92C22×C32—62no—
93C22×C32—62no—
94C22×C32—62no—
95C22×C32C22×D312yes—
96C22×C32C22×D311yescharacteristic
97C2×C3⋊D3—32no—
98C2×C3⋊D3—34no—
99C2×C3⋊D3—62no—
100C2×C3⋊D3—64no—
101C2×C3⋊D3—61no—
102C2×C3⋊D3—61no—
103C2×C3⋊D3—62no—
104C2×C3⋊D3—62no—
105C2×C3⋊D3—62no—
106C2×C3⋊D3—62no—
107C2×C3⋊D3—122no—
108C2×C3⋊D3—122no—
109C2×C3⋊D3—122no—
110C2×C3⋊D3—122no—
111C2×C3⋊D3—122no—
112C2×C3×D3—24no—
113C2×C3×D3—22no—
114C2×C3×D3—22no—
115C2×C3×D3—22no—
116C2×C3×D3—22no—
117C2×C3×D3—22no—
118C2×C3×D3—22no—
119C2×C3×D3—42no—
120C2×C3×D3—42no—
121C32×C4—61no—
122C3⋊Dic3—22no—
123C3⋊Dic3#6712yescomplemented
124C3×Dic3—22no—
125C3×Dic3—22no—
126C3×Dic3—34no—
127C3×Dic3—62no—
128C3×Dic3—64no—
129C3×Dic3—62no—
130C3×Dic3—62no—
131#121—271noSylow
132C33#12111yescharacteristic, complemented, Sylow
133C23×C3—32no—
134C23×C3—32no—
135C23×C3—62no—
136C22×D3—92no—
137C22×D3—91no—
138C22×D3—181no—
139C22×D3—182no—
140C22×D3—361no—
141C22×D3D3212yes—
142C3×D4—182no—
143C2×C3×C4—32no—
144C2×C3×C4—61no—
145C2×C3×C4—91no—
146C2×C3×C4—92no—
147#67—64no—
148#67—184no—
149#67—184no—
150#67—188no—
151C2×Dic3—32no—
152C2×Dic3—32no—
153C2×Dic3—32no—
154C2×Dic3—61no—
155C2×Dic3—62no—
156C2×Dic3D3211yescharacteristic
157D12—184no—
158C4×D3—182no—
159C4×D3—184no—
160C4×D3—184no—
161C2×C32—21no—
162C2×C32—21no—
163C2×C32—22no—
164C2×C32—22no—
165C2×C32—21no—
166C2×C32—22no—
167C2×C32—41no—
168C2×C32—41no—
169C2×C32—41no—
170C2×C32—122no—
171C2×C32#23512yes—
172C2×C32D3×D412yes—
173C2×C32D3×D412yes—
174C2×C32D3×D411yescharacteristic
175C2×C32#23711yescharacteristic
176C2×C32C2×#6711yescharacteristic
177C3⋊D3—62no—
178C3⋊D3—64no—
179C3⋊D3—64no—
180C3⋊D3—124no—
181C3⋊D3—124no—
182C3×D3—44no—
183C3×D3—42no—
184C3×D3—42no—
185C2×D4—91no—
186C2×D4—272no—
187C22×C4—271no—
188C4⋊C4—271no—
189C22⋊C4—92no—
190C22×C3—21no—
191C22×C3—22no—
192C22×C3—41no—
193C22×C3—64no—
194C22×C3—62no—
195C22×C3—62no—
196C22×C3—62no—
197C22×C3—62no—
198C22×C3—62no—
199C22×C3—62no—
200C22×C3—122no—
201C22×C3—122no—
202C22×C3C2×D3211yescharacteristic
203C22×C3C2×D3212yes—
204C2×D3—22no—
205C2×D3—22no—
206C2×D3—22no—
207C2×D3—92no—
208C2×D3—94no—
209C2×D3—182no—
210C2×D3—184no—
211C2×D3—182no—
212C2×D3—182no—
213C2×D3—181no—
214C2×D3—182no—
215C2×D3—182no—
216C2×D3—181no—
217C2×D3—362no—
218C2×D3—362no—
219C2×D3—362no—
220C2×D3—362no—
221C2×D3—362no—
222C3×C4—62no—
223C3×C4—62no—
224C3×C4—92no—
225C3×C4—182no—
226Dic3—21no—
227Dic3—34no—
228Dic3—62no—
229Dic3—64no—
230Dic3—62no—
231Dic3—62no—
232C32—21no—
233C32—22no—
234C32—41no—
235C32#88412yescomplemented
236C32#93811yescharacteristic, complemented
237C23—32no—
238C23—271no—
239D4—182no—
240D4—544no—
241C2×C4—31no—
242C2×C4—91no—
243C2×C4—92no—
244C2×C4—542no—
245C2×C3—21no—
246C2×C3—22no—
247C2×C3—22no—
248C2×C3—21no—
249C2×C3—21no—
250C2×C3—22no—
251C2×C3—41no—
252C2×C3—41no—
253C2×C3—41no—
254C2×C3—124no—
255C2×C3—122no—
256C2×C3—122no—
257C2×C3#379811yescharacteristic
258C2×C3#380211yescharacteristic
259C2×C3#380412yes—
260C2×C3D3×#6712yes—
261C2×C3#381111yescharacteristic
262C2×C3#381112yes—
263D3—42no—
264D3—184no—
265D3—182no—
266D3—184no—
267D3—364no—
268D3—364no—
269C22—62no—
270C22—62no—
271C22—62no—
272C22—272no—
273C22—541no—
274C22—541no—
275C22D3311yescharacteristic
276C4—61no—
277C4—92no—
278C4—182no—
279C3—21no—
280C3—22no—
281C3—41no—
282C3#6391211yescharacteristic, complemented
283C3#6395812yescomplemented
284C2—122no—
285C2—542no—
286C2#8895211yescharacteristic
287C2#8895411yescharacteristic
288C2#8895911yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, cosocleC22×C33C23
hypocenterC33#121
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#6291C2semidirect
2. C2×#6289C2semidirect
3. C2×C3×#625C2semidirect
4. #88815C2semidirect
5. C3×(C2×D3)⋊Dic3C2semidirect
22. C2×#625D3semidirect
29. (C2×D3)⋊Dic3D3semidirect
33. C2×C3⋊(C3⋊D3)D4semidirect
38. C3×C3⋊Dic3D4semidirect
123. C3⋊Dic3#67semidirect
132. C33#121semidirect
235. C32#884semidirect
236. C32#938semidirect
282. C3#63912semidirect
283. C3#63958semidirect

Maximal subgroups

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

subgroupindex
C2×#62912
C2×#62892
C2×C3×#6252
#888152
C3×(C2×D3)⋊Dic32
#639583
#639123