Finite Group Databasebeta

← #11723539

Subgroups — #11723539

Order 864.

1008 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 1008 conjugacy classes of subgroups — too many to draw. Only its 134 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
1C22×D3×C3⋊D3C211yesmaximal, characteristic
2C2×C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C2×#6289C211yesmaximal, characteristic, complemented
4C3×#3752C211yesmaximal, characteristic, complemented
5C32×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
6#88803C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
8C22⋊C4×C3⋊D3—31nomaximal
9#63921—34nomaximal
10C22×C3⋊(C3⋊D3)C2211yescharacteristic
11C22×C3×C3⋊D3C2211yescharacteristic
12C22×C32×D3C2211yescharacteristic
13C2×D3×C3⋊D3—22no—
14C2×D3×C3⋊D3—22no—
15C2×D3×C3⋊D3C414yescomplemented
16C2×C33×C4C2211yescharacteristic, complemented
17C2×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
18C4×C3⋊(C3⋊D3)—22no—
19C2×C3×C3⋊Dic3C2211yescharacteristic, complemented
20C2×C32×Dic3C2211yescharacteristic, complemented
21#6289—22no—
22C23×C3⋊D3—31no—
23C22×D32—34no—
24C2×C4×C3⋊D3—31no—
25C2×C4×C3⋊D3—34no—
26C2×C4×C3⋊D3—31no—
27C2×C4×C3⋊D3—64no—
28C2×#624—34no—
29C32×C22⋊C4—31no—
30#3757—31no—
31#3752—31no—
32#3752D311yescharacteristic, complemented
33C3×(C2×D3)⋊C4—34no—
34C3×(C2×D3)⋊C4D314yescomplemented
35(C2×D3)⋊Dic3—34no—
36(C2×D3)⋊Dic3—34no—
37C22×C33C2311yescharacteristic
38C2×C3⋊(C3⋊D3)C2×C412yes—
39C2×C3⋊(C3⋊D3)D412yes—
40C2×C3⋊(C3⋊D3)D412yes—
41C2×C3×C3⋊D3—21no—
42C2×C3×C3⋊D3—21no—
43C2×C3×C3⋊D3C2×C412yescomplemented
44C2×C32×D3—21no—
45C2×C32×D3—21no—
46C2×C32×D3C2×C412yescomplemented
47D3×C3⋊D3—28no—
48C33×C4—21no—
49C3⋊(C3⋊Dic3)—21no—
50C3×C3⋊Dic3—21no—
51C32×Dic3—21no—
52D3×C22⋊C4—94no—
53D3×C22⋊C4—91no—
54C23×C32—31no—
55C22×C3⋊D3—31no—
56C22×C3⋊D3—34no—
57C22×C3⋊D3—34no—
58C22×C3⋊D3—64no—
59C22×C3⋊D3—62no—
60C22×C3⋊D3—62no—
61C22×C3⋊D3C2×D311yescharacteristic
62C22×C3×D3—34no—
63C22×C3×D3C2×D314yes—
64C2×D32—316no—
65C2×D32—68no—
66C2×D32—68no—
67C2×C32×C4—24no—
68C2×C32×C4—31no—
69C2×C32×C4C2×D314yescomplemented
70C2×C32×C4C2×D311yescharacteristic, complemented
71C2×C3⋊Dic3—31no—
72C2×C3⋊Dic3—34no—
73C2×C3⋊Dic3—64no—
74C2×C3⋊Dic3C2×D311yescharacteristic, complemented
75C4×C3⋊D3—62no—
76C4×C3⋊D3—62no—
77C4×C3⋊D3—68no—
78C4×C3⋊D3—616no—
79C2×C3×Dic3—34no—
80C2×C3×Dic3C2×D314yescomplemented
81#624—68no—
82C2×C33C22×C411yescharacteristic
83C2×C33C2×D411yescharacteristic
84C2×C33C2×D411yescharacteristic
85C3⋊(C3⋊D3)C22⋊C414yescomplemented
86C3×C3⋊D3—22no—
87C32×D3—22no—
88C23×D3—94no—
89C23×D3—91no—
90C2×C4×D3—94no—
91C2×C4×D3—94no—
92C2×C4×D3—91no—
93C2×C4×D3—91no—
94C2×C4×D3—184no—
95C3×C22⋊C4—34no—
96C3×C22⋊C4—91no—
97#217—94no—
98#217—91no—
99(C2×D3)⋊C4—34no—
100(C2×D3)⋊C4—94no—
101(C2×D3)⋊C4—91no—
102(C2×D3)⋊C4C3⋊D311yescharacteristic, complemented
103C22×C32—24no—
104C22×C32—32no—
105C22×C32—61no—
106C22×C32—61no—
107C22×C32C22×D314yes—
108C22×C32C22×D311yescharacteristic
109C2×C3⋊D3—21no—
110C2×C3⋊D3—21no—
111C2×C3⋊D3—32no—
112C2×C3⋊D3—32no—
113C2×C3⋊D3—32no—
114C2×C3⋊D3—38no—
115C2×C3⋊D3—38no—
116C2×C3⋊D3—38no—
117C2×C3⋊D3—68no—
118C2×C3⋊D3—68no—
119C2×C3⋊D3—68no—
120C2×C3⋊D3—68no—
121C2×C3⋊D3C4×D312yescomplemented
122C2×C3×D3—24no—
123C2×C3×D3—24no—
124C2×C3×D3—38no—
125C2×C3×D3—64no—
126C2×C3×D3—64no—
127C2×C3×D3C4×D318yescomplemented
128D32—632no—
129C32×C4—28no—
130C32×C4—24no—
131C32×C4—21no—
132C32×C4—61no—
133C3⋊Dic3—21no—
134C3⋊Dic3—68no—
135C3⋊Dic3—64no—
136C3⋊Dic3—61no—
137C3×Dic3—24no—
138C3×Dic3—64no—
139C2×C22⋊C4—271noSylow
140C33C2×C22⋊C411yescharacteristic, complemented, Sylow
141C23×C3—34no—
142C23×C3—91no—
143C22×D3—34no—
144C22×D3—916no—
145C22×D3—94no—
146C22×D3—91no—
147C22×D3—94no—
148C22×D3—184no—
149C22×D3—182no—
150C22×D3—182no—
151C22×D3—188no—
152C22×D3—188no—
153C22×D3C2×C3⋊D311yescharacteristic
154C2×C3×C4—24no—
155C2×C3×C4—34no—
156C2×C3×C4—91no—
157C2×C3×C4D3214yescomplemented
158C2×C3×C4C2×C3⋊D311yescharacteristic, complemented
159C2×Dic3—34no—
160C2×Dic3—94no—
161C2×Dic3—91no—
162C2×Dic3—184no—
163C2×Dic3C2×C3⋊D311yescharacteristic, complemented
164C4×D3—182no—
165C4×D3—188no—
166C4×D3—188no—
167C4×D3—182no—
168C4×D3—1816no—
169C2×C32—24no—
170C2×C32—24no—
171C2×C32—24no—
172C2×C32—62no—
173C2×C32C2×C4×D314yes—
174C2×C32C2×C4×D311yescharacteristic
175C2×C32D3×D414yes—
176C2×C32D3×D414yes—
177C2×C32D3×D411yescharacteristic
178C2×C32D3×D411yescharacteristic
179C3⋊D3—22no—
180C3⋊D3—34no—
181C3⋊D3—316no—
182C3⋊D3—616no—
183C3×D3—28no—
184C3×D3—68no—
185C24—271no—
186C22×C4—271no—
187C22×C4—271no—
188C22⋊C4—31no—
189C22⋊C4—91no—
190C22⋊C4—271no—
191C22⋊C4—271no—
192C22×C3—24no—
193C22×C3—38no—
194C22×C3—64no—
195C22×C3—64no—
196C22×C3—92no—
197C22×C3—181no—
198C22×C3—181no—
199C22×C3C2×D3214yes—
200C22×C3C22×C3⋊D311yescharacteristic
201C2×D3—21no—
202C2×D3—21no—
203C2×D3—38no—
204C2×D3—64no—
205C2×D3—64no—
206C2×D3—98no—
207C2×D3—98no—
208C2×D3—98no—
209C2×D3—92no—
210C2×D3—92no—
211C2×D3—92no—
212C2×D3—188no—
213C2×D3—188no—
214C2×D3—188no—
215C2×D3—188no—
216C2×D3—1832no—
217C2×D3C4×C3⋊D312yescomplemented
218C3×C4—28no—
219C3×C4—24no—
220C3×C4—21no—
221C3×C4—64no—
222C3×C4—181no—
223Dic3—21no—
224Dic3—64no—
225Dic3—188no—
226Dic3—184no—
227Dic3—181no—
228C32—24no—
229C32D3×C22⋊C414yescomplemented
230C32D3×C22⋊C411yescharacteristic, complemented
231C23—31no—
232C23—91no—
233C23—274no—
234C23—271no—
235C23—542no—
236C23—542no—
237C2×C4—31no—
238C2×C4—91no—
239C2×C4—271no—
240C2×C4—542no—
241C2×C4—542no—
242C2×C4D3×C3⋊D311yescharacteristic, complemented
243C2×C3—24no—
244C2×C3—24no—
245C2×C3—24no—
246C2×C3—68no—
247C2×C3—182no—
248C2×C3C4×D3214yes—
249C2×C3#380214yes—
250C2×C3#381114yes—
251C2×C3C2×C4×C3⋊D311yescharacteristic
252C2×C3D4×C3⋊D311yescharacteristic
253C2×C3D4×C3⋊D311yescharacteristic
254D3—22no—
255D3—68no—
256D3—94no—
257D3—916no—
258D3—1816no—
259C22—32no—
260C22—61no—
261C22—61no—
262C22—92no—
263C22—181no—
264C22—181no—
265C22—272no—
266C22—272no—
267C22—272no—
268C22—548no—
269C22C2×D3×C3⋊D311yescharacteristic
270C4—21no—
271C4—61no—
272C4—181no—
273C4—541no—
274C3—24no—
275C3#6392114yescomplemented
276C3C22⋊C4×C3⋊D311yescharacteristic, complemented
277C2—62no—
278C2—182no—
279C2—274no—
280C2C4×D3×C3⋊D311yescharacteristic
281C2#8902611yescharacteristic
282C2#8903911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×C3⋊D3
commutatorC2×C33C22×C4
FittingC2×C33×C4C22
socle, cosocleC22×C33C23
prime radicalC22×D3×C3⋊D3C2
hypercenterC2×C4D3×C3⋊D3
hypocenterC33C2×C22⋊C4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C3⋊(C3⋊D3)C2semidirect
3. C2×#6289C2semidirect
4. C3×#3752C2semidirect
5. C32×(C2×D3)⋊C4C2semidirect
6. #88803C2semidirect
7. (C2×D3)⋊(C3⋊Dic3)C2semidirect
15. C2×D3×C3⋊D3C4semidirect
16. C2×C33×C4C22semidirect
17. C2×C3⋊(C3⋊Dic3)C22semidirect
19. C2×C3×C3⋊Dic3C22semidirect
20. C2×C32×Dic3C22semidirect
32. #3752D3semidirect
34. C3×(C2×D3)⋊C4D3semidirect
43. C2×C3×C3⋊D3C2×C4semidirect
46. C2×C32×D3C2×C4semidirect
69. C2×C32×C4C2×D3semidirect
70. C2×C32×C4C2×D3semidirect
74. C2×C3⋊Dic3C2×D3semidirect
80. C2×C3×Dic3C2×D3semidirect
85. C3⋊(C3⋊D3)C22⋊C4semidirect
102. (C2×D3)⋊C4C3⋊D3semidirect
121. C2×C3⋊D3C4×D3semidirect
127. C2×C3×D3C4×D3semidirect
140. C33C2×C22⋊C4semidirect
157. C2×C3×C4D32semidirect
158. C2×C3×C4C2×C3⋊D3semidirect
163. C2×Dic3C2×C3⋊D3semidirect
217. C2×D3C4×C3⋊D3semidirect
229. C32D3×C22⋊C4semidirect
230. C32D3×C22⋊C4semidirect
242. C2×C4D3×C3⋊D3semidirect
275. C3#63921semidirect
276. C3C22⋊C4×C3⋊D3semidirect

Maximal subgroups

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

subgroupindex
C22×D3×C3⋊D32
C2×C4×C3⋊(C3⋊D3)2
C2×#62892
C3×#37522
C32×(C2×D3)⋊C42
#888032
(C2×D3)⋊(C3⋊Dic3)2
C22⋊C4×C3⋊D33
#639213