Finite Group Databasebeta

← #13179769

Subgroups — #13179769

Order 1728.

3500 conjugacy classes of subgroups; only the 667 normal subgroups are shown.

normal subgroups only

This group has 3500 conjugacy classes of subgroups — too many to draw. Only its 667 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
1C33×C22⋊Q8C211yesmaximal, characteristic, complemented
2#11723988C212yesmaximal, complemented
3#11723971C211yesmaximal, characteristic, complemented
4C2×#88859C211yesmaximal, characteristic, complemented
5#11723939C212yesmaximal, complemented
6#11723936C212yesmaximal, complemented
7#11723935C212yesmaximal, complemented
8#11723923C212yesmaximal, complemented
9#11723922C212yesmaximal, complemented
10#10596115—313nomaximal
11C23×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C33×Q8C2211yescharacteristic, complemented
13C22×C33×C4C2211yescharacteristic
14C2×#6312C2212yes—
15C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C4⋊(C3⋊(C3⋊D3))C2212yes—
17C2×C4⋊(C3⋊(C3⋊D3))C2212yes—
18C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
19C2×C4×C3⋊(C3⋊D3)C2212yescomplemented
20C33×C4⋊C4C2211yescharacteristic, complemented
21C33×C4⋊C4C2212yescomplemented
22C33×C22⋊C4C2212yes—
23#88859C2212yes—
24#88859C2212yes—
25#88859C2212yes—
26#88859C2212yes—
27#88859C2212yescomplemented
28#88859C2212yescomplemented
29C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
30#88857C2212yescomplemented
31C4×C3⋊(C3⋊Dic3)C2212yescomplemented
32C32×C22⋊Q8D3113yescomplemented
33#64156—326no—
34#64139—313no—
35C2×#3752—313no—
36#64107—326no—
37#64104—326no—
38#64103—326no—
39#64091—326no—
40#64090—326no—
41C23×C33C2311yescharacteristic
42C22×C3⋊(C3⋊D3)—22no—
43C22×C3⋊(C3⋊D3)—41no—
44C22×C3⋊(C3⋊D3)—41no—
45C22×C3⋊(C3⋊D3)C2312yes—
46C22×C3⋊(C3⋊D3)C2312yes—
47C33×Q8—41no—
48C2×C33×C4—41no—
49C2×C33×C4C2312yes—
50C2×C33×C4C2312yes—
51C2×C33×C4C2311yescharacteristic
52C2×C33×C4C2311yescharacteristic
53#6312—24no—
54C2×C3⋊(C3⋊Dic3)—41no—
55C2×C3⋊(C3⋊Dic3)C2312yes—
56C2×C3⋊(C3⋊Dic3)C2312yes—
57C4⋊(C3⋊(C3⋊D3))—42no—
58C4⋊(C3⋊(C3⋊D3))—42no—
59C4⋊(C3⋊(C3⋊D3))—41no—
60C4×C3⋊(C3⋊D3)—42no—
61#5644—913no—
62C23×C3⋊D3—313no—
63C2×C32×Q8C2×D3113yescomplemented
64C22×C32×C4C2×D3113yes—
65C2×#638—326no—
66C22×C3⋊Dic3—313no—
67C2×C4⋊(C3⋊D3)—326no—
68C2×C4⋊(C3⋊D3)—326no—
69C2×C4⋊(C3⋊D3)—313no—
70C2×C4×C3⋊D3—326no—
71C32×C4⋊C4C2×D3113yescomplemented
72C32×C4⋊C4C2×D3126yescomplemented
73C32×C22⋊C4C2×D3126yes—
74#3752—326no—
75#3752—326no—
76#3752—326no—
77#3752—326no—
78#3752—326no—
79#3752—326no—
80C4⋊(C3⋊Dic3)—313no—
81#3750—326no—
82C4×C3⋊Dic3—326no—
83C22×C33—21no—
84C22×C33—21no—
85C22×C33#4112yes—
86C22×C33C2411yescharacteristic
87C2×C3⋊(C3⋊D3)—22no—
88C2×C3⋊(C3⋊D3)—22no—
89C2×C3⋊(C3⋊D3)—22no—
90C2×C3⋊(C3⋊D3)—22no—
91C2×C3⋊(C3⋊D3)—22no—
92C2×C3⋊(C3⋊D3)—22no—
93C2×C3⋊(C3⋊D3)—44no—
94C33×C4—22no—
95C33×C4—21no—
96C33×C4—22no—
97C33×C4—21no—
98C3⋊(C3⋊Dic3)—22no—
99C3⋊(C3⋊Dic3)—22no—
100C3×C22⋊Q8C3⋊D3113yescomplemented
101#947—926no—
102#930—913no—
103C2×(C2×D3)⋊C4—913no—
104#898—926no—
105#895—926no—
106#894—926no—
107#882—926no—
108#67⋊C4—926no—
109C23×C32C22×D3113yes—
110C22×C3⋊D3—326no—
111C22×C3⋊D3—326no—
112C22×C3⋊D3—626no—
113C22×C3⋊D3—1213no—
114C22×C3⋊D3—1213no—
115C32×Q8—413no—
116C2×C32×C4—413no—
117C2×C32×C4C22×D3126yes—
118C2×C32×C4C22×D3126yes—
119C2×C32×C4C22×D3113yes—
120C2×C32×C4C22×D3113yes—
121#638—652no—
122C2×C3⋊Dic3—326no—
123C2×C3⋊Dic3—326no—
124C2×C3⋊Dic3—1213no—
125C4⋊(C3⋊D3)—1213no—
126C4⋊(C3⋊D3)—1226no—
127C4⋊(C3⋊D3)—1226no—
128C4×C3⋊D3—1226no—
129#538—271noSylow
130C2×C33—22no—
131C2×C33C2×#4111yescharacteristic
132C2×C33H(5,𝔽2)11yescharacteristic
133C2×C33H(5,𝔽2)11yescharacteristic
134C3⋊(C3⋊D3)—42no—
135C3⋊(C3⋊D3)—42no—
136C23×D3—913no—
137C2×C3×Q8C2×C3⋊D3113yescomplemented
138C22×C3×C4C2×C3⋊D3113yes—
139C2×#67—926no—
140C22×Dic3—913no—
141C2×D12—926no—
142C2×D12—926no—
143C2×D12—913no—
144C2×C4×D3—926no—
145C3×C4⋊C4C2×C3⋊D3113yescomplemented
146C3×C4⋊C4C2×C3⋊D3126yescomplemented
147C3×C22⋊C4C2×C3⋊D3126yes—
148(C2×D3)⋊C4—926no—
149(C2×D3)⋊C4—926no—
150(C2×D3)⋊C4—926no—
151(C2×D3)⋊C4—926no—
152(C2×D3)⋊C4—926no—
153(C2×D3)⋊C4—926no—
154C4⋊Dic3—913no—
155Dic3⋊C4—926no—
156C4×Dic3—926no—
157C22×C32—213no—
158C22×C32—213no—
159C22×C32#239126yes—
160C22×C32C23×D3113yes—
161C2×C3⋊D3—626no—
162C2×C3⋊D3—626no—
163C2×C3⋊D3—626no—
164C2×C3⋊D3—626no—
165C2×C3⋊D3—626no—
166C2×C3⋊D3—626no—
167C2×C3⋊D3—1252no—
168C32×C4—213no—
169C32×C4—226no—
170C32×C4—213no—
171C32×C4—226no—
172C3⋊Dic3—626no—
173C3⋊Dic3—626no—
174#126—272no—
175#124—272no—
176#123—272no—
177C22⋊Q8C3⋊(C3⋊D3)11yescharacteristic, complemented
178#121—271no—
179#121—272no—
180#120—272no—
181C4×D4—272no—
182C2×C22⋊C4—271no—
183C33#53811yescharacteristic, complemented, Sylow
184C23×C3C22×C3⋊D3113yes—
185C22×D3—926no—
186C22×D3—926no—
187C22×D3—1826no—
188C22×D3—3613no—
189C22×D3—3613no—
190C3×Q8—413no—
191C2×C3×C4—413no—
192C2×C3×C4C22×C3⋊D3126yes—
193C2×C3×C4C22×C3⋊D3126yes—
194C2×C3×C4C22×C3⋊D3113yes—
195C2×C3×C4C22×C3⋊D3113yes—
196#67—1852no—
197C2×Dic3—926no—
198C2×Dic3—926no—
199C2×Dic3—3613no—
200D12—3613no—
201D12—3626no—
202D12—3626no—
203C4×D3—3626no—
204C2×C32—226no—
205C2×C32#1004113yes—
206C2×C32C2×#239113yes—
207C2×C32#1009113yes—
208C3⋊D3—1226no—
209C3⋊D3—1226no—
210C24—271no—
211C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
212C2×D4—272no—
213C2×D4—272no—
214C2×D4—272no—
215C2×D4—271no—
216C22×C4—271no—
217C22×C4—272no—
218C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
219C4⋊C4—272no—
220C4⋊C4—271no—
221C4⋊C4C2×C3⋊(C3⋊D3)12yescomplemented
222C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
223C22⋊C4—272no—
224C22⋊C4—272no—
225C22⋊C4—272no—
226C22⋊C4—272no—
227C22⋊C4—272no—
228C22⋊C4—272no—
229C22⋊C4C2×C3⋊(C3⋊D3)12yes—
230C42—272no—
231C22×C3—213no—
232C22×C3—213no—
233C22×C3#3832126yes—
234C22×C3C23×C3⋊D3113yes—
235C2×D3—1826no—
236C2×D3—1826no—
237C2×D3—1826no—
238C2×D3—1826no—
239C2×D3—1826no—
240C2×D3—1826no—
241C2×D3—3652no—
242C3×C4—213no—
243C3×C4—226no—
244C3×C4—213no—
245C3×C4—226no—
246Dic3—1826no—
247Dic3—1826no—
248C32#5644113yescomplemented
249C23—272no—
250C23—272no—
251C23—542no—
252C23—1081no—
253C23—1081no—
254C23C22×C3⋊(C3⋊D3)11yescharacteristic
255Q8—41no—
256D4—544no—
257D4—1081no—
258D4—1082no—
259D4—1082no—
260C2×C4—41no—
261C2×C4—272no—
262C2×C4—272no—
263C2×C4—1082no—
264C2×C4—1081no—
265C2×C4C22×C3⋊(C3⋊D3)12yes—
266C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
267C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
268C2×C4C22×C3⋊(C3⋊D3)12yes—
269C2×C3—226no—
270C2×C3#64361113yes—
271C2×C3C2×#3832113yes—
272C2×C3#64366113yes—
273D3—3626no—
274D3—3626no—
275C22—21no—
276C22—21no—
277C22—542no—
278C22—542no—
279C22—542no—
280C22—542no—
281C22—542no—
282C22—542no—
283C22—1084no—
284C22#8908012yes—
285C22C23×C3⋊(C3⋊D3)11yescharacteristic
286C4—21no—
287C4—21no—
288C4—22no—
289C4—22no—
290C4—542no—
291C4—542no—
292C3#10596115113yescomplemented
293C2—22no—
294C2—1082no—
295C2—1082no—
296C2#1172467911yescharacteristic
297C2C2×#8908011yescharacteristic
298C2#1172468411yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊(C3⋊D3)
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊Q8C2
hypercenterC22⋊Q8C3⋊(C3⋊D3)
hypocenterC33#538
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C33×C22⋊Q8C2semidirect
2. #11723988C2semidirect
3. #11723971C2semidirect
4. C2×#88859C2semidirect
5. #11723939C2semidirect
6. #11723936C2semidirect
7. #11723935C2semidirect
8. #11723923C2semidirect
9. #11723922C2semidirect
12. C2×C33×Q8C22semidirect
18. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
19. C2×C4×C3⋊(C3⋊D3)C22semidirect
20. C33×C4⋊C4C22semidirect
21. C33×C4⋊C4C22semidirect
27. #88859C22semidirect
28. #88859C22semidirect
29. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
30. #88857C22semidirect
31. C4×C3⋊(C3⋊Dic3)C22semidirect
32. C32×C22⋊Q8D3semidirect
63. C2×C32×Q8C2×D3semidirect
71. C32×C4⋊C4C2×D3semidirect
72. C32×C4⋊C4C2×D3semidirect
100. C3×C22⋊Q8C3⋊D3semidirect
137. C2×C3×Q8C2×C3⋊D3semidirect
145. C3×C4⋊C4C2×C3⋊D3semidirect
146. C3×C4⋊C4C2×C3⋊D3semidirect
177. C22⋊Q8C3⋊(C3⋊D3)semidirect
183. C33#538semidirect
211. C2×Q8C2×C3⋊(C3⋊D3)semidirect
221. C4⋊C4C2×C3⋊(C3⋊D3)semidirect
222. C4⋊C4C2×C3⋊(C3⋊D3)semidirect
248. C32#5644semidirect
292. C3#10596115semidirect

Maximal subgroups

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

subgroupindex
C33×C22⋊Q82
#117239882
#117239712
C2×#888592
#117239392
#117239362
#117239352
#117239232
#117239222
#105961153