Finite Group Databasebeta

← #13179742

Subgroups — #13179742

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×#121C211yesmaximal, characteristic, complemented
2#11723980C212yesmaximal, complemented
3#11723978C212yesmaximal, complemented
4D4×C3⋊(C3⋊Dic3)C212yesmaximal, complemented
5C2×#88859C211yesmaximal, characteristic, complemented
6#11723965C211yesmaximal, characteristic, complemented
7#11723939C212yesmaximal, complemented
8#11723927C212yesmaximal, complemented
9#11723926C212yesmaximal, complemented
10#10596088—313nomaximal
11C23×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C33×D4C2212yes—
13C2×C33×D4C2211yescharacteristic, complemented
14C22×C33×C4C2211yescharacteristic
15C2×#6312C2212yes—
16C2×#6312C2212yes—
17C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
18C22×C3⋊(C3⋊Dic3)C2212yescomplemented
19C2×#6308C2211yescharacteristic, complemented
20C33×C4⋊C4C2211yescharacteristic, complemented
21C33×C22⋊C4C2212yescomplemented
22#88864C2212yes—
23#88864C2212yescomplemented
24#88859C2212yes—
25#88859C2212yes—
26#88859C2212yes—
27#88859C2212yes—
28C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
29C4⋊(C3⋊(C3⋊Dic3))C2212yescomplemented
30#88857C2212yescomplemented
31C4×C3⋊(C3⋊Dic3)C2212yescomplemented
32C32×#121D3113yescomplemented
33#64148—326no—
34#64146—326no—
35D4×C3⋊Dic3—326no—
36C2×#3752—313no—
37#64133—313no—
38#64107—326no—
39#64095—326no—
40#64094—326no—
41C23×C33C2312yes—
42C23×C33C2311yescharacteristic
43C22×C3⋊(C3⋊D3)—22no—
44C22×C3⋊(C3⋊D3)—41no—
45C22×C3⋊(C3⋊D3)—41no—
46C22×C3⋊(C3⋊D3)C2312yes—
47C33×D4—24no—
48C33×D4—41no—
49C2×C33×C4—41no—
50C2×C33×C4C2312yes—
51C2×C33×C4C2311yescharacteristic
52C2×C33×C4C2311yescharacteristic
53#6312—42no—
54#6312—42no—
55C2×C3⋊(C3⋊Dic3)—41no—
56C2×C3⋊(C3⋊Dic3)—42no—
57C2×C3⋊(C3⋊Dic3)C2312yes—
58C2×C3⋊(C3⋊Dic3)C2312yes—
59C2×C3⋊(C3⋊Dic3)C2312yes—
60#6308—41no—
61#5617—913no—
62C23×C3⋊D3—313no—
63C2×C32×D4C2×D3126yes—
64C2×C32×D4C2×D3113yescomplemented
65C22×C32×C4C2×D3113yes—
66C2×#638—326no—
67C2×#638—326no—
68C22×C3⋊Dic3—326no—
69C22×C3⋊Dic3—313no—
70C2×#634—313no—
71C32×C4⋊C4C2×D3113yescomplemented
72C32×C22⋊C4C2×D3126yescomplemented
73#3757—326no—
74#3757—326no—
75#3752—326no—
76#3752—326no—
77#3752—326no—
78#3752—326no—
79C4⋊(C3⋊Dic3)—313no—
80C4⋊(C3⋊Dic3)—326no—
81#3750—326no—
82C4×C3⋊Dic3—326no—
83C22×C33—21no—
84C22×C33—21no—
85C22×C33—22no—
86C22×C33—22no—
87C22×C33—22no—
88C22×C33#4112yes—
89C22×C33C2411yescharacteristic
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—21no—
97C3⋊(C3⋊Dic3)—22no—
98C3⋊(C3⋊Dic3)—22no—
99C3⋊(C3⋊Dic3)—22no—
100C3×#121C3⋊D3113yescomplemented
101#939—926no—
102#937—926no—
103D4×Dic3—926no—
104C2×(C2×D3)⋊C4—913no—
105#924—913no—
106#898—926no—
107#886—926no—
108#885—926no—
109C23×C32C22×D3126yes—
110C23×C32C22×D3113yes—
111C22×C3⋊D3—326no—
112C22×C3⋊D3—626no—
113C22×C3⋊D3—1213no—
114C22×C3⋊D3—1213no—
115C32×D4—252no—
116C32×D4—413no—
117C2×C32×C4—413no—
118C2×C32×C4C22×D3126yes—
119C2×C32×C4C22×D3113yes—
120C2×C32×C4C22×D3113yes—
121#638—1226no—
122#638—1226no—
123C2×C3⋊Dic3—326no—
124C2×C3⋊Dic3—326no—
125C2×C3⋊Dic3—326no—
126C2×C3⋊Dic3—1213no—
127C2×C3⋊Dic3—1226no—
128#634—1213no—
129#538—271noSylow
130C2×C33—22no—
131C2×C33—42no—
132C2×C33C2×#4111yescharacteristic
133C2×C33H(5,𝔽2)11yescharacteristic
134C2×C33H(5,𝔽2)11yescharacteristic
135C3⋊(C3⋊D3)—42no—
136C23×D3—913no—
137C2×C3×D4C2×C3⋊D3126yes—
138C2×C3×D4C2×C3⋊D3113yescomplemented
139C22×C3×C4C2×C3⋊D3113yes—
140C2×#67—926no—
141C2×#67—926no—
142C22×Dic3—926no—
143C22×Dic3—913no—
144C2×Dic6—913no—
145C3×C4⋊C4C2×C3⋊D3113yescomplemented
146C3×C22⋊C4C2×C3⋊D3126yescomplemented
147#217—926no—
148#217—926no—
149(C2×D3)⋊C4—926no—
150(C2×D3)⋊C4—926no—
151(C2×D3)⋊C4—926no—
152(C2×D3)⋊C4—926no—
153C4⋊Dic3—913no—
154C4⋊Dic3—926no—
155Dic3⋊C4—926no—
156C4×Dic3—926no—
157C22×C32—213no—
158C22×C32—213no—
159C22×C32—226no—
160C22×C32—226no—
161C22×C32—226no—
162C22×C32#237126yes—
163C22×C32C23×D3113yes—
164C2×C3⋊D3—626no—
165C2×C3⋊D3—626no—
166C2×C3⋊D3—626no—
167C2×C3⋊D3—1252no—
168C32×C4—226no—
169C32×C4—213no—
170C32×C4—213no—
171C3⋊Dic3—626no—
172C3⋊Dic3—626no—
173C3⋊Dic3—626no—
174#126—272no—
175#124—272no—
176#123—272no—
177C22⋊Q8—271no—
178#121—272no—
179#121C3⋊(C3⋊D3)11yescharacteristic, complemented
180#120—272no—
181C4×D4—272no—
182C2×C22⋊C4—271no—
183C33#53811yescharacteristic, complemented, Sylow
184C23×C3C22×C3⋊D3126yes—
185C23×C3C22×C3⋊D3113yes—
186C22×D3—926no—
187C22×D3—1826no—
188C22×D3—3613no—
189C22×D3—3613no—
190C3×D4—252no—
191C3×D4—413no—
192C2×C3×C4—413no—
193C2×C3×C4C22×C3⋊D3126yes—
194C2×C3×C4C22×C3⋊D3113yes—
195C2×C3×C4C22×C3⋊D3113yes—
196#67—3626no—
197#67—3626no—
198C2×Dic3—926no—
199C2×Dic3—926no—
200C2×Dic3—926no—
201C2×Dic3—3613no—
202C2×Dic3—3626no—
203Dic6—3613no—
204C2×C32—226no—
205C2×C32—426no—
206C2×C32C2×#237113yes—
207C2×C32#1004113yes—
208C2×C32#1009113yes—
209C3⋊D3—1226no—
210C24—271no—
211C2×Q8—271no—
212C2×D4—272no—
213C2×D4—272no—
214C2×D4C2×C3⋊(C3⋊D3)12yes—
215C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
216C22×C4—271no—
217C22×C4—272no—
218C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
219C4⋊C4—272no—
220C4⋊C4—272no—
221C4⋊C4—271no—
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)12yescomplemented
230C42—272no—
231C22×C3—213no—
232C22×C3—213no—
233C22×C3—226no—
234C22×C3—226no—
235C22×C3—226no—
236C22×C3#3830126yes—
237C22×C3C23×C3⋊D3113yes—
238C2×D3—1826no—
239C2×D3—1826no—
240C2×D3—1826no—
241C2×D3—3652no—
242C3×C4—226no—
243C3×C4—213no—
244C3×C4—213no—
245Dic3—1826no—
246Dic3—1826no—
247Dic3—1826no—
248C32#5617113yescomplemented
249C23—272no—
250C23—542no—
251C23—1081no—
252C23—1081no—
253C23C22×C3⋊(C3⋊D3)12yes—
254C23C22×C3⋊(C3⋊D3)11yescharacteristic
255Q8—1081no—
256D4—24no—
257D4—41no—
258D4—1082no—
259D4—1082no—
260C2×C4—41no—
261C2×C4—272no—
262C2×C4—272no—
263C2×C4—272no—
264C2×C4—1081no—
265C2×C4—1082no—
266C2×C4C22×C3⋊(C3⋊D3)12yes—
267C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
268C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
269C2×C3—226no—
270C2×C3—426no—
271C2×C3C2×#3830113yes—
272C2×C3#64361113yes—
273C2×C3#64366113yes—
274D3—3626no—
275C22—21no—
276C22—21no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—542no—
281C22—542no—
282C22—542no—
283C22—1084no—
284C22#8907812yes—
285C22C23×C3⋊(C3⋊D3)11yescharacteristic
286C4—22no—
287C4—21no—
288C4—21no—
289C4—542no—
290C4—542no—
291C4—542no—
292C3#10596088113yescomplemented
293C2—22no—
294C2—42no—
295C2—1082no—
296C2C2×#8907811yescharacteristic
297C2#1172467911yescharacteristic
298C2#1172468411yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊(C3⋊D3)
commutator, socle, cosocleC22×C33C24
FittingC33×#121C2
hypercenter#121C3⋊(C3⋊D3)
hypocenterC33#538
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C33×#121C2semidirect
2. #11723980C2semidirect
3. #11723978C2semidirect
4. D4×C3⋊(C3⋊Dic3)C2semidirect
5. C2×#88859C2semidirect
6. #11723965C2semidirect
7. #11723939C2semidirect
8. #11723927C2semidirect
9. #11723926C2semidirect
13. C2×C33×D4C22semidirect
18. C22×C3⋊(C3⋊Dic3)C22semidirect
19. C2×#6308C22semidirect
20. C33×C4⋊C4C22semidirect
21. C33×C22⋊C4C22semidirect
23. #88864C22semidirect
28. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
29. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
30. #88857C22semidirect
31. C4×C3⋊(C3⋊Dic3)C22semidirect
32. C32×#121D3semidirect
64. C2×C32×D4C2×D3semidirect
71. C32×C4⋊C4C2×D3semidirect
72. C32×C22⋊C4C2×D3semidirect
100. C3×#121C3⋊D3semidirect
138. C2×C3×D4C2×C3⋊D3semidirect
145. C3×C4⋊C4C2×C3⋊D3semidirect
146. C3×C22⋊C4C2×C3⋊D3semidirect
179. #121C3⋊(C3⋊D3)semidirect
183. C33#538semidirect
215. C2×D4C2×C3⋊(C3⋊D3)semidirect
222. C4⋊C4C2×C3⋊(C3⋊D3)semidirect
229. C22⋊C4C2×C3⋊(C3⋊D3)semidirect
248. C32#5617semidirect
292. C3#10596088semidirect

Maximal subgroups

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

subgroupindex
C33×#1212
#117239802
#117239782
D4×C3⋊(C3⋊Dic3)2
C2×#888592
#117239652
#117239392
#117239272
#117239262
#105960883