Finite Group Databasebeta

← #13179716

Subgroups — #13179716

Order 1728.

5992 conjugacy classes of subgroups; only the 739 normal subgroups are shown.

normal subgroups only

This group has 5992 conjugacy classes of subgroups — too many to draw. Only its 739 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×#6312C212yesmaximal, complemented
2C2×D4×C3⋊(C3⋊D3)C212yesmaximal, complemented
3C33×#120C211yesmaximal, characteristic, complemented
4#11723981C211yesmaximal, characteristic, complemented
5#11723980C212yesmaximal, complemented
6#11723978C211yesmaximal, characteristic, complemented
7#11723925C212yesmaximal, complemented
8#11723923C212yesmaximal, complemented
9#11723922C212yesmaximal, complemented
10#10596061—313nomaximal
11C24×C33C2211yescharacteristic, complemented
12C23×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
13C23×C3⋊(C3⋊D3)C2212yescomplemented
14C2×C33×D4C2212yescomplemented
15C2×C33×D4C2211yescharacteristic, complemented
16C2×#6312—28no—
17C2×#6312C2212yes—
18C2×#6312C2212yescomplemented
19C2×#6312C2212yescomplemented
20C2×#6312C2212yescomplemented
21C2×#6312C2212yescomplemented
22C22×C3⋊(C3⋊Dic3)C2212yescomplemented
23D4×C3⋊(C3⋊D3)—28no—
24C2×C4⋊(C3⋊(C3⋊D3))C2212yescomplemented
25C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
26C2×C4×C3⋊(C3⋊D3)C2212yescomplemented
27C33×C22⋊C4C2212yescomplemented
28C33×C22⋊C4C2211yescharacteristic, complemented
29#88864C2211yescharacteristic, complemented
30#88859C2212yescomplemented
31#88859C2212yescomplemented
32#88857C2212yescomplemented
33C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
34C22×#638—326no—
35C2×D4×C3⋊D3—326no—
36C32×#120D3113yescomplemented
37#64149—313no—
38#64148—326no—
39#64146—313no—
40#64093—326no—
41#64091—326no—
42#64090—326no—
43C23×C33—24no—
44C23×C33—41no—
45C23×C33C2311yescharacteristic
46C23×C33C2311yescharacteristic
47C23×C33C2312yes—
48C22×C3⋊(C3⋊D3)—24no—
49C22×C3⋊(C3⋊D3)—24no—
50C22×C3⋊(C3⋊D3)—24no—
51C22×C3⋊(C3⋊D3)—41no—
52C22×C3⋊(C3⋊D3)—42no—
53C22×C3⋊(C3⋊D3)C2312yes—
54C22×C3⋊(C3⋊D3)C2312yes—
55C33×D4—24no—
56C33×D4—41no—
57C2×C33×C4C2311yescharacteristic
58C2×C33×C4C2312yes—
59#6312—24no—
60#6312—24no—
61#6312—24no—
62#6312—24no—
63#6312D418yescomplemented
64C2×C3⋊(C3⋊Dic3)—24no—
65C2×C3⋊(C3⋊Dic3)C2312yes—
66C2×C3⋊(C3⋊Dic3)C2312yes—
67C4⋊(C3⋊(C3⋊D3))—24no—
68C4⋊(C3⋊(C3⋊D3))—41no—
69C4×C3⋊(C3⋊D3)—24no—
70#5590—913no—
71C24×C32C2×D3113yescomplemented
72C23×C3⋊D3—326no—
73C23×C3⋊D3—313no—
74C2×C32×D4C2×D3126yescomplemented
75C2×C32×D4C2×D3113yescomplemented
76C2×#638—326no—
77C2×#638—326no—
78C2×#638—326no—
79C2×#638—326no—
80C2×#638—326no—
81C2×#638—6104no—
82C22×C3⋊Dic3—326no—
83D4×C3⋊D3—6104no—
84C2×C4⋊(C3⋊D3)—326no—
85C2×C4⋊(C3⋊D3)—313no—
86C2×C4×C3⋊D3—326no—
87C32×C22⋊C4C2×D3126yescomplemented
88C32×C22⋊C4C2×D3113yescomplemented
89#3757—313no—
90#3752—326no—
91#3752—326no—
92#3750—326no—
93C4×C3⋊Dic3—313no—
94C22×C33—22no—
95C22×C33—22no—
96C22×C33—22no—
97C22×C33—22no—
98C22×C33—21no—
99C22×C33—21no—
100C22×C33—44no—
101C22×C33C2×D414yescomplemented
102C22×C33C2411yescharacteristic
103C2×C3⋊(C3⋊D3)—22no—
104C2×C3⋊(C3⋊D3)—22no—
105C2×C3⋊(C3⋊D3)—22no—
106C2×C3⋊(C3⋊D3)—22no—
107C2×C3⋊(C3⋊D3)—22no—
108C2×C3⋊(C3⋊D3)—44no—
109C2×C3⋊(C3⋊D3)—48no—
110C2×C3⋊(C3⋊D3)C2×D414yescomplemented
111C33×C4—22no—
112C33×C4—21no—
113C3⋊(C3⋊Dic3)—22no—
114C3⋊(C3⋊Dic3)C2×D414yescomplemented
115C22×#67—926no—
116C2×D3×D4—926no—
117C3×#120C3⋊D3113yescomplemented
118#940—913no—
119#939—926no—
120#937—913no—
121#884—926no—
122#882—926no—
123#67⋊C4—926no—
124C23×C32—252no—
125C23×C32—413no—
126C23×C32C22×D3113yes—
127C23×C32C22×D3113yes—
128C23×C32C22×D3126yes—
129C22×C3⋊D3—326no—
130C22×C3⋊D3—326no—
131C22×C3⋊D3—652no—
132C22×C3⋊D3—652no—
133C22×C3⋊D3—652no—
134C22×C3⋊D3—1213no—
135C22×C3⋊D3—1226no—
136C32×D4—252no—
137C32×D4—413no—
138C2×C32×C4C22×D3113yes—
139C2×C32×C4C22×D3126yes—
140#638—3104no—
141#638—652no—
142#638—652no—
143#638—652no—
144#638—652no—
145C2×C3⋊Dic3—326no—
146C2×C3⋊Dic3—326no—
147C2×C3⋊Dic3—652no—
148C4⋊(C3⋊D3)—652no—
149C4⋊(C3⋊D3)—1213no—
150C4×C3⋊D3—652no—
151D42—271noSylow
152C2×C33—24no—
153C2×C33—41no—
154C2×C33—41no—
155C2×C33C22×D412yes—
156C2×C33H(5,𝔽2)11yescharacteristic
157C3⋊(C3⋊D3)—24no—
158C3⋊(C3⋊D3)—42no—
159C24×C3C2×C3⋊D3113yescomplemented
160C23×D3—926no—
161C23×D3—913no—
162C2×C3×D4C2×C3⋊D3126yescomplemented
163C2×C3×D4C2×C3⋊D3113yescomplemented
164C2×#67—926no—
165C2×#67—926no—
166C2×#67—926no—
167C2×#67—926no—
168C2×#67—926no—
169C2×#67—18104no—
170C22×Dic3—926no—
171D3×D4—18104no—
172C2×D12—926no—
173C2×D12—913no—
174C2×C4×D3—926no—
175C3×C22⋊C4C2×C3⋊D3126yescomplemented
176C3×C22⋊C4C2×C3⋊D3113yescomplemented
177#217—913no—
178(C2×D3)⋊C4—926no—
179(C2×D3)⋊C4—926no—
180Dic3⋊C4—926no—
181C4×Dic3—913no—
182C22×C32—226no—
183C22×C32—226no—
184C22×C32—226no—
185C22×C32—226no—
186C22×C32—213no—
187C22×C32—213no—
188C22×C32—452no—
189C22×C32D3×D4152yescomplemented
190C22×C32C23×D3113yes—
191C2×C3⋊D3—352no—
192C2×C3⋊D3—626no—
193C2×C3⋊D3—626no—
194C2×C3⋊D3—626no—
195C2×C3⋊D3—626no—
196C2×C3⋊D3—626no—
197C2×C3⋊D3—1252no—
198C2×C3⋊D3—12104no—
199C32×C4—226no—
200C32×C4—213no—
201C3⋊Dic3—352no—
202C3⋊Dic3—626no—
203C22×D4—272no—
204C22×D4—272no—
205#127—271no—
206#121—272no—
207#121—272no—
208#120—271no—
209#120—272no—
210#120C3⋊(C3⋊D3)11yescharacteristic, complemented
211C4×D4—272no—
212C33D4211yescharacteristic, complemented, Sylow
213C23×C3—252no—
214C23×C3—413no—
215C23×C3C22×C3⋊D3113yes—
216C23×C3C22×C3⋊D3113yes—
217C23×C3C22×C3⋊D3126yes—
218C22×D3—926no—
219C22×D3—926no—
220C22×D3—1852no—
221C22×D3—1852no—
222C22×D3—1852no—
223C22×D3—3613no—
224C22×D3—3626no—
225C3×D4—252no—
226C3×D4—413no—
227C2×C3×C4C22×C3⋊D3113yes—
228C2×C3×C4C22×C3⋊D3126yes—
229#67—9104no—
230#67—1852no—
231#67—1852no—
232#67—1852no—
233#67—1852no—
234C2×Dic3—926no—
235C2×Dic3—926no—
236C2×Dic3—1852no—
237D12—1852no—
238D12—3613no—
239C4×D3—1852no—
240C2×C32—252no—
241C2×C32—413no—
242C2×C32—413no—
243C2×C32C2×D3×D4126yes—
244C2×C32#1004113yes—
245C3⋊D3—652no—
246C3⋊D3—1226no—
247C24—271no—
248C24—272no—
249C24C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
250C2×D4—272no—
251C2×D4—272no—
252C2×D4—272no—
253C2×D4—272no—
254C2×D4—272no—
255C2×D4—272no—
256C2×D4—271no—
257C2×D4—548no—
258C2×D4—548no—
259C2×D4C2×C3⋊(C3⋊D3)12yescomplemented
260C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
261C22×C4—272no—
262C22×C4—272no—
263C4⋊C4—272no—
264C22⋊C4—271no—
265C22⋊C4—272no—
266C22⋊C4—272no—
267C22⋊C4C2×C3⋊(C3⋊D3)12yescomplemented
268C22⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
269C42—271no—
270C22×C3—226no—
271C22×C3—226no—
272C22×C3—226no—
273C22×C3—226no—
274C22×C3—213no—
275C22×C3—213no—
276C22×C3—452no—
277C22×C3D4×C3⋊D3152yescomplemented
278C22×C3C23×C3⋊D3113yes—
279C2×D3—952no—
280C2×D3—1826no—
281C2×D3—1826no—
282C2×D3—1826no—
283C2×D3—1826no—
284C2×D3—1826no—
285C2×D3—3652no—
286C2×D3—36104no—
287C3×C4—226no—
288C3×C4—213no—
289Dic3—952no—
290Dic3—1826no—
291C32#5590113yescomplemented
292C23—24no—
293C23—41no—
294C23—272no—
295C23—272no—
296C23—544no—
297C23—544no—
298C23—544no—
299C23—1082no—
300C23—1081no—
301C23C22×C3⋊(C3⋊D3)11yescharacteristic
302C23C22×C3⋊(C3⋊D3)11yescharacteristic
303C23C22×C3⋊(C3⋊D3)12yes—
304D4—24no—
305D4—41no—
306D4—278no—
307D4—544no—
308D4—544no—
309D4—544no—
310D4—544no—
311D4—544no—
312D4—1081no—
313C2×C4—272no—
314C2×C4—272no—
315C2×C4—544no—
316C2×C4—544no—
317C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
318C2×C4C22×C3⋊(C3⋊D3)12yes—
319C2×C3—252no—
320C2×C3—413no—
321C2×C3—413no—
322C2×C3C2×D4×C3⋊D3126yes—
323C2×C3#64361113yes—
324D3—1852no—
325D3—3626no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—21no—
331C22—21no—
332C22—44no—
333C22—274no—
334C22—542no—
335C22—542no—
336C22—542no—
337C22—542no—
338C22—542no—
339C22—1088no—
340C22—1084no—
341C22D4×C3⋊(C3⋊D3)14yescomplemented
342C22C23×C3⋊(C3⋊D3)11yescharacteristic
343C4—22no—
344C4—21no—
345C4—274no—
346C4—542no—
347C3#10596061113yescomplemented
348C2—24no—
349C2—41no—
350C2—41no—
351C2—544no—
352C2—1082no—
353C2C2×D4×C3⋊(C3⋊D3)12yes—
354C2#1172467911yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊(C3⋊D3)
commutator, socle, cosocleC22×C33C24
FittingC33×#120C2
hypercenter#120C3⋊(C3⋊D3)
hypocenterC33D42
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×#6312C2semidirect
2. C2×D4×C3⋊(C3⋊D3)C2semidirect
3. C33×#120C2semidirect
4. #11723981C2semidirect
5. #11723980C2semidirect
6. #11723978C2semidirect
7. #11723925C2semidirect
8. #11723923C2semidirect
9. #11723922C2semidirect
11. C24×C33C22semidirect
12. C23×C3⋊(C3⋊D3)C22semidirect
13. C23×C3⋊(C3⋊D3)C22semidirect
14. C2×C33×D4C22semidirect
15. C2×C33×D4C22semidirect
18. C2×#6312C22semidirect
19. C2×#6312C22semidirect
20. C2×#6312C22semidirect
21. C2×#6312C22semidirect
22. C22×C3⋊(C3⋊Dic3)C22semidirect
24. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
25. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
26. C2×C4×C3⋊(C3⋊D3)C22semidirect
27. C33×C22⋊C4C22semidirect
28. C33×C22⋊C4C22semidirect
29. #88864C22semidirect
30. #88859C22semidirect
31. #88859C22semidirect
32. #88857C22semidirect
33. C4×C3⋊(C3⋊Dic3)C22semidirect
36. C32×#120D3semidirect
63. #6312D4semidirect
71. C24×C32C2×D3semidirect
74. C2×C32×D4C2×D3semidirect
75. C2×C32×D4C2×D3semidirect
87. C32×C22⋊C4C2×D3semidirect
88. C32×C22⋊C4C2×D3semidirect
101. C22×C33C2×D4semidirect
110. C2×C3⋊(C3⋊D3)C2×D4semidirect
114. C3⋊(C3⋊Dic3)C2×D4semidirect
117. C3×#120C3⋊D3semidirect
159. C24×C3C2×C3⋊D3semidirect
162. C2×C3×D4C2×C3⋊D3semidirect
163. C2×C3×D4C2×C3⋊D3semidirect
175. C3×C22⋊C4C2×C3⋊D3semidirect
176. C3×C22⋊C4C2×C3⋊D3semidirect
189. C22×C32D3×D4semidirect
210. #120C3⋊(C3⋊D3)semidirect
212. C33D42semidirect
249. C24C2×C3⋊(C3⋊D3)semidirect
259. C2×D4C2×C3⋊(C3⋊D3)semidirect
260. C2×D4C2×C3⋊(C3⋊D3)semidirect
267. C22⋊C4C2×C3⋊(C3⋊D3)semidirect
268. C22⋊C4C2×C3⋊(C3⋊D3)semidirect
277. C22×C3D4×C3⋊D3semidirect
291. C32#5590semidirect
341. C22D4×C3⋊(C3⋊D3)semidirect
347. C3#10596061semidirect

Maximal subgroups

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

subgroupindex
C22×#63122
C2×D4×C3⋊(C3⋊D3)2
C33×#1202
#117239812
#117239802
#117239782
#117239252
#117239232
#117239222
#105960613