Finite Group Databasebeta

← #86467

Subgroups — #86467

Order 384.

1040 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#540C211yesmaximal, characteristic, complemented
2#5719C211yesmaximal, characteristic, complemented
3#5716C211yesmaximal, characteristic, complemented
4#5698C211yesmaximal, characteristic, complemented
5#5688C211yesmaximal, characteristic, complemented
6#5660C214yesmaximal, complemented
7#5649C214yesmaximal, complemented
8#5613C214yesmaximal, complemented
9#5606C212yesmaximal, complemented
10#5603C212yesmaximal, complemented
11#5600C214yesmaximal, complemented
12#5561C212yesmaximal, complemented
13C4×#237C212yesmaximal, complemented
14D3×#117C211yesmaximal, characteristic, complemented
15#5518C211yesmaximal, characteristic, complemented
16#3455—31nomaximal, Sylow
17C2×#237C2212yes—
18C22×C4×D3C2211yescharacteristic
19C3×#127C2211yescharacteristic, complemented
20C3×#125C2211yescharacteristic, complemented
21C3×#123C2214yescomplemented
22C3×#121C2214yes—
23C3×#121C2212yes—
24C3×C4×D4C2212yescomplemented
25C3×#117C2211yescharacteristic
26#939C2214yes—
27#938C2214yes—
28#938C2212yes—
29#937C2214yes—
30#936C2214yes—
31#936C2212yescomplemented
32#935C2214yes—
33#935C2214yescomplemented
34D4×Dic3C2214yes—
35D4×Dic3C2214yescomplemented
36D4×Dic3C2212yescomplemented
37#929C2214yes—
38C4×#67C2214yes—
39#926C2211yescharacteristic
40#924C2212yes—
41#924C2214yescomplemented
42C2×C4×Dic3C2214yescomplemented
43#898C2214yes—
44#897C2212yes—
45#896C2214yes—
46#892C2212yes—
47#892C2212yes—
48D3×C4⋊C4C2212yes—
49#890C2214yescomplemented
50#889C2214yescomplemented
51#888C2212yescomplemented
52#887C2214yescomplemented
53#887C2212yescomplemented
54#885C2214yes—
55#883C2214yes—
56#67⋊C4C2214yes—
57#67⋊C4C2214yes—
58D3×C22⋊C4C2212yes—
59#879C2214yes—
60#879C2214yescomplemented
61#879C2214yescomplemented
62#878C2214yescomplemented
63#877C2212yes—
64#877C2214yescomplemented
65#872C2211yescharacteristic
66#872C2211yescharacteristic
67C42×D3C2211yescharacteristic
68C42×D3C2211yescharacteristic
69#870C2211yescharacteristic, complemented
70#869C2211yescharacteristic, complemented
71C4×Dic6C2212yescomplemented
72#542—32no—
73#542—32no—
74#541—31no—
75#541—31no—
76#540—31no—
77#540D311yescharacteristic, complemented
78#539—34no—
79#538—34no—
80#533—31no—
81#532—31no—
82#529—34no—
83#529—34no—
84#525—32no—
85C4×#41—32no—
86C2×#117—31no—
87C23×D3C2311yescharacteristic
88C2×C3×D4C2312yes—
89C2×C3×D4C2314yes—
90C2×C3×D4C2314yes—
91C22×C3×C4C2311yescharacteristic
92C22×C3×C4C2314yes—
93C2×#67C2314yes—
94C2×#67C2314yes—
95C22×Dic3C2311yescharacteristic
96C22×Dic3C2314yes—
97C22×Dic3C2314yes—
98#237—28no—
99C2×C4×D3—24no—
100C2×C4×D3C2311yescharacteristic
101C2×C4×D3C2311yescharacteristic
102C2×C4×D3C2311yescharacteristic
103C2×C4×D3C2312yes—
104C2×C4×D3C2312yes—
105C2×C4×D3C2311yescharacteristic
106C2×Dic6C2312yes—
107C2×Dic6C2314yes—
108C3×C4⋊C4C2312yes—
109C3×C4⋊C4C2314yes—
110C3×C4⋊C4C2312yes—
111C3×C22⋊C4C2312yes—
112C3×C22⋊C4C2314yes—
113C3×C22⋊C4C2314yes—
114C3×C42C2311yescharacteristic
115C3×C42C2311yescharacteristic
116#217C2312yes—
117#217C2314yes—
118#217C2314yes—
119#217C2314yes—
120#217C2314yes—
121#217C2314yes—
122(C2×D3)⋊C4C2312yes—
123(C2×D3)⋊C4C2312yes—
124(C2×D3)⋊C4C2314yes—
125(C2×D3)⋊C4C2314yes—
126C4⋊Dic3C2312yes—
127C4⋊Dic3C2314yes—
128C4⋊Dic3C2312yes—
129C4⋊Dic3C2314yes—
130Dic3⋊C4C2314yes—
131Dic3⋊C4C2314yes—
132Dic3⋊C4C2312yes—
133Dic3⋊C4C2312yes—
134Dic3⋊C4C2314yes—
135Dic3⋊C4C2314yes—
136Dic3⋊C4C2314yes—
137C4×Dic3C2314yes—
138C4×Dic3C2311yescharacteristic
139C4×Dic3C2311yescharacteristic
140C4×Dic3C2312yes—
141C4×Dic3C2312yes—
142C4×Dic3C2314yes—
143C4×Dic3C2314yes—
144C2×#41—32no—
145C23×C4—31no—
146C4⋊Q8—32no—
147C4⋊Q8—31no—
148#127C2×D311yescharacteristic, complemented
149#126—34no—
150#126—34no—
151#126—34no—
152#126—34no—
153#125—34no—
154#125—34no—
155#125—31no—
156#125C2×D311yescharacteristic, complemented
157#124—34no—
158#124—32no—
159#124—34no—
160#123—34no—
161#123—34no—
162#123—34no—
163#123—34no—
164#123C2×D314yescomplemented
165C22⋊Q8—32no—
166C22⋊Q8—34no—
167C22⋊Q8—32no—
168C22⋊Q8—34no—
169C22⋊Q8—34no—
170#121—32no—
171#121—34no—
172#121—34no—
173#121C2×D312yes—
174#121C2×D314yes—
175#120—34no—
176C4×Q8—32no—
177C4×Q8—34no—
178C4×Q8—32no—
179C4×D4—32no—
180C4×D4—34no—
181C4×D4—34no—
182C4×D4—34no—
183C4×D4—34no—
184C4×D4—34no—
185C4×D4C2×D312yescomplemented
186#117—31no—
187#117—32no—
188#117—32no—
189#117—32no—
190#117—34no—
191#117—31no—
192#117—31no—
193#117C2×D311yescharacteristic
194C2×C4⋊C4—32no—
195C2×C22⋊C4—32no—
196C2×C42—34no—
197C2×C42—31no—
198C2×C42—31no—
199C23×C3C2411yescharacteristic
200C23×C3C2414yes—
201C22×D3—22no—
202C22×D3—42no—
203C22×D3C2411yescharacteristic
204C22×D3C2411yescharacteristic
205C3×D4—24no—
206C3×D4—44no—
207C3×D4—44no—
208C2×C3×C4—22no—
209C2×C3×C4—44no—
210C2×C3×C4C2412yes—
211C2×C3×C4C2414yes—
212C2×C3×C4C2412yes—
213C2×C3×C4C2411yescharacteristic
214C2×C3×C4C2411yescharacteristic
215#67—28no—
216#67—44no—
217C2×Dic3—22no—
218C2×Dic3—28no—
219C2×Dic3—44no—
220C2×Dic3C2412yes—
221C2×Dic3C2414yes—
222C2×Dic3C2414yes—
223C2×Dic3C2412yes—
224C2×Dic3C2411yescharacteristic
225C2×Dic3C2411yescharacteristic
226C4×D3—22no—
227C4×D3—22no—
228C4×D3—22no—
229C4×D3—42no—
230C4×D3—42no—
231C4×D3#4114yes—
232Dic6—24no—
233Dic6—44no—
234C24—31no—
235#41—68no—
236C2×Q8—32no—
237C2×Q8—34no—
238C2×D4—34no—
239C2×D4—34no—
240C2×D4C22×D312yes—
241C2×D4C22×D314yes—
242C2×D4C22×D314yes—
243C22×C4—31no—
244C22×C4—34no—
245C22×C4—31no—
246C22×C4—34no—
247C22×C4—31no—
248C22×C4—31no—
249C22×C4—31no—
250C22×C4—32no—
251C22×C4—32no—
252C22×C4—64no—
253C22×C4C22×D311yescharacteristic
254C22×C4C22×D314yes—
255C4⋊C4—32no—
256C4⋊C4—32no—
257C4⋊C4—32no—
258C4⋊C4—32no—
259C4⋊C4—34no—
260C4⋊C4—34no—
261C4⋊C4—34no—
262C4⋊C4—34no—
263C4⋊C4—34no—
264C4⋊C4—34no—
265C4⋊C4—34no—
266C4⋊C4C22×D312yes—
267C4⋊C4C22×D312yes—
268C4⋊C4C22×D314yes—
269C22⋊C4—32no—
270C22⋊C4—34no—
271C22⋊C4—34no—
272C22⋊C4—34no—
273C22⋊C4—34no—
274C22⋊C4—34no—
275C22⋊C4—32no—
276C22⋊C4—32no—
277C22⋊C4—34no—
278C22⋊C4—34no—
279C22⋊C4C22×D312yes—
280C22⋊C4C22×D314yes—
281C22⋊C4C22×D314yes—
282C42—32no—
283C42—32no—
284C42—34no—
285C42—31no—
286C42—31no—
287C42—34no—
288C42—34no—
289C42C22×D311yescharacteristic
290C42C22×D311yescharacteristic
291C22×C3—24no—
292C22×C3—21no—
293C22×C3—24no—
294C22×C3—24no—
295C22×C3—22no—
296C22×C3C2511yescharacteristic
297C2×D3—22no—
298C2×D3—21no—
299C2×D3—22no—
300C2×D3—44no—
301C2×D3C2×#4112yes—
302C3×C4—21no—
303C3×C4—24no—
304C3×C4—22no—
305C3×C4—22no—
306C3×C4C2×#4112yes—
307Dic3—24no—
308Dic3—21no—
309Dic3—22no—
310Dic3—24no—
311Dic3—22no—
312Dic3C2×#4112yes—
313C23—31no—
314C23—31no—
315C23—62no—
316C23—122no—
317C23C23×D311yescharacteristic
318C23C23×D314yes—
319Q8—64no—
320Q8—124no—
321D4—24no—
322D4—44no—
323D4—44no—
324D4—68no—
325D4—124no—
326C2×C4—22no—
327C2×C4—31no—
328C2×C4—34no—
329C2×C4—32no—
330C2×C4—32no—
331C2×C4—34no—
332C2×C4—34no—
333C2×C4—31no—
334C2×C4—44no—
335C2×C4—62no—
336C2×C4—62no—
337C2×C4—62no—
338C2×C4—68no—
339C2×C4—62no—
340C2×C4—122no—
341C2×C4—122no—
342C2×C4—124no—
343C2×C4C23×D312yes—
344C2×C4C23×D314yes—
345C2×C4C23×D312yes—
346C2×C4C23×D311yescharacteristic
347C2×C4C23×D311yescharacteristic
348C2×C3—41no—
349C2×C3—44no—
350C2×C3C22×#4111yescharacteristic
351C2×C3#58512yes—
352D3—22no—
353D3—41no—
354C22—21no—
355C22—24no—
356C22—24no—
357C22—24no—
358C22—22no—
359C22—32no—
360C22—62no—
361C22—61no—
362C22—62no—
363C22—124no—
364C22C24×D311yescharacteristic
365C4—21no—
366C4—22no—
367C4—24no—
368C4—22no—
369C4—32no—
370C4—61no—
371C4—64no—
372C4—62no—
373C4—64no—
374C4—62no—
375C4D3×#4112yes—
376C3#345511yescharacteristic, complemented, Sylow
377C2—41no—
378C2—44no—
379C2—62no—
380C2—121no—
381C2C2×D3×#4111yescharacteristic
382C2#596612yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#540C2
hypercenter#540D3
hypocenterC3#3455
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #540

Complements

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

NH ≅ G/Ntype
1. C3×#540C2semidirect
2. #5719C2semidirect
3. #5716C2semidirect
4. #5698C2semidirect
5. #5688C2semidirect
6. #5660C2semidirect
7. #5649C2semidirect
8. #5613C2semidirect
9. #5606C2semidirect
10. #5603C2semidirect
11. #5600C2semidirect
12. #5561C2semidirect
13. C4×#237C2semidirect
14. D3×#117C2semidirect
15. #5518C2semidirect
19. C3×#127C22semidirect
20. C3×#125C22semidirect
21. C3×#123C22semidirect
24. C3×C4×D4C22semidirect
31. #936C22semidirect
33. #935C22semidirect
35. D4×Dic3C22semidirect
36. D4×Dic3C22semidirect
41. #924C22semidirect
42. C2×C4×Dic3C22semidirect
49. #890C22semidirect
50. #889C22semidirect
51. #888C22semidirect
52. #887C22semidirect
53. #887C22semidirect
60. #879C22semidirect
61. #879C22semidirect
62. #878C22semidirect
64. #877C22semidirect
69. #870C22semidirect
70. #869C22semidirect
71. C4×Dic6C22semidirect
77. #540D3semidirect
148. #127C2×D3semidirect
156. #125C2×D3semidirect
164. #123C2×D3semidirect
185. C4×D4C2×D3semidirect
376. C3#3455semidirect

Maximal subgroups

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

subgroupindex
C3×#5402
#57192
#57162
#56982
#56882
#56602
#56492
#56132
#56062
#56032
#56002
#55612
C4×#2372
D3×#1172
#55182
#34553