Finite Group Databasebeta

← #86472

Subgroups — #86472

Order 384.

1050 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#540C211yesmaximal, characteristic, complemented
2#5718C212yesmaximal, complemented
3#5694C212yesmaximal, complemented
4#5665C214yesmaximal, complemented
5#5663C214yesmaximal, complemented
6#5620C214yesmaximal, complemented
7#5618C214yesmaximal, complemented
8#5608C212yesmaximal, complemented
9#5606C212yesmaximal, complemented
10#5558C212yesmaximal, complemented
11#5555C212yesmaximal, complemented
12#5524C211yesmaximal, characteristic, complemented
13#5518C211yesmaximal, characteristic, complemented
14#3539—31nomaximal, Sylow
15C2×#237C2212yes—
16C2×#237C2212yescomplemented
17C2×#235C2211yescharacteristic
18C22×C4×D3C2211yescharacteristic
19C3×#127C2211yescharacteristic, complemented
20C3×#125C2211yescharacteristic, complemented
21C3×#123C2214yescomplemented
22C3×#121C2212yes—
23C3×#121C2214yes—
24C3×C4×D4C2212yescomplemented
25C3×#117C2211yescharacteristic
26#940C2214yes—
27#939C2214yes—
28#939C2214yes—
29#939C2214yescomplemented
30#938C2212yes—
31#938C2214yes—
32#938C2212yescomplemented
33#937C2214yes—
34D4×Dic3C2214yes—
35D4×Dic3C2214yescomplemented
36#930C2214yes—
37#929C2214yes—
38C4×#67C2214yescomplemented
39C2×C4⋊Dic3C2214yescomplemented
40#924C2212yes—
41#898C2214yes—
42#897C2212yes—
43#897C2212yes—
44#897C2212yescomplemented
45#895C2212yes—
46#894C2214yes—
47#892C2212yes—
48#892C2214yescomplemented
49D3×C4⋊C4C2212yescomplemented
50#890C2214yescomplemented
51#889C2214yescomplemented
52#886C2214yes—
53#886C2212yes—
54#886C2214yescomplemented
55#884C2214yes—
56#884C2214yescomplemented
57#883C2214yes—
58#882C2212yes—
59#67⋊C4C2214yes—
60#879C2214yes—
61#879C2214yescomplemented
62#878C2214yescomplemented
63C4×D12C2212yes—
64C4×D12C2212yes—
65#870C2211yescharacteristic, complemented
66#869C2211yescharacteristic, complemented
67C4×Dic6C2212yescomplemented
68#560—34no—
69#553—34no—
70#553—32no—
71#552—32no—
72#550—32no—
73#547—32no—
74#541—31no—
75#540—34no—
76#540D311yescharacteristic, complemented
77#539—34no—
78#537—32no—
79#525—32no—
80#525—31no—
81C23×D3C2311yescharacteristic
82C2×C3×D4C2312yes—
83C2×C3×D4C2314yes—
84C2×C3×D4C2314yes—
85C22×C3×C4C2311yescharacteristic
86C22×C3×C4C2314yes—
87C2×#67C2314yes—
88C2×#67C2312yes—
89C2×#67C2314yes—
90C2×#67C2314yes—
91C2×#67C2314yes—
92C22×Dic3C2314yes—
93C22×Dic3C2314yes—
94C22×Dic3C2311yescharacteristic
95#237—44no—
96#237—44no—
97#235—42no—
98C2×D12C2312yes—
99C2×D12C2312yes—
100C2×D12C2311yescharacteristic
101C2×C4×D3—42no—
102C2×C4×D3C2312yes—
103C2×C4×D3C2312yes—
104C2×C4×D3C2312yes—
105C2×C4×D3C2314yes—
106C2×Dic6C2312yes—
107C2×Dic6C2311yescharacteristic
108C3×C4⋊C4C2312yes—
109C3×C4⋊C4C2314yes—
110C3×C4⋊C4C2312yes—
111C3×C22⋊C4C2314yes—
112C3×C22⋊C4C2312yes—
113C3×C22⋊C4C2314yes—
114C3×C42C2311yescharacteristic
115C3×C42C2311yescharacteristic
116#217C2314yes—
117#217C2314yes—
118#217C2314yes—
119(C2×D3)⋊C4C2312yes—
120(C2×D3)⋊C4C2312yes—
121(C2×D3)⋊C4C2312yes—
122(C2×D3)⋊C4C2312yes—
123(C2×D3)⋊C4C2314yes—
124(C2×D3)⋊C4C2314yes—
125(C2×D3)⋊C4C2314yes—
126C4⋊Dic3C2312yes—
127C4⋊Dic3C2314yes—
128C4⋊Dic3C2312yes—
129C4⋊Dic3C2312yes—
130C4⋊Dic3C2314yes—
131C4⋊Dic3C2314yes—
132Dic3⋊C4C2314yes—
133Dic3⋊C4C2314yes—
134Dic3⋊C4C2314yes—
135Dic3⋊C4C2314yes—
136C4×Dic3C2314yes—
137C4×Dic3C2314yes—
138C2×#41—32no—
139C2×#41—32no—
140C2×#41—31no—
141C23×C4—31no—
142C4⋊Q8—31no—
143#127—34no—
144#127C2×D311yescharacteristic, complemented
145#126—34no—
146#126—34no—
147#126—34no—
148#125—34no—
149#125—34no—
150#125—31no—
151#125C2×D311yescharacteristic, complemented
152#123—34no—
153#123—34no—
154#123—34no—
155#123—32no—
156#123—34no—
157#123—34no—
158#123C2×D314yescomplemented
159C22⋊Q8—32no—
160C22⋊Q8—32no—
161C22⋊Q8—32no—
162C22⋊Q8—32no—
163C22⋊Q8—34no—
164#121—32no—
165#121—32no—
166#121—34no—
167#121—34no—
168#121—34no—
169#121—34no—
170#121—34no—
171#121—32no—
172#121—34no—
173#121—34no—
174#121C2×D312yes—
175#121C2×D314yes—
176#120—34no—
177#120—32no—
178C4×Q8—32no—
179C4×D4—34no—
180C4×D4—34no—
181C4×D4—34no—
182C4×D4—34no—
183C4×D4—32no—
184C4×D4—32no—
185C4×D4C2×D312yescomplemented
186#117—32no—
187#117—34no—
188#117C2×D311yescharacteristic
189C2×C4⋊C4—34no—
190C2×C4⋊C4—32no—
191C23×C3C2411yescharacteristic
192C23×C3C2414yes—
193C22×D3—41no—
194C22×D3—42no—
195C22×D3C2412yes—
196C22×D3C2412yes—
197C3×D4—24no—
198C3×D4—44no—
199C3×D4—44no—
200C2×C3×C4—22no—
201C2×C3×C4—44no—
202C2×C3×C4C2412yes—
203C2×C3×C4C2414yes—
204C2×C3×C4C2411yescharacteristic
205C2×C3×C4C2412yes—
206C2×C3×C4C2411yescharacteristic
207#67—42no—
208#67—44no—
209#67—44no—
210#67—44no—
211#67—44no—
212C2×Dic3—41no—
213C2×Dic3—44no—
214C2×Dic3—44no—
215C2×Dic3C2414yes—
216C2×Dic3C2412yes—
217C2×Dic3C2414yes—
218C2×Dic3C2412yes—
219D12—42no—
220D12—42no—
221D12—41no—
222C4×D3—24no—
223C4×D3—24no—
224C4×D3—42no—
225C4×D3—44no—
226Dic6—24no—
227Dic6—41no—
228C24—31no—
229#41—122no—
230#41—124no—
231#41—124no—
232C2×Q8—32no—
233C2×Q8—31no—
234C2×D4—34no—
235C2×D4—34no—
236C2×D4—32no—
237C2×D4—34no—
238C2×D4—34no—
239C2×D4—32no—
240C2×D4—32no—
241C2×D4—31no—
242C2×D4C22×D312yes—
243C2×D4C22×D314yes—
244C2×D4C22×D314yes—
245C22×C4—34no—
246C22×C4—34no—
247C22×C4—31no—
248C22×C4—32no—
249C22×C4—32no—
250C22×C4—34no—
251C22×C4—32no—
252C22×C4—122no—
253C22×C4C22×D311yescharacteristic
254C22×C4C22×D314yes—
255C4⋊C4—32no—
256C4⋊C4—34no—
257C4⋊C4—32no—
258C4⋊C4—34no—
259C4⋊C4—34no—
260C4⋊C4—34no—
261C4⋊C4—32no—
262C4⋊C4—34no—
263C4⋊C4—34no—
264C4⋊C4—34no—
265C4⋊C4C22×D312yes—
266C4⋊C4C22×D314yes—
267C4⋊C4C22×D312yes—
268C22⋊C4—34no—
269C22⋊C4—34no—
270C22⋊C4—34no—
271C22⋊C4—32no—
272C22⋊C4—34no—
273C22⋊C4—34no—
274C22⋊C4—32no—
275C22⋊C4—32no—
276C22⋊C4—34no—
277C22⋊C4—32no—
278C22⋊C4C22×D312yes—
279C22⋊C4C22×D314yes—
280C22⋊C4C22×D314yes—
281C42—34no—
282C42—34no—
283C42C22×D311yescharacteristic
284C42C22×D311yescharacteristic
285C22×C3—24no—
286C22×C3—21no—
287C22×C3—24no—
288C22×C3—24no—
289C22×C3—22no—
290C22×C3C2511yescharacteristic
291C2×D3—22no—
292C2×D3—22no—
293C2×D3—22no—
294C2×D3—22no—
295C2×D3—22no—
296C2×D3—22no—
297C2×D3—44no—
298C3×C4—21no—
299C3×C4—24no—
300C3×C4—22no—
301C3×C4—22no—
302C3×C4H(5,𝔽2)12yes—
303Dic3—24no—
304Dic3—22no—
305Dic3—22no—
306Dic3—24no—
307C23—32no—
308C23—32no—
309C23—121no—
310C23—122no—
311C23C23×D311yescharacteristic
312C23C23×D314yes—
313Q8—64no—
314Q8—121no—
315D4—24no—
316D4—44no—
317D4—44no—
318D4—121no—
319D4—122no—
320D4—122no—
321D4—122no—
322D4—124no—
323D4—124no—
324D4—124no—
325D4—124no—
326C2×C4—22no—
327C2×C4—34no—
328C2×C4—32no—
329C2×C4—34no—
330C2×C4—32no—
331C2×C4—44no—
332C2×C4—64no—
333C2×C4—64no—
334C2×C4—122no—
335C2×C4—124no—
336C2×C4—121no—
337C2×C4—124no—
338C2×C4—124no—
339C2×C4C23×D312yes—
340C2×C4C23×D314yes—
341C2×C4C23×D312yes—
342C2×C4C23×D311yescharacteristic
343C2×C4C23×D311yescharacteristic
344C2×C3—41no—
345C2×C3—44no—
346C2×C3C2×H(5,𝔽2)11yescharacteristic
347C2×C3#58512yes—
348D3—42no—
349D3—42no—
350C22—21no—
351C22—24no—
352C22—24no—
353C22—24no—
354C22—22no—
355C22—62no—
356C22—62no—
357C22—62no—
358C22—62no—
359C22—62no—
360C22—62no—
361C22—124no—
362C22C24×D311yescharacteristic
363C4—21no—
364C4—22no—
365C4—24no—
366C4—22no—
367C4—62no—
368C4—62no—
369C4—64no—
370C4—64no—
371C4#100912yes—
372C3#353911yescharacteristic, complemented, Sylow
373C2—41no—
374C2—44no—
375C2—122no—
376C2—122no—
377C2C2×#100911yescharacteristic
378C2#596612yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#540C2
hypercenter#540D3
hypocenterC3#3539
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. #5718C2semidirect
3. #5694C2semidirect
4. #5665C2semidirect
5. #5663C2semidirect
6. #5620C2semidirect
7. #5618C2semidirect
8. #5608C2semidirect
9. #5606C2semidirect
10. #5558C2semidirect
11. #5555C2semidirect
12. #5524C2semidirect
13. #5518C2semidirect
16. C2×#237C22semidirect
19. C3×#127C22semidirect
20. C3×#125C22semidirect
21. C3×#123C22semidirect
24. C3×C4×D4C22semidirect
29. #939C22semidirect
32. #938C22semidirect
35. D4×Dic3C22semidirect
38. C4×#67C22semidirect
39. C2×C4⋊Dic3C22semidirect
44. #897C22semidirect
48. #892C22semidirect
49. D3×C4⋊C4C22semidirect
50. #890C22semidirect
51. #889C22semidirect
54. #886C22semidirect
56. #884C22semidirect
61. #879C22semidirect
62. #878C22semidirect
65. #870C22semidirect
66. #869C22semidirect
67. C4×Dic6C22semidirect
76. #540D3semidirect
144. #127C2×D3semidirect
151. #125C2×D3semidirect
158. #123C2×D3semidirect
185. C4×D4C2×D3semidirect
372. C3#3539semidirect

Maximal subgroups

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

subgroupindex
C3×#5402
#57182
#56942
#56652
#56632
#56202
#56182
#56082
#56062
#55582
#55552
#55242
#55182
#35393