Finite Group Databasebeta

← #86040

Subgroups — #86040

Order 384.

1560 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1009C211yesmaximal, characteristic, complemented
2C22×#235C211yesmaximal, characteristic, complemented
3C3×#525C211yesmaximal, characteristic, complemented
4#5825C211yesmaximal, characteristic, complemented
5#5791C211yesmaximal, characteristic, complemented
6D3×#123C214yesmaximal, complemented
7#5631C214yesmaximal, complemented
8#5612C214yesmaximal, complemented
9#5589C214yesmaximal, complemented
10#5551C214yesmaximal, complemented
11#5550C214yesmaximal, complemented
12#5526C211yesmaximal, characteristic, complemented
13#5521C211yesmaximal, characteristic, complemented
14#3485—31nomaximal, Sylow
15C2×C3×#41C2211yescharacteristic
16C23×C3×C4C2211yescharacteristic, complemented
17C22×#67C2212yescomplemented
18#1009—28no—
19C2×#239C2212yescomplemented
20C2×D3×D4C2214yes—
21C2×D3×D4C2212yescomplemented
22C2×#235—28no—
23C2×#235C2211yescharacteristic
24C2×#235C2214yescomplemented
25C2×#235C2214yescomplemented
26C2×#235C2212yescomplemented
27C22×D12C2211yescharacteristic
28C22×D12C2212yescomplemented
29C22×C4×D3C2212yescomplemented
30C22×Dic6C2211yescharacteristic, complemented
31C3×#123C2212yescomplemented
32C3×C22⋊Q8C2212yescomplemented
33C3×#121C2212yescomplemented
34C3×#120C2212yescomplemented
35C3×C4×D4C2214yescomplemented
36C3×#117C2211yescharacteristic, complemented
37#953C2212yescomplemented
38#944C2212yescomplemented
39#938C2218yescomplemented
40#937C2218yescomplemented
41#936C2212yescomplemented
42#935C2214yescomplemented
43#930C2212yescomplemented
44#929C2212yescomplemented
45C4×#67C2214yescomplemented
46#926C2211yescharacteristic, complemented
47#924C2212yescomplemented
48#924C2214yescomplemented
49#897C2214yescomplemented
50#894C2218yescomplemented
51#893C2214yescomplemented
52D3×C4⋊C4C2218yescomplemented
53#886C2214yescomplemented
54#884C2218yescomplemented
55#883C2218yescomplemented
56#883C2218yescomplemented
57D3×C22⋊C4C2214yescomplemented
58D3×C22⋊C4C2218yescomplemented
59#875C2212yescomplemented
60C4×D12C2214yescomplemented
61#869C2212yescomplemented
62C2×H(5,𝔽2)—31no—
63C22×#41—31no—
64#547—34no—
65#547—34no—
66#546—34no—
67#546—34no—
68#536—31no—
69#536—31no—
70#534—34no—
71#525—31no—
72#525D311yescharacteristic, complemented
73C2×#123—34no—
74#518—31no—
75C24×C3C2311yescharacteristic
76C23×D3C2314yes—
77C23×D3C2312yes—
78C3×#41—24no—
79C2×C3×Q8C2311yescharacteristic
80C2×C3×D4C2312yes—
81C2×C3×D4C2311yescharacteristic
82C2×C3×D4C2314yes—
83C22×C3×C4—24no—
84C22×C3×C4C2312yes—
85C22×C3×C4C2312yes—
86C22×C3×C4C2312yes—
87C22×C3×C4C2311yescharacteristic
88C22×C3×C4C2311yescharacteristic
89C2×#67—28no—
90C2×#67C2314yes—
91C2×#67C2318yes—
92C2×#67C2312yes—
93C2×#67C2314yes—
94C22×Dic3C2312yes—
95#239—28no—
96D3×D4—216no—
97D3×D4—28no—
98#235—24no—
99#235—216no—
100#235—216no—
101#235—28no—
102C2×D12—28no—
103C2×D12D418yescomplemented
104C2×D12C2311yescharacteristic
105C2×D12C2314yes—
106C2×D12C2312yes—
107C2×D12C2312yes—
108C2×C4×D3—28no—
109C2×C4×D3C2318yes—
110C2×C4×D3C2314yes—
111C2×C4×D3C2312yes—
112C2×C4×D3C2318yes—
113C2×Dic6—24no—
114C2×Dic6C2311yescharacteristic
115C2×Dic6C2312yes—
116C2×Dic6C2312yes—
117C3×C4⋊C4C2314yes—
118C3×C4⋊C4C2312yes—
119C3×C22⋊C4C2314yes—
120C3×C22⋊C4C2312yes—
121C3×C22⋊C4C2314yes—
122C3×C42C2312yes—
123#217C2312yes—
124#217C2318yes—
125#217C2314yes—
126(C2×D3)⋊C4C2318yes—
127(C2×D3)⋊C4C2318yes—
128(C2×D3)⋊C4C2314yes—
129C4⋊Dic3C2318yes—
130C4⋊Dic3C2312yes—
131Dic3⋊C4C2314yes—
132Dic3⋊C4C2318yes—
133C4×Dic3C2312yes—
134H(5,𝔽2)—68no—
135C2×#41—32no—
136C2×#41—31no—
137C2×#41—34no—
138C2×#41—34no—
139C2×#41—32no—
140C2×#41—68no—
141C2×#41C2×D311yescharacteristic
142C22×Q8—31no—
143C22×D4—32no—
144C22×D4—34no—
145C22×D4—32no—
146C22×D4—31no—
147C22×D4—32no—
148C23×C4—32no—
149C23×C4C2×D311yescharacteristic, complemented
150C4⋊Q8—32no—
151C4⋊Q8—32no—
152#124—32no—
153#124—32no—
154#123—34no—
155#123—32no—
156#123—38no—
157#123—34no—
158#123—38no—
159#123—38no—
160#123C2×D312yescomplemented
161C22⋊Q8—32no—
162C22⋊Q8—34no—
163C22⋊Q8—34no—
164C22⋊Q8C2×D312yescomplemented
165#121—38no—
166#121—32no—
167#121—38no—
168#121C2×D312yescomplemented
169#120—32no—
170#120—38no—
171#120C2×D312yescomplemented
172C4×D4—34no—
173C4×D4—34no—
174C4×D4—34no—
175C4×D4C2×D314yescomplemented
176#117—31no—
177#117C2×D311yescharacteristic, complemented
178C2×C4⋊C4—38no—
179C2×C22⋊C4—34no—
180C2×C22⋊C4—38no—
181C23×C3—22no—
182C23×C3—42no—
183C23×C3C2412yes—
184C23×C3C2412yes—
185C23×C3C2411yescharacteristic
186C22×D3—28no—
187C22×D3—44no—
188C22×D3—48no—
189C22×D3C2×D418yes—
190C22×D3C2412yes—
191C22×D3C2414yes—
192C3×Q8—22no—
193C3×D4—24no—
194C3×D4—22no—
195C3×D4—44no—
196C2×C3×C4—22no—
197C2×C3×C4—24no—
198C2×C3×C4—24no—
199C2×C3×C4—24no—
200C2×C3×C4C2×D414yes—
201C2×C3×C4C2412yes—
202C2×C3×C4C2412yes—
203C2×C3×C4C2414yes—
204C2×C3×C4C2411yescharacteristic
205C2×C3×C4C2411yescharacteristic
206#67—24no—
207#67—216no—
208#67—28no—
209#67—44no—
210C2×Dic3—24no—
211C2×Dic3C2414yes—
212C2×Dic3C2412yes—
213C2×Dic3C2414yes—
214D12—28no—
215D12—22no—
216D12C2×D418yes—
217D12C2×D418yes—
218C4×D3—216no—
219C4×D3—28no—
220C4×D3—24no—
221C4×D3—48no—
222Dic6—24no—
223Dic6—24no—
224Dic6—22no—
225C24—32no—
226C24—34no—
227C24C22×D311yescharacteristic
228#41—24no—
229#41—616no—
230#41—616no—
231#41—68no—
232#41—68no—
233#41—64no—
234C2×Q8—32no—
235C2×Q8—32no—
236C2×Q8—31no—
237C2×Q8—64no—
238C2×Q8C22×D311yescharacteristic
239C2×D4—38no—
240C2×D4—32no—
241C2×D4—34no—
242C2×D4—34no—
243C2×D4—38no—
244C2×D4—32no—
245C2×D4—32no—
246C2×D4—34no—
247C2×D4—31no—
248C2×D4—68no—
249C2×D4—68no—
250C2×D4—616no—
251C2×D4—68no—
252C2×D4C22×D312yes—
253C2×D4C22×D311yescharacteristic
254C2×D4C22×D314yes—
255C22×C4—24no—
256C22×C4—32no—
257C22×C4—38no—
258C22×C4—34no—
259C22×C4—32no—
260C22×C4—38no—
261C22×C4—68no—
262C22×C4C22×D312yes—
263C22×C4C22×D312yes—
264C22×C4C22×D312yes—
265C22×C4C22×D311yescharacteristic
266C22×C4C22×D311yescharacteristic
267C4⋊C4—34no—
268C4⋊C4—38no—
269C4⋊C4—38no—
270C4⋊C4—32no—
271C4⋊C4C22×D314yes—
272C4⋊C4C22×D312yes—
273C22⋊C4—32no—
274C22⋊C4—38no—
275C22⋊C4—34no—
276C22⋊C4—34no—
277C22⋊C4—38no—
278C22⋊C4—38no—
279C22⋊C4C22×D312yes—
280C22⋊C4C22×D314yes—
281C22⋊C4C22×D314yes—
282C42—32no—
283C42C22×D312yes—
284C22×C3—22no—
285C22×C3—22no—
286C22×C3—22no—
287C22×C3—22no—
288C22×C3—22no—
289C22×C3—24no—
290C22×C3—44no—
291C22×C3C22×D412yes—
292C22×C3C2511yescharacteristic
293C2×D3—28no—
294C2×D3—22no—
295C2×D3—24no—
296C2×D3—48no—
297C2×D3—88no—
298C2×D3C22×D418yes—
299C3×C4—22no—
300C3×C4—22no—
301C3×C4—24no—
302C3×C4C22×D412yes—
303C3×C4C22×D412yes—
304Dic3—24no—
305Dic3—22no—
306Dic3—24no—
307C23—22no—
308C23—34no—
309C23—38no—
310C23—32no—
311C23—42no—
312C23—68no—
313C23—124no—
314C23—128no—
315C23C23×D312yes—
316C23C23×D312yes—
317C23C23×D311yescharacteristic
318Q8—22no—
319Q8—64no—
320Q8—64no—
321Q8—62no—
322D4—24no—
323D4—22no—
324D4—38no—
325D4—38no—
326D4—44no—
327D4—68no—
328D4—68no—
329D4—616no—
330D4—64no—
331D4—62no—
332D4—124no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—34no—
338C2×C4—34no—
339C2×C4—32no—
340C2×C4—616no—
341C2×C4—68no—
342C2×C4—64no—
343C2×C4—64no—
344C2×C4—128no—
345C2×C4D3×D414yes—
346C2×C4C23×D314yes—
347C2×C4C23×D312yes—
348C2×C4C23×D312yes—
349C2×C4C23×D311yescharacteristic
350C2×C4C23×D311yescharacteristic
351C2×C3—22no—
352C2×C3—42no—
353C2×C3—42no—
354C2×C3C23×D411yescharacteristic
355C2×C3#58512yes—
356D3—28no—
357D3—42no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—24no—
363C22—22no—
364C22—38no—
365C22—44no—
366C22—68no—
367C22—62no—
368C22—64no—
369C22—128no—
370C22—248no—
371C22C2×D3×D412yes—
372C22C24×D311yescharacteristic
373C4—22no—
374C4—22no—
375C4—24no—
376C4—62no—
377C4—64no—
378C4—64no—
379C4C2×D3×D412yes—
380C4C2×D3×D412yes—
381C3#348511yescharacteristic, complemented, Sylow
382C2—22no—
383C2—42no—
384C2—42no—
385C2—68no—
386C2—122no—
387C2C22×D3×D411yescharacteristic
388C2#596412yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#525C2
hypercenter#525D3
hypocenterC3#3485
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#1009C2semidirect
2. C22×#235C2semidirect
3. C3×#525C2semidirect
4. #5825C2semidirect
5. #5791C2semidirect
6. D3×#123C2semidirect
7. #5631C2semidirect
8. #5612C2semidirect
9. #5589C2semidirect
10. #5551C2semidirect
11. #5550C2semidirect
12. #5526C2semidirect
13. #5521C2semidirect
16. C23×C3×C4C22semidirect
17. C22×#67C22semidirect
19. C2×#239C22semidirect
21. C2×D3×D4C22semidirect
24. C2×#235C22semidirect
25. C2×#235C22semidirect
26. C2×#235C22semidirect
28. C22×D12C22semidirect
29. C22×C4×D3C22semidirect
30. C22×Dic6C22semidirect
31. C3×#123C22semidirect
32. C3×C22⋊Q8C22semidirect
33. C3×#121C22semidirect
34. C3×#120C22semidirect
35. C3×C4×D4C22semidirect
36. C3×#117C22semidirect
37. #953C22semidirect
38. #944C22semidirect
39. #938C22semidirect
40. #937C22semidirect
41. #936C22semidirect
42. #935C22semidirect
43. #930C22semidirect
44. #929C22semidirect
45. C4×#67C22semidirect
46. #926C22semidirect
47. #924C22semidirect
48. #924C22semidirect
49. #897C22semidirect
50. #894C22semidirect
51. #893C22semidirect
52. D3×C4⋊C4C22semidirect
53. #886C22semidirect
54. #884C22semidirect
55. #883C22semidirect
56. #883C22semidirect
57. D3×C22⋊C4C22semidirect
58. D3×C22⋊C4C22semidirect
59. #875C22semidirect
60. C4×D12C22semidirect
61. #869C22semidirect
72. #525D3semidirect
103. C2×D12D4semidirect
149. C23×C4C2×D3semidirect
160. #123C2×D3semidirect
164. C22⋊Q8C2×D3semidirect
168. #121C2×D3semidirect
171. #120C2×D3semidirect
175. C4×D4C2×D3semidirect
177. #117C2×D3semidirect
381. C3#3485semidirect

Maximal subgroups

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

subgroupindex
C2×#10092
C22×#2352
C3×#5252
#58252
#57912
D3×#1232
#56312
#56122
#55892
#55512
#55502
#55262
#55212
#34853