Finite Group Databasebeta

← #86023

Subgroups — #86023

Order 384.

1536 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1010C211yesmaximal, characteristic, complemented
2C22×#235C211yesmaximal, characteristic, complemented
3C3×#525C211yesmaximal, characteristic, complemented
4#5830C211yesmaximal, characteristic, complemented
5#5791C211yesmaximal, characteristic, complemented
6#5656C214yesmaximal, complemented
7#5633C214yesmaximal, complemented
8#5599C214yesmaximal, complemented
9#5592C214yesmaximal, complemented
10#5553C214yesmaximal, complemented
11#5552C214yesmaximal, complemented
12#5525C211yesmaximal, characteristic, complemented
13#5516C211yesmaximal, characteristic, complemented
14#3487—31nomaximal, Sylow
15C2×C3×#41C2211yescharacteristic
16C23×C3×C4C2211yescharacteristic
17C22×#67C2212yes—
18#1010—28no—
19C2×D3×Q8C2212yescomplemented
20C2×#237C2214yes—
21C2×#237C2212yescomplemented
22C2×#235—28no—
23C2×#235C2214yes—
24C2×#235C2214yes—
25C2×#235C2211yescharacteristic
26C2×#235C2212yescomplemented
27C22×D12C2211yescharacteristic
28C22×C4×D3C2212yes—
29C22×Dic6C2211yescharacteristic
30C22×Dic6C2212yescomplemented
31C3×#123C2212yes—
32C3×C22⋊Q8C2212yescomplemented
33C3×#121C2212yescomplemented
34C3×#120C2212yes—
35C3×C4×D4C2214yescomplemented
36C3×#117C2211yescharacteristic
37#953C2212yes—
38#947C2212yescomplemented
39#940C2218yescomplemented
40#940C2212yescomplemented
41#939C2218yescomplemented
42#937C2214yes—
43#930C2212yescomplemented
44#930C2214yescomplemented
45#929C2212yes—
46C4×#67C2214yescomplemented
47#926C2211yescharacteristic
48#924C2212yescomplemented
49#896C2218yescomplemented
50#895C2214yescomplemented
51#893C2218yescomplemented
52#887C2214yescomplemented
53#885C2218yescomplemented
54#884C2218yescomplemented
55#884C2218yescomplemented
56#882C2214yes—
57#67⋊C4C2218yescomplemented
58#877C2214yes—
59#875C2212yescomplemented
60#874C2212yescomplemented
61C4×Dic6C2214yescomplemented
62C2×2−1+4—31no—
63C22×#41—31no—
64#550—34no—
65#550—34no—
66#548—34no—
67#548—34no—
68#535—31no—
69#535—34no—
70#535—31no—
71#525—31no—
72#525—34no—
73#525D311yescharacteristic, complemented
74#519—31no—
75C24×C3C2311yescharacteristic
76C23×D3C2312yes—
77C3×#41—24no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4C2312yes—
80C2×C3×D4C2311yescharacteristic
81C2×C3×D4C2314yes—
82C22×C3×C4—24no—
83C22×C3×C4C2312yes—
84C22×C3×C4C2312yes—
85C22×C3×C4C2311yescharacteristic
86C22×C3×C4C2312yes—
87C22×C3×C4C2311yescharacteristic
88C2×#67—28no—
89C2×#67C2312yes—
90C2×#67C2314yes—
91C2×#67C2318yes—
92C2×#67C2318yes—
93C2×#67C2314yes—
94C22×Dic3C2314yes—
95C22×Dic3C2312yes—
96D3×Q8—28no—
97#237—216no—
98#237—28no—
99#235—24no—
100#235—216no—
101#235—216no—
102#235—28no—
103C2×D12—24no—
104C2×D12C2311yescharacteristic
105C2×D12C2312yes—
106C2×D12C2312yes—
107C2×D12C2318yes—
108C2×C4×D3—28no—
109C2×C4×D3C2318yes—
110C2×C4×D3C2314yes—
111C2×C4×D3C2312yes—
112C2×Dic6—28no—
113C2×Dic6D418yescomplemented
114C2×Dic6C2311yescharacteristic
115C2×Dic6C2314yes—
116C2×Dic6C2312yes—
117C2×Dic6C2312yes—
118C3×C4⋊C4C2312yes—
119C3×C4⋊C4C2314yes—
120C3×C22⋊C4C2312yes—
121C3×C22⋊C4C2314yes—
122C3×C22⋊C4C2314yes—
123C3×C42C2312yes—
124#217C2312yes—
125#217C2314yes—
126(C2×D3)⋊C4C2318yes—
127(C2×D3)⋊C4C2314yes—
128(C2×D3)⋊C4C2318yes—
129C4⋊Dic3C2312yes—
130Dic3⋊C4C2318yes—
131Dic3⋊C4C2314yes—
132C4×Dic3C2318yes—
133C4×Dic3C2312yes—
1342−1+4—68no—
135C2×#41—34no—
136C2×#41—32no—
137C2×#41—31no—
138C2×#41—34no—
139C2×#41—34no—
140C2×#41—32no—
141C2×#41—68no—
142C2×#41C2×D311yescharacteristic
143C22×Q8—32no—
144C22×Q8—31no—
145C22×Q8—32no—
146C22×D4—32no—
147C22×D4—31no—
148C23×C4—32no—
149C23×C4C2×D311yescharacteristic
150#127—38no—
151#127—32no—
152#127—32no—
153#124—32no—
154#124—38no—
155#124—32no—
156#123—32no—
157#123C2×D312yes—
158C22⋊Q8—32no—
159C22⋊Q8—38no—
160C22⋊Q8C2×D312yescomplemented
161#121—38no—
162#121—32no—
163#121—34no—
164#121—34no—
165#121—38no—
166#121—38no—
167#121C2×D312yescomplemented
168#120—32no—
169#120—34no—
170#120—34no—
171#120C2×D312yes—
172C4×Q8—34no—
173C4×Q8—34no—
174C4×D4—34no—
175C4×D4—38no—
176C4×D4—38no—
177C4×D4C2×D314yescomplemented
178#117—31no—
179#117—34no—
180#117C2×D311yescharacteristic
181C23×C3—22no—
182C23×C3—42no—
183C23×C3C2412yes—
184C23×C3C2412yes—
185C23×C3C2411yescharacteristic
186C22×D3—24no—
187C22×D3—44no—
188C22×D3C2414yes—
189C22×D3C2412yes—
190C3×Q8—22no—
191C3×D4—24no—
192C3×D4—22no—
193C3×D4—44no—
194C2×C3×C4—22no—
195C2×C3×C4—24no—
196C2×C3×C4—24no—
197C2×C3×C4—24no—
198C2×C3×C4C2×D414yes—
199C2×C3×C4C2414yes—
200C2×C3×C4C2412yes—
201C2×C3×C4C2412yes—
202C2×C3×C4C2411yescharacteristic
203C2×C3×C4C2411yescharacteristic
204#67—24no—
205#67—216no—
206#67—28no—
207#67—44no—
208#67—48no—
209C2×Dic3—28no—
210C2×Dic3C2×D418yes—
211C2×Dic3C2414yes—
212C2×Dic3C2412yes—
213C2×Dic3C2414yes—
214D12—24no—
215D12—24no—
216D12—22no—
217D12—48no—
218C4×D3—24no—
219C4×D3—216no—
220C4×D3—28no—
221Dic6—22no—
222Dic6—28no—
223Dic6C2×D418yes—
224Dic6C2×D418yes—
225C24—32no—
226C24C22×D311yescharacteristic
227#41—24no—
228#41—616no—
229#41—616no—
230#41—68no—
231#41—616no—
232#41—68no—
233#41—64no—
234C2×Q8—38no—
235C2×Q8—32no—
236C2×Q8—32no—
237C2×Q8—34no—
238C2×Q8—31no—
239C2×Q8—68no—
240C2×Q8—68no—
241C2×Q8C22×D311yescharacteristic
242C2×D4—32no—
243C2×D4—38no—
244C2×D4—34no—
245C2×D4—38no—
246C2×D4—34no—
247C2×D4—32no—
248C2×D4—32no—
249C2×D4—31no—
250C2×D4—38no—
251C2×D4—68no—
252C2×D4—64no—
253C2×D4C22×D312yes—
254C2×D4C22×D311yescharacteristic
255C2×D4C22×D314yes—
256C22×C4—24no—
257C22×C4—32no—
258C22×C4—34no—
259C22×C4—34no—
260C22×C4—32no—
261C22×C4—38no—
262C22×C4—68no—
263C22×C4C22×D312yes—
264C22×C4C22×D312yes—
265C22×C4C22×D312yes—
266C22×C4C22×D311yescharacteristic
267C22×C4C22×D311yescharacteristic
268C4⋊C4—38no—
269C4⋊C4—32no—
270C4⋊C4—34no—
271C4⋊C4C22×D312yes—
272C4⋊C4C22×D314yes—
273C22⋊C4—32no—
274C22⋊C4—34no—
275C22⋊C4—34no—
276C22⋊C4—38no—
277C22⋊C4—38no—
278C22⋊C4C22×D312yes—
279C22⋊C4C22×D314yes—
280C22⋊C4C22×D314yes—
281C42—38no—
282C42—32no—
283C42C22×D312yes—
284C22×C3—22no—
285C22×C3—22no—
286C22×C3—22no—
287C22×C3—24no—
288C22×C3—22no—
289C22×C3—22no—
290C22×C3—44no—
291C22×C3C22×D412yes—
292C22×C3C2511yescharacteristic
293C2×D3—22no—
294C2×D3—24no—
295C2×D3—24no—
296C2×D3—28no—
297C2×D3—48no—
298C3×C4—22no—
299C3×C4—22no—
300C3×C4—24no—
301C3×C4C22×D412yes—
302C3×C4C22×D412yes—
303Dic3—22no—
304Dic3—24no—
305Dic3C22×D418yes—
306C23—22no—
307C23—34no—
308C23—32no—
309C23—42no—
310C23—64no—
311C23—124no—
312C23C23×D312yes—
313C23C23×D312yes—
314C23C23×D311yescharacteristic
315Q8—22no—
316Q8—38no—
317Q8—38no—
318Q8—68no—
319Q8—62no—
320D4—24no—
321D4—22no—
322D4—44no—
323D4—616no—
324D4—68no—
325D4—64no—
326D4—64no—
327D4—64no—
328D4—62no—
329D4—128no—
330D4—128no—
331D4—124no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—38no—
337C2×C4—34no—
338C2×C4—34no—
339C2×C4—32no—
340C2×C4—616no—
341C2×C4—68no—
342C2×C4—68no—
343C2×C4—64no—
344C2×C4D3×D414yes—
345C2×C4C23×D314yes—
346C2×C4C23×D312yes—
347C2×C4C23×D312yes—
348C2×C4C23×D311yescharacteristic
349C2×C4C23×D311yescharacteristic
350C2×C3—22no—
351C2×C3—42no—
352C2×C3—42no—
353C2×C3C23×D411yescharacteristic
354C2×C3#58512yes—
355D3—42no—
356D3—44no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—24no—
362C22—22no—
363C22—44no—
364C22—62no—
365C22—64no—
366C22—64no—
367C22—68no—
368C22—128no—
369C22C2×D3×D412yes—
370C22C24×D311yescharacteristic
371C4—22no—
372C4—22no—
373C4—24no—
374C4—38no—
375C4—62no—
376C4—64no—
377C4C2×D3×D412yes—
378C4C2×D3×D412yes—
379C3#348711yescharacteristic, complemented, Sylow
380C2—22no—
381C2—42no—
382C2—42no—
383C2—124no—
384C2—122no—
385C2C22×D3×D411yescharacteristic
386C2#596412yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#525C2
hypercenter#525D3
hypocenterC3#3487
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#1010C2semidirect
2. C22×#235C2semidirect
3. C3×#525C2semidirect
4. #5830C2semidirect
5. #5791C2semidirect
6. #5656C2semidirect
7. #5633C2semidirect
8. #5599C2semidirect
9. #5592C2semidirect
10. #5553C2semidirect
11. #5552C2semidirect
12. #5525C2semidirect
13. #5516C2semidirect
19. C2×D3×Q8C22semidirect
21. C2×#237C22semidirect
26. C2×#235C22semidirect
30. C22×Dic6C22semidirect
32. C3×C22⋊Q8C22semidirect
33. C3×#121C22semidirect
35. C3×C4×D4C22semidirect
38. #947C22semidirect
39. #940C22semidirect
40. #940C22semidirect
41. #939C22semidirect
43. #930C22semidirect
44. #930C22semidirect
46. C4×#67C22semidirect
48. #924C22semidirect
49. #896C22semidirect
50. #895C22semidirect
51. #893C22semidirect
52. #887C22semidirect
53. #885C22semidirect
54. #884C22semidirect
55. #884C22semidirect
57. #67⋊C4C22semidirect
59. #875C22semidirect
60. #874C22semidirect
61. C4×Dic6C22semidirect
73. #525D3semidirect
113. C2×Dic6D4semidirect
160. C22⋊Q8C2×D3semidirect
167. #121C2×D3semidirect
177. C4×D4C2×D3semidirect
379. C3#3487semidirect

Maximal subgroups

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

subgroupindex
C2×#10102
C22×#2352
C3×#5252
#58302
#57912
#56562
#56332
#55992
#55922
#55532
#55522
#55252
#55162
#34873