Finite Group Databasebeta

← #86359

Subgroups — #86359

Order 384.

1014 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#536C211yesmaximal, characteristic, complemented
2#5828C211yesmaximal, characteristic
3#5825C211yesmaximal, characteristic, complemented
4Q8×#67C212yesmaximal, complemented
5#5729C214yesmaximal, complemented
6#5679C212yesmaximal, complemented
7#5672C212yesmaximal, complemented
8#5660C214yesmaximal, complemented
9#5659C212yesmaximal, complemented
10#5651C212yesmaximal, complemented
11#5639C212yesmaximal, complemented
12#5632C212yesmaximal, complemented
13D3×C22⋊Q8C212yesmaximal, complemented
14#5625C212yesmaximal, complemented
15#5529C212yesmaximal, complemented
16#3538—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×Q8C2211yescharacteristic, complemented
19C2×D3×Q8C2212yes—
20C22×C4×D3C2211yescharacteristic
21C22×Dic6C2211yescharacteristic, complemented
22C3×C4⋊Q8C2212yescomplemented
23C3×#124C2212yes—
24C3×#123C2212yes—
25C3×#123C2212yescomplemented
26C3×C22⋊Q8C2212yes—
27C3×C22⋊Q8C2212yescomplemented
28C3×#117C2211yescharacteristic, complemented
29#946C2212yes—
30#946C2214yes—
31#946C2212yes—
32Q8×Dic3C2212yescomplemented
33#944C2212yescomplemented
34#944C2214yescomplemented
35#944C2212yescomplemented
36#938C2212yes—
37#937C2211yescharacteristic
38#937C2211yescharacteristic
39#936C2212yes—
40#935C2211yescharacteristic
41#935C2211yescharacteristic
42#935C2212yes—
43#935C2212yescomplemented
44C4×#67C2212yes—
45C4×#67C2214yes—
46C4×#67C2212yes—
47#926C2211yescharacteristic
48#926C2211yescharacteristic, complemented
49#924C2212yes—
50#924C2214yescomplemented
51#924C2212yescomplemented
52#897C2212yes—
53#896C2214yes—
54#896C2214yes—
55#892C2214yes—
56D3×C4⋊C4C2212yes—
57D3×C4⋊C4C2214yes—
58#890C2212yescomplemented
59#890C2214yescomplemented
60#889C2214yescomplemented
61#889C2214yescomplemented
62#888C2214yescomplemented
63#887C2212yescomplemented
64#883C2214yes—
65#883C2214yes—
66#67⋊C4C2214yes—
67D3×C22⋊C4C2214yes—
68#879C2214yes—
69#879C2214yes—
70#879C2214yescomplemented
71#879C2214yescomplemented
72#878C2214yes—
73#878C2214yescomplemented
74#872C2214yes—
75#870C2214yescomplemented
76#563—32no—
77#563—32no—
78#552—32no—
79#552—32no—
80#551—32no—
81#551—32no—
82D4×Q8—32no—
83D4×Q8—32no—
84#544—34no—
85#541—32no—
86#539—34no—
87#536—31no—
88#536D311yescharacteristic, complemented
89#525—31no—
90C2×C22⋊Q8—32no—
91C23×D3C2311yescharacteristic
92C3×#41—42no—
93C2×C3×Q8—42no—
94C2×C3×Q8C2312yes—
95C2×C3×Q8C2312yes—
96C2×C3×Q8C2311yescharacteristic
97C2×C3×D4C2311yescharacteristic
98C2×C3×D4C2311yescharacteristic
99C2×C3×D4C2311yescharacteristic
100C22×C3×C4C2311yescharacteristic
101C22×C3×C4C2311yescharacteristic
102C22×C3×C4C2311yescharacteristic
103C22×C3×C4C2312yes—
104C2×#67C2312yes—
105C2×#67C2312yes—
106C22×Dic3C2311yescharacteristic
107C22×Dic3C2312yes—
108D3×Q8—44no—
109C2×C4×D3—42no—
110C2×C4×D3C2314yes—
111C2×C4×D3C2312yes—
112C2×C4×D3C2314yes—
113C2×C4×D3C2312yes—
114C2×Dic6—42no—
115C2×Dic6C2312yes—
116C2×Dic6C2314yes—
117C2×Dic6C2312yes—
118C3×C4⋊C4C2312yes—
119C3×C4⋊C4C2314yes—
120C3×C4⋊C4C2314yes—
121C3×C22⋊C4C2314yes—
122C3×C22⋊C4C2314yes—
123C3×C22⋊C4C2312yes—
124C3×C42C2312yes—
125#217C2314yes—
126#217C2312yes—
127#217C2311yescharacteristic
128#217C2311yescharacteristic
129#217C2312yes—
130#217C2312yes—
131#217C2314yes—
132(C2×D3)⋊C4C2312yes—
133(C2×D3)⋊C4C2314yes—
134(C2×D3)⋊C4C2314yes—
135(C2×D3)⋊C4C2312yes—
136C4⋊Dic3C2314yes—
137C4⋊Dic3C2314yes—
138C4⋊Dic3C2314yes—
139C4⋊Dic3C2312yes—
140C4⋊Dic3C2312yes—
141Dic3⋊C4C2312yes—
142Dic3⋊C4C2314yes—
143Dic3⋊C4C2314yes—
144Dic3⋊C4C2314yes—
145Dic3⋊C4C2314yes—
146Dic3⋊C4C2314yes—
147Dic3⋊C4C2314yes—
148Dic3⋊C4C2314yes—
149Dic3⋊C4C2312yes—
150Dic3⋊C4C2314yes—
151C4×Dic3C2314yes—
152C4×Dic3C2314yes—
153C4×Dic3C2312yes—
154C4×Dic3C2312yes—
155C2×#41C2×D311yescharacteristic
156C22×Q8—32no—
157C22×Q8—31no—
158C22×Q8C2×D311yescharacteristic, complemented
159C23×C4—31no—
160C4⋊Q8—32no—
161C4⋊Q8—32no—
162C4⋊Q8—34no—
163C4⋊Q8—34no—
164C4⋊Q8C2×D312yescomplemented
165#126—34no—
166#126—34no—
167#126—34no—
168#126—34no—
169#125—34no—
170#125—34no—
171#125—32no—
172#125—34no—
173#125—34no—
174#124—32no—
175#124C2×D312yes—
176#123—31no—
177#123—31no—
178#123—32no—
179#123—32no—
180#123—34no—
181#123—34no—
182#123C2×D312yes—
183#123C2×D312yescomplemented
184C22⋊Q8—32no—
185C22⋊Q8—32no—
186C22⋊Q8—34no—
187C22⋊Q8—32no—
188C22⋊Q8—32no—
189C22⋊Q8—34no—
190C22⋊Q8—34no—
191C22⋊Q8—32no—
192C22⋊Q8—34no—
193C22⋊Q8—34no—
194C22⋊Q8—34no—
195C22⋊Q8C2×D312yes—
196C22⋊Q8C2×D312yescomplemented
197#121—32no—
198#120—31no—
199#120—31no—
200C4×Q8—32no—
201C4×Q8—32no—
202C4×D4—32no—
203C4×D4—32no—
204C4×D4—34no—
205C4×D4—34no—
206#117—31no—
207#117—31no—
208#117—34no—
209#117—34no—
210#117C2×D311yescharacteristic, complemented
211C2×C4⋊C4—34no—
212C2×C4⋊C4—32no—
213C2×C22⋊C4—34no—
214C23×C3C2411yescharacteristic
215C23×C3C2411yescharacteristic
216C23×C3C2411yescharacteristic
217C22×D3—24no—
218C22×D3—41no—
219C22×D3C2412yes—
220C3×Q8—24no—
221C3×Q8—42no—
222C3×Q8—41no—
223C3×D4—41no—
224C3×D4—41no—
225C3×D4—41no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—42no—
230C2×C3×C4C2411yescharacteristic
231C2×C3×C4C2414yes—
232C2×C3×C4C2414yes—
233C2×C3×C4C2411yescharacteristic
234C2×C3×C4C2411yescharacteristic
235C2×C3×C4C2411yescharacteristic
236#67—24no—
237#67—24no—
238C2×Dic3—42no—
239C2×Dic3—41no—
240C2×Dic3C2412yes—
241C2×Dic3C2412yes—
242C2×Dic3C2414yes—
243C2×Dic3C2414yes—
244C2×Dic3C2412yes—
245C4×D3—28no—
246C4×D3—24no—
247C4×D3—42no—
248C4×D3—44no—
249Dic6—24no—
250Dic6—42no—
251Dic6—44no—
252C24—31no—
253#41—42no—
254C2×Q8—32no—
255C2×Q8—34no—
256C2×Q8—32no—
257C2×Q8—42no—
258C2×Q8—122no—
259C2×Q8—124no—
260C2×Q8C22×D312yes—
261C2×Q8C22×D312yes—
262C2×Q8C22×D311yescharacteristic
263C2×D4—32no—
264C2×D4—32no—
265C2×D4C22×D311yescharacteristic
266C2×D4C22×D311yescharacteristic
267C2×D4C22×D311yescharacteristic
268C22×C4—32no—
269C22×C4—31no—
270C22×C4—34no—
271C22×C4—32no—
272C22×C4—34no—
273C22×C4—32no—
274C22×C4—122no—
275C22×C4C22×D311yescharacteristic
276C22×C4C22×D311yescharacteristic
277C22×C4C22×D311yescharacteristic
278C22×C4C22×D312yes—
279C4⋊C4—32no—
280C4⋊C4—34no—
281C4⋊C4—34no—
282C4⋊C4—34no—
283C4⋊C4—34no—
284C4⋊C4—32no—
285C4⋊C4—34no—
286C4⋊C4—34no—
287C4⋊C4—34no—
288C4⋊C4—34no—
289C4⋊C4—34no—
290C4⋊C4—32no—
291C4⋊C4—32no—
292C4⋊C4—34no—
293C4⋊C4—34no—
294C4⋊C4C22×D312yes—
295C4⋊C4C22×D314yes—
296C4⋊C4C22×D314yes—
297C22⋊C4—34no—
298C22⋊C4—31no—
299C22⋊C4—31no—
300C22⋊C4—32no—
301C22⋊C4—32no—
302C22⋊C4—32no—
303C22⋊C4—34no—
304C22⋊C4—34no—
305C22⋊C4—34no—
306C22⋊C4—32no—
307C22⋊C4—32no—
308C22⋊C4C22×D314yes—
309C22⋊C4C22×D312yes—
310C22⋊C4C22×D314yes—
311C42—32no—
312C42—32no—
313C42—34no—
314C42—34no—
315C42C22×D312yes—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—22no—
320C22×C3—22no—
321C22×C3—22no—
322C22×C3C2511yescharacteristic
323C2×D3—22no—
324C2×D3—22no—
325C2×D3—44no—
326C2×D32−1+414yes—
327C3×C4—21no—
328C3×C4—24no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—24no—
332C3×C4H(5,𝔽2)12yes—
333Dic3—24no—
334Dic3—22no—
335Dic3—24no—
336Dic3—22no—
337Dic3—22no—
338C23—32no—
339C23—64no—
340C23—121no—
341C23C23×D311yescharacteristic
342C23C23×D311yescharacteristic
343C23C23×D311yescharacteristic
344Q8—24no—
345Q8—42no—
346Q8—41no—
347Q8—64no—
348Q8—122no—
349Q8—124no—
350D4—41no—
351D4—41no—
352D4—41no—
353D4—64no—
354D4—64no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—32no—
359C2×C4—32no—
360C2×C4—34no—
361C2×C4—34no—
362C2×C4—32no—
363C2×C4—42no—
364C2×C4—68no—
365C2×C4—64no—
366C2×C4—122no—
367C2×C4—124no—
368C2×C4—121no—
369C2×C4—122no—
370C2×C4C23×D311yescharacteristic
371C2×C4C23×D314yes—
372C2×C4C23×D314yes—
373C2×C4C23×D311yescharacteristic
374C2×C4C23×D311yescharacteristic
375C2×C4C23×D311yescharacteristic
376C2×C3—41no—
377C2×C3—41no—
378C2×C3—41no—
379C2×C3C2×H(5,𝔽2)11yescharacteristic
380C2×C3C2×2−1+412yes—
381D3—24no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—34no—
389C22—62no—
390C22—62no—
391C22—124no—
392C22C24×D311yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—24no—
397C4—24no—
398C4—62no—
399C4—64no—
400C4—64no—
401C4—62no—
402C4—62no—
403C4#100412yes—
404C3#353811yescharacteristic, complemented, Sylow
405C2—41no—
406C2—41no—
407C2—41no—
408C2—64no—
409C2C2×#100411yescharacteristic
410C2D3×2−1+412yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#536C2
prime radical#5828C2
hypercenter#536D3
hypocenterC3#3538
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C3×#536C2semidirect
3. #5825C2semidirect
4. Q8×#67C2semidirect
5. #5729C2semidirect
6. #5679C2semidirect
7. #5672C2semidirect
8. #5660C2semidirect
9. #5659C2semidirect
10. #5651C2semidirect
11. #5639C2semidirect
12. #5632C2semidirect
13. D3×C22⋊Q8C2semidirect
14. #5625C2semidirect
15. #5529C2semidirect
18. C22×C3×Q8C22semidirect
21. C22×Dic6C22semidirect
22. C3×C4⋊Q8C22semidirect
25. C3×#123C22semidirect
27. C3×C22⋊Q8C22semidirect
28. C3×#117C22semidirect
32. Q8×Dic3C22semidirect
33. #944C22semidirect
34. #944C22semidirect
35. #944C22semidirect
43. #935C22semidirect
48. #926C22semidirect
50. #924C22semidirect
51. #924C22semidirect
58. #890C22semidirect
59. #890C22semidirect
60. #889C22semidirect
61. #889C22semidirect
62. #888C22semidirect
63. #887C22semidirect
70. #879C22semidirect
71. #879C22semidirect
73. #878C22semidirect
75. #870C22semidirect
88. #536D3semidirect
158. C22×Q8C2×D3semidirect
164. C4⋊Q8C2×D3semidirect
183. #123C2×D3semidirect
196. C22⋊Q8C2×D3semidirect
210. #117C2×D3semidirect
404. C3#3538semidirect

Maximal subgroups

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

subgroupindex
C3×#5362
#58282
#58252
Q8×#672
#57292
#56792
#56722
#56602
#56592
#56512
#56392
#56322
D3×C22⋊Q82
#56252
#55292
#35383