Finite Group Databasebeta

← #86001

Subgroups — #86001

Order 384.

1052 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
1C2×C3×#123C211yesmaximal, characteristic, complemented
2#5804C211yesmaximal, characteristic
3#5798C211yesmaximal, characteristic, complemented
4C2×#929C211yesmaximal, characteristic, complemented
5C2×#926C211yesmaximal, characteristic, complemented
6#5665C214yesmaximal, complemented
7#5658C214yesmaximal, complemented
8#5650C214yesmaximal, complemented
9#5649C214yesmaximal, complemented
10#5513C214yesmaximal, complemented
11#5495C214yesmaximal, complemented
12#5486C211yesmaximal, characteristic, complemented
13#5483C211yesmaximal, characteristic, complemented
14#3491—31nomaximal, Sylow
15C22×C3×D4C2211yescharacteristic
16C23×C3×C4C2211yescharacteristic
17C22×#67C2211yescharacteristic
18C3×#123C2214yes—
19C3×#123C2212yescomplemented
20C3×#123C2212yescomplemented
21C2×C3×C4⋊C4C2212yescomplemented
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4C2212yescomplemented
24#953C2212yes—
25C2×#217C2211yescharacteristic
26C2×#217C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×#217C2212yescomplemented
29#939C2214yes—
30#937C2212yes—
31#935C2212yes—
32D4×Dic3C2214yes—
33D4×Dic3C2214yescomplemented
34#929C2214yes—
35#929C2212yes—
36#929C2212yescomplemented
37#929C2212yescomplemented
38C4×#67C2214yes—
39C2×(C2×D3)⋊C4C2212yescomplemented
40#926C2214yescomplemented
41#926C2214yescomplemented
42#926C2212yescomplemented
43C2×C4⋊Dic3C2212yescomplemented
44#924C2214yescomplemented
45C2×Dic3⋊C4C2212yescomplemented
46C2×C4×Dic3C2212yescomplemented
47#898C2214yescomplemented
48#898C2214yescomplemented
49#897C2214yescomplemented
50#892C2214yescomplemented
51#890C2214yescomplemented
52#889C2214yescomplemented
53#889C2214yescomplemented
54#887C2214yescomplemented
55#886C2214yescomplemented
56#885C2214yescomplemented
57#883C2214yescomplemented
58#67⋊C4C2214yescomplemented
59#67⋊C4C2214yescomplemented
60D3×C22⋊C4C2212yescomplemented
61#879C2214yescomplemented
62#879C2214yescomplemented
63#878C2214yescomplemented
64#878C2214yescomplemented
65#878C2214yescomplemented
66#877C2214yescomplemented
67#877C2212yescomplemented
68#552—34no—
69#552—34no—
70#551—34no—
71#551—34no—
72#543—31no—
73#539—34no—
74#534—31no—
75#529—34no—
76C2×#123—31no—
77C2×#123D311yescharacteristic, complemented
78#518—31no—
79#518—31no—
80C2×#117—31no—
81C24×C3C2311yescharacteristic
82C24×C3C2311yescharacteristic
83C23×D3C2311yescharacteristic
84C2×C3×D4—24no—
85C2×C3×D4C2312yes—
86C2×C3×D4C2312yes—
87C22×C3×C4—24no—
88C22×C3×C4C2311yescharacteristic
89C22×C3×C4C2312yes—
90C22×C3×C4C2311yescharacteristic
91C22×C3×C4C2312yes—
92C22×C3×C4C2311yescharacteristic
93C22×C3×C4C2312yes—
94C2×#67—24no—
95C2×#67C2312yes—
96C2×#67C2312yes—
97C2×#67C2314yes—
98C22×Dic3C2311yescharacteristic
99C22×Dic3C2314yes—
100C22×Dic3C2312yes—
101C22×Dic3C2312yes—
102C22×Dic3C2312yes—
103C2×C4×D3C2314yes—
104C2×Dic6C2314yes—
105C3×C4⋊C4C2314yes—
106C3×C4⋊C4C2314yes—
107C3×C22⋊C4C2312yes—
108C3×C22⋊C4C2314yes—
109C3×C22⋊C4C2312yes—
110C3×C22⋊C4C2314yes—
111#217C2314yes—
112#217C2312yes—
113#217C2312yes—
114#217C2312yes—
115#217C2314yes—
116#217C2312yes—
117#217C2312yes—
118#217C2312yes—
119(C2×D3)⋊C4C2314yes—
120(C2×D3)⋊C4C2314yes—
121(C2×D3)⋊C4C2314yes—
122(C2×D3)⋊C4C2314yes—
123C4⋊Dic3C2314yes—
124C4⋊Dic3C2314yes—
125C4⋊Dic3C2314yes—
126C4⋊Dic3C2314yes—
127Dic3⋊C4C2314yes—
128Dic3⋊C4C2314yes—
129Dic3⋊C4C2314yes—
130Dic3⋊C4C2314yes—
131Dic3⋊C4C2314yes—
132Dic3⋊C4C2314yes—
133C4×Dic3C2314yes—
134C4×Dic3C2314yes—
135C4×Dic3C2314yes—
136C4×Dic3C2314yes—
137C22×D4—31no—
138C22×D4C2×D311yescharacteristic
139C23×C4C2×D311yescharacteristic
140#126—34no—
141#126—34no—
142#126—34no—
143#126—34no—
144#125—34no—
145#125—34no—
146#125—34no—
147#124—34no—
148#123—32no—
149#123—32no—
150#123—32no—
151#123—34no—
152#123—32no—
153#123—34no—
154#123—34no—
155#123C2×D314yes—
156#123C2×D312yescomplemented
157#123C2×D312yescomplemented
158C22⋊Q8—34no—
159C22⋊Q8—34no—
160C22⋊Q8—34no—
161C22⋊Q8—34no—
162C22⋊Q8—34no—
163#121—34no—
164#120—32no—
165#120—32no—
166C4×Q8—34no—
167C4×D4—34no—
168C4×D4—34no—
169C4×D4—34no—
170C4×D4—34no—
171C4×D4—34no—
172#117—34no—
173#117—34no—
174#117—32no—
175#117—34no—
176#117—32no—
177#117—34no—
178C2×C4⋊C4—32no—
179C2×C4⋊C4—32no—
180C2×C4⋊C4C2×D312yescomplemented
181C2×C22⋊C4—31no—
182C2×C22⋊C4—31no—
183C2×C22⋊C4—31no—
184C2×C22⋊C4—32no—
185C2×C22⋊C4—32no—
186C2×C22⋊C4—32no—
187C2×C22⋊C4C2×D311yescharacteristic
188C2×C22⋊C4C2×D312yescomplemented
189C2×C42—32no—
190C23×C3—22no—
191C23×C3—42no—
192C23×C3—42no—
193C23×C3#4114yes—
194C23×C3C2412yes—
195C23×C3C2411yescharacteristic
196C23×C3C2411yescharacteristic
197C23×C3C2411yescharacteristic
198C22×D3—22no—
199C22×D3—42no—
200C22×D3C2412yes—
201C3×D4—24no—
202C3×D4—24no—
203C2×C3×C4—24no—
204C2×C3×C4—24no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—42no—
209C2×C3×C4C2412yes—
210C2×C3×C4C2412yes—
211C2×C3×C4C2414yes—
212C2×C3×C4C2412yes—
213#67—24no—
214#67—24no—
215#67—44no—
216C2×Dic3—22no—
217C2×Dic3—42no—
218C2×Dic3—44no—
219C2×Dic3—42no—
220C2×Dic3—42no—
221C2×Dic3C2414yes—
222C2×Dic3C2412yes—
223C2×Dic3C2414yes—
224C2×Dic3C2414yes—
225C4×D3—44no—
226Dic6—44no—
227C24—31no—
228C24C22×D311yescharacteristic
229C24C22×D311yescharacteristic
230C2×Q8—34no—
231C2×D4—24no—
232C2×D4—32no—
233C2×D4—32no—
234C2×D4—34no—
235C2×D4—64no—
236C2×D4C22×D312yes—
237C2×D4C22×D312yes—
238C22×C4—24no—
239C22×C4—31no—
240C22×C4—32no—
241C22×C4—32no—
242C22×C4—32no—
243C22×C4—34no—
244C22×C4—34no—
245C22×C4C22×D311yescharacteristic
246C22×C4C22×D312yes—
247C22×C4C22×D312yes—
248C22×C4C22×D311yescharacteristic
249C22×C4C22×D311yescharacteristic
250C22×C4C22×D312yes—
251C4⋊C4—34no—
252C4⋊C4—34no—
253C4⋊C4—34no—
254C4⋊C4—34no—
255C4⋊C4—34no—
256C4⋊C4—34no—
257C4⋊C4—34no—
258C4⋊C4—34no—
259C4⋊C4—34no—
260C4⋊C4—34no—
261C4⋊C4C22×D314yes—
262C4⋊C4C22×D314yes—
263C22⋊C4—32no—
264C22⋊C4—32no—
265C22⋊C4—32no—
266C22⋊C4—32no—
267C22⋊C4—32no—
268C22⋊C4—32no—
269C22⋊C4—34no—
270C22⋊C4—34no—
271C22⋊C4—34no—
272C22⋊C4—34no—
273C22⋊C4—34no—
274C22⋊C4—34no—
275C22⋊C4C22×D312yes—
276C22⋊C4C22×D312yes—
277C22⋊C4C22×D314yes—
278C22⋊C4C22×D314yes—
279C42—34no—
280C42—34no—
281C42—34no—
282C42—34no—
283C22×C3—22no—
284C22×C3—22no—
285C22×C3—22no—
286C22×C3—22no—
287C22×C3—24no—
288C22×C3—44no—
289C22×C3—82no—
290C22×C3C2×#4112yes—
291C22×C3C2×#4112yes—
292C22×C3C2×#4112yes—
293C22×C3C2511yescharacteristic
294C2×D3—22no—
295C2×D3—24no—
296C2×D3—82no—
297C3×C4—22no—
298C3×C4—22no—
299C3×C4—22no—
300C3×C4—24no—
301Dic3—22no—
302Dic3—24no—
303Dic3—24no—
304Dic3—24no—
305C23—22no—
306C23—32no—
307C23—42no—
308C23—42no—
309C23—62no—
310C23—122no—
311C23#23714yes—
312C23C23×D312yes—
313C23C23×D311yescharacteristic
314C23C23×D311yescharacteristic
315C23C23×D311yescharacteristic
316Q8—124no—
317D4—24no—
318D4—24no—
319D4—64no—
320D4—64no—
321D4—124no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—34no—
328C2×C4—34no—
329C2×C4—32no—
330C2×C4—34no—
331C2×C4—42no—
332C2×C4—62no—
333C2×C4—124no—
334C2×C4—122no—
335C2×C4—124no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4C23×D312yes—
339C2×C4C23×D312yes—
340C2×C4C23×D314yes—
341C2×C4C23×D312yes—
342C2×C3—22no—
343C2×C3—22no—
344C2×C3—22no—
345C2×C3—42no—
346C2×C3C22×#4111yescharacteristic
347C2×C3#58512yes—
348D3—42no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—24no—
354C22—44no—
355C22—62no—
356C22—64no—
357C22—82no—
358C22—242no—
359C22C2×#23712yes—
360C22C2×#23712yes—
361C22C2×#23712yes—
362C22C24×D311yescharacteristic
363C4—22no—
364C4—22no—
365C4—22no—
366C4—24no—
367C4—62no—
368C4—64no—
369C4—64no—
370C4—64no—
371C3#349111yescharacteristic, complemented, Sylow
372C2—22no—
373C2—22no—
374C2—22no—
375C2—42no—
376C2—122no—
377C2C22×#23711yescharacteristic
378C2#596412yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC2×C3×#123C2
prime radical#5804C2
hypercenterC2×#123D3
hypocenterC3#3491
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×C3×#123C2semidirect
3. #5798C2semidirect
4. C2×#929C2semidirect
5. C2×#926C2semidirect
6. #5665C2semidirect
7. #5658C2semidirect
8. #5650C2semidirect
9. #5649C2semidirect
10. #5513C2semidirect
11. #5495C2semidirect
12. #5486C2semidirect
13. #5483C2semidirect
19. C3×#123C22semidirect
20. C3×#123C22semidirect
21. C2×C3×C4⋊C4C22semidirect
23. C2×C3×C22⋊C4C22semidirect
28. C2×#217C22semidirect
33. D4×Dic3C22semidirect
36. #929C22semidirect
37. #929C22semidirect
39. C2×(C2×D3)⋊C4C22semidirect
40. #926C22semidirect
41. #926C22semidirect
42. #926C22semidirect
43. C2×C4⋊Dic3C22semidirect
44. #924C22semidirect
45. C2×Dic3⋊C4C22semidirect
46. C2×C4×Dic3C22semidirect
47. #898C22semidirect
48. #898C22semidirect
49. #897C22semidirect
50. #892C22semidirect
51. #890C22semidirect
52. #889C22semidirect
53. #889C22semidirect
54. #887C22semidirect
55. #886C22semidirect
56. #885C22semidirect
57. #883C22semidirect
58. #67⋊C4C22semidirect
59. #67⋊C4C22semidirect
60. D3×C22⋊C4C22semidirect
61. #879C22semidirect
62. #879C22semidirect
63. #878C22semidirect
64. #878C22semidirect
65. #878C22semidirect
66. #877C22semidirect
67. #877C22semidirect
77. C2×#123D3semidirect
156. #123C2×D3semidirect
157. #123C2×D3semidirect
180. C2×C4⋊C4C2×D3semidirect
188. C2×C22⋊C4C2×D3semidirect
371. C3#3491semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1232
#58042
#57982
C2×#9292
C2×#9262
#56652
#56582
#56502
#56492
#55132
#54952
#54862
#54832
#34913