Finite Group Databasebeta

← #86079

Subgroups — #86079

Order 384.

1060 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×#124C211yesmaximal, characteristic, complemented
2#5816C212yesmaximal, complemented
3#5804C212yesmaximal, complemented
4#5684C212yesmaximal, complemented
5#5683C211yesmaximal, characteristic, complemented
6#5679C212yesmaximal, complemented
7#5678C214yesmaximal, complemented
8#5672C211yesmaximal, characteristic, complemented
9#5670C212yesmaximal, complemented
10#5669C214yesmaximal, complemented
11#5495C214yesmaximal, complemented
12#5490C214yesmaximal, complemented
13#5479C212yesmaximal, complemented
14#3545—31nomaximal, Sylow
15C22×C3×Q8C2211yescharacteristic, complemented
16C22×C3×D4C2211yescharacteristic
17C22×#67C2212yescomplemented
18C2×#235C2212yes—
19C3×#124C2211yescharacteristic
20C3×#124C2211yescharacteristic
21C3×#124C2214yescomplemented
22C3×#124C2211yescharacteristic, complemented
23C3×#124C2211yescharacteristic, complemented
24C2×C3×C22⋊C4C2214yescomplemented
25C2×C3×C42C2211yescharacteristic, complemented
26#953C2214yes—
27C2×#217C2212yescomplemented
28#947C2212yescomplemented
29#947C2212yescomplemented
30#946C2214yescomplemented
31#946C2214yescomplemented
32#944C2212yescomplemented
33#944C2212yescomplemented
34#939C2214yes—
35#939C2214yescomplemented
36#937C2212yes—
37#937C2212yescomplemented
38#935C2212yes—
39#935C2212yescomplemented
40#930C2212yes—
41#929C2214yescomplemented
42#929C2214yescomplemented
43C4×#67C2214yescomplemented
44#926C2212yescomplemented
45#924C2212yescomplemented
46#886C2214yescomplemented
47#885C2214yescomplemented
48#885C2214yescomplemented
49#884C2214yescomplemented
50#884C2214yescomplemented
51#883C2214yescomplemented
52#883C2214yescomplemented
53#882C2214yescomplemented
54#67⋊C4C2214yescomplemented
55#67⋊C4C2214yescomplemented
56D3×C22⋊C4C2214yescomplemented
57#879C2214yescomplemented
58#879C2214yescomplemented
59#878C2214yescomplemented
60#878C2214yescomplemented
61#877C2214yescomplemented
62#876C2214yescomplemented
63#875C2212yescomplemented
64C4×D12C2212yescomplemented
65#872C2214yescomplemented
66#870C2212yescomplemented
67C4×Dic6C2212yescomplemented
68#563—31no—
69#562—32no—
70H(3,𝔽4)—31no—
71#555—32no—
72#551—32no—
73#551—34no—
74#546—34no—
75#542—34no—
76#538—34no—
77#536—32no—
78#534—32no—
79#529—32no—
80C2×#124D311yescharacteristic, complemented
81C24×C3C2312yes—
82C23×D3C2312yes—
83C2×C3×Q8—42no—
84C2×C3×Q8C2311yescharacteristic
85C2×C3×Q8C2312yes—
86C2×C3×Q8C2311yescharacteristic
87C2×C3×D4—42no—
88C2×C3×D4C2311yescharacteristic
89C2×C3×D4C2312yes—
90C2×C3×D4C2311yescharacteristic
91C22×C3×C4C2312yes—
92C22×C3×C4C2311yescharacteristic
93C22×C3×C4C2312yes—
94C2×#67—44no—
95C2×#67C2314yes—
96C2×#67C2314yes—
97C2×#67C2314yes—
98C2×#67C2314yes—
99C22×Dic3C2314yes—
100C22×Dic3C2312yes—
101#235—44no—
102C2×D12C2312yes—
103C2×D12C2312yes—
104C2×C4×D3C2314yes—
105C2×C4×D3C2314yes—
106C2×Dic6C2312yes—
107C2×Dic6C2312yes—
108C3×C22⋊C4C2314yes—
109C3×C22⋊C4C2314yes—
110C3×C22⋊C4C2314yes—
111C3×C22⋊C4C2314yes—
112C3×C42C2312yes—
113C3×C42C2311yescharacteristic
114C3×C42C2311yescharacteristic
115#217C2314yes—
116#217C2314yes—
117#217C2312yes—
118#217C2312yes—
119#217C2314yes—
120(C2×D3)⋊C4C2314yes—
121(C2×D3)⋊C4C2314yes—
122(C2×D3)⋊C4C2314yes—
123(C2×D3)⋊C4C2314yes—
124(C2×D3)⋊C4C2314yes—
125(C2×D3)⋊C4C2314yes—
126C4⋊Dic3C2312yes—
127C4⋊Dic3C2312yes—
128C4⋊Dic3C2314yes—
129Dic3⋊C4C2314yes—
130Dic3⋊C4C2314yes—
131Dic3⋊C4C2314yes—
132Dic3⋊C4C2314yes—
133Dic3⋊C4C2314yes—
134Dic3⋊C4C2314yes—
135C4×Dic3C2314yes—
136C4×Dic3C2312yes—
137C4×Dic3C2312yes—
138C2×#41—32no—
139C22×Q8C2×D311yescharacteristic, complemented
140C22×D4—32no—
141C22×D4C2×D311yescharacteristic
142C4⋊Q8—32no—
143C4⋊Q8—32no—
144#126—34no—
145#126—34no—
146#126—34no—
147#125—32no—
148#124—32no—
149#124—32no—
150#124—34no—
151#124—34no—
152#124—32no—
153#124C2×D311yescharacteristic
154#124C2×D311yescharacteristic
155#124C2×D314yescomplemented
156#124C2×D311yescharacteristic, complemented
157#124C2×D311yescharacteristic, complemented
158#123—32no—
159#123—32no—
160#123—34no—
161#123—34no—
162#123—34no—
163#123—34no—
164#123—34no—
165C22⋊Q8—34no—
166C22⋊Q8—34no—
167C22⋊Q8—32no—
168C22⋊Q8—34no—
169C22⋊Q8—34no—
170#121—34no—
171#121—34no—
172#121—32no—
173#121—34no—
174#121—34no—
175#120—34no—
176#120—32no—
177#120—32no—
178#120—34no—
179C4×Q8—32no—
180C4×D4—34no—
181C4×D4—34no—
182C4×D4—34no—
183C4×D4—32no—
184#117—32no—
185#117—34no—
186#117—34no—
187C2×C22⋊C4—32no—
188C2×C22⋊C4—34no—
189C2×C22⋊C4C2×D314yescomplemented
190C2×C42C2×D311yescharacteristic, complemented
191C23×C3—24no—
192C23×C3—44no—
193C23×C3C2412yes—
194C23×C3C2412yes—
195C23×C3C2411yescharacteristic
196C22×D3—42no—
197C22×D3—44no—
198C22×D3C2414yes—
199C3×Q8—41no—
200C3×Q8—42no—
201C3×Q8—41no—
202C3×D4—41no—
203C3×D4—42no—
204C3×D4—41no—
205C2×C3×C4—24no—
206C2×C3×C4—22no—
207C2×C3×C4—42no—
208C2×C3×C4C2412yes—
209C2×C3×C4C2412yes—
210C2×C3×C4C2411yescharacteristic
211C2×C3×C4C2411yescharacteristic
212C2×C3×C4C2412yes—
213C2×C3×C4C2412yes—
214#67—28no—
215#67—44no—
216#67—44no—
217#67—44no—
218C2×Dic3—44no—
219C2×Dic3—42no—
220C2×Dic3C2414yes—
221C2×Dic3C2414yes—
222C2×Dic3C2414yes—
223D12—42no—
224D12—42no—
225C4×D3—44no—
226C4×D3—44no—
227Dic6—42no—
228Dic6—42no—
229C24—32no—
230C24C22×D312yes—
231#41—124no—
232C2×Q8—32no—
233C2×Q8—32no—
234C2×Q8—42no—
235C2×Q8C22×D311yescharacteristic
236C2×Q8C22×D312yes—
237C2×Q8C22×D311yescharacteristic
238C2×D4—34no—
239C2×D4—34no—
240C2×D4—34no—
241C2×D4—34no—
242C2×D4—32no—
243C2×D4—32no—
244C2×D4—42no—
245C2×D4—124no—
246C2×D4C22×D311yescharacteristic
247C2×D4C22×D312yes—
248C2×D4C22×D311yescharacteristic
249C22×C4—34no—
250C22×C4—32no—
251C22×C4—34no—
252C22×C4—34no—
253C22×C4C22×D311yescharacteristic
254C22×C4C22×D312yes—
255C22×C4C22×D312yes—
256C4⋊C4—34no—
257C4⋊C4—34no—
258C4⋊C4—34no—
259C4⋊C4—34no—
260C4⋊C4—34no—
261C4⋊C4—34no—
262C4⋊C4—32no—
263C4⋊C4—32no—
264C4⋊C4—34no—
265C22⋊C4—34no—
266C22⋊C4—34no—
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⋊C4—34no—
276C22⋊C4C22×D314yes—
277C22⋊C4C22×D314yes—
278C22⋊C4C22×D314yes—
279C22⋊C4C22×D314yes—
280C42—32no—
281C42—34no—
282C42—32no—
283C42C22×D311yescharacteristic
284C42C22×D312yes—
285C42C22×D311yescharacteristic
286C22×C3—22no—
287C22×C3—22no—
288C22×C3—22no—
289C22×C3—24no—
290C22×C3—24no—
291C22×C3—84no—
292C22×C3H(5,𝔽2)12yes—
293C22×C3C2511yescharacteristic
294C2×D3—24no—
295C2×D3—24no—
296C2×D3—24no—
297C2×D3—84no—
298C3×C4—22no—
299C3×C4—22no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—22no—
303C3×C4—22no—
304Dic3—24no—
305Dic3—24no—
306Dic3—24no—
307C23—24no—
308C23—34no—
309C23—44no—
310C23—122no—
311C23—124no—
312C23C23×D312yes—
313C23C23×D312yes—
314C23C23×D311yescharacteristic
315Q8—41no—
316Q8—42no—
317Q8—41no—
318Q8—122no—
319Q8—122no—
320D4—41no—
321D4—42no—
322D4—41no—
323D4—68no—
324D4—122no—
325D4—122no—
326D4—124no—
327D4—124no—
328D4—124no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—34no—
332C2×C4—34no—
333C2×C4—34no—
334C2×C4—42no—
335C2×C4—124no—
336C2×C4—124no—
337C2×C4—122no—
338C2×C4—124no—
339C2×C4C23×D312yes—
340C2×C4C23×D312yes—
341C2×C4C23×D311yescharacteristic
342C2×C4C23×D311yescharacteristic
343C2×C4C23×D312yes—
344C2×C4C23×D312yes—
345C2×C3—22no—
346C2×C3—42no—
347C2×C3—42no—
348C2×C3C2×H(5,𝔽2)11yescharacteristic
349C2×C3#58512yes—
350D3—44no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—24no—
355C22—24no—
356C22—64no—
357C22—64no—
358C22—64no—
359C22—84no—
360C22—244no—
361C22#100412yes—
362C22C24×D311yescharacteristic
363C4—22no—
364C4—21no—
365C4—21no—
366C4—22no—
367C4—22no—
368C4—22no—
369C4—64no—
370C4—64no—
371C4—64no—
372C3#354511yescharacteristic, complemented, Sylow
373C2—22no—
374C2—42no—
375C2—42no—
376C2—124no—
377C2C2×#100411yescharacteristic
378C2#596412yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC2×C3×#124C2
hypercenterC2×#124D3
hypocenterC3#3545
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×C3×#124C2semidirect
2. #5816C2semidirect
3. #5804C2semidirect
4. #5684C2semidirect
5. #5683C2semidirect
6. #5679C2semidirect
7. #5678C2semidirect
8. #5672C2semidirect
9. #5670C2semidirect
10. #5669C2semidirect
11. #5495C2semidirect
12. #5490C2semidirect
13. #5479C2semidirect
15. C22×C3×Q8C22semidirect
17. C22×#67C22semidirect
21. C3×#124C22semidirect
22. C3×#124C22semidirect
23. C3×#124C22semidirect
24. C2×C3×C22⋊C4C22semidirect
25. C2×C3×C42C22semidirect
27. C2×#217C22semidirect
28. #947C22semidirect
29. #947C22semidirect
30. #946C22semidirect
31. #946C22semidirect
32. #944C22semidirect
33. #944C22semidirect
35. #939C22semidirect
37. #937C22semidirect
39. #935C22semidirect
41. #929C22semidirect
42. #929C22semidirect
43. C4×#67C22semidirect
44. #926C22semidirect
45. #924C22semidirect
46. #886C22semidirect
47. #885C22semidirect
48. #885C22semidirect
49. #884C22semidirect
50. #884C22semidirect
51. #883C22semidirect
52. #883C22semidirect
53. #882C22semidirect
54. #67⋊C4C22semidirect
55. #67⋊C4C22semidirect
56. D3×C22⋊C4C22semidirect
57. #879C22semidirect
58. #879C22semidirect
59. #878C22semidirect
60. #878C22semidirect
61. #877C22semidirect
62. #876C22semidirect
63. #875C22semidirect
64. C4×D12C22semidirect
65. #872C22semidirect
66. #870C22semidirect
67. C4×Dic6C22semidirect
80. C2×#124D3semidirect
139. C22×Q8C2×D3semidirect
155. #124C2×D3semidirect
156. #124C2×D3semidirect
157. #124C2×D3semidirect
189. C2×C22⋊C4C2×D3semidirect
190. C2×C42C2×D3semidirect
372. C3#3545semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1242
#58162
#58042
#56842
#56832
#56792
#56782
#56722
#56702
#56692
#54952
#54902
#54792
#35453