Finite Group Databasebeta

← #85776

Subgroups — #85776

Order 384.

1160 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D4C211yesmaximal, characteristic, complemented
2#5804C212yesmaximal, complemented
3#5791C212yesmaximal, complemented
4C2×C4×#67C212yesmaximal, complemented
5#5564C214yesmaximal, complemented
6#5560C214yesmaximal, complemented
7#5547C214yesmaximal, complemented
8#5538C214yesmaximal, complemented
9#5515C212yesmaximal, complemented
10#5494C212yesmaximal, complemented
11#5493C211yesmaximal, characteristic, complemented
12#5483C211yesmaximal, characteristic, complemented
13#5479C212yesmaximal, complemented
14#3459—31nomaximal, Sylow
15C22×C3×D4C2211yescharacteristic
16C23×C3×C4C2211yescharacteristic
17C23×C3×C4C2211yescharacteristic, complemented
18C22×#67C2212yes—
19C2×#235C2212yescomplemented
20C22×C4×D3C2212yescomplemented
21C3×C4×D4C2214yescomplemented
22C3×C4×D4C2214yescomplemented
23C2×C3×C4⋊C4C2211yescharacteristic, complemented
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic, complemented
26C2×C3×C42C2211yescharacteristic, complemented
27#953C2214yescomplemented
28C2×#217C2212yes—
29#939C2214yes—
30#939C2214yes—
31#937C2214yescomplemented
32#935C2214yescomplemented
33#930C2212yes—
34#930C2212yes—
35#930C2212yescomplemented
36#929C2214yescomplemented
37#929C2214yescomplemented
38C4×#67C2214yescomplemented
39C4×#67C2214yescomplemented
40C4×#67C2214yescomplemented
41C4×#67C2214yescomplemented
42C4×#67C2214yescomplemented
43C2×(C2×D3)⋊C4C2212yescomplemented
44#926C2212yescomplemented
45#924C2212yescomplemented
46#924C2212yescomplemented
47#924C2212yescomplemented
48C2×Dic3⋊C4C2212yescomplemented
49C2×C4×Dic3C2212yescomplemented
50#898C2214yescomplemented
51#896C2214yescomplemented
52#894C2214yescomplemented
53#889C2214yescomplemented
54#886C2214yescomplemented
55#885C2214yescomplemented
56#884C2214yescomplemented
57#883C2214yescomplemented
58#882C2214yescomplemented
59#879C2214yescomplemented
60#878C2214yescomplemented
61#878C2214yescomplemented
62#876C2214yescomplemented
63#875C2212yescomplemented
64C4×D12C2212yescomplemented
65#872C2214yescomplemented
66#870C2212yescomplemented
67C4×Dic6C2212yescomplemented
68#543—31no—
69#539—34no—
70#539—32no—
71#538—34no—
72#538—32no—
73#534—32no—
74#534—31no—
75#529—34no—
76#529—32no—
77#525—32no—
78#525—34no—
79C2×C4×D4—32no—
80C2×C4×D4D311yescharacteristic, complemented
81C24×C3C2311yescharacteristic
82C24×C3C2311yescharacteristic
83C23×D3C2312yes—
84C2×C3×D4—24no—
85C2×C3×D4C2314yes—
86C22×C3×C4—42no—
87C22×C3×C4—42no—
88C22×C3×C4C2312yes—
89C22×C3×C4C2312yes—
90C22×C3×C4C2311yescharacteristic
91C22×C3×C4C2311yescharacteristic
92C22×C3×C4C2312yes—
93C22×C3×C4C2311yescharacteristic
94C22×C3×C4C2311yescharacteristic
95C22×C3×C4C2312yes—
96C22×C3×C4C2311yescharacteristic
97C2×#67—28no—
98C2×#67C2314yes—
99C2×#67C2314yes—
100C2×#67C2314yes—
101C2×#67C2314yes—
102C22×Dic3C2312yes—
103C22×Dic3C2312yes—
104C22×Dic3C2312yes—
105#235—44no—
106C2×D12C2312yes—
107C2×D12C2312yes—
108C2×C4×D3—44no—
109C2×C4×D3C2314yes—
110C2×C4×D3C2314yes—
111C2×Dic6C2312yes—
112C2×Dic6C2312yes—
113C3×C4⋊C4C2314yes—
114C3×C22⋊C4C2314yes—
115C3×C22⋊C4C2314yes—
116C3×C42C2312yes—
117C3×C42C2312yes—
118#217C2314yes—
119#217C2314yes—
120#217C2314yes—
121#217C2314yes—
122(C2×D3)⋊C4C2314yes—
123(C2×D3)⋊C4C2314yes—
124(C2×D3)⋊C4C2314yes—
125(C2×D3)⋊C4C2314yes—
126(C2×D3)⋊C4C2314yes—
127(C2×D3)⋊C4C2314yes—
128C4⋊Dic3C2312yes—
129C4⋊Dic3C2312yes—
130C4⋊Dic3C2312yes—
131C4⋊Dic3C2312yes—
132Dic3⋊C4C2314yes—
133Dic3⋊C4C2314yes—
134Dic3⋊C4C2314yes—
135Dic3⋊C4C2314yes—
136Dic3⋊C4C2314yes—
137Dic3⋊C4C2314yes—
138C4×Dic3C2314yes—
139C4×Dic3C2314yes—
140C2×#41—32no—
141C22×D4—32no—
142C22×D4C2×D311yescharacteristic
143C23×C4—32no—
144C23×C4C2×D311yescharacteristic
145C23×C4C2×D311yescharacteristic, complemented
146#126—34no—
147#126—34no—
148#126—34no—
149#125—34no—
150#125—32no—
151#124—34no—
152#124—32no—
153#123—34no—
154#123—34no—
155#123—34no—
156#123—34no—
157#123—34no—
158#123—34no—
159C22⋊Q8—32no—
160C22⋊Q8—32no—
161C22⋊Q8—32no—
162C22⋊Q8—34no—
163C22⋊Q8—34no—
164C22⋊Q8—34no—
165#121—34no—
166#121—34no—
167#121—32no—
168#121—32no—
169#121—32no—
170#121—34no—
171#120—34no—
172#120—34no—
173#120—34no—
174C4×Q8—32no—
175C4×D4—34no—
176C4×D4—34no—
177C4×D4—34no—
178C4×D4—34no—
179C4×D4—34no—
180C4×D4—32no—
181C4×D4C2×D314yescomplemented
182C4×D4C2×D314yescomplemented
183#117—32no—
184#117—34no—
185C2×C4⋊C4—32no—
186C2×C4⋊C4C2×D311yescharacteristic, complemented
187C2×C22⋊C4—32no—
188C2×C22⋊C4—32no—
189C2×C22⋊C4C2×D311yescharacteristic
190C2×C22⋊C4C2×D311yescharacteristic, complemented
191C2×C42—32no—
192C2×C42C2×D311yescharacteristic, complemented
193C23×C3—22no—
194C23×C3—41no—
195C23×C3—41no—
196C23×C3—41no—
197C23×C3—41no—
198C23×C3#4114yescomplemented
199C23×C3C2412yes—
200C23×C3C2411yescharacteristic
201C23×C3C2412yes—
202C22×D3—24no—
203C22×D3—42no—
204C22×D3—42no—
205C22×D3C2414yes—
206C3×D4—28no—
207C2×C3×C4—22no—
208C2×C3×C4—24no—
209C2×C3×C4—24no—
210C2×C3×C4—22no—
211C2×C3×C4—41no—
212C2×C3×C4—42no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4—41no—
216C2×C3×C4C2412yes—
217C2×C3×C4C2412yes—
218C2×C3×C4C2411yescharacteristic
219C2×C3×C4C2412yes—
220C2×C3×C4C2412yes—
221C2×C3×C4C2411yescharacteristic
222#67—28no—
223#67—28no—
224#67—44no—
225#67—44no—
226C2×Dic3—24no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3C2414yes—
230C2×Dic3C2414yes—
231C2×Dic3C2414yes—
232D12—42no—
233D12—42no—
234C4×D3—28no—
235C4×D3—44no—
236Dic6—42no—
237Dic6—42no—
238C24—32no—
239C24C22×D311yescharacteristic
240C24C22×D311yescharacteristic
241#41—124no—
242C2×Q8—32no—
243C2×Q8—32no—
244C2×D4—24no—
245C2×D4—34no—
246C2×D4—34no—
247C2×D4—34no—
248C2×D4—34no—
249C2×D4—32no—
250C2×D4—32no—
251C2×D4—68no—
252C2×D4C22×D314yes—
253C22×C4—32no—
254C22×C4—32no—
255C22×C4—32no—
256C22×C4—34no—
257C22×C4—34no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—124no—
261C22×C4C22×D311yescharacteristic
262C22×C4C22×D312yes—
263C22×C4C22×D311yescharacteristic
264C22×C4C22×D312yes—
265C22×C4C22×D311yescharacteristic
266C22×C4C22×D312yes—
267C22×C4C22×D311yescharacteristic
268C22×C4C22×D311yescharacteristic
269C22×C4C22×D312yes—
270C4⋊C4—34no—
271C4⋊C4—34no—
272C4⋊C4—34no—
273C4⋊C4—32no—
274C4⋊C4—34no—
275C4⋊C4—32no—
276C4⋊C4—34no—
277C4⋊C4—32no—
278C4⋊C4—34no—
279C4⋊C4—32no—
280C4⋊C4C22×D314yes—
281C22⋊C4—34no—
282C22⋊C4—34no—
283C22⋊C4—34no—
284C22⋊C4—34no—
285C22⋊C4—34no—
286C22⋊C4—34no—
287C22⋊C4—34no—
288C22⋊C4—34no—
289C22⋊C4—34no—
290C22⋊C4—34no—
291C22⋊C4C22×D314yes—
292C22⋊C4C22×D314yes—
293C42—34no—
294C42—34no—
295C42C22×D312yes—
296C42C22×D312yes—
297C22×C3—22no—
298C22×C3—21no—
299C22×C3—22no—
300C22×C3—21no—
301C22×C3—22no—
302C22×C3—22no—
303C22×C3—22no—
304C22×C3—44no—
305C22×C3—44no—
306C22×C3C2×#4114yes—
307C22×C3C2×#4112yes—
308C22×C3C2511yescharacteristic
309C2×D3—24no—
310C2×D3—24no—
311C2×D3—24no—
312C2×D3—48no—
313C3×C4—22no—
314C3×C4—22no—
315C3×C4—22no—
316C3×C4—21no—
317C3×C4—22no—
318C3×C4H(5,𝔽2)12yes—
319Dic3—24no—
320Dic3—24no—
321Dic3—24no—
322C23—22no—
323C23—34no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—64no—
329C23—122no—
330C23—122no—
331C23#23514yescomplemented
332C23C23×D312yes—
333C23C23×D311yescharacteristic
334C23C23×D312yes—
335Q8—122no—
336Q8—122no—
337D4—28no—
338D4—68no—
339D4—68no—
340D4—122no—
341D4—122no—
342D4—124no—
343D4—124no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—34no—
349C2×C4—34no—
350C2×C4—34no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—64no—
357C2×C4—68no—
358C2×C4—124no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4C23×D312yes—
362C2×C4C23×D312yes—
363C2×C4C23×D311yescharacteristic
364C2×C4C23×D312yes—
365C2×C4C23×D312yes—
366C2×C4C23×D311yescharacteristic
367C2×C3—24no—
368C2×C3—22no—
369C2×C3—42no—
370C2×C3C22×#4111yescharacteristic
371C2×C3C2×H(5,𝔽2)11yescharacteristic
372C2×C3#58511yescharacteristic
373D3—44no—
374C22—21no—
375C22—22no—
376C22—21no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—44no—
382C22—44no—
383C22—64no—
384C22—64no—
385C22—64no—
386C22—128no—
387C22C2×#23512yes—
388C22C2×#23514yes—
389C22C24×D311yescharacteristic
390C4—22no—
391C4—21no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—64no—
396C4—64no—
397C4—64no—
398C4#100412yes—
399C3#345911yescharacteristic, complemented, Sylow
400C2—22no—
401C2—24no—
402C2—42no—
403C2—124no—
404C2C22×#23511yescharacteristic
405C2C2×#100411yescharacteristic
406C2#596411yescharacteristic
subgroupG/H
centerC2×C4C23×D3
commutator, socle, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×C4×D4C2
hypercenterC2×C4×D4D3
hypocenterC3#3459
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×D4C2semidirect
2. #5804C2semidirect
3. #5791C2semidirect
4. C2×C4×#67C2semidirect
5. #5564C2semidirect
6. #5560C2semidirect
7. #5547C2semidirect
8. #5538C2semidirect
9. #5515C2semidirect
10. #5494C2semidirect
11. #5493C2semidirect
12. #5483C2semidirect
13. #5479C2semidirect
17. C23×C3×C4C22semidirect
19. C2×#235C22semidirect
20. C22×C4×D3C22semidirect
21. C3×C4×D4C22semidirect
22. C3×C4×D4C22semidirect
23. C2×C3×C4⋊C4C22semidirect
25. C2×C3×C22⋊C4C22semidirect
26. C2×C3×C42C22semidirect
27. #953C22semidirect
31. #937C22semidirect
32. #935C22semidirect
35. #930C22semidirect
36. #929C22semidirect
37. #929C22semidirect
38. C4×#67C22semidirect
39. C4×#67C22semidirect
40. C4×#67C22semidirect
41. C4×#67C22semidirect
42. C4×#67C22semidirect
43. C2×(C2×D3)⋊C4C22semidirect
44. #926C22semidirect
45. #924C22semidirect
46. #924C22semidirect
47. #924C22semidirect
48. C2×Dic3⋊C4C22semidirect
49. C2×C4×Dic3C22semidirect
50. #898C22semidirect
51. #896C22semidirect
52. #894C22semidirect
53. #889C22semidirect
54. #886C22semidirect
55. #885C22semidirect
56. #884C22semidirect
57. #883C22semidirect
58. #882C22semidirect
59. #879C22semidirect
60. #878C22semidirect
61. #878C22semidirect
62. #876C22semidirect
63. #875C22semidirect
64. C4×D12C22semidirect
65. #872C22semidirect
66. #870C22semidirect
67. C4×Dic6C22semidirect
80. C2×C4×D4D3semidirect
145. C23×C4C2×D3semidirect
181. C4×D4C2×D3semidirect
182. C4×D4C2×D3semidirect
186. C2×C4⋊C4C2×D3semidirect
190. C2×C22⋊C4C2×D3semidirect
192. C2×C42C2×D3semidirect
198. C23×C3#41semidirect
331. C23#235semidirect
399. C3#3459semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
#58042
#57912
C2×C4×#672
#55642
#55602
#55472
#55382
#55152
#54942
#54932
#54832
#54792
#34593