Finite Group Databasebeta

← #60146

Subgroups — #60146

Order 256.

1378 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#568C211yesmaximal, characteristic, complemented
2C22×#131C211yesmaximal, characteristic, complemented
3D4×#41C211yesmaximal, characteristic
4#2935C213yesmaximal, complemented
5D4×M16C213yesmaximal, complemented
6#2933C213yesmaximal, complemented
7#2933C213yesmaximal, complemented
8#2932C213yesmaximal, complemented
9#2930C213yesmaximal, complemented
10#2929C213yesmaximal, complemented
11C2×#435C213yesmaximal, complemented
12#2906C211yesmaximal, characteristic, complemented
13#2880C211yesmaximal, characteristic, complemented
14#2880C211yesmaximal, characteristic, complemented
15#2875C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×#41C2211yescharacteristic
18#568—28no—
19C2×#131—28no—
20C2×#131C2211yescharacteristic
21C2×#131C2211yescharacteristic
22C2×#131C2211yescharacteristic
23C2×#131C2211yescharacteristic
24C2×#131C2213yescomplemented
25C2×#131C2213yescomplemented
26C2×#131C2213yescomplemented
27C2×#131C2213yescomplemented
28C22×M16C2213yescomplemented
29C22×M16C2213yescomplemented
30C22×M16C2213yescomplemented
31C23×C8C2213yescomplemented
32#550C411yescharacteristic
33D4×Q8C411yescharacteristic
34#548C411yescharacteristic
35#548C411yescharacteristic
36#547C413yes—
37#546C413yes—
38#546C413yes—
39D42C413yes—
40#532C2213yes—
41#525C2213yes—
42#525C2213yes—
43C4×#41C2211yescharacteristic
44C2×C4×D4C2213yes—
45#436C2216yescomplemented
46#436C2216yescomplemented
47#435C2211yescharacteristic
48#435C2213yes—
49#435C2213yescomplemented
50#435C2213yescomplemented
51#435C2213yescomplemented
52#435C2213yescomplemented
53#435C2213yescomplemented
54#435C2213yescomplemented
55#435C2216yescomplemented
56#435C2216yescomplemented
57#435C2216yescomplemented
58C8×D4C2216yescomplemented
59C8×D4C2216yescomplemented
60#424C2213yescomplemented
61#424C2213yescomplemented
62#423C2213yescomplemented
63C2×C4⋊C8C2213yescomplemented
64#408C2213yescomplemented
65#408C2213yescomplemented
66#408C2213yescomplemented
67#408C2213yescomplemented
68#407C2213yescomplemented
69#407C2213yescomplemented
70C2×C22⋊C8C2213yescomplemented
71C2×C22⋊C8C2213yescomplemented
72#405C2213yescomplemented
73#405C2213yescomplemented
74C4×M16C2213yescomplemented
75C2×#97C2213yescomplemented
76C2×#41—24no—
77C2×#41—24no—
78C2×#41—24no—
79C2×#41—24no—
80C2×#41C2×C413yes—
81C2×#41C2×C413yes—
82C2×#41C2×C411yescharacteristic
83C2×#41C2×C411yescharacteristic
84C2×#41C2313yes—
85C2×#41C2313yes—
86C2×#41C2311yescharacteristic
87C22×Q8C2×C411yescharacteristic
88C22×Q8C2×C411yescharacteristic
89C22×D4C2×C413yes—
90C22×D4C2×C413yes—
91C22×D4C2×C413yes—
92C23×C4C2313yes—
93C23×C4C2313yes—
94#131—24no—
95#131—24no—
96#131—24no—
97#131—212no—
98#131—212no—
99#131—212no—
100#131—212no—
101#131D418yescomplemented
102C2×M16—212no—
103C2×M16—212no—
104C2×M16—212no—
105C2×M16C2311yescharacteristic
106C2×M16C2313yes—
107C2×M16C2313yes—
108C2×M16C2313yes—
109C2×M16C2313yes—
110C2×M16C2313yes—
111C2×M16C2316yes—
112C2×M16C2313yes—
113C2×M16C2313yes—
114C22×C8—212no—
115C22×C8C2311yescharacteristic
116C22×C8C2313yes—
117C22×C8C2313yes—
118C22×C8C2313yes—
119C22×C8C2313yes—
120C22×C8C2313yes—
121C22×C8C2316yes—
122C22×C8C2313yes—
123C22×C8C2313yes—
124C4⋊Q8C2×C413yes—
125#127C2×C413yes—
126#124C2×C413yes—
127#124C2×C413yes—
128#123C2×C416yes—
129#123C2×C416yes—
130C22⋊Q8C2×C413yes—
131C22⋊Q8C2×C413yes—
132C22⋊Q8C2×C413yes—
133C22⋊Q8C2×C413yes—
134#121C2×C413yes—
135#121C2×C413yes—
136#121C2×C413yes—
137#121C2×C413yes—
138#121C2×C416yes—
139#121C2×C416yes—
140#120C2×C416yes—
141#120C2×C416yes—
142C4×Q8C2×C411yescharacteristic
143C4×Q8C2×C411yescharacteristic
144C4×D4C2×C413yes—
145C4×D4C2×C413yes—
146C4×D4C2×C413yes—
147C4×D4C2×C413yes—
148C4×D4C2×C413yes—
149C4×D4C2×C413yes—
150C4×D4C2311yescharacteristic
151C4×D4C2313yes—
152C4×D4C2313yes—
153C4×D4C2313yes—
154C4×D4C2316yes—
155#117C2313yes—
156C2×C4⋊C4C2×C413yes—
157C2×C22⋊C4C2×C413yes—
158C2×C22⋊C4C2×C413yes—
159C2×C42C2313yes—
160C4⋊C8C2311yescharacteristic
161C4⋊C8C2313yes—
162C4⋊C8C2313yes—
163C4⋊C8C2313yes—
164C4⋊C8C2316yes—
165C22⋊C8C2311yescharacteristic
166C22⋊C8C2311yescharacteristic
167C22⋊C8C2313yes—
168C22⋊C8C2313yes—
169C22⋊C8C2313yes—
170C22⋊C8C2313yes—
171C22⋊C8C2313yes—
172C22⋊C8C2313yes—
173C22⋊C8C2316yes—
174C22⋊C8C2316yes—
175#97C2311yescharacteristic
176#97C2313yes—
177#97C2313yes—
178#97C2313yes—
179C4×C8C2316yes—
180C24C22×C413yes—
181C24C22×C413yes—
182#41—24no—
183#41—24no—
184#41—212no—
185#41—212no—
186#41—26no—
187#41—26no—
188#41—26no—
189#41—26no—
190#41C2×D414yes—
191#41#4114yes—
192C2×Q8—24no—
193C2×Q8—24no—
194C2×Q8C22×C413yes—
195C2×Q8C22×C413yes—
196C2×Q8C22×C411yescharacteristic
197C2×D4—212no—
198C2×D4—212no—
199C2×D4—46no—
200C2×D4C22×C411yescharacteristic
201C2×D4C22×C413yes—
202C2×D4C22×C416yes—
203C2×D4C22×C416yes—
204C2×D4C22×C413yes—
205C2×D4C22×C413yes—
206C2×D4C22×C413yes—
207C2×D4C22×C413yes—
208C2×D4C22×C416yes—
209C2×D4C22×C413yes—
210C22×C4—26no—
211C22×C4—26no—
212C22×C4—26no—
213C22×C4—26no—
214C22×C4C22×C413yes—
215C22×C4C22×C413yes—
216C22×C4C22×C413yes—
217C22×C4C22×C413yes—
218C22×C4C22×C413yes—
219C22×C4C22×C413yes—
220C22×C4C22×C413yes—
221C22×C4C22×C413yes—
222C22×C4C2411yescharacteristic
223C22×C4C2413yes—
224C22×C4C2413yes—
225C22×C4C2411yescharacteristic
226C22×C4C2413yes—
227M16—26no—
228M16—26no—
229M16—26no—
230M16—22no—
231M16—26no—
232M16—26no—
233M16—212no—
234M16—26no—
235M16C2×D4112yescomplemented
236C2×C8—26no—
237C2×C8—26no—
238C2×C8—26no—
239C2×C8—22no—
240C2×C8—26no—
241C2×C8—26no—
242C2×C8—26no—
243C2×C8—212no—
244C2×C8C2×D4112yescomplemented
245C2×C8C2413yes—
246C2×C8C2411yescharacteristic
247C2×C8C2411yescharacteristic
248C2×C8C2411yescharacteristic
249C2×C8C2413yes—
250C2×C8C2413yes—
251C2×C8C2413yes—
252C2×C8C2411yescharacteristic
253C4⋊C4C22×C411yescharacteristic
254C4⋊C4C22×C413yes—
255C4⋊C4C22×C413yes—
256C4⋊C4C22×C413yes—
257C4⋊C4C22×C413yes—
258C4⋊C4C22×C413yes—
259C4⋊C4C22×C416yes—
260C22⋊C4C22×C411yescharacteristic
261C22⋊C4C22×C411yescharacteristic
262C22⋊C4C22×C413yes—
263C22⋊C4C22×C413yes—
264C22⋊C4C22×C413yes—
265C22⋊C4C22×C413yes—
266C22⋊C4C22×C413yes—
267C22⋊C4C22×C413yes—
268C22⋊C4C22×C416yes—
269C22⋊C4C22×C416yes—
270C22⋊C4C22×C416yes—
271C42C22×C413yes—
272C42C22×C413yes—
273C42C2413yes—
274C42C2411yescharacteristic
275C23—26no—
276C23—26no—
277C23—26no—
278C23—43no—
279C23—43no—
280C23—43no—
281C23C23×C411yescharacteristic
282C23C23×C413yes—
283C23C23×C413yes—
284C23C23×C413yes—
285C23C23×C411yescharacteristic
286Q8—26no—
287Q8—26no—
288Q8C4×D414yes—
289D4—212no—
290D4—212no—
291D4—22no—
292D4—26no—
293D4—26no—
294D4—26no—
295D4—43no—
296D4—43no—
297D4—46no—
298D4C4×D4112yes—
299C2×C4—26no—
300C2×C4—26no—
301C2×C4—26no—
302C2×C4—26no—
303C2×C4—26no—
304C2×C4—26no—
305C2×C4—21no—
306C2×C4—23no—
307C2×C4—23no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—23no—
312C2×C4—23no—
313C2×C4—43no—
314C2×C4—43no—
315C2×C4C23×C411yescharacteristic
316C2×C4C23×C411yescharacteristic
317C2×C4C23×C413yes—
318C2×C4C23×C413yes—
319C2×C4C23×C411yescharacteristic
320C2×C4C23×C411yescharacteristic
321C2×C4C23×C413yes—
322C2×C4C23×C413yes—
323C2×C4C23×C413yes—
324C2×C4C22×D416yes—
325C2×C4C2×#4116yes—
326C2×C4C2511yescharacteristic
327C8—21no—
328C8—23no—
329C8—21no—
330C8—23no—
331C8—21no—
332C8—23no—
333C8C22×D412yes—
334C8C22×D416yes—
335C22—21no—
336C22—23no—
337C22—23no—
338C22—21no—
339C22—21no—
340C22—23no—
341C22—23no—
342C22—21no—
343C22—23no—
344C22—23no—
345C22—23no—
346C22—23no—
347C22—21no—
348C22—412no—
349C22—86no—
350C22C2×C4×D416yes—
351C22#56812yescomplemented
352C22C24×C411yescharacteristic
353C4—21no—
354C4—21no—
355C4—23no—
356C4—23no—
357C4—21no—
358C4—23no—
359C4—21no—
360C4—23no—
361C4C2×C4×D416yes—
362C4C23×D411yescharacteristic
363C4C22×#4111yescharacteristic
364C2—22no—
365C2—26no—
366C2—43no—
367C2—43no—
368C2—41no—
369C2C22×C4×D411yescharacteristic
370C2C2×#56811yescharacteristic
371C2#357311yescharacteristic
subgroupG/H
center, Frattini, cosocleC2×C4C25
commutator, socleC22C24×C4
prime radicalD4×#41C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#568C2semidirect
2. C22×#131C2semidirect
4. #2935C2semidirect
5. D4×M16C2semidirect
6. #2933C2semidirect
7. #2933C2semidirect
8. #2932C2semidirect
9. #2930C2semidirect
10. #2929C2semidirect
11. C2×#435C2semidirect
12. #2906C2semidirect
13. #2880C2semidirect
14. #2880C2semidirect
15. #2875C2semidirect
24. C2×#131C22semidirect
25. C2×#131C22semidirect
26. C2×#131C22semidirect
27. C2×#131C22semidirect
28. C22×M16C22semidirect
29. C22×M16C22semidirect
30. C22×M16C22semidirect
31. C23×C8C22semidirect
45. #436C22semidirect
46. #436C22semidirect
49. #435C22semidirect
50. #435C22semidirect
51. #435C22semidirect
52. #435C22semidirect
53. #435C22semidirect
54. #435C22semidirect
55. #435C22semidirect
56. #435C22semidirect
57. #435C22semidirect
58. C8×D4C22semidirect
59. C8×D4C22semidirect
60. #424C22semidirect
61. #424C22semidirect
62. #423C22semidirect
63. C2×C4⋊C8C22semidirect
64. #408C22semidirect
65. #408C22semidirect
66. #408C22semidirect
67. #408C22semidirect
68. #407C22semidirect
69. #407C22semidirect
70. C2×C22⋊C8C22semidirect
71. C2×C22⋊C8C22semidirect
72. #405C22semidirect
73. #405C22semidirect
74. C4×M16C22semidirect
75. C2×#97C22semidirect
101. #131D4semidirect
235. M16C2×D4semidirect
244. C2×C8C2×D4semidirect
351. C22#568semidirect

Maximal subgroups

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

subgroupindex
C2×#5682
C22×#1312
D4×#412
#29352
D4×M162
#29332
#29322
#29302
#29292
C2×#4352
#29062
#28802
#28752