Finite Group Databasebeta

← #35359

Subgroups — #35359

Order 256.

1385 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#543C211yesmaximal, characteristic, complemented
2C2×#518C211yesmaximal, characteristic
3#2564C214yesmaximal, complemented
4#2450C214yesmaximal, complemented
5#2449C214yesmaximal, complemented
6#2438C214yesmaximal, complemented
7#2437C214yesmaximal, complemented
8#2417C214yesmaximal, complemented
9#2315C211yesmaximal, characteristic, complemented
10#2308C211yesmaximal, characteristic
11C2×#386C211yesmaximal, characteristic
12C2×#385C211yesmaximal, characteristic, complemented
13C2×#380C211yesmaximal, characteristic, complemented
14C23×D4C2211yescharacteristic
15C24×C4C2211yescharacteristic
16#543—28no—
17C2×C22⋊Q8C2214yes—
18C2×C22⋊Q8C2214yescomplemented
19C2×C22⋊Q8C2214yescomplemented
20C2×C22⋊Q8C2214yescomplemented
21#518—28no—
22C2×C4×D4C2214yes—
23C2×C4×D4C2214yes—
24C2×#117C2212yes—
25C22×C4⋊C4C2212yescomplemented
26C22×C4⋊C4C2212yescomplemented
27C22×C22⋊C4C2211yescharacteristic
28C22×C22⋊C4C2211yescharacteristic
29C22×C22⋊C4C2211yescharacteristic
30C22×C22⋊C4C2211yescharacteristic
31C22×C22⋊C4C2212yes—
32C22×C22⋊C4C2212yes—
33C22×C22⋊C4C2211yescharacteristic
34#400C2214yescomplemented
35#399C2214yescomplemented
36#398C2214yescomplemented
37#398C2214yescomplemented
38#396C2214yescomplemented
39#396C2214yescomplemented
40#395C2214yescomplemented
41#393C2214yescomplemented
42#391C2214yescomplemented
43#389C2214yescomplemented
44#387C2214yescomplemented
45#387C2214yescomplemented
46#386C2212yes—
47#386C2212yes—
48#386C2214yes—
49#386C2214yes—
50#386C2212yes—
51#386C2212yes—
52#385C2214yes—
53#385C2214yes—
54#385C2214yescomplemented
55#385C2214yescomplemented
56#385C2214yescomplemented
57#385C2212yescomplemented
58#385C2212yescomplemented
59#381C2212yes—
60#380C2214yes—
61#380C2212yescomplemented
62#380C2214yescomplemented
63#380C2212yescomplemented
64#380C2214yescomplemented
65#379C2212yes—
66C2×#95C2212yes—
67C2×#95C2212yescomplemented
68C2×#95C2212yescomplemented
69C25C2311yescharacteristic
70C25C2311yescharacteristic
71C22×Q8C2314yes—
72C22×D4—24no—
73C22×D4—24no—
74C22×D4Q818yescomplemented
75C22×D4C2312yes—
76C22×D4C2312yes—
77C23×C4—24no—
78C23×C4—44no—
79C23×C4C2311yescharacteristic
80C23×C4C2312yes—
81C23×C4C2312yes—
82C23×C4C2311yescharacteristic
83C23×C4C2312yes—
84C23×C4C2311yescharacteristic
85C23×C4C2312yes—
86C23×C4C2312yes—
87C23×C4C2314yes—
88C23×C4C2311yescharacteristic
89C23×C4C2311yescharacteristic
90C23×C4C2312yes—
91C22⋊Q8—216no—
92C22⋊Q8—216no—
93C22⋊Q8—216no—
94C22⋊Q8—48no—
95C4×D4—216no—
96C4×D4—216no—
97#117—28no—
98C2×C4⋊C4—44no—
99C2×C4⋊C4—44no—
100C2×C4⋊C4C2314yes—
101C2×C4⋊C4C2314yes—
102C2×C4⋊C4C2314yes—
103C2×C4⋊C4C2314yes—
104C2×C4⋊C4C2314yes—
105C2×C4⋊C4C2314yes—
106C2×C4⋊C4C2314yes—
107C2×C4⋊C4C2314yes—
108C2×C4⋊C4C2314yes—
109C2×C4⋊C4C2314yes—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—28no—
115C2×C22⋊C4—44no—
116C2×C22⋊C4D418yes—
117C2×C22⋊C4C2312yes—
118C2×C22⋊C4C2312yes—
119C2×C22⋊C4C2314yes—
120C2×C22⋊C4C2312yes—
121C2×C22⋊C4C2312yes—
122C2×C22⋊C4C2312yes—
123C2×C22⋊C4C2312yes—
124C2×C22⋊C4C2314yes—
125C2×C22⋊C4C2312yes—
126C2×C22⋊C4C2312yes—
127C2×C22⋊C4C2314yes—
128C2×C22⋊C4C2314yes—
129C2×C22⋊C4C2314yes—
130C2×C22⋊C4C2312yes—
131C2×C22⋊C4C2312yes—
132C2×C42C2314yes—
133#95C2314yes—
134#95C2314yes—
135#95C2314yes—
136#95C2314yes—
137#95C2314yes—
138#95C2314yes—
139#95C2314yes—
140#95C2314yes—
141#95C2314yes—
142#95C2314yes—
143C24—22no—
144C24—22no—
145C24—42no—
146C24—42no—
147C24—42no—
148C24—42no—
149C24C2×D414yes—
150C24C2×Q814yes—
151C24C2×Q814yes—
152C24#4114yes—
153C24C2412yes—
154C24C2411yescharacteristic
155C24C2411yescharacteristic
156C24C2411yescharacteristic
157C2×Q8—28no—
158C2×Q8—44no—
159C2×Q8—44no—
160C2×D4—232no—
161C2×D4—24no—
162C2×D4—24no—
163C2×D4—24no—
164C2×D4—24no—
165C2×D4C2×Q818yes—
166C2×D4C2×Q818yes—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—22no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—28no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—416no—
198C22×C4C2×D418yes—
199C22×C4C2×Q814yes—
200C22×C4C2412yes—
201C22×C4C2412yes—
202C22×C4C2414yes—
203C22×C4C2412yes—
204C22×C4C2414yes—
205C22×C4C2414yes—
206C22×C4C2414yes—
207C22×C4C2412yes—
208C22×C4C2412yes—
209C4⋊C4—28no—
210C4⋊C4—28no—
211C4⋊C4—28no—
212C4⋊C4—28no—
213C4⋊C4—28no—
214C4⋊C4—28no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C22⋊C4—24no—
220C22⋊C4—24no—
221C22⋊C4—24no—
222C22⋊C4—24no—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4—24no—
226C22⋊C4—24no—
227C22⋊C4—28no—
228C22⋊C4—28no—
229C22⋊C4—44no—
230C22⋊C4—44no—
231C22⋊C4—44no—
232C22⋊C4C2×D418yes—
233C22⋊C4C2×D418yes—
234C42—28no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—24no—
244C23—24no—
245C23—28no—
246C23—44no—
247C23—48no—
248C23—48no—
249C23—44no—
250C23—48no—
251C23—82no—
252C23—82no—
253C23—84no—
254C23C22⋊Q8116yescomplemented
255C23C22×D412yes—
256C23C22×D412yes—
257C23C22×D412yes—
258C23C22×Q814yes—
259C23C22×Q812yes—
260C23C22×Q812yes—
261C23C22×Q812yes—
262C23C2×#4112yes—
263C23C2×#4112yes—
264C23C2×#4112yes—
265C23C2511yescharacteristic
266Q8—88no—
267D4—216no—
268D4—216no—
269C2×C4—28no—
270C2×C4—28no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—48no—
294C2×C4—48no—
295C2×C4—48no—
296C2×C4—44no—
297C2×C4—88no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—84no—
301C2×C4—84no—
302C2×C4—84no—
303C2×C4C22×D418yes—
304C2×C4C22×Q814yes—
305C2×C4H(5,𝔽2)14yes—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—24no—
313C22—24no—
314C22—24no—
315C22—24no—
316C22—24no—
317C22—24no—
318C22—24no—
319C22—28no—
320C22—416no—
321C22—42no—
322C22—42no—
323C22—44no—
324C22—88no—
325C22C2×C22⋊Q814yes—
326C22C2×C22⋊Q814yes—
327C22C2×C22⋊Q814yes—
328C22C23×D411yescharacteristic
329C22C23×Q811yescharacteristic
330C22C22×#4111yescharacteristic
331C22C2×H(5,𝔽2)12yes—
332C22#58512yes—
333C4—24no—
334C4—24no—
335C4—28no—
336C4—42no—
337C4—42no—
338C4—42no—
339C4—44no—
340C4—44no—
341C4—44no—
342C2—24no—
343C2—24no—
344C2—24no—
345C2—44no—
346C2C22×C22⋊Q811yescharacteristic
347C2#349011yescharacteristic
348C2#349111yescharacteristic
349C2#349612yes—
350C2#350112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radicalC23×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#543C2semidirect
3. #2564C2semidirect
4. #2450C2semidirect
5. #2449C2semidirect
6. #2438C2semidirect
7. #2437C2semidirect
8. #2417C2semidirect
9. #2315C2semidirect
12. C2×#385C2semidirect
13. C2×#380C2semidirect
18. C2×C22⋊Q8C22semidirect
19. C2×C22⋊Q8C22semidirect
20. C2×C22⋊Q8C22semidirect
25. C22×C4⋊C4C22semidirect
26. C22×C4⋊C4C22semidirect
34. #400C22semidirect
35. #399C22semidirect
36. #398C22semidirect
37. #398C22semidirect
38. #396C22semidirect
39. #396C22semidirect
40. #395C22semidirect
41. #393C22semidirect
42. #391C22semidirect
43. #389C22semidirect
44. #387C22semidirect
45. #387C22semidirect
54. #385C22semidirect
55. #385C22semidirect
56. #385C22semidirect
57. #385C22semidirect
58. #385C22semidirect
61. #380C22semidirect
62. #380C22semidirect
63. #380C22semidirect
64. #380C22semidirect
67. C2×#95C22semidirect
68. C2×#95C22semidirect
74. C22×D4Q8semidirect
254. C23C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5432
C2×#5182
#25642
#24502
#24492
#24382
#24372
#24172
#23152
#23082
C2×#3862
C2×#3852
C2×#3802