Finite Group Databasebeta

← #36706

Subgroups — #36706

Order 256.

1519 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#538C211yesmaximal, characteristic, complemented
2C22×#121C211yesmaximal, characteristic, complemented
3#2848C211yesmaximal, characteristic, complemented
4#2847C211yesmaximal, characteristic, complemented
5#2691C214yesmaximal, complemented
6#2676C214yesmaximal, complemented
7#2675C214yesmaximal, complemented
8#2671C214yesmaximal, complemented
9#2626C211yesmaximal, characteristic, complemented
10#2617C211yesmaximal, characteristic, complemented
11#2579C212yesmaximal, complemented
12#2410C212yesmaximal, complemented
13#2410C212yesmaximal, complemented
14#2408C212yesmaximal, complemented
15#2315C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C23×D4C2211yescharacteristic
19C24×C4C2211yescharacteristic
20#538—28no—
21C2×#126C2212yescomplemented
22C2×#124C2212yescomplemented
23C2×#123C2212yescomplemented
24C2×C22⋊Q8C2211yescharacteristic
25C2×C22⋊Q8C2212yescomplemented
26C2×#121—28no—
27C2×#121C2211yescharacteristic
28C2×#121C2214yes—
29C2×#121C2214yes—
30C2×#121C2212yescomplemented
31C2×#121C2212yescomplemented
32C2×#121C2214yescomplemented
33C2×#121C2214yescomplemented
34C2×#121C2214yescomplemented
35C2×#120C2212yes—
36C2×#120C2212yes—
37C2×#120C2212yes—
38C2×C4×D4C2212yes—
39C22×C4⋊C4C2211yescharacteristic
40C22×C22⋊C4C2211yescharacteristic
41C22×C22⋊C4C2212yes—
42C22×C22⋊C4C2211yescharacteristic
43#399C2212yescomplemented
44#398C2214yescomplemented
45#397C2212yescomplemented
46#396C2214yescomplemented
47#396C2214yescomplemented
48#394C2214yescomplemented
49#394C2214yescomplemented
50#394C2214yescomplemented
51#394C2214yescomplemented
52#393C2212yescomplemented
53#392C2212yescomplemented
54#392C2214yescomplemented
55#392C2214yescomplemented
56#392C2214yescomplemented
57#390C2214yescomplemented
58#387C2214yescomplemented
59#386C2212yes—
60#386C2212yes—
61#386C2214yescomplemented
62#386C2214yescomplemented
63#386C2214yescomplemented
64#385C2212yes—
65#385C2212yes—
66#385C2214yescomplemented
67#385C2214yescomplemented
68#385C2214yescomplemented
69#385C2214yescomplemented
70#380C2212yes—
71#380C2211yescharacteristic
72#380C2211yescharacteristic
73#380C2212yescomplemented
74#380C2212yescomplemented
75#379C2212yes—
76#379C2212yes—
77C25C2311yescharacteristic
78C25C2311yescharacteristic
79C25C2311yescharacteristic
80C22×Q8C2311yescharacteristic
81C22×D4—24no—
82C22×D4—24no—
83C22×D4—24no—
84C22×D4—44no—
85C22×D4—44no—
86C22×D4—44no—
87C22×D4C2312yes—
88C22×D4C2314yes—
89C22×D4C2314yes—
90C22×D4C2314yes—
91C22×D4C2312yes—
92C22×D4C2312yes—
93C22×D4C2314yes—
94C22×D4C2311yescharacteristic
95C22×D4C2314yes—
96C23×C4—24no—
97C23×C4—44no—
98C23×C4C2311yescharacteristic
99C23×C4C2312yes—
100C23×C4C2312yes—
101C23×C4C2311yescharacteristic
102C23×C4C2312yes—
103C23×C4C2314yes—
104C23×C4C2311yescharacteristic
105C23×C4C2311yescharacteristic
106#126—28no—
107#124—28no—
108#123—28no—
109C22⋊Q8—24no—
110C22⋊Q8—44no—
111#121—24no—
112#121—216no—
113#121—216no—
114#121—28no—
115#121—44no—
116#121—48no—
117#121—48no—
118#121—48no—
119#120—28no—
120#120—44no—
121#120—44no—
122C4×D4—28no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4C2314yes—
125C2×C4⋊C4C2312yes—
126C2×C4⋊C4C2311yescharacteristic
127C2×C4⋊C4C2312yes—
128C2×C4⋊C4C2314yes—
129C2×C4⋊C4C2311yescharacteristic
130C2×C4⋊C4C2312yes—
131C2×C4⋊C4C2314yes—
132C2×C4⋊C4C2314yes—
133C2×C4⋊C4C2312yes—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—44no—
136C2×C22⋊C4D418yescomplemented
137C2×C22⋊C4C2312yes—
138C2×C22⋊C4C2312yes—
139C2×C22⋊C4C2312yes—
140C2×C22⋊C4C2312yes—
141C2×C22⋊C4C2312yes—
142C2×C22⋊C4C2312yes—
143C2×C22⋊C4C2312yes—
144C2×C22⋊C4C2312yes—
145C2×C22⋊C4C2312yes—
146C2×C22⋊C4C2314yes—
147C2×C22⋊C4C2314yes—
148C2×C22⋊C4C2314yes—
149C2×C22⋊C4C2314yes—
150C2×C22⋊C4C2314yes—
151C2×C22⋊C4C2314yes—
152C2×C22⋊C4C2314yes—
153C2×C22⋊C4C2314yes—
154C2×C22⋊C4C2314yes—
155C2×C22⋊C4C2312yes—
156C2×C22⋊C4C2312yes—
157C2×C42C2312yes—
158#95C2312yes—
159#95C2314yes—
160#95C2312yes—
161#95C2314yes—
162#95C2314yes—
163#95C2314yes—
164#95C2314yes—
165C24—22no—
166C24—24no—
167C24—22no—
168C24—22no—
169C24—22no—
170C24—22no—
171C24—22no—
172C24—22no—
173C24—22no—
174C24—42no—
175C24—42no—
176C24—42no—
177C24—42no—
178C24—42no—
179C24C2×D414yes—
180C24C2412yes—
181C24C2412yes—
182C24C2412yes—
183C24C2412yes—
184C24C2411yescharacteristic
185C2×Q8—22no—
186C2×Q8—42no—
187C2×D4—24no—
188C2×D4—28no—
189C2×D4—28no—
190C2×D4—24no—
191C2×D4—28no—
192C2×D4—28no—
193C2×D4—24no—
194C2×D4—24no—
195C2×D4—24no—
196C2×D4—28no—
197C2×D4—28no—
198C2×D4—22no—
199C2×D4—28no—
200C2×D4—44no—
201C2×D4—44no—
202C2×D4—44no—
203C2×D4—416no—
204C2×D4—44no—
205C2×D4—44no—
206C2×D4—42no—
207C2×D4—44no—
208C2×D4—44no—
209C2×D4—44no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4—416no—
213C2×D4—416no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—416no—
232C22×C4C2×D414yes—
233C22×C4C2×D414yes—
234C22×C4C2412yes—
235C22×C4C2414yes—
236C22×C4C2414yes—
237C22×C4C2412yes—
238C22×C4C2412yes—
239C22×C4C2412yes—
240C22×C4C2412yes—
241C22×C4C2411yescharacteristic
242C22×C4C2411yescharacteristic
243C22×C4C2412yes—
244C4⋊C4—28no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—28no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—42no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—24no—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—28no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—24no—
263C22⋊C4—24no—
264C22⋊C4—28no—
265C22⋊C4—44no—
266C22⋊C4—44no—
267C22⋊C4—44no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C22⋊C4—44no—
271C22⋊C4—44no—
272C22⋊C4C2×D418yes—
273C22⋊C4C2×D418yes—
274C42—24no—
275C23—22no—
276C23—22no—
277C23—28no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—24no—
284C23—24no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—28no—
289C23—22no—
290C23—22no—
291C23—24no—
292C23—22no—
293C23—24no—
294C23—24no—
295C23—24no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—44no—
300C23—44no—
301C23—48no—
302C23—48no—
303C23—44no—
304C23—44no—
305C23—44no—
306C23—44no—
307C23—44no—
308C23—84no—
309C23—84no—
310C23—84no—
311C23—84no—
312C23C22×D414yes—
313C23C22×D412yes—
314C23H(5,𝔽2)14yes—
315C23H(5,𝔽2)12yes—
316C232−1+412yes—
317C23C2511yescharacteristic
318Q8—82no—
319D4—416no—
320D4—416no—
321D4—416no—
322D4—48no—
323D4—82no—
324D4—84no—
325D4—84no—
326D4—88no—
327D4—88no—
328C2×C4—28no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—48no—
352C2×C4—48no—
353C2×C4—44no—
354C2×C4—84no—
355C2×C4—88no—
356C2×C4C22×D414yes—
357C2×C4C22×D412yes—
358C2×C4C22×D412yes—
359C2×C4H(5,𝔽2)14yes—
360C22—22no—
361C22—22no—
362C22—24no—
363C22—24no—
364C22—24no—
365C22—24no—
366C22—24no—
367C22—24no—
368C22—24no—
369C22—24no—
370C22—416no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—44no—
375C22—44no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—44no—
384C22—44no—
385C22—44no—
386C22—44no—
387C22—88no—
388C22—88no—
389C22#53714yes—
390C22C23×D411yescharacteristic
391C22C2×H(5,𝔽2)11yescharacteristic
392C22C2×H(5,𝔽2)12yes—
393C22C2×H(5,𝔽2)12yes—
394C22C2×2−1+411yescharacteristic
395C4—24no—
396C4—22no—
397C4—22no—
398C4—24no—
399C4—42no—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—44no—
404C4—44no—
405C2—24no—
406C2—44no—
407C2—44no—
408C2—82no—
409C2—82no—
410C2C2×#53711yescharacteristic
411C2#348411yescharacteristic
412C2#348411yescharacteristic
413C2#353612yes—
414C2#353712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
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×#538C2semidirect
2. C22×#121C2semidirect
3. #2848C2semidirect
4. #2847C2semidirect
5. #2691C2semidirect
6. #2676C2semidirect
7. #2675C2semidirect
8. #2671C2semidirect
9. #2626C2semidirect
10. #2617C2semidirect
11. #2579C2semidirect
12. #2410C2semidirect
13. #2410C2semidirect
14. #2408C2semidirect
15. #2315C2semidirect
21. C2×#126C22semidirect
22. C2×#124C22semidirect
23. C2×#123C22semidirect
25. C2×C22⋊Q8C22semidirect
30. C2×#121C22semidirect
31. C2×#121C22semidirect
32. C2×#121C22semidirect
33. C2×#121C22semidirect
34. C2×#121C22semidirect
43. #399C22semidirect
44. #398C22semidirect
45. #397C22semidirect
46. #396C22semidirect
47. #396C22semidirect
48. #394C22semidirect
49. #394C22semidirect
50. #394C22semidirect
51. #394C22semidirect
52. #393C22semidirect
53. #392C22semidirect
54. #392C22semidirect
55. #392C22semidirect
56. #392C22semidirect
57. #390C22semidirect
58. #387C22semidirect
61. #386C22semidirect
62. #386C22semidirect
63. #386C22semidirect
66. #385C22semidirect
67. #385C22semidirect
68. #385C22semidirect
69. #385C22semidirect
73. #380C22semidirect
74. #380C22semidirect
136. C2×C22⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5382
C22×#1212
#28482
#28472
#26912
#26762
#26752
#26712
#26262
#26172
#25792
#24102
#24082
#23152