Finite Group Databasebeta

← #35252

Subgroups — #35252

Order 256.

1801 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#535C211yesmaximal, characteristic, complemented
2C2×#525C211yesmaximal, characteristic, complemented
3C22×#121C211yesmaximal, characteristic, complemented
4C22×#117C211yesmaximal, characteristic, complemented
5#2856C212yesmaximal, complemented
6#2850C212yesmaximal, complemented
7#2620C212yesmaximal, complemented
8#2619C212yesmaximal, complemented
9#2416C214yesmaximal, complemented
10#2414C214yesmaximal, complemented
11#2411C214yesmaximal, complemented
12#2410C214yesmaximal, complemented
13#2317C211yesmaximal, characteristic, complemented
14#2311C211yesmaximal, characteristic, complemented
15#2309C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C23×D4C2212yes—
19C24×C4C2211yescharacteristic
20#535—28no—
21C2×#127C2212yescomplemented
22C2×#124C2212yescomplemented
23#525—28no—
24C2×#123C2212yes—
25C2×C22⋊Q8C2211yescharacteristic, complemented
26C2×C22⋊Q8C2211yescharacteristic, complemented
27C2×#121—28no—
28C2×#121C2212yes—
29C2×#121C2212yes—
30C2×#121C2212yes—
31C2×#121C2212yes—
32C2×#121C2211yescharacteristic, complemented
33C2×#121C2211yescharacteristic, complemented
34C2×#121C2214yescomplemented
35C2×#121C2218yescomplemented
36C2×#120C2212yes—
37C2×#120C2214yes—
38C2×C4×D4C2212yescomplemented
39C2×C4×D4C2212yescomplemented
40C2×#117—28no—
41C2×#117C2211yescharacteristic
42C2×#117C2211yescharacteristic
43C2×#117C2212yes—
44C2×#117C2211yescharacteristic
45C2×#117C2212yescomplemented
46C2×#117C2212yescomplemented
47C22×C4⋊C4C2211yescharacteristic
48C22×C4⋊C4C2212yescomplemented
49C22×C22⋊C4C2212yes—
50C22×C22⋊C4C2212yes—
51C22×C42C2212yescomplemented
52#396C2218yescomplemented
53#394C2218yescomplemented
54#394C2218yescomplemented
55#392C2218yescomplemented
56#391C2212yescomplemented
57#390C2212yescomplemented
58#390C2212yescomplemented
59#390C2214yescomplemented
60#390C2212yescomplemented
61#390C2218yescomplemented
62#389C2214yescomplemented
63#388C2218yescomplemented
64#386C2214yes—
65#386C2218yescomplemented
66#385C2214yes—
67#384C2212yescomplemented
68#382C2212yescomplemented
69#380C2214yescomplemented
70#379C2214yes—
71C25C2311yescharacteristic
72C25C2312yes—
73C2×#41—24no—
74C2×#41—24no—
75C2×#41—24no—
76C22×Q8C2311yescharacteristic
77C22×D4—28no—
78C22×D4—24no—
79C22×D4—24no—
80C22×D4—24no—
81C22×D4—48no—
82C22×D4C2312yes—
83C22×D4C2312yes—
84C22×D4C2314yes—
85C22×D4C2312yes—
86C22×D4C2312yes—
87C22×D4C2311yescharacteristic
88C22×D4C2312yes—
89C22×D4C2318yes—
90C22×D4C2314yes—
91C23×C4—24no—
92C23×C4—24no—
93C23×C4D418yescomplemented
94C23×C4C2312yes—
95C23×C4C2314yes—
96C23×C4C2311yescharacteristic
97C23×C4C2311yescharacteristic
98C23×C4C2311yescharacteristic
99C23×C4C2311yescharacteristic
100C23×C4C2311yescharacteristic
101C23×C4C2312yes—
102C23×C4C2311yescharacteristic
103#127—28no—
104#124—28no—
105#123—28no—
106C22⋊Q8—24no—
107C22⋊Q8—24no—
108#121—28no—
109#121—28no—
110#121—28no—
111#121—28no—
112#121—24no—
113#121—24no—
114#121—216no—
115#121—416no—
116#120—28no—
117#120—216no—
118C4×D4—28no—
119C4×D4—28no—
120#117—24no—
121#117—24no—
122#117—28no—
123#117—28no—
124#117—28no—
125#117D418yescomplemented
126C2×C4⋊C4—24no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4C2312yes—
129C2×C4⋊C4C2312yes—
130C2×C4⋊C4C2312yes—
131C2×C4⋊C4C2314yes—
132C2×C4⋊C4C2311yescharacteristic
133C2×C4⋊C4C2311yescharacteristic
134C2×C4⋊C4C2312yes—
135C2×C4⋊C4C2312yes—
136C2×C4⋊C4C2318yes—
137C2×C4⋊C4C2312yes—
138C2×C22⋊C4—28no—
139C2×C22⋊C4—28no—
140C2×C22⋊C4C2312yes—
141C2×C22⋊C4C2314yes—
142C2×C22⋊C4C2314yes—
143C2×C22⋊C4C2314yes—
144C2×C22⋊C4C2312yes—
145C2×C22⋊C4C2312yes—
146C2×C22⋊C4C2312yes—
147C2×C22⋊C4C2312yes—
148C2×C22⋊C4C2312yes—
149C2×C22⋊C4C2318yes—
150C2×C22⋊C4C2318yes—
151C2×C22⋊C4C2318yes—
152C2×C22⋊C4C2318yes—
153C2×C22⋊C4C2312yes—
154C2×C42—28no—
155C2×C42C2314yes—
156C2×C42C2312yes—
157C2×C42C2312yes—
158C2×C42C2312yes—
159#95C2318yes—
160#95C2318yes—
161C24—22no—
162C24—22no—
163C24—24no—
164C24—24no—
165C24—24no—
166C24—24no—
167C24—42no—
168C24—44no—
169C24—44no—
170C24C2×D414yes—
171C24C2414yes—
172C24C2412yes—
173C24C2411yescharacteristic
174C24C2411yescharacteristic
175C24C2411yescharacteristic
176#41—416no—
177C2×Q8—22no—
178C2×Q8—22no—
179C2×Q8—22no—
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—24no—
183C2×D4—24no—
184C2×D4—28no—
185C2×D4—28no—
186C2×D4—24no—
187C2×D4—28no—
188C2×D4—216no—
189C2×D4—24no—
190C2×D4—24no—
191C2×D4—24no—
192C2×D4—22no—
193C2×D4—24no—
194C2×D4—22no—
195C2×D4—216no—
196C2×D4—24no—
197C2×D4—24no—
198C2×D4—22no—
199C2×D4—48no—
200C2×D4—44no—
201C2×D4—48no—
202C2×D4—48no—
203C2×D4—432no—
204C2×D4—416no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—232no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—28no—
218C22×C4—28no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4C2×D414yes—
225C22×C4C2×D414yes—
226C22×C4C2×D414yes—
227C22×C4C2×D414yes—
228C22×C4C2×D414yes—
229C22×C4C2×D414yes—
230C22×C4C2×D414yes—
231C22×C4#4114yes—
232C22×C4C2412yes—
233C22×C4C2412yes—
234C22×C4C2414yes—
235C22×C4C2414yes—
236C22×C4C2412yes—
237C22×C4C2414yes—
238C22×C4C2411yescharacteristic
239C22×C4C2411yescharacteristic
240C22×C4C2411yescharacteristic
241C22×C4C2411yescharacteristic
242C4⋊C4—28no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—48no—
251C4⋊C4C2×D418yescomplemented
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—28no—
257C22⋊C4—28no—
258C22⋊C4—28no—
259C22⋊C4—24no—
260C22⋊C4—24no—
261C22⋊C4—216no—
262C22⋊C4—48no—
263C22⋊C4—48no—
264C22⋊C4—48no—
265C22⋊C4C2×D418yes—
266C42—28no—
267C42—24no—
268C42—24no—
269C42C2×D418yescomplemented
270C23—21no—
271C23—21no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—28no—
276C23—28no—
277C23—21no—
278C23—21no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—24no—
283C23—24no—
284C23—22no—
285C23—22no—
286C23—24no—
287C23—24no—
288C23—28no—
289C23—28no—
290C23—24no—
291C23—44no—
292C23—48no—
293C23—48no—
294C23—48no—
295C23—48no—
296C23—48no—
297C23—88no—
298C23—88no—
299C23C22×D412yes—
300C23C22×D412yes—
301C23C22×D412yes—
302C23C22×D412yes—
303C23C2×#4112yes—
304C23H(5,𝔽2)14yes—
305C23C2511yescharacteristic
306Q8—44no—
307D4—48no—
308D4—416no—
309D4—48no—
310D4—48no—
311D4—416no—
312D4—48no—
313D4—48no—
314D4—44no—
315D4—816no—
316C2×C4—28no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—28no—
324C2×C4—28no—
325C2×C4—28no—
326C2×C4—28no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—28no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—416no—
340C2×C4—48no—
341C2×C4—48no—
342C2×C4#121116yescomplemented
343C2×C4C22×D418yes—
344C2×C4C22×D412yes—
345C2×C4C22×D412yes—
346C2×C4C22×D412yes—
347C2×C4C22×D412yes—
348C2×C4C22×D412yes—
349C2×C4C22×D412yes—
350C2×C4C2×#4112yes—
351C2×C4C2×#4112yes—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—24no—
357C22—24no—
358C22—24no—
359C22—24no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—416no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—44no—
368C22—44no—
369C22—42no—
370C22—42no—
371C22—44no—
372C22—44no—
373C22—48no—
374C22—44no—
375C22—48no—
376C22—816no—
377C22C2×#12114yes—
378C22#53514yescomplemented
379C22C23×D411yescharacteristic
380C22C23×D411yescharacteristic
381C22C22×#4111yescharacteristic
382C22C2×H(5,𝔽2)12yes—
383C22#58512yes—
384C4—22no—
385C4—22no—
386C4—28no—
387C4—42no—
388C4—44no—
389C4—42no—
390C4—42no—
391C4—44no—
392C4C2×#12114yes—
393C4C2×#12114yes—
394C2—24no—
395C2—24no—
396C2—42no—
397C2—82no—
398C2—84no—
399C2C22×#12111yescharacteristic
400C2C2×#53511yescharacteristic
401C2#344911yescharacteristic
402C2#348812yes—
403C2#349412yes—
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×#535C2semidirect
2. C2×#525C2semidirect
3. C22×#121C2semidirect
4. C22×#117C2semidirect
5. #2856C2semidirect
6. #2850C2semidirect
7. #2620C2semidirect
8. #2619C2semidirect
9. #2416C2semidirect
10. #2414C2semidirect
11. #2411C2semidirect
12. #2410C2semidirect
13. #2317C2semidirect
14. #2311C2semidirect
15. #2309C2semidirect
21. C2×#127C22semidirect
22. C2×#124C22semidirect
25. C2×C22⋊Q8C22semidirect
26. C2×C22⋊Q8C22semidirect
32. C2×#121C22semidirect
33. C2×#121C22semidirect
34. C2×#121C22semidirect
35. C2×#121C22semidirect
38. C2×C4×D4C22semidirect
39. C2×C4×D4C22semidirect
45. C2×#117C22semidirect
46. C2×#117C22semidirect
48. C22×C4⋊C4C22semidirect
51. C22×C42C22semidirect
52. #396C22semidirect
53. #394C22semidirect
54. #394C22semidirect
55. #392C22semidirect
56. #391C22semidirect
57. #390C22semidirect
58. #390C22semidirect
59. #390C22semidirect
60. #390C22semidirect
61. #390C22semidirect
62. #389C22semidirect
63. #388C22semidirect
65. #386C22semidirect
67. #384C22semidirect
68. #382C22semidirect
69. #380C22semidirect
93. C23×C4D4semidirect
125. #117D4semidirect
251. C4⋊C4C2×D4semidirect
269. C42C2×D4semidirect
342. C2×C4#121semidirect
378. C22#535semidirect

Maximal subgroups

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

subgroupindex
C2×#5352
C2×#5252
C22×#1212
C22×#1172
#28562
#28502
#26202
#26192
#24162
#24142
#24112
#24102
#23172
#23112
#23092