Finite Group Databasebeta

← #45759

Subgroups — #45759

Order 256.

727 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2851C212yesmaximal
2#2835C211yesmaximal, characteristic
3#2834C211yesmaximal, characteristic
4#2802C212yesmaximal, complemented
5#2802C212yesmaximal, complemented
6#2792C212yesmaximal, complemented
7#2776C212yesmaximal, complemented
8#2747C212yesmaximal
9#2742C211yesmaximal, characteristic
10#2722C211yesmaximal, characteristic
11#2611C211yesmaximal, characteristic
12#2599C211yesmaximal, characteristic
13#2528C212yesmaximal, complemented
14#2524C212yesmaximal, complemented
15#2506C212yesmaximal, complemented
16#2478C212yesmaximal
17#2462C212yesmaximal, complemented
18#2383C212yesmaximal
19#2296C211yesmaximal, characteristic
20C2×#126C2212yes—
21C2×#125C2212yes—
22C2×#125C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×#117C2211yescharacteristic
25C2×#117C2211yescharacteristic
26C22×C42C2211yescharacteristic
27#401C2212yes—
28#400C2212yes—
29#400C2212yes—
30#400C2212yes—
31#400C2211yescharacteristic
32#400C2212yes—
33#400C2212yes—
34#400C2212yes—
35#399C2211yescharacteristic
36#398C2212yes—
37#398C2212yes—
38#398C2212yes—
39#398C2211yescharacteristic
40#398C2211yescharacteristic
41#398C2212yes—
42#398C2212yes—
43#398C2212yes—
44#398C2212yes—
45#398C2212yes—
46#398C2212yes—
47#397C2211yescharacteristic
48#397C2212yes—
49#397C2211yescharacteristic
50#397C2212yes—
51#395C2212yes—
52#395C2211yescharacteristic
53#395C2212yes—
54#393C2211yescharacteristic
55#389C2212yes—
56#389C2212yes—
57#389C2212yes—
58#389C2212yes—
59#389C2212yes—
60#389C2212yes—
61#389C2212yes—
62#389C2212yes—
63#389C2212yes—
64#389C2212yes—
65#389C2212yes—
66#389C2212yes—
67#388C2212yes—
68#388C2212yes—
69#388C2212yes—
70#387C2212yes—
71#387C2212yes—
72#387C2212yes—
73#387C2212yes—
74#387C2212yes—
75#387C2212yes—
76#387C2212yes—
77#387C2212yes—
78#387C2212yes—
79#387C2212yes—
80#387C2212yes—
81#387C2212yes—
82#387C2212yes—
83#387C2212yes—
84#387C2212yes—
85#387C2212yes—
86#387C2212yes—
87#387C2212yes—
88#387C2212yes—
89#385C2211yescharacteristic
90#385C2212yes—
91#385C2212yes—
92#385C2212yes—
93#385C2212yes—
94#385C2211yescharacteristic
95#385C2211yescharacteristic
96#385C2211yescharacteristic
97#384C2212yes—
98#383C2211yescharacteristic
99#383C2212yes—
100#382C2212yes—
101#380C2212yes—
102#380C2212yes—
103C4×C4⋊C4C2212yes—
104C4×C4⋊C4C2212yes—
105C4×C4⋊C4C2212yes—
106#376C2211yescharacteristic
107C22×Q8C2311yescharacteristic
108C23×C4C2311yescharacteristic
109C23×C4C2311yescharacteristic
110C23×C4C2311yescharacteristic
111#126—28no—
112#125—44no—
113#125—42no—
114C4×Q8—42no—
115#117—24no—
116#117—42no—
117C2×C4⋊C4C2312yes—
118C2×C4⋊C4C2312yes—
119C2×C4⋊C4C2312yes—
120C2×C4⋊C4C2312yes—
121C2×C4⋊C4C2312yes—
122C2×C4⋊C4C2312yes—
123C2×C4⋊C4C2312yes—
124C2×C4⋊C4C2312yes—
125C2×C4⋊C4C2312yes—
126C2×C4⋊C4C2312yes—
127C2×C4⋊C4C2312yes—
128C2×C4⋊C4C2312yes—
129C2×C4⋊C4C2311yescharacteristic
130C2×C4⋊C4C2311yescharacteristic
131C2×C4⋊C4C2311yescharacteristic
132C2×C4⋊C4C2312yes—
133C2×C4⋊C4C2312yes—
134C2×C4⋊C4C2312yes—
135C2×C4⋊C4C2312yes—
136C2×C4⋊C4C2312yes—
137C2×C4⋊C4C2311yescharacteristic
138C2×C4⋊C4C2311yescharacteristic
139C2×C4⋊C4C2312yes—
140C2×C4⋊C4C2312yes—
141C2×C4⋊C4C2312yes—
142C2×C4⋊C4C2312yes—
143C2×C4⋊C4C2312yes—
144C2×C4⋊C4C2312yes—
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2312yes—
147C2×C4⋊C4C2312yes—
148C2×C22⋊C4C2311yescharacteristic
149C2×C22⋊C4C2312yes—
150C2×C22⋊C4C2312yes—
151C2×C22⋊C4C2311yescharacteristic
152C2×C22⋊C4C2311yescharacteristic
153C2×C22⋊C4C2312yes—
154C2×C22⋊C4C2312yes—
155C2×C22⋊C4C2311yescharacteristic
156C2×C42—42no—
157C2×C42C2311yescharacteristic
158C2×C42C2311yescharacteristic
159C2×C42C2312yes—
160C2×C42C2312yes—
161C2×C42C2312yes—
162C2×C42C2312yes—
163C2×C42C2312yes—
164C2×C42C2312yes—
165C2×C42C2312yes—
166#95C2312yes—
167#95C2311yescharacteristic
168#95C2311yescharacteristic
169#95C2312yes—
170#95C2312yes—
171#95C2312yes—
172#95C2312yes—
173#95C2312yes—
174#95C2312yes—
175#95C2312yes—
176#95C2312yes—
177#95C2312yes—
178#95C2312yes—
179#95C2312yes—
180#95C2312yes—
181#95C2312yes—
182#95C2312yes—
183#95C2312yes—
184#95C2312yes—
185#95C2311yescharacteristic
186#95C2311yescharacteristic
187#95C2312yes—
188#95C2312yes—
189#95C2312yes—
190#95C2312yes—
191#95C2312yes—
192#95C2312yes—
193#95C2312yes—
194#95C2312yes—
195#95C2312yes—
196#95C2312yes—
197#95C2312yes—
198#95C2312yes—
199#95C2312yes—
200#95C2312yes—
201C24C2411yescharacteristic
202C2×Q8—41no—
203C2×Q8—42no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—42no—
211C22×C4C2412yes—
212C22×C4C2412yes—
213C22×C4C2411yescharacteristic
214C22×C4C2412yes—
215C22×C4C2412yes—
216C22×C4C2412yes—
217C22×C4C2411yescharacteristic
218C22×C4C2412yes—
219C22×C4C2412yes—
220C22×C4C2412yes—
221C22×C4C2411yescharacteristic
222C22×C4C2412yes—
223C22×C4C2411yescharacteristic
224C22×C4C2412yes—
225C22×C4C2412yes—
226C22×C4C2411yescharacteristic
227C22×C4C2412yes—
228C22×C4C2411yescharacteristic
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C22⋊C4—22no—
261C22⋊C4—24no—
262C22⋊C4—24no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4#4114yes—
268C42—24no—
269C42—24no—
270C42—24no—
271C42—22no—
272C42—22no—
273C42—42no—
274C42—42no—
275C42—42no—
276C42—42no—
277C23—21no—
278C23—21no—
279C23—22no—
280C23—22no—
281C23C2×#4112yes—
282C23C2511yescharacteristic
283Q8—44no—
284C2×C4—28no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4C2×#4114yes—
334C2×C4H(5,𝔽2)12yes—
335C2×C42−1+412yes—
336C2×C42−1+414yes—
337C22—22no—
338C22—24no—
339C22—41no—
340C22—41no—
341C22—42no—
342C22C22×#4111yescharacteristic
343C22C2×H(5,𝔽2)11yescharacteristic
344C22C2×2−1+411yescharacteristic
345C22C2×2−1+412yes—
346C22#58512yes—
347C4—22no—
348C4—22no—
349C4—24no—
350C4—24no—
351C4—42no—
352C4—42no—
353C4—42no—
354C4—42no—
355C4—42no—
356C4—41no—
357C4—42no—
358C4—42no—
359C4—41no—
360C4—42no—
361C4—41no—
362C4—41no—
363C4—42no—
364C4—42no—
365C2—42no—
366C2#349211yescharacteristic
367C2#351512yes—
368C2#352911yescharacteristic
369C2#353811yescharacteristic
370C2#355512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radicalC24C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #2802C2semidirect
5. #2802C2semidirect
6. #2792C2semidirect
7. #2776C2semidirect
13. #2528C2semidirect
14. #2524C2semidirect
15. #2506C2semidirect
17. #2462C2semidirect

Maximal subgroups

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

subgroupindex
#28512
#28352
#28342
#28022
#27922
#27762
#27472
#27422
#27222
#26112
#25992
#25282
#25242
#25062
#24782
#24622
#23832
#22962