Finite Group Databasebeta

← #48800

Subgroups — #48800

Order 256.

681 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2810C212yesmaximal, complemented
2#2776C212yesmaximal, complemented
3#2771C212yesmaximal
4#2769C212yesmaximal, complemented
5#2765C212yesmaximal
6#2762C212yesmaximal, complemented
7#2667C211yesmaximal, characteristic
8#2625C211yesmaximal, characteristic
9#2590C212yesmaximal, complemented
10#2585C212yesmaximal, complemented
11#2583C211yesmaximal, characteristic
12#2575C212yesmaximal
13#2520C212yesmaximal, complemented
14#2515C212yesmaximal
15#2508C212yesmaximal, complemented
16#2484C211yesmaximal, characteristic
17#2378C211yesmaximal, characteristic
18#2347C211yesmaximal, characteristic
19#2313C211yesmaximal, characteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23#401C2212yes—
24#400C2212yes—
25#400C2212yes—
26#400C2212yes—
27#400C2211yescharacteristic
28#400C2211yescharacteristic
29#400C2212yes—
30#400C2212yes—
31#400C2212yes—
32#400C2212yes—
33#400C2212yes—
34#400C2212yes—
35#399C2212yes—
36#398C2211yescharacteristic
37#398C2211yescharacteristic
38#398C2212yes—
39#398C2212yes—
40#398C2211yescharacteristic
41#398C2212yes—
42#398C2212yes—
43#398C2212yes—
44#398C2211yescharacteristic
45#398C2212yes—
46#397C2212yes—
47#397C2212yes—
48#396C2212yes—
49#395C2211yescharacteristic
50#395C2211yescharacteristic
51#391C2212yes—
52#391C2212yes—
53#389C2212yes—
54#389C2212yes—
55#389C2212yes—
56#389C2212yes—
57#389C2212yes—
58#389C2212yes—
59#389C2212yes—
60#389C2212yes—
61#389C2212yes—
62#388C2212yes—
63#388C2212yes—
64#388C2212yes—
65#388C2212yes—
66#387C2212yes—
67#387C2212yes—
68#387C2212yes—
69#387C2212yes—
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#385C2212yes—
87#385C2212yes—
88#385C2212yes—
89#383C2212yes—
90#383C2212yes—
91#382C2212yes—
92#382C2212yes—
93#382C2212yes—
94#381C2211yescharacteristic
95#381C2211yescharacteristic
96#381C2212yes—
97#381C2211yescharacteristic
98#380C2211yescharacteristic
99#380C2211yescharacteristic
100C4×C4⋊C4C2212yes—
101C4×C4⋊C4C2212yes—
102C4×C4⋊C4C2212yes—
103C4×C4⋊C4C2212yes—
104C4×C22⋊C4C2211yescharacteristic
105C4×C22⋊C4C2211yescharacteristic
106C4×C22⋊C4C2211yescharacteristic
107C22×Q8C2311yescharacteristic
108C23×C4C2311yescharacteristic
109C23×C4C2311yescharacteristic
110C23×C4C2311yescharacteristic
111C22⋊Q8—24no—
112#117—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
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⋊C4C2312yes—
130C2×C4⋊C4C2312yes—
131C2×C4⋊C4C2312yes—
132C2×C4⋊C4C2312yes—
133C2×C4⋊C4C2312yes—
134C2×C4⋊C4C2312yes—
135C2×C4⋊C4C2312yes—
136C2×C4⋊C4C2312yes—
137C2×C4⋊C4C2312yes—
138C2×C4⋊C4C2312yes—
139C2×C4⋊C4C2312yes—
140C2×C4⋊C4C2312yes—
141C2×C4⋊C4C2312yes—
142C2×C22⋊C4C2312yes—
143C2×C22⋊C4C2312yes—
144C2×C22⋊C4C2312yes—
145C2×C22⋊C4C2311yescharacteristic
146C2×C22⋊C4C2311yescharacteristic
147C2×C22⋊C4C2311yescharacteristic
148C2×C22⋊C4C2311yescharacteristic
149C2×C22⋊C4C2312yes—
150C2×C42C2312yes—
151C2×C42C2312yes—
152C2×C42C2312yes—
153C2×C42C2312yes—
154C2×C42C2312yes—
155C2×C42C2312yes—
156C2×C42C2312yes—
157C2×C42C2312yes—
158#95C2311yescharacteristic
159#95C2312yes—
160#95C2311yescharacteristic
161#95C2312yes—
162#95C2312yes—
163#95C2312yes—
164#95C2312yes—
165#95C2312yes—
166#95C2312yes—
167#95C2312yes—
168#95C2312yes—
169#95C2312yes—
170#95C2312yes—
171#95C2312yes—
172#95C2312yes—
173#95C2312yes—
174#95C2312yes—
175#95C2312yes—
176#95C2312yes—
177#95C2311yescharacteristic
178#95C2311yescharacteristic
179#95C2312yes—
180#95C2312yes—
181#95C2312yes—
182#95C2312yes—
183#95C2312yes—
184#95C2312yes—
185#95C2312yes—
186#95C2312yes—
187#95C2312yes—
188#95C2312yes—
189#95C2312yes—
190#95C2312yes—
191#95C2312yes—
192#95C2312yes—
193#95C2312yes—
194#95C2312yes—
195#95C2312yes—
196#95C2312yes—
197C24C2411yescharacteristic
198C2×Q8—22no—
199C2×Q8—42no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—42no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4#4114yes—
207C22×C4C2412yes—
208C22×C4C2412yes—
209C22×C4C2412yes—
210C22×C4C2411yescharacteristic
211C22×C4C2411yescharacteristic
212C22×C4C2412yes—
213C22×C4C2412yes—
214C22×C4C2412yes—
215C22×C4C2412yes—
216C22×C4C2412yes—
217C22×C4C2412yes—
218C22×C4C2411yescharacteristic
219C22×C4C2411yescharacteristic
220C22×C4C2412yes—
221C22×C4C2412yes—
222C22×C4C2411yescharacteristic
223C22×C4C2411yescharacteristic
224C22×C4C2412yes—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
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—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—42no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C42—24no—
262C42—42no—
263C42—42no—
264C42—42no—
265C42—42no—
266C42—42no—
267C42—42no—
268C42—42no—
269C23—22no—
270C23—21no—
271C23—21no—
272C23—22no—
273C23C2×#4112yes—
274C23C2511yescharacteristic
275Q8—82no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
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—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—82no—
326C2×C4C2×#4112yes—
327C2×C4C2×#4112yes—
328C2×C42−1+414yes—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—42no—
333C22—42no—
334C22C22×#4111yescharacteristic
335C22C2×2−1+412yes—
336C22#58512yes—
337C22#58511yescharacteristic
338C22#58511yescharacteristic
339C4—22no—
340C4—22no—
341C4—24no—
342C4—41no—
343C4—41no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—41no—
348C4—41no—
349C4—42no—
350C4—42no—
351C4—42no—
352C4—42no—
353C4—42no—
354C4—42no—
355C4—42no—
356C4—42no—
357C2—42no—
358C2#349111yescharacteristic
359C2#349311yescharacteristic
360C2#350911yescharacteristic
361C2#355312yes—
362C2#355712yes—
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
1. #2810C2semidirect
2. #2776C2semidirect
4. #2769C2semidirect
6. #2762C2semidirect
9. #2590C2semidirect
10. #2585C2semidirect
13. #2520C2semidirect
15. #2508C2semidirect

Maximal subgroups

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

subgroupindex
#28102
#27762
#27712
#27692
#27652
#27622
#26672
#26252
#25902
#25852
#25832
#25752
#25202
#25152
#25082
#24842
#23782
#23472
#23132