Finite Group Databasebeta

← #41394

Subgroups — #41394

Order 256.

695 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2839C212yesmaximal, complemented
2#2794C212yesmaximal, complemented
3#2794C212yesmaximal, complemented
4#2789C212yesmaximal, complemented
5#2758C212yesmaximal, complemented
6#2655C212yesmaximal, complemented
7#2655C212yesmaximal, complemented
8#2627C211yesmaximal, characteristic
9#2569C211yesmaximal, characteristic
10#2569C212yesmaximal
11#2524C212yesmaximal, complemented
12#2505C212yesmaximal
13#2502C211yesmaximal, characteristic
14#2493C212yesmaximal
15#2493C212yesmaximal
16C2×#400C211yesmaximal, characteristic
17#2368C211yesmaximal, characteristic
18#2345C211yesmaximal, characteristic
19#2313C211yesmaximal, characteristic
20C2×C22⋊Q8C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23#401C2212yes—
24#401C2212yes—
25#400C2212yes—
26#400C2212yes—
27#400C2212yes—
28#400C2211yescharacteristic
29#400C2212yes—
30#400C2212yes—
31#400C2212yes—
32#400C2212yes—
33#400C2212yes—
34#400C2212yes—
35#400C2211yescharacteristic
36#400C2211yescharacteristic
37#400C2211yescharacteristic
38#400C2211yescharacteristic
39#400C2212yes—
40#400C2211yescharacteristic
41#400C2212yes—
42#400C2212yes—
43#400C2212yes—
44#400C2212yes—
45#398C2212yes—
46#398C2212yes—
47#398C2212yes—
48#398C2212yes—
49#398C2212yes—
50#398C2212yes—
51#398C2212yes—
52#398C2212yes—
53#395C2211yescharacteristic
54#395C2211yescharacteristic
55#391C2212yes—
56#391C2212yes—
57#389C2212yes—
58#389C2212yes—
59#387C2212yes—
60#387C2212yes—
61#387C2212yes—
62#387C2212yes—
63#387C2212yes—
64#387C2212yes—
65#387C2212yes—
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#385C2212yes—
82#385C2212yes—
83#385C2212yes—
84#385C2212yes—
85#385C2212yes—
86#384C2212yes—
87#383C2212yes—
88#383C2212yes—
89#382C2212yes—
90#381C2211yescharacteristic
91#381C2211yescharacteristic
92#381C2212yes—
93#380C2211yescharacteristic
94#380C2211yescharacteristic
95#380C2212yes—
96#380C2211yescharacteristic
97#380C2212yes—
98C4×C4⋊C4C2212yes—
99C4×C4⋊C4C2212yes—
100C4×C22⋊C4C2211yescharacteristic
101C4×C22⋊C4C2211yescharacteristic
102C4×C22⋊C4C2212yes—
103C4×C22⋊C4C2212yes—
104C2×#95C2211yescharacteristic
105C2×#95C2212yes—
106C2×#95C2212yes—
107C22×Q8C2311yescharacteristic
108C23×C4C2311yescharacteristic
109C23×C4C2312yes—
110C23×C4C2311yescharacteristic
111C23×C4C2312yes—
112C23×C4C2311yescharacteristic
113C22⋊Q8—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2312yes—
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
121C2×C4⋊C4C2311yescharacteristic
122C2×C4⋊C4C2311yescharacteristic
123C2×C4⋊C4C2312yes—
124C2×C4⋊C4C2311yescharacteristic
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⋊C4C2311yescharacteristic
144C2×C22⋊C4C2311yescharacteristic
145C2×C22⋊C4C2311yescharacteristic
146C2×C22⋊C4C2312yes—
147C2×C22⋊C4C2311yescharacteristic
148C2×C42C2312yes—
149C2×C42C2312yes—
150C2×C42C2312yes—
151C2×C42C2312yes—
152C2×C42C2312yes—
153C2×C42C2312yes—
154#95C2311yescharacteristic
155#95C2312yes—
156#95C2312yes—
157#95C2312yes—
158#95C2311yescharacteristic
159#95C2312yes—
160#95C2312yes—
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#95C2311yescharacteristic
177#95C2311yescharacteristic
178#95C2312yes—
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—22no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4#4114yes—
213C22×C4C2411yescharacteristic
214C22×C4C2411yescharacteristic
215C22×C4C2412yes—
216C22×C4C2412yes—
217C22×C4C2412yes—
218C22×C4C2412yes—
219C22×C4C2412yes—
220C22×C4C2412yes—
221C22×C4C2412yes—
222C22×C4C2412yes—
223C22×C4C2412yes—
224C22×C4C2412yes—
225C22×C4C2411yescharacteristic
226C22×C4C2411yescharacteristic
227C22×C4C2412yes—
228C22×C4C2412yes—
229C22×C4C2411yescharacteristic
230C22×C4C2411yescharacteristic
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—24no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—24no—
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—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C22⋊C4—24no—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—24no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C42—42no—
264C42—42no—
265C42—42no—
266C42—42no—
267C42—42no—
268C42—42no—
269C23—22no—
270C23—22no—
271C23C2×#4112yes—
272C232−1+412yes—
273C232−1+412yes—
274C23C2511yescharacteristic
275Q8—82no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
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—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—82no—
327C2×C4—84no—
328C2×C4—84no—
329C2×C4C2×#4112yes—
330C2×C4C2×#4112yes—
331C22—22no—
332C22—22no—
333C22—24no—
334C22—24no—
335C22#54114yes—
336C22C22×#4111yescharacteristic
337C22C2×2−1+411yescharacteristic
338C22C2×2−1+411yescharacteristic
339C22#58512yes—
340C22#58512yes—
341C4—22no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—42no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—42no—
352C4—42no—
353C4—42no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4—42no—
358C4—42no—
359C2—24no—
360C2C2×#54111yescharacteristic
361C2#349111yescharacteristic
362C2#351011yescharacteristic
363C2#355312yes—
364C2#356412yes—
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. #2839C2semidirect
2. #2794C2semidirect
3. #2794C2semidirect
4. #2789C2semidirect
5. #2758C2semidirect
6. #2655C2semidirect
7. #2655C2semidirect
11. #2524C2semidirect

Maximal subgroups

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

subgroupindex
#28392
#27942
#27892
#27582
#26552
#26272
#25692
#25242
#25052
#25022
#24932
C2×#4002
#23682
#23452
#23132