Finite Group Databasebeta

← #53563

Subgroups — #53563

Order 256.

906 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2835C211yesmaximal, characteristic, complemented
2#2831C212yesmaximal, complemented
3#2788C212yesmaximal, complemented
4#2755C212yesmaximal, complemented
5#2722C211yesmaximal, characteristic, complemented
6#2696C212yesmaximal, complemented
7#2695C212yesmaximal, complemented
8#2693C211yesmaximal, characteristic, complemented
9#2691C212yesmaximal, complemented
10#2687C212yesmaximal, complemented
11#2683C212yesmaximal, complemented
12#2659C212yesmaximal, complemented
13#2638C211yesmaximal, characteristic, complemented
14#2535C211yesmaximal, characteristic
15#2463C211yesmaximal, characteristic, complemented
16#2450C212yesmaximal, complemented
17#2425C212yesmaximal, complemented
18#2409C211yesmaximal, characteristic, complemented
19#2409C212yesmaximal, complemented
20C23×Q8C2211yescharacteristic
21C2×#126C2212yes—
22C2×#123C2212yes—
23C2×#123C2212yes—
24C2×C22⋊Q8C2211yescharacteristic
25C2×C22⋊Q8C2212yes—
26C2×C22⋊Q8C2212yes—
27C2×C22⋊Q8C2212yes—
28C2×C22⋊Q8C2212yes—
29C2×#121C2212yes—
30C2×C4×D4C2211yescharacteristic
31C2×C4×D4C2212yes—
32C2×C4×D4C2211yescharacteristic
33#400C2212yes—
34#400C2212yes—
35#400C2212yes—
36#400C2212yes—
37#400C2212yes—
38#398C2212yes—
39#398C2212yes—
40#398C2211yescharacteristic
41#398C2212yes—
42#398C2212yes—
43#398C2212yes—
44#398C2212yes—
45#397C2211yescharacteristic
46#397C2212yes—
47#397C2212yes—
48#397C2212yes—
49#397C2211yescharacteristic
50#397C2211yescharacteristic
51#396C2212yes—
52#396C2212yes—
53#396C2212yes—
54#396C2212yes—
55#395C2212yes—
56#395C2212yes—
57#395C2211yescharacteristic
58#395C2212yes—
59#395C2212yes—
60#394C2212yes—
61#394C2212yes—
62#394C2212yes—
63#393C2212yes—
64#393C2211yescharacteristic
65#393C2212yes—
66#393C2212yes—
67#392C2211yescharacteristic
68#392C2211yescharacteristic
69#391C2212yes—
70#391C2212yes—
71#391C2212yes—
72#390C2212yes—
73#389C2212yes—
74#389C2212yes—
75#389C2212yes—
76#388C2212yes—
77#388C2212yes—
78#388C2212yes—
79#387C2212yes—
80#387C2212yes—
81#387C2212yes—
82#387C2212yes—
83#386C2211yescharacteristic
84#386C2211yescharacteristic
85#386C2212yes—
86#386C2212yes—
87#386C2212yes—
88#386C2212yes—
89#386C2212yes—
90#386C2212yes—
91#385C2212yes—
92#385C2211yescharacteristic
93#385C2212yes—
94#385C2211yescharacteristic
95#385C2212yes—
96#385C2212yes—
97#385C2212yes—
98#382C2211yescharacteristic
99#381C2211yescharacteristic
100#380C2212yes—
101#380C2212yes—
102#380C2212yes—
103#380C2212yes—
104#380C2212yes—
105#380C2211yescharacteristic
106C2×#95C2212yes—
107C22×Q8—44no—
108C22×Q8—42no—
109C22×Q8C2312yes—
110C22×Q8C2312yes—
111C22×Q8C2311yescharacteristic
112C22×D4C2311yescharacteristic
113C22×D4C2312yes—
114C22×D4C2311yescharacteristic
115C22×D4C2312yes—
116C23×C4C2311yescharacteristic
117C23×C4C2312yes—
118C23×C4C2312yes—
119C23×C4C2311yescharacteristic
120C23×C4C2311yescharacteristic
121C23×C4C2312yes—
122C23×C4C2311yescharacteristic
123#126—44no—
124#123—44no—
125#123—44no—
126C22⋊Q8—44no—
127C22⋊Q8—44no—
128C22⋊Q8—44no—
129C22⋊Q8—42no—
130C22⋊Q8—44no—
131#121—44no—
132C4×D4—42no—
133C4×D4—42no—
134C4×D4—44no—
135C2×C4⋊C4C2312yes—
136C2×C4⋊C4C2312yes—
137C2×C4⋊C4C2312yes—
138C2×C4⋊C4C2312yes—
139C2×C4⋊C4C2312yes—
140C2×C4⋊C4C2311yescharacteristic
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2312yes—
143C2×C4⋊C4C2312yes—
144C2×C4⋊C4C2312yes—
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2312yes—
147C2×C4⋊C4C2312yes—
148C2×C4⋊C4C2312yes—
149C2×C4⋊C4C2312yes—
150C2×C4⋊C4C2311yescharacteristic
151C2×C4⋊C4C2312yes—
152C2×C4⋊C4C2312yes—
153C2×C4⋊C4C2312yes—
154C2×C4⋊C4C2312yes—
155C2×C4⋊C4C2312yes—
156C2×C4⋊C4C2312yes—
157C2×C22⋊C4C2312yes—
158C2×C22⋊C4C2312yes—
159C2×C22⋊C4C2312yes—
160C2×C22⋊C4C2312yes—
161C2×C22⋊C4C2311yescharacteristic
162C2×C22⋊C4C2312yes—
163C2×C22⋊C4C2311yescharacteristic
164C2×C22⋊C4C2312yes—
165C2×C22⋊C4C2311yescharacteristic
166C2×C22⋊C4C2312yes—
167C2×C22⋊C4C2311yescharacteristic
168C2×C22⋊C4C2312yes—
169C2×C22⋊C4C2312yes—
170C2×C22⋊C4C2312yes—
171C2×C22⋊C4C2312yes—
172C2×C22⋊C4C2312yes—
173C2×C22⋊C4C2312yes—
174C2×C22⋊C4C2312yes—
175C2×C22⋊C4C2312yes—
176C2×C22⋊C4C2312yes—
177C2×C22⋊C4C2312yes—
178C2×C42C2311yescharacteristic
179C2×C42C2312yes—
180C2×C42C2312yes—
181#95C2312yes—
182#95C2312yes—
183#95C2312yes—
184#95C2312yes—
185#95C2311yescharacteristic
186#95C2312yes—
187#95C2312yes—
188#95C2312yes—
189#95C2312yes—
190#95C2312yes—
191#95C2312yes—
192#95C2312yes—
193#95C2312yes—
194#95C2311yescharacteristic
195#95C2312yes—
196#95C2311yescharacteristic
197#95C2311yescharacteristic
198#95C2312yes—
199#95C2312yes—
200#95C2312yes—
201#95C2312yes—
202#95C2312yes—
203#95C2312yes—
204#95C2312yes—
205#95C2312yes—
206#95C2312yes—
207#95C2312yes—
208C24C2411yescharacteristic
209C24C2411yescharacteristic
210C24C2412yes—
211C2×Q8—24no—
212C2×Q8—24no—
213C2×Q8—24no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—41no—
219C2×Q8—416no—
220C2×D4—22no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—24no—
224C2×D4—24no—
225C2×D4—22no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4C2411yescharacteristic
249C22×C4C2412yes—
250C22×C4C2412yes—
251C22×C4C2412yes—
252C22×C4C2412yes—
253C22×C4C2412yes—
254C22×C4C2412yes—
255C22×C4C2411yescharacteristic
256C22×C4C2412yes—
257C22×C4C2412yes—
258C22×C4C2412yes—
259C22×C4C2412yes—
260C22×C4C2411yescharacteristic
261C22×C4C2412yes—
262C22×C4C2411yescharacteristic
263C22×C4C2411yescharacteristic
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C22⋊C4—24no—
287C22⋊C4—22no—
288C22⋊C4—24no—
289C22⋊C4—22no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C42—22no—
308C42—42no—
309C42—42no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—21no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—21no—
319C23—22no—
320C23—22no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—22no—
325C23H(5,𝔽2)12yes—
326C232−1+414yes—
327C23C2511yescharacteristic
328Q8—48no—
329Q8—48no—
330Q8—82no—
331D4—44no—
332D4—48no—
333D4—44no—
334D4—84no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—84no—
380C2×C4—82no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4H(5,𝔽2)12yes—
384C2×C4H(5,𝔽2)14yes—
385C2×C42−1+412yes—
386C22—24no—
387C22—22no—
388C22—24no—
389C22—24no—
390C22—24no—
391C22—41no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—41no—
396C22—42no—
397C22—42no—
398C22—41no—
399C22—41no—
400C22—42no—
401C22—41no—
402C22—42no—
403C22C2×H(5,𝔽2)11yescharacteristic
404C22C2×H(5,𝔽2)12yes—
405C22C2×H(5,𝔽2)11yescharacteristic
406C22C2×2−1+412yes—
407C22C2×2−1+411yescharacteristic
408C4—24no—
409C4—22no—
410C4—22no—
411C4—41no—
412C4—42no—
413C4—42no—
414C4—41no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4—41no—
423C4—42no—
424C2—44no—
425C2—42no—
426C2—81no—
427C2#353211yescharacteristic
428C2#353611yescharacteristic
429C2#353612yes—
430C2#353811yescharacteristic
431C2#355212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radical#2535C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2835C2semidirect
2. #2831C2semidirect
3. #2788C2semidirect
4. #2755C2semidirect
5. #2722C2semidirect
6. #2696C2semidirect
7. #2695C2semidirect
8. #2693C2semidirect
9. #2691C2semidirect
10. #2687C2semidirect
11. #2683C2semidirect
12. #2659C2semidirect
13. #2638C2semidirect
15. #2463C2semidirect
16. #2450C2semidirect
17. #2425C2semidirect
18. #2409C2semidirect
19. #2409C2semidirect

Maximal subgroups

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

subgroupindex
#28352
#28312
#27882
#27552
#27222
#26962
#26952
#26932
#26912
#26872
#26832
#26592
#26382
#25352
#24632
#24502
#24252
#24092