Finite Group Databasebeta

← #54848

Subgroups — #54848

Order 256.

842 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2843C211yesmaximal, characteristic, complemented
2#2816C212yesmaximal, complemented
3#2804C211yesmaximal, characteristic, complemented
4#2795C211yesmaximal, characteristic, complemented
5#2756C212yesmaximal, complemented
6#2743C212yesmaximal, complemented
7#2738C212yesmaximal, complemented
8#2699C212yesmaximal, complemented
9#2696C212yesmaximal, complemented
10#2672C212yesmaximal, complemented
11#2671C212yesmaximal, complemented
12#2632C212yesmaximal, complemented
13#2627C211yesmaximal, characteristic, complemented
14#2591C211yesmaximal, characteristic, complemented
15#2572C212yesmaximal, complemented
16#2555C211yesmaximal, characteristic, complemented
17#2501C211yesmaximal, characteristic, complemented
18#2458C212yesmaximal, complemented
19#2423C212yesmaximal, complemented
20C2×#124C2212yes—
21C2×#123C2212yes—
22C2×#123C2212yes—
23C2×C22⋊Q8C2212yes—
24C2×C22⋊Q8C2211yescharacteristic
25C2×#121C2212yes—
26C2×#121C2211yescharacteristic
27C2×#121C2212yes—
28C2×#121C2211yescharacteristic
29C2×C4×Q8C2212yes—
30C22×C4⋊C4C2211yescharacteristic
31#400C2212yes—
32#400C2211yescharacteristic
33#400C2212yes—
34#400C2211yescharacteristic
35#400C2211yescharacteristic
36#400C2211yescharacteristic
37#399C2212yes—
38#399C2212yes—
39#399C2212yes—
40#399C2212yes—
41#398C2212yes—
42#398C2212yes—
43#398C2212yes—
44#398C2212yes—
45#398C2212yes—
46#397C2212yes—
47#396C2212yes—
48#395C2211yescharacteristic
49#395C2212yes—
50#395C2211yescharacteristic
51#394C2211yescharacteristic
52#394C2212yes—
53#394C2212yes—
54#394C2212yes—
55#394C2212yes—
56#394C2212yes—
57#394C2212yes—
58#394C2211yescharacteristic
59#394C2212yes—
60#394C2212yes—
61#393C2212yes—
62#393C2212yes—
63#393C2212yes—
64#393C2212yes—
65#392C2212yes—
66#392C2212yes—
67#391C2212yes—
68#390C2212yes—
69#390C2211yescharacteristic
70#390C2211yescharacteristic
71#389C2212yes—
72#388C2212yes—
73#388C2212yes—
74#388C2212yes—
75#388C2212yes—
76#388C2212yes—
77#388C2212yes—
78#388C2212yes—
79#388C2212yes—
80#388C2212yes—
81#388C2212yes—
82#387C2212yes—
83#387C2212yes—
84#387C2212yes—
85#386C2212yes—
86#386C2212yes—
87#386C2212yes—
88#386C2212yes—
89#386C2212yes—
90#386C2212yes—
91#385C2212yes—
92#385C2212yes—
93#385C2212yes—
94#384C2211yescharacteristic
95#384C2211yescharacteristic
96#383C2212yes—
97#383C2212yes—
98#381C2212yes—
99#381C2212yes—
100#381C2212yes—
101#380C2211yescharacteristic
102#380C2212yes—
103#380C2212yes—
104C4×C22⋊C4C2211yescharacteristic
105C4×C22⋊C4C2212yes—
106C4×C22⋊C4C2211yescharacteristic
107C22×Q8C2312yes—
108C22×Q8C2311yescharacteristic
109C22×D4C2312yes—
110C22×D4C2312yes—
111C22×D4C2311yescharacteristic
112C22×D4C2311yescharacteristic
113C22×D4C2312yes—
114C22×D4C2312yes—
115C23×C4C2311yescharacteristic
116C23×C4C2311yescharacteristic
117C23×C4C2311yescharacteristic
118C23×C4C2312yes—
119C23×C4C2312yes—
120#124—44no—
121#123—44no—
122#123—44no—
123C22⋊Q8—42no—
124C22⋊Q8—44no—
125#121—44no—
126#121—44no—
127#121—42no—
128#121—42no—
129C4×Q8—44no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4C2312yes—
132C2×C4⋊C4C2312yes—
133C2×C4⋊C4C2311yescharacteristic
134C2×C4⋊C4C2311yescharacteristic
135C2×C4⋊C4C2311yescharacteristic
136C2×C4⋊C4C2311yescharacteristic
137C2×C4⋊C4C2312yes—
138C2×C4⋊C4C2312yes—
139C2×C4⋊C4C2311yescharacteristic
140C2×C4⋊C4C2311yescharacteristic
141C2×C4⋊C4C2312yes—
142C2×C4⋊C4C2312yes—
143C2×C4⋊C4C2312yes—
144C2×C4⋊C4C2312yes—
145C2×C4⋊C4C2311yescharacteristic
146C2×C4⋊C4C2312yes—
147C2×C4⋊C4C2312yes—
148C2×C4⋊C4C2312yes—
149C2×C4⋊C4C2312yes—
150C2×C4⋊C4C2312yes—
151C2×C22⋊C4C2312yes—
152C2×C22⋊C4C2312yes—
153C2×C22⋊C4C2312yes—
154C2×C22⋊C4C2312yes—
155C2×C22⋊C4C2312yes—
156C2×C22⋊C4C2312yes—
157C2×C22⋊C4C2312yes—
158C2×C22⋊C4C2312yes—
159C2×C22⋊C4C2312yes—
160C2×C22⋊C4C2312yes—
161C2×C22⋊C4C2312yes—
162C2×C22⋊C4C2312yes—
163C2×C22⋊C4C2312yes—
164C2×C22⋊C4C2312yes—
165C2×C22⋊C4C2312yes—
166C2×C22⋊C4C2312yes—
167C2×C22⋊C4C2312yes—
168C2×C22⋊C4C2312yes—
169C2×C22⋊C4C2312yes—
170C2×C22⋊C4C2312yes—
171C2×C22⋊C4C2312yes—
172C2×C22⋊C4C2312yes—
173C2×C22⋊C4C2312yes—
174C2×C22⋊C4C2312yes—
175C2×C42C2312yes—
176C2×C42C2311yescharacteristic
177C2×C42C2312yes—
178C2×C42C2312yes—
179C2×C42C2311yescharacteristic
180#95C2312yes—
181#95C2312yes—
182#95C2312yes—
183#95C2311yescharacteristic
184#95C2311yescharacteristic
185#95C2312yes—
186#95C2312yes—
187#95C2312yes—
188#95C2311yescharacteristic
189#95C2312yes—
190#95C2311yescharacteristic
191#95C2312yes—
192#95C2312yes—
193#95C2312yes—
194#95C2312yes—
195#95C2312yes—
196#95C2312yes—
197#95C2312yes—
198#95C2312yes—
199#95C2312yes—
200#95C2312yes—
201#95C2312yes—
202#95C2312yes—
203#95C2312yes—
204#95C2312yes—
205C24C2412yes—
206C24C2412yes—
207C24C2411yescharacteristic
208C2×Q8—24no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—41no—
212C2×Q8—42no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—22no—
216C2×D4—41no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—41no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—41no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4C2411yescharacteristic
244C22×C4C2412yes—
245C22×C4C2412yes—
246C22×C4C2412yes—
247C22×C4C2412yes—
248C22×C4C2412yes—
249C22×C4C2412yes—
250C22×C4C2412yes—
251C22×C4C2412yes—
252C22×C4C2411yescharacteristic
253C22×C4C2411yescharacteristic
254C22×C4C2411yescharacteristic
255C22×C4C2412yes—
256C22×C4C2411yescharacteristic
257C22×C4C2412yes—
258C22×C4C2411yescharacteristic
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C42—24no—
304C42—22no—
305C42—41no—
306C42—42no—
307C42—42no—
308C23—22no—
309C23—21no—
310C23—21no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23H(5,𝔽2)12yes—
327C23C2511yescharacteristic
328Q8—48no—
329Q8—82no—
330D4—82no—
331D4—84no—
332D4—84no—
333D4—84no—
334D4—82no—
335D4—84no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
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—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—21no—
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—22no—
377C2×C4—22no—
378C2×C4—44no—
379C2×C4—84no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—84no—
383C2×C4H(5,𝔽2)14yes—
384C2×C42−1+412yes—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22C2×H(5,𝔽2)12yes—
405C22C2×H(5,𝔽2)11yescharacteristic
406C22C2×2−1+411yescharacteristic
407C22#58511yescharacteristic
408C22#58512yes—
409C4—24no—
410C4—22no—
411C4—42no—
412C4—42no—
413C4—42no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4—42no—
423C4—42no—
424C4—42no—
425C2—42no—
426C2—82no—
427C2—82no—
428C2#353312yes—
429C2#353511yescharacteristic
430C2#354311yescharacteristic
431C2#354511yescharacteristic
432C2#354712yes—
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. #2843C2semidirect
2. #2816C2semidirect
3. #2804C2semidirect
4. #2795C2semidirect
5. #2756C2semidirect
6. #2743C2semidirect
7. #2738C2semidirect
8. #2699C2semidirect
9. #2696C2semidirect
10. #2672C2semidirect
11. #2671C2semidirect
12. #2632C2semidirect
13. #2627C2semidirect
14. #2591C2semidirect
15. #2572C2semidirect
16. #2555C2semidirect
17. #2501C2semidirect
18. #2458C2semidirect
19. #2423C2semidirect

Maximal subgroups

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

subgroupindex
#28432
#28162
#28042
#27952
#27562
#27432
#27382
#26992
#26962
#26722
#26712
#26322
#26272
#25912
#25722
#25552
#25012
#24582
#24232