Finite Group Databasebeta

← #43432

Subgroups — #43432

Order 256.

919 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2794C211yesmaximal, characteristic, complemented
2#2741C212yesmaximal, complemented
3#2728C211yesmaximal, characteristic, complemented
4#2688C212yesmaximal, complemented
5#2688C212yesmaximal, complemented
6#2598C211yesmaximal, characteristic, complemented
7#2577C212yesmaximal, complemented
8#2576C212yesmaximal, complemented
9#2572C211yesmaximal, characteristic, complemented
10#2537C211yesmaximal, characteristic, complemented
11#2510C212yesmaximal, complemented
12#2499C211yesmaximal, characteristic, complemented
13#2493C212yesmaximal, complemented
14#2463C212yesmaximal, complemented
15#2450C212yesmaximal, complemented
16#2392C211yesmaximal, characteristic, complemented
17#2364C212yesmaximal, complemented
18#2364C212yesmaximal, complemented
19#2325C212yesmaximal, complemented
20C2×C22⋊Q8C2212yes—
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121C2212yes—
23C2×#121C2212yes—
24C2×#121C2212yes—
25C2×C4×D4C2211yescharacteristic
26C2×C4×D4C2212yes—
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2212yes—
29C2×C4×D4C2212yes—
30C2×#117C2211yescharacteristic
31#401C2212yes—
32#400C2211yescharacteristic
33#400C2212yes—
34#400C2212yes—
35#400C2212yes—
36#400C2212yes—
37#399C2211yescharacteristic
38#398C2212yes—
39#398C2211yescharacteristic
40#397C2212yes—
41#397C2212yes—
42#397C2211yescharacteristic
43#397C2212yes—
44#397C2212yes—
45#396C2212yes—
46#394C2211yescharacteristic
47#394C2212yes—
48#394C2211yescharacteristic
49#394C2212yes—
50#394C2212yes—
51#393C2212yes—
52#392C2211yescharacteristic
53#392C2211yescharacteristic
54#392C2212yes—
55#391C2211yescharacteristic
56#391C2211yescharacteristic
57#388C2212yes—
58#388C2212yes—
59#388C2212yes—
60#388C2212yes—
61#388C2212yes—
62#388C2212yes—
63#388C2212yes—
64#387C2212yes—
65#387C2212yes—
66#387C2212yes—
67#387C2212yes—
68#387C2212yes—
69#387C2212yes—
70#387C2212yes—
71#387C2212yes—
72#387C2212yes—
73#386C2212yes—
74#386C2212yes—
75#386C2212yes—
76#386C2212yes—
77#386C2212yes—
78#386C2212yes—
79#386C2212yes—
80#386C2212yes—
81#386C2212yes—
82#386C2212yes—
83#386C2212yes—
84#386C2212yes—
85#385C2212yes—
86#385C2212yes—
87#385C2212yes—
88#385C2212yes—
89#385C2212yes—
90#384C2211yescharacteristic
91#383C2212yes—
92#383C2211yescharacteristic
93#382C2211yescharacteristic
94#381C2212yes—
95#381C2212yes—
96#380C2212yes—
97#380C2212yes—
98#380C2212yes—
99C4×C4⋊C4C2211yescharacteristic
100C4×C22⋊C4C2212yes—
101C4×C22⋊C4C2212yes—
102C4×C22⋊C4C2212yes—
103C4×C22⋊C4C2211yescharacteristic
104C4×C22⋊C4C2212yes—
105C2×#95C2212yes—
106C2×#95C2212yes—
107C22×Q8C2311yescharacteristic
108C22×D4C2311yescharacteristic
109C22×D4C2312yes—
110C22×D4C2311yescharacteristic
111C22×D4C2312yes—
112C22×D4C2312yes—
113C22×D4C2312yes—
114C23×C4C2311yescharacteristic
115C23×C4C2312yes—
116C23×C4C2312yes—
117C23×C4C2312yes—
118C23×C4C2312yes—
119C23×C4C2312yes—
120C23×C4C2312yes—
121C22⋊Q8—44no—
122C22⋊Q8—42no—
123#121—28no—
124#121—28no—
125#121—28no—
126C4×D4—42no—
127C4×D4—44no—
128C4×D4—42no—
129C4×D4—44no—
130C4×D4—44no—
131#117—42no—
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⋊C4C2311yescharacteristic
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2312yes—
143C2×C4⋊C4C2311yescharacteristic
144C2×C4⋊C4C2312yes—
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2312yes—
147C2×C22⋊C4C2312yes—
148C2×C22⋊C4C2312yes—
149C2×C22⋊C4C2312yes—
150C2×C22⋊C4C2312yes—
151C2×C22⋊C4C2312yes—
152C2×C22⋊C4C2312yes—
153C2×C22⋊C4C2312yes—
154C2×C22⋊C4C2312yes—
155C2×C22⋊C4C2312yes—
156C2×C22⋊C4C2311yescharacteristic
157C2×C22⋊C4C2311yescharacteristic
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×C42C2311yescharacteristic
170C2×C42C2311yescharacteristic
171C2×C42C2311yescharacteristic
172C2×C42C2312yes—
173C2×C42C2312yes—
174C2×C42C2311yescharacteristic
175C2×C42C2312yes—
176#95C2312yes—
177#95C2312yes—
178#95C2312yes—
179#95C2312yes—
180#95C2312yes—
181#95C2312yes—
182#95C2312yes—
183#95C2312yes—
184#95C2312yes—
185#95C2312yes—
186#95C2312yes—
187#95C2312yes—
188#95C2312yes—
189#95C2311yescharacteristic
190#95C2311yescharacteristic
191#95C2311yescharacteristic
192#95C2312yes—
193#95C2312yes—
194#95C2311yescharacteristic
195#95C2312yes—
196#95C2312yes—
197#95C2312yes—
198#95C2312yes—
199#95C2312yes—
200#95C2312yes—
201#95C2312yes—
202#95C2312yes—
203#95C2312yes—
204#95C2312yes—
205C24C2411yescharacteristic
206C24C2412yes—
207C24C2412yes—
208C2×Q8—22no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—24no—
217C2×D4—42no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4#4118yes—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
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×C4C2412yes—
249C22×C4C2412yes—
250C22×C4C2411yescharacteristic
251C22×C4C2412yes—
252C22×C4C2412yes—
253C22×C4C2411yescharacteristic
254C22×C4C2411yescharacteristic
255C22×C4C2412yes—
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—42no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
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—24no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—41no—
305C42—42no—
306C42—42no—
307C42—42no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—21no—
322C23—21no—
323C23—22no—
324C23—21no—
325C23C2×#4114yes—
326C23C2×#4114yes—
327C23C2511yescharacteristic
328Q8—82no—
329D4—48no—
330D4—44no—
331D4—44no—
332D4—48no—
333D4—48no—
334D4—84no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
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—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
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—84no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4C2×#4114yes—
384C2×C4H(5,𝔽2)12yes—
385C2×C4H(5,𝔽2)12yes—
386C22—24no—
387C22—24no—
388C22—24no—
389C22—24no—
390C22—24no—
391C22—24no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—41no—
401C22—42no—
402C22—42no—
403C22—41no—
404C22—41no—
405C22C22×#4112yes—
406C22C2×H(5,𝔽2)11yescharacteristic
407C22C2×H(5,𝔽2)11yescharacteristic
408C22#58512yes—
409C22#58511yescharacteristic
410C4—24no—
411C4—22no—
412C4—22no—
413C4—41no—
414C4—41no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4—42no—
423C4—41no—
424C4—42no—
425C4—41no—
426C2—44no—
427C2—44no—
428C2—81no—
429C2#348111yescharacteristic
430C2#350512yes—
431C2#352112yes—
432C2#353411yescharacteristic
433C2#355111yescharacteristic
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. #2794C2semidirect
2. #2741C2semidirect
3. #2728C2semidirect
4. #2688C2semidirect
5. #2688C2semidirect
6. #2598C2semidirect
7. #2577C2semidirect
8. #2576C2semidirect
9. #2572C2semidirect
10. #2537C2semidirect
11. #2510C2semidirect
12. #2499C2semidirect
13. #2493C2semidirect
14. #2463C2semidirect
15. #2450C2semidirect
16. #2392C2semidirect
17. #2364C2semidirect
18. #2364C2semidirect
19. #2325C2semidirect

Maximal subgroups

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

subgroupindex
#27942
#27412
#27282
#26882
#25982
#25772
#25762
#25722
#25372
#25102
#24992
#24932
#24632
#24502
#23922
#23642
#23252