Finite Group Databasebeta

← #58959

Subgroups — #58959

Order 256.

857 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2832C211yesmaximal, characteristic, complemented
2#2820C212yesmaximal, complemented
3#2811C211yesmaximal, characteristic, complemented
4#2775C212yesmaximal, complemented
5#2720C212yesmaximal, complemented
6#2712C212yesmaximal, complemented
7#2710C212yesmaximal, complemented
8#2692C212yesmaximal, complemented
9#2683C211yesmaximal, characteristic, complemented
10#2674C212yesmaximal, complemented
11#2672C212yesmaximal, complemented
12#2632C211yesmaximal, characteristic, complemented
13#2620C212yesmaximal, complemented
14#2567C211yesmaximal, characteristic, complemented
15#2556C212yesmaximal, complemented
16#2555C211yesmaximal, characteristic
17#2545C211yesmaximal, characteristic, complemented
18#2474C212yesmaximal, complemented
19#2439C212yesmaximal, complemented
20C2×C4⋊Q8C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic
22C2×#125C2212yes—
23C2×#124C2212yes—
24C2×#124C2212yes—
25C2×#124C2211yescharacteristic
26C2×#124C2211yescharacteristic
27C2×#124C2211yescharacteristic
28C2×#123C2212yes—
29C2×C22⋊Q8C2212yes—
30C2×C22⋊Q8C2212yes—
31C2×C22⋊Q8C2212yes—
32C2×#121C2212yes—
33C2×C4×Q8C2211yescharacteristic
34C2×#117C2212yes—
35#400C2212yes—
36#399C2212yes—
37#399C2212yes—
38#399C2212yes—
39#398C2212yes—
40#397C2212yes—
41#397C2212yes—
42#397C2212yes—
43#396C2212yes—
44#396C2212yes—
45#396C2212yes—
46#396C2212yes—
47#396C2212yes—
48#396C2212yes—
49#396C2212yes—
50#395C2211yescharacteristic
51#395C2212yes—
52#395C2211yescharacteristic
53#395C2211yescharacteristic
54#395C2211yescharacteristic
55#395C2211yescharacteristic
56#395C2211yescharacteristic
57#395C2212yes—
58#395C2212yes—
59#394C2212yes—
60#394C2212yes—
61#394C2212yes—
62#394C2211yescharacteristic
63#394C2211yescharacteristic
64#394C2212yes—
65#394C2212yes—
66#393C2212yes—
67#393C2212yes—
68#393C2212yes—
69#393C2212yes—
70#392C2212yes—
71#391C2212yes—
72#391C2211yescharacteristic
73#391C2212yes—
74#391C2212yes—
75#391C2212yes—
76#391C2212yes—
77#390C2212yes—
78#390C2211yescharacteristic
79#389C2212yes—
80#389C2212yes—
81#389C2212yes—
82#389C2212yes—
83#388C2212yes—
84#388C2212yes—
85#388C2212yes—
86#388C2212yes—
87#388C2212yes—
88#388C2212yes—
89#388C2212yes—
90#388C2212yes—
91#387C2212yes—
92#387C2212yes—
93#387C2212yes—
94#386C2212yes—
95#386C2212yes—
96#386C2212yes—
97#385C2212yes—
98#385C2212yes—
99#385C2212yes—
100#384C2212yes—
101#383C2211yescharacteristic
102#382C2212yes—
103#380C2212yes—
104C4×C4⋊C4C2211yescharacteristic
105C4×C4⋊C4C2211yescharacteristic
106C4×C22⋊C4C2212yes—
107C22×Q8C2311yescharacteristic
108C22×Q8C2312yes—
109C22×Q8C2311yescharacteristic
110C22×Q8C2311yescharacteristic
111C22×Q8C2311yescharacteristic
112C22×Q8C2311yescharacteristic
113C22×D4C2312yes—
114C22×D4C2311yescharacteristic
115C22×D4C2311yescharacteristic
116C22×D4C2312yes—
117C23×C4C2312yes—
118C23×C4C2312yes—
119C4⋊Q8—42no—
120C4⋊Q8—42no—
121#125—44no—
122#124—44no—
123#124—42no—
124#124—42no—
125#124—44no—
126#124—42no—
127#123—44no—
128C22⋊Q8—44no—
129C22⋊Q8—44no—
130C22⋊Q8—44no—
131#121—44no—
132C4×Q8—42no—
133#117—44no—
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×C4⋊C4C2311yescharacteristic
143C2×C4⋊C4C2311yescharacteristic
144C2×C4⋊C4C2311yescharacteristic
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2312yes—
147C2×C4⋊C4C2312yes—
148C2×C4⋊C4C2312yes—
149C2×C4⋊C4C2312yes—
150C2×C4⋊C4C2312yes—
151C2×C4⋊C4C2312yes—
152C2×C4⋊C4C2312yes—
153C2×C4⋊C4C2312yes—
154C2×C4⋊C4C2312yes—
155C2×C4⋊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×C22⋊C4C2312yes—
176C2×C22⋊C4C2312yes—
177C2×C22⋊C4C2312yes—
178C2×C42C2312yes—
179C2×C42C2311yescharacteristic
180C2×C42C2311yescharacteristic
181C2×C42C2311yescharacteristic
182C2×C42C2312yes—
183C2×C42C2312yes—
184C2×C42C2311yescharacteristic
185C2×C42C2311yescharacteristic
186#95C2312yes—
187#95C2312yes—
188#95C2312yes—
189#95C2311yescharacteristic
190#95C2311yescharacteristic
191#95C2312yes—
192#95C2312yes—
193#95C2312yes—
194#95C2312yes—
195#95C2312yes—
196#95C2312yes—
197#95C2312yes—
198#95C2311yescharacteristic
199#95C2311yescharacteristic
200#95C2312yes—
201#95C2312yes—
202#95C2312yes—
203#95C2312yes—
204#95C2312yes—
205#95C2312yes—
206#95C2312yes—
207#95C2312yes—
208#95C2312yes—
209C24C2412yes—
210C24C2412yes—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—24no—
215C2×Q8—41no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—42no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×D4—22no—
227C2×D4—24no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—42no—
236C2×D4—41no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4C2412yes—
244C22×C4C2411yescharacteristic
245C22×C4C2412yes—
246C22×C4C2412yes—
247C22×C4C2412yes—
248C22×C4C2412yes—
249C22×C4C2411yescharacteristic
250C22×C4C2412yes—
251C22×C4C2411yescharacteristic
252C22×C4C2412yes—
253C22×C4C2412yes—
254C22×C4C2411yescharacteristic
255C22×C4C2411yescharacteristic
256C22×C4C2411yescharacteristic
257C22×C4C2412yes—
258C22×C4C2411yescharacteristic
259C22×C4C2412yes—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—42no—
268C4⋊C4—41no—
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—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—24no—
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—
303C22⋊C4—42no—
304C42—24no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—24no—
309C42—42no—
310C42—41no—
311C42—41no—
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—
326C23C2511yescharacteristic
327Q8—44no—
328Q8—84no—
329Q8—82no—
330Q8—82no—
331Q8—82no—
332Q8—82no—
333D4—84no—
334D4—82no—
335D4—84no—
336D4—82no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—22no—
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—21no—
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—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—48no—
380C2×C4—84no—
381C2×C4H(5,𝔽2)12yes—
382C2×C4H(5,𝔽2)12yes—
383C2×C4H(5,𝔽2)14yes—
384C2×C42−1+412yes—
385C2×C42−1+414yes—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22C2×H(5,𝔽2)11yescharacteristic
401C22C2×H(5,𝔽2)11yescharacteristic
402C22C2×H(5,𝔽2)12yes—
403C22C2×2−1+412yes—
404C22C2×2−1+411yescharacteristic
405C4—24no—
406C4—24no—
407C4—22no—
408C4—22no—
409C4—22no—
410C4—42no—
411C4—41no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—41no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—41no—
420C4—41no—
421C4—42no—
422C2—82no—
423C2—82no—
424C2#353612yes—
425C2#353611yescharacteristic
426C2#354411yescharacteristic
427C2#354911yescharacteristic
428C2#355012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radical#2555C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2832C2semidirect
2. #2820C2semidirect
3. #2811C2semidirect
4. #2775C2semidirect
5. #2720C2semidirect
6. #2712C2semidirect
7. #2710C2semidirect
8. #2692C2semidirect
9. #2683C2semidirect
10. #2674C2semidirect
11. #2672C2semidirect
12. #2632C2semidirect
13. #2620C2semidirect
14. #2567C2semidirect
15. #2556C2semidirect
17. #2545C2semidirect
18. #2474C2semidirect
19. #2439C2semidirect

Maximal subgroups

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

subgroupindex
#28322
#28202
#28112
#27752
#27202
#27122
#27102
#26922
#26832
#26742
#26722
#26322
#26202
#25672
#25562
#25552
#25452
#24742
#24392