Finite Group Databasebeta

← #52799

Subgroups — #52799

Order 256.

884 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2843C211yesmaximal, characteristic, complemented
2#2838C212yesmaximal, complemented
3#2828C211yesmaximal, characteristic, complemented
4#2827C212yesmaximal, complemented
5#2826C212yesmaximal, complemented
6#2795C211yesmaximal, characteristic, complemented
7#2790C212yesmaximal, complemented
8#2784C211yesmaximal, characteristic, complemented
9#2749C212yesmaximal, complemented
10#2746C212yesmaximal, complemented
11#2618C212yesmaximal, complemented
12#2543C212yesmaximal, complemented
13#2510C212yesmaximal, complemented
14#2501C211yesmaximal, characteristic, complemented
15#2430C212yesmaximal, complemented
16#2425C212yesmaximal, complemented
17#2419C212yesmaximal, complemented
18#2406C211yesmaximal, characteristic, complemented
19C2×#400C211yesmaximal, characteristic, complemented
20C22×#41C2211yescharacteristic
21C2×#123C2212yes—
22C2×#123C2212yes—
23C2×#123C2212yes—
24C2×#121C2212yes—
25C2×#121C2212yes—
26C2×#121C2211yescharacteristic
27C2×C4×D4C2212yes—
28C22×C4⋊C4C2211yescharacteristic
29C22×C4⋊C4C2211yescharacteristic
30#401C2212yes—
31#400C2212yes—
32#400C2212yes—
33#400C2211yescharacteristic
34#400C2212yes—
35#400C2211yescharacteristic
36#400C2211yescharacteristic
37#400C2212yes—
38#400C2211yescharacteristic
39#399C2212yes—
40#399C2212yes—
41#398C2212yes—
42#398C2212yes—
43#398C2212yes—
44#398C2212yes—
45#398C2212yes—
46#397C2212yes—
47#397C2212yes—
48#397C2212yes—
49#397C2212yes—
50#397C2212yes—
51#397C2212yes—
52#396C2212yes—
53#396C2212yes—
54#396C2212yes—
55#396C2212yes—
56#396C2212yes—
57#395C2211yescharacteristic
58#395C2212yes—
59#395C2211yescharacteristic
60#394C2211yescharacteristic
61#394C2212yes—
62#394C2212yes—
63#394C2212yes—
64#394C2212yes—
65#394C2212yes—
66#394C2211yescharacteristic
67#393C2212yes—
68#392C2212yes—
69#392C2212yes—
70#390C2211yescharacteristic
71#390C2211yescharacteristic
72#389C2212yes—
73#388C2212yes—
74#388C2212yes—
75#388C2212yes—
76#388C2212yes—
77#388C2212yes—
78#387C2212yes—
79#387C2212yes—
80#387C2212yes—
81#386C2212yes—
82#386C2212yes—
83#386C2212yes—
84#386C2212yes—
85#386C2212yes—
86#386C2212yes—
87#386C2212yes—
88#386C2212yes—
89#386C2212yes—
90#385C2212yes—
91#385C2212yes—
92#385C2212yes—
93#385C2212yes—
94#385C2212yes—
95#385C2212yes—
96#384C2211yescharacteristic
97#384C2211yescharacteristic
98#383C2212yes—
99#381C2211yescharacteristic
100#381C2212yes—
101#380C2212yes—
102#380C2212yes—
103C4×C22⋊C4C2211yescharacteristic
104C2×#95C2212yes—
105C2×#95C2211yescharacteristic
106C2×#95C2212yes—
107C2×#41—42no—
108C2×#41—44no—
109C22×Q8C2311yescharacteristic
110C22×D4C2312yes—
111C22×D4C2312yes—
112C22×D4C2311yescharacteristic
113C22×D4C2312yes—
114C22×D4C2312yes—
115C22×D4C2311yescharacteristic
116C23×C4C2311yescharacteristic
117C23×C4C2311yescharacteristic
118C23×C4C2311yescharacteristic
119C23×C4C2312yes—
120C23×C4C2312yes—
121C23×C4C2312yes—
122C23×C4C2312yes—
123#123—44no—
124#123—44no—
125#123—44no—
126#121—42no—
127#121—44no—
128#121—44no—
129C4×D4—44no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4C2311yescharacteristic
133C2×C4⋊C4C2311yescharacteristic
134C2×C4⋊C4C2311yescharacteristic
135C2×C4⋊C4C2311yescharacteristic
136C2×C4⋊C4C2311yescharacteristic
137C2×C4⋊C4C2312yes—
138C2×C4⋊C4C2311yescharacteristic
139C2×C4⋊C4C2312yes—
140C2×C4⋊C4C2311yescharacteristic
141C2×C4⋊C4C2312yes—
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⋊C4C2312yes—
151C2×C4⋊C4C2312yes—
152C2×C4⋊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×C42C2311yescharacteristic
176C2×C42C2312yes—
177C2×C42C2311yescharacteristic
178#95C2311yescharacteristic
179#95C2312yes—
180#95C2312yes—
181#95C2312yes—
182#95C2312yes—
183#95C2312yes—
184#95C2312yes—
185#95C2312yes—
186#95C2312yes—
187#95C2311yescharacteristic
188#95C2312yes—
189#95C2312yes—
190#95C2312yes—
191#95C2312yes—
192#95C2312yes—
193#95C2312yes—
194#95C2311yescharacteristic
195#95C2312yes—
196#95C2311yescharacteristic
197#95C2312yes—
198#95C2312yes—
199#95C2312yes—
200#95C2312yes—
201#95C2312yes—
202#95C2312yes—
203#95C2312yes—
204#95C2312yes—
205C24C2412yes—
206C24C2412yes—
207C24C2411yescharacteristic
208#41—416no—
209C2×Q8—42no—
210C2×Q8—41no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—24no—
217C2×D4—22no—
218C2×D4—42no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—41no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—41no—
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×C4C2412yes—
248C22×C4C2411yescharacteristic
249C22×C4C2412yes—
250C22×C4C2412yes—
251C22×C4C2412yes—
252C22×C4C2412yes—
253C22×C4C2412yes—
254C22×C4C2412yes—
255C22×C4C2411yescharacteristic
256C22×C4C2411yescharacteristic
257C22×C4C2412yes—
258C22×C4C2412yes—
259C22×C4C2412yes—
260C22×C4C2411yescharacteristic
261C22×C4C2411yescharacteristic
262C22×C4C2411yescharacteristic
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—24no—
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—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C42—22no—
307C42—41no—
308C42—42no—
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)14yes—
327C23H(5,𝔽2)12yes—
328C23C2511yescharacteristic
329Q8—44no—
330D4—44no—
331D4—48no—
332D4—48no—
333D4—82no—
334D4—84no—
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—21no—
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—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—21no—
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—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—44no—
380C2×C4—48no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C42−1+412yes—
387C22—22no—
388C22—22no—
389C22—24no—
390C22—24no—
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—
404C22—42no—
405C22#56014yes—
406C22C2×H(5,𝔽2)11yescharacteristic
407C22C2×H(5,𝔽2)12yes—
408C22C2×2−1+411yescharacteristic
409C22#58511yescharacteristic
410C22#58512yes—
411C4—22no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—42no—
417C4—42no—
418C4—41no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4—42no—
423C4—42no—
424C4—42no—
425C4—42no—
426C4—42no—
427C2—24no—
428C2—82no—
429C2—82no—
430C2C2×#56011yescharacteristic
431C2#353511yescharacteristic
432C2#354511yescharacteristic
433C2#355912yes—
434C2#356012yes—
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. #2838C2semidirect
3. #2828C2semidirect
4. #2827C2semidirect
5. #2826C2semidirect
6. #2795C2semidirect
7. #2790C2semidirect
8. #2784C2semidirect
9. #2749C2semidirect
10. #2746C2semidirect
11. #2618C2semidirect
12. #2543C2semidirect
13. #2510C2semidirect
14. #2501C2semidirect
15. #2430C2semidirect
16. #2425C2semidirect
17. #2419C2semidirect
18. #2406C2semidirect
19. C2×#400C2semidirect

Maximal subgroups

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

subgroupindex
#28432
#28382
#28282
#28272
#28262
#27952
#27902
#27842
#27492
#27462
#26182
#25432
#25102
#25012
#24302
#24252
#24192
#24062
C2×#4002