Finite Group Databasebeta

← #36670

Subgroups — #36670

Order 256.

897 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#2833C211yesmaximal, characteristic, complemented
2#2832C211yesmaximal, characteristic, complemented
3#2831C212yesmaximal, complemented
4#2830C211yesmaximal, characteristic, complemented
5#2829C212yesmaximal, complemented
6#2828C211yesmaximal, characteristic, complemented
7#2827C212yesmaximal, complemented
8#2826C212yesmaximal, complemented
9#2825C212yesmaximal, complemented
10#2824C212yesmaximal, complemented
11#2401C212yesmaximal, complemented
12#2398C212yesmaximal, complemented
13#2396C211yesmaximal, characteristic, complemented
14#2395C212yesmaximal, complemented
15#2392C212yesmaximal, complemented
16#2392C211yesmaximal, characteristic, complemented
17#2388C212yesmaximal, complemented
18#2297C211yesmaximal, characteristic, complemented
19#2290C212yesmaximal, complemented
20C22×#41C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4C2212yes—
23C2×C4×D4C2212yes—
24C2×C4×D4C2212yes—
25C2×C4×D4C2211yescharacteristic
26C2×#117C2212yes—
27C2×#117C2211yescharacteristic
28C22×C42C2211yescharacteristic
29#401C2212yes—
30#400C2212yes—
31#400C2212yes—
32#400C2212yes—
33#400C2212yes—
34#400C2212yes—
35#400C2212yes—
36#400C2211yescharacteristic
37#399C2211yescharacteristic
38#399C2212yes—
39#398C2212yes—
40#398C2212yes—
41#398C2212yes—
42#398C2211yescharacteristic
43#397C2212yes—
44#397C2212yes—
45#397C2212yes—
46#397C2212yes—
47#397C2212yes—
48#397C2212yes—
49#397C2212yes—
50#397C2211yescharacteristic
51#397C2212yes—
52#397C2212yes—
53#396C2212yes—
54#396C2212yes—
55#396C2212yes—
56#396C2212yes—
57#396C2212yes—
58#395C2211yescharacteristic
59#395C2212yes—
60#394C2212yes—
61#394C2212yes—
62#394C2212yes—
63#394C2212yes—
64#394C2212yes—
65#394C2212yes—
66#394C2212yes—
67#394C2212yes—
68#394C2212yes—
69#394C2212yes—
70#394C2211yescharacteristic
71#394C2212yes—
72#393C2211yescharacteristic
73#393C2212yes—
74#393C2212yes—
75#393C2212yes—
76#392C2212yes—
77#392C2211yescharacteristic
78#392C2212yes—
79#392C2212yes—
80#392C2212yes—
81#391C2211yescharacteristic
82#391C2211yescharacteristic
83#390C2211yescharacteristic
84#390C2211yescharacteristic
85#388C2212yes—
86#388C2212yes—
87#387C2212yes—
88#387C2212yes—
89#386C2212yes—
90#386C2212yes—
91#386C2212yes—
92#386C2212yes—
93#385C2212yes—
94#385C2212yes—
95C4×C4⋊C4C2212yes—
96C4×C4⋊C4C2211yescharacteristic
97C4×C4⋊C4C2211yescharacteristic
98C4×C4⋊C4C2212yes—
99C4×C22⋊C4C2212yes—
100C4×C22⋊C4C2212yes—
101C4×C22⋊C4C2212yes—
102C4×C22⋊C4C2212yes—
103C4×C22⋊C4C2212yes—
104C4×C22⋊C4C2212yes—
105C4×C22⋊C4C2212yes—
106#376C2212yes—
107C2×#41—24no—
108C2×#41—42no—
109C2×#41—42no—
110C22×Q8C2311yescharacteristic
111C22×Q8C2311yescharacteristic
112C22×D4C2312yes—
113C22×D4C2312yes—
114C22×D4C2311yescharacteristic
115C22×D4C2312yes—
116C22×D4C2312yes—
117C22×D4C2311yescharacteristic
118C23×C4C2312yes—
119C23×C4C2312yes—
120C23×C4C2311yescharacteristic
121C23×C4C2312yes—
122C4×Q8—42no—
123C4×D4—28no—
124C4×D4—44no—
125C4×D4—42no—
126C4×D4—44no—
127#117—44no—
128#117—42no—
129C2×C4⋊C4C2312yes—
130C2×C4⋊C4C2312yes—
131C2×C4⋊C4C2312yes—
132C2×C4⋊C4C2311yescharacteristic
133C2×C4⋊C4C2311yescharacteristic
134C2×C4⋊C4C2312yes—
135C2×C4⋊C4C2312yes—
136C2×C4⋊C4C2311yescharacteristic
137C2×C4⋊C4C2311yescharacteristic
138C2×C4⋊C4C2312yes—
139C2×C4⋊C4C2312yes—
140C2×C4⋊C4C2312yes—
141C2×C4⋊C4C2312yes—
142C2×C4⋊C4C2312yes—
143C2×C4⋊C4C2312yes—
144C2×C4⋊C4C2312yes—
145C2×C22⋊C4C2312yes—
146C2×C22⋊C4C2312yes—
147C2×C22⋊C4C2311yescharacteristic
148C2×C22⋊C4C2311yescharacteristic
149C2×C22⋊C4C2312yes—
150C2×C22⋊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×C42—42no—
171C2×C42C2312yes—
172C2×C42C2311yescharacteristic
173C2×C42C2311yescharacteristic
174C2×C42C2312yes—
175C2×C42C2312yes—
176C2×C42C2312yes—
177C2×C42C2311yescharacteristic
178C2×C42C2311yescharacteristic
179#95C2312yes—
180#95C2312yes—
181#95C2312yes—
182#95C2312yes—
183#95C2312yes—
184#95C2312yes—
185#95C2312yes—
186#95C2312yes—
187#95C2312yes—
188#95C2312yes—
189#95C2312yes—
190#95C2312yes—
191#95C2312yes—
192#95C2312yes—
193#95C2312yes—
194#95C2312yes—
195#95C2312yes—
196#95C2312yes—
197#95C2312yes—
198#95C2312yes—
199#95C2312yes—
200#95C2311yescharacteristic
201#95C2311yescharacteristic
202#95C2312yes—
203#95C2311yescharacteristic
204#95C2311yescharacteristic
205C24C2412yes—
206C24C2412yes—
207C24C2411yescharacteristic
208#41—416no—
209C2×Q8—22no—
210C2×Q8—41no—
211C2×Q8—42no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—22no—
217C2×D4—42no—
218C2×D4—41no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—42no—
230C2×D4—42no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4#4114yes—
243C22×C4C2412yes—
244C22×C4C2412yes—
245C22×C4C2412yes—
246C22×C4C2412yes—
247C22×C4C2412yes—
248C22×C4C2412yes—
249C22×C4C2411yescharacteristic
250C22×C4C2412yes—
251C22×C4C2411yescharacteristic
252C22×C4C2412yes—
253C22×C4C2411yescharacteristic
254C22×C4C2412yes—
255C22×C4C2411yescharacteristic
256C22×C4C2412yes—
257C22×C4C2411yescharacteristic
258C22×C4C2411yescharacteristic
259C4⋊C4—24no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—42no—
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—41no—
294C22⋊C4—41no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C42—24no—
301C42—24no—
302C42—24no—
303C42—22no—
304C42—22no—
305C42—24no—
306C42—41no—
307C42—41no—
308C23—21no—
309C23—21no—
310C23—22no—
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—
326C23C2×#4112yes—
327C23C2511yescharacteristic
328Q8—44no—
329Q8—44no—
330D4—44no—
331D4—48no—
332D4—48no—
333D4—48no—
334D4—44no—
335D4—48no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—28no—
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—21no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
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—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—22no—
377C2×C4—48no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4C2×#4112yes—
381C2×C4C2×#4112yes—
382C2×C4H(5,𝔽2)12yes—
383C2×C4H(5,𝔽2)12yes—
384C22—22no—
385C22—24no—
386C22—42no—
387C22—41no—
388C22—41no—
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—
403C22C22×#4111yescharacteristic
404C22C2×H(5,𝔽2)11yescharacteristic
405C22C2×H(5,𝔽2)11yescharacteristic
406C22#58512yes—
407C22#58512yes—
408C4—22no—
409C4—42no—
410C4—42no—
411C4—42no—
412C4—42no—
413C4—41no—
414C4—42no—
415C4—41no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—41no—
420C4—42no—
421C4—42no—
422C4—41no—
423C4#53814yes—
424C2—42no—
425C2—82no—
426C2—82no—
427C2C2×#53811yescharacteristic
428C2#345512yes—
429C2#352911yescharacteristic
430C2#352912yes—
431C2#352911yescharacteristic
subgroupG/H
centerC22×C4C24
commutator, Frattini, socle, cosocleC23C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2833C2semidirect
2. #2832C2semidirect
3. #2831C2semidirect
4. #2830C2semidirect
5. #2829C2semidirect
6. #2828C2semidirect
7. #2827C2semidirect
8. #2826C2semidirect
9. #2825C2semidirect
10. #2824C2semidirect
11. #2401C2semidirect
12. #2398C2semidirect
13. #2396C2semidirect
14. #2395C2semidirect
15. #2392C2semidirect
16. #2392C2semidirect
17. #2388C2semidirect
18. #2297C2semidirect
19. #2290C2semidirect

Maximal subgroups

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

subgroupindex
#28332
#28322
#28312
#28302
#28292
#28282
#28272
#28262
#28252
#28242
#24012
#23982
#23962
#23952
#23922
#23882
#22972
#22902