Finite Group Databasebeta

← #52279

Subgroups — #52279

Order 256.

883 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2810C211yesmaximal, characteristic, complemented
2#2809C211yesmaximal, characteristic, complemented
3#2787C212yesmaximal, complemented
4#2787C212yesmaximal, complemented
5#2771C212yesmaximal, complemented
6#2750C212yesmaximal, complemented
7#2732C211yesmaximal, characteristic, complemented
8#2728C212yesmaximal, complemented
9#2724C211yesmaximal, characteristic, complemented
10#2708C212yesmaximal
11#2402C212yesmaximal, complemented
12#2401C212yesmaximal, complemented
13#2395C212yesmaximal, complemented
14C4×C22⋊Q8C211yesmaximal, characteristic, complemented
15C4×C22⋊Q8C212yesmaximal, complemented
16C4×#123C212yesmaximal, complemented
17C4×#121C211yesmaximal, characteristic
18#2296C211yesmaximal, characteristic, complemented
19#2290C212yesmaximal, complemented
20C2×#123C2212yes—
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2212yes—
23C2×#121C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×C4×D4C2212yes—
26C2×C4×D4C2211yescharacteristic
27C2×#117C2211yescharacteristic
28C2×#117C2211yescharacteristic
29C22×C42C2212yes—
30C22×C42C2211yescharacteristic
31#401C2212yes—
32#400C2212yes—
33#400C2212yes—
34#400C2212yes—
35#400C2212yes—
36#400C2212yes—
37#400C2212yes—
38#400C2212yes—
39#400C2212yes—
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#394C2212yes—
54#394C2212yes—
55#391C2212yes—
56#391C2211yescharacteristic
57#390C2211yescharacteristic
58#389C2211yescharacteristic
59#389C2211yescharacteristic
60#389C2211yescharacteristic
61#389C2211yescharacteristic
62#389C2212yes—
63#389C2211yescharacteristic
64#389C2211yescharacteristic
65#388C2212yes—
66#388C2212yes—
67#388C2212yes—
68#388C2212yes—
69#388C2212yes—
70#387C2212yes—
71#387C2212yes—
72#387C2212yes—
73#387C2212yes—
74#387C2212yes—
75#387C2212yes—
76#387C2212yes—
77#386C2212yes—
78#386C2212yes—
79#386C2212yes—
80#386C2212yes—
81#385C2212yes—
82#385C2212yes—
83#385C2212yes—
84#385C2212yes—
85#385C2212yes—
86#385C2212yes—
87#385C2212yes—
88#385C2212yes—
89#381C2212yes—
90#380C2212yes—
91#380C2211yescharacteristic
92#380C2211yescharacteristic
93C4×C4⋊C4C2212yes—
94C4×C4⋊C4C2212yes—
95C4×C4⋊C4C2211yescharacteristic
96C4×C4⋊C4C2212yes—
97C4×C4⋊C4C2211yescharacteristic
98C4×C4⋊C4C2212yes—
99C4×C4⋊C4C2212yes—
100C4×C4⋊C4C2212yes—
101C4×C22⋊C4C2212yes—
102C4×C22⋊C4C2212yes—
103C4×C22⋊C4C2212yes—
104C4×C22⋊C4C2212yes—
105C4×C22⋊C4C2212yes—
106#376C2212yes—
107C22×Q8C2311yescharacteristic
108C22×D4C2312yes—
109C22×D4C2311yescharacteristic
110C23×C4C2312yes—
111C23×C4C2311yescharacteristic
112C23×C4C2311yescharacteristic
113C23×C4C2311yescharacteristic
114C23×C4C2312yes—
115C23×C4C2312yes—
116#123—28no—
117C22⋊Q8—24no—
118C22⋊Q8—28no—
119#121—24no—
120C4×Q8—42no—
121C4×D4—28no—
122C4×D4—42no—
123#117—24no—
124#117—42no—
125C2×C4⋊C4C2311yescharacteristic
126C2×C4⋊C4C2311yescharacteristic
127C2×C4⋊C4C2312yes—
128C2×C4⋊C4C2311yescharacteristic
129C2×C4⋊C4C2312yes—
130C2×C4⋊C4C2312yes—
131C2×C4⋊C4C2312yes—
132C2×C4⋊C4C2311yescharacteristic
133C2×C4⋊C4C2312yes—
134C2×C4⋊C4C2312yes—
135C2×C4⋊C4C2312yes—
136C2×C4⋊C4C2311yescharacteristic
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×C4⋊C4C2312yes—
148C2×C4⋊C4C2312yes—
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×C42—44no—
165C2×C42—42no—
166C2×C42C2312yes—
167C2×C42C2312yes—
168C2×C42C2312yes—
169C2×C42C2311yescharacteristic
170C2×C42C2312yes—
171C2×C42C2311yescharacteristic
172C2×C42C2312yes—
173C2×C42C2312yes—
174C2×C42C2311yescharacteristic
175C2×C42C2312yes—
176C2×C42C2311yescharacteristic
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#95C2312yes—
190#95C2312yes—
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—
206C24C2411yescharacteristic
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8#4114yes—
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—42no—
215C2×D4#4114yes—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4#4114yes—
234C22×C4C2412yes—
235C22×C4C2412yes—
236C22×C4C2412yes—
237C22×C4C2412yes—
238C22×C4C2411yescharacteristic
239C22×C4C2412yes—
240C22×C4C2412yes—
241C22×C4C2411yescharacteristic
242C22×C4C2412yes—
243C22×C4C2412yes—
244C22×C4C2412yes—
245C22×C4C2412yes—
246C22×C4C2411yescharacteristic
247C22×C4C2412yes—
248C22×C4C2411yescharacteristic
249C22×C4C2411yescharacteristic
250C22×C4C2411yescharacteristic
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—24no—
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—
290C42—24no—
291C42—24no—
292C42—24no—
293C42—22no—
294C42—24no—
295C42—22no—
296C42—24no—
297C42—41no—
298C42—42no—
299C42—41no—
300C42—42no—
301C23—21no—
302C23—22no—
303C23—22no—
304C23—21no—
305C23—21no—
306C23—22no—
307C23—21no—
308C23—22no—
309C23—21no—
310C23—22no—
311C23—22no—
312C23—21no—
313C23C2×#4114yes—
314C23C2×#4112yes—
315C23C2511yescharacteristic
316Q8—44no—
317D4—44no—
318D4—48no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—28no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
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—21no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—48no—
367C2×C4—48no—
368C2×C4—48no—
369C2×C4C2×#4112yes—
370C2×C4C2×#4114yes—
371C2×C4C2×#4112yes—
372C2×C4C2×#4112yes—
373C2×C4C2×#4112yes—
374C2×C4H(5,𝔽2)12yes—
375C2×C42−1+412yes—
376C22—22no—
377C22—24no—
378C22—22no—
379C22—24no—
380C22—22no—
381C22—24no—
382C22—42no—
383C22—41no—
384C22—42no—
385C22—41no—
386C22—42no—
387C22—41no—
388C22—41no—
389C22—42no—
390C22C22×#4111yescharacteristic
391C22C22×#4111yescharacteristic
392C22C22×#4111yescharacteristic
393C22C2×H(5,𝔽2)11yescharacteristic
394C22C2×2−1+411yescharacteristic
395C22#58511yescharacteristic
396C22#58511yescharacteristic
397C4—22no—
398C4—22no—
399C4—22no—
400C4—24no—
401C4—42no—
402C4—41no—
403C4—42no—
404C4—41no—
405C4—42no—
406C4—42no—
407C4—42no—
408C4—42no—
409C4—42no—
410C4—42no—
411C4—42no—
412C4—42no—
413C4#53914yes—
414C2—42no—
415C2—44no—
416C2C2×#53911yescharacteristic
417C2#345911yescharacteristic
418C2#345911yescharacteristic
419C2#347511yescharacteristic
420C2#348011yescharacteristic
421C2#348211yescharacteristic
422C2#353111yescharacteristic
subgroupG/H
centerC22×C4C24
commutator, Frattini, socle, cosocleC23C25
prime radicalC2×#121C22
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. #2810C2semidirect
2. #2809C2semidirect
3. #2787C2semidirect
4. #2787C2semidirect
5. #2771C2semidirect
6. #2750C2semidirect
7. #2732C2semidirect
8. #2728C2semidirect
9. #2724C2semidirect
11. #2402C2semidirect
12. #2401C2semidirect
13. #2395C2semidirect
14. C4×C22⋊Q8C2semidirect
15. C4×C22⋊Q8C2semidirect
16. C4×#123C2semidirect
18. #2296C2semidirect
19. #2290C2semidirect

Maximal subgroups

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

subgroupindex
#28102
#28092
#27872
#27712
#27502
#27322
#27282
#27242
#27082
#24022
#24012
#23952
C4×C22⋊Q82
C4×#1232
C4×#1212
#22962
#22902