Finite Group Databasebeta

← #53183

Subgroups — #53183

Order 256.

798 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2857C211yesmaximal, characteristic, complemented
2#2854C211yesmaximal, characteristic
3#2851C211yesmaximal, characteristic, complemented
4#2808C211yesmaximal, characteristic, complemented
5#2805C211yesmaximal, characteristic, complemented
6#2796C212yesmaximal, complemented
7#2793C212yesmaximal, complemented
8#2792C212yesmaximal, complemented
9#2773C212yesmaximal, complemented
10#2747C212yesmaximal, complemented
11#2741C212yesmaximal, complemented
12#2734C211yesmaximal, characteristic, complemented
13#2732C211yesmaximal, characteristic, complemented
14#2720C212yesmaximal, complemented
15#2717C212yesmaximal, complemented
16#2680C212yesmaximal, complemented
17#2677C212yesmaximal, complemented
18#2674C212yesmaximal, complemented
19#2672C212yesmaximal
20C2×C4⋊Q8C2211yescharacteristic
21C2×C4⋊Q8C2212yes—
22C2×#125C2211yescharacteristic
23C2×#124C2212yes—
24C2×#123C2212yes—
25C2×#123C2212yes—
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×C22⋊Q8C2212yes—
29C2×C22⋊Q8C2212yes—
30C2×#121C2211yescharacteristic
31C22×C42C2211yescharacteristic
32#400C2212yes—
33#400C2212yes—
34#400C2212yes—
35#399C2212yes—
36#399C2212yes—
37#399C2212yes—
38#398C2212yes—
39#398C2212yes—
40#398C2212yes—
41#398C2212yes—
42#398C2212yes—
43#397C2212yes—
44#397C2212yes—
45#397C2212yes—
46#397C2212yes—
47#397C2212yes—
48#397C2212yes—
49#396C2212yes—
50#396C2212yes—
51#395C2212yes—
52#395C2212yes—
53#395C2212yes—
54#395C2212yes—
55#395C2212yes—
56#394C2212yes—
57#394C2212yes—
58#394C2212yes—
59#393C2212yes—
60#391C2211yescharacteristic
61#391C2211yescharacteristic
62#390C2211yescharacteristic
63#390C2211yescharacteristic
64#389C2211yescharacteristic
65#389C2211yescharacteristic
66#389C2211yescharacteristic
67#389C2211yescharacteristic
68#389C2212yes—
69#389C2212yes—
70#389C2212yes—
71#389C2212yes—
72#388C2212yes—
73#388C2212yes—
74#388C2212yes—
75#388C2212yes—
76#388C2212yes—
77#388C2212yes—
78#388C2212yes—
79#388C2212yes—
80#388C2212yes—
81#387C2212yes—
82#387C2212yes—
83#387C2212yes—
84#387C2212yes—
85#387C2212yes—
86#387C2212yes—
87#387C2212yes—
88#387C2212yes—
89#387C2212yes—
90#387C2212yes—
91#387C2212yes—
92#386C2212yes—
93#386C2212yes—
94#385C2212yes—
95#385C2212yes—
96#385C2212yes—
97#385C2212yes—
98#385C2212yes—
99#384C2211yescharacteristic
100#384C2212yes—
101#382C2211yescharacteristic
102#380C2211yescharacteristic
103#380C2211yescharacteristic
104#380C2211yescharacteristic
105#380C2212yes—
106#380C2212yes—
107C22×Q8C2311yescharacteristic
108C22×Q8C2311yescharacteristic
109C22×Q8C2312yes—
110C22×D4C2312yes—
111C22×D4C2311yescharacteristic
112C23×C4C2311yescharacteristic
113C23×C4C2311yescharacteristic
114C23×C4C2311yescharacteristic
115C23×C4C2312yes—
116C4⋊Q8—42no—
117C4⋊Q8—44no—
118#125—42no—
119#124—44no—
120#123—44no—
121#123—44no—
122C22⋊Q8—42no—
123C22⋊Q8—42no—
124C22⋊Q8—44no—
125C22⋊Q8—44no—
126#121—42no—
127C2×C4⋊C4C2312yes—
128C2×C4⋊C4C2312yes—
129C2×C4⋊C4C2312yes—
130C2×C4⋊C4C2312yes—
131C2×C4⋊C4C2311yescharacteristic
132C2×C4⋊C4C2311yescharacteristic
133C2×C4⋊C4C2312yes—
134C2×C4⋊C4C2311yescharacteristic
135C2×C4⋊C4C2312yes—
136C2×C4⋊C4C2311yescharacteristic
137C2×C4⋊C4C2312yes—
138C2×C4⋊C4C2312yes—
139C2×C4⋊C4C2311yescharacteristic
140C2×C4⋊C4C2311yescharacteristic
141C2×C4⋊C4C2312yes—
142C2×C4⋊C4C2312yes—
143C2×C4⋊C4C2312yes—
144C2×C4⋊C4C2312yes—
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2312yes—
147C2×C4⋊C4C2311yescharacteristic
148C2×C4⋊C4C2311yescharacteristic
149C2×C4⋊C4C2312yes—
150C2×C4⋊C4C2312yes—
151C2×C4⋊C4C2312yes—
152C2×C4⋊C4C2312yes—
153C2×C4⋊C4C2312yes—
154C2×C4⋊C4C2311yescharacteristic
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×C42—42no—
174C2×C42C2311yescharacteristic
175C2×C42C2311yescharacteristic
176C2×C42C2312yes—
177C2×C42C2312yes—
178C2×C42C2312yes—
179C2×C42C2311yescharacteristic
180C2×C42C2311yescharacteristic
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—
205#95C2312yes—
206C24C2412yes—
207C24C2411yescharacteristic
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—42no—
217C2×D4—24no—
218C2×D4—22no—
219C2×D4—41no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—41no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4C2412yes—
236C22×C4C2412yes—
237C22×C4C2412yes—
238C22×C4C2412yes—
239C22×C4C2412yes—
240C22×C4C2412yes—
241C22×C4C2412yes—
242C22×C4C2412yes—
243C22×C4C2412yes—
244C22×C4C2412yes—
245C22×C4C2412yes—
246C22×C4C2411yescharacteristic
247C22×C4C2411yescharacteristic
248C22×C4C2411yescharacteristic
249C22×C4C2411yescharacteristic
250C22×C4C2411yescharacteristic
251C22×C4C2411yescharacteristic
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—24no—
259C4⋊C4—22no—
260C4⋊C4—24no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
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—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C42—22no—
299C42—22no—
300C42—41no—
301C42—41no—
302C42—42no—
303C42—42no—
304C42—42no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23H(5,𝔽2)12yes—
319C23C2511yescharacteristic
320Q8—82no—
321Q8—82no—
322Q8—84no—
323D4—82no—
324D4—84no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—22no—
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—22no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
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—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—84no—
375C2×C4H(5,𝔽2)12yes—
376C2×C4H(5,𝔽2)12yes—
377C2×C42−1+412yes—
378C2×C42−1+412yes—
379C2×C42−1+412yes—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22C2×H(5,𝔽2)11yescharacteristic
395C22C2×H(5,𝔽2)11yescharacteristic
396C22C2×H(5,𝔽2)11yescharacteristic
397C22C2×2−1+411yescharacteristic
398C22C2×2−1+411yescharacteristic
399C22C2×2−1+411yescharacteristic
400C22#58511yescharacteristic
401C4—22no—
402C4—22no—
403C4—22no—
404C4—22no—
405C4—22no—
406C4—42no—
407C4—41no—
408C4—42no—
409C4—42no—
410C4—42no—
411C4—42no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—42no—
418C2—42no—
419C2—82no—
420C2#353511yescharacteristic
421C2#353711yescharacteristic
422C2#353811yescharacteristic
423C2#354411yescharacteristic
424C2#355011yescharacteristic
425C2#355411yescharacteristic
426C2#355911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radicalC2×#121C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2857C2semidirect
3. #2851C2semidirect
4. #2808C2semidirect
5. #2805C2semidirect
6. #2796C2semidirect
7. #2793C2semidirect
8. #2792C2semidirect
9. #2773C2semidirect
10. #2747C2semidirect
11. #2741C2semidirect
12. #2734C2semidirect
13. #2732C2semidirect
14. #2720C2semidirect
15. #2717C2semidirect
16. #2680C2semidirect
17. #2677C2semidirect
18. #2674C2semidirect

Maximal subgroups

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

subgroupindex
#28572
#28542
#28512
#28082
#28052
#27962
#27932
#27922
#27732
#27472
#27412
#27342
#27322
#27202
#27172
#26802
#26772
#26742
#26722