Finite Group Databasebeta

← #53357

Subgroups — #53357

Order 256.

809 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2862C211yesmaximal, characteristic, complemented
2#2861C211yesmaximal, characteristic, complemented
3#2860C211yesmaximal, characteristic, complemented
4#2806C211yesmaximal, characteristic, complemented
5#2804C211yesmaximal, characteristic, complemented
6#2788C212yesmaximal, complemented
7#2773C212yesmaximal, complemented
8#2754C212yesmaximal, complemented
9#2733C211yesmaximal, characteristic
10#2732C211yesmaximal, characteristic, complemented
11#2730C212yesmaximal, complemented
12#2715C212yesmaximal, complemented
13#2712C212yesmaximal, complemented
14#2706C212yesmaximal, complemented
15#2706C212yesmaximal, complemented
16#2704C212yesmaximal, complemented
17#2702C212yesmaximal, complemented
18#2674C212yesmaximal, complemented
19#2674C212yesmaximal, complemented
20C2×C4⋊Q8C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic
22C2×#126C2212yes—
23C2×#125C2211yescharacteristic
24C2×#124C2211yescharacteristic
25C2×#124C2212yes—
26C2×#124C2212yes—
27C2×#123C2212yes—
28C2×#123C2212yes—
29C2×C22⋊Q8C2212yes—
30C2×C22⋊Q8C2212yes—
31#400C2212yes—
32#400C2212yes—
33#399C2212yes—
34#399C2212yes—
35#399C2212yes—
36#398C2212yes—
37#398C2212yes—
38#398C2212yes—
39#398C2212yes—
40#397C2212yes—
41#397C2212yes—
42#397C2212yes—
43#397C2212yes—
44#396C2212yes—
45#396C2212yes—
46#396C2212yes—
47#396C2212yes—
48#395C2212yes—
49#395C2212yes—
50#394C2212yes—
51#394C2212yes—
52#394C2212yes—
53#394C2212yes—
54#393C2212yes—
55#393C2212yes—
56#393C2212yes—
57#392C2212yes—
58#392C2212yes—
59#391C2211yescharacteristic
60#391C2211yescharacteristic
61#391C2211yescharacteristic
62#391C2212yes—
63#391C2211yescharacteristic
64#391C2212yes—
65#391C2211yescharacteristic
66#391C2211yescharacteristic
67#391C2212yes—
68#390C2211yescharacteristic
69#390C2211yescharacteristic
70#390C2211yescharacteristic
71#390C2211yescharacteristic
72#390C2211yescharacteristic
73#389C2211yescharacteristic
74#389C2211yescharacteristic
75#389C2211yescharacteristic
76#389C2212yes—
77#389C2212yes—
78#389C2212yes—
79#388C2212yes—
80#388C2212yes—
81#388C2212yes—
82#388C2212yes—
83#388C2212yes—
84#388C2212yes—
85#388C2212yes—
86#388C2212yes—
87#388C2212yes—
88#388C2212yes—
89#388C2212yes—
90#387C2212yes—
91#387C2212yes—
92#387C2212yes—
93#387C2212yes—
94#386C2212yes—
95#386C2212yes—
96#386C2212yes—
97#386C2212yes—
98#385C2212yes—
99#385C2212yes—
100#385C2212yes—
101#385C2212yes—
102#382C2212yes—
103#382C2212yes—
104#380C2212yes—
105#380C2212yes—
106C43C2211yescharacteristic
107C22×Q8C2311yescharacteristic
108C22×Q8C2312yes—
109C22×Q8C2311yescharacteristic
110C22×Q8C2311yescharacteristic
111C22×D4C2311yescharacteristic
112C22×D4C2311yescharacteristic
113C22×D4C2312yes—
114C22×D4C2312yes—
115C23×C4C2312yes—
116C23×C4C2312yes—
117C4⋊Q8—42no—
118C4⋊Q8—42no—
119#126—44no—
120#125—42no—
121#124—42no—
122#124—44no—
123#124—44no—
124#123—44no—
125#123—44no—
126C22⋊Q8—44no—
127C22⋊Q8—44no—
128C2×C4⋊C4C2312yes—
129C2×C4⋊C4C2312yes—
130C2×C4⋊C4C2312yes—
131C2×C4⋊C4C2312yes—
132C2×C4⋊C4C2311yescharacteristic
133C2×C4⋊C4C2312yes—
134C2×C4⋊C4C2312yes—
135C2×C4⋊C4C2312yes—
136C2×C4⋊C4C2312yes—
137C2×C4⋊C4C2312yes—
138C2×C4⋊C4C2312yes—
139C2×C4⋊C4C2312yes—
140C2×C4⋊C4C2311yescharacteristic
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2312yes—
143C2×C4⋊C4C2312yes—
144C2×C4⋊C4C2311yescharacteristic
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2311yescharacteristic
147C2×C4⋊C4C2311yescharacteristic
148C2×C4⋊C4C2312yes—
149C2×C4⋊C4C2312yes—
150C2×C4⋊C4C2311yescharacteristic
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×C22⋊C4C2312yes—
171C2×C22⋊C4C2312yes—
172C2×C22⋊C4C2312yes—
173C2×C42C2311yescharacteristic
174C2×C42C2311yescharacteristic
175C2×C42C2311yescharacteristic
176C2×C42C2311yescharacteristic
177C2×C42C2311yescharacteristic
178C2×C42C2312yes—
179C2×C42C2312yes—
180C2×C42C2311yescharacteristic
181C2×C42C2311yescharacteristic
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—
206C24C2412yes—
207C2×Q8—22no—
208C2×Q8—24no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—42no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—42no—
218C2×Q8—41no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—41no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—42no—
230C2×D4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4C2412yes—
238C22×C4C2411yescharacteristic
239C22×C4C2412yes—
240C22×C4C2412yes—
241C22×C4C2412yes—
242C22×C4C2412yes—
243C22×C4C2412yes—
244C22×C4C2411yescharacteristic
245C22×C4C2412yes—
246C22×C4C2412yes—
247C22×C4C2412yes—
248C22×C4C2412yes—
249C22×C4C2411yescharacteristic
250C22×C4C2411yescharacteristic
251C22×C4C2411yescharacteristic
252C22×C4C2411yescharacteristic
253C22×C4C2411yescharacteristic
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—22no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—41no—
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—
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—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—41no—
305C42—41no—
306C42—42no—
307C42—42no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23C2511yescharacteristic
323Q8—82no—
324Q8—84no—
325Q8—82no—
326Q8—82no—
327D4—82no—
328D4—82no—
329D4—84no—
330D4—84no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
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—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
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—84no—
378C2×C4—84no—
379C2×C4H(5,𝔽2)12yes—
380C2×C4H(5,𝔽2)12yes—
381C2×C4H(5,𝔽2)12yes—
382C2×C42−1+412yes—
383C2×C42−1+412yes—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22C2×H(5,𝔽2)11yescharacteristic
399C22C2×H(5,𝔽2)11yescharacteristic
400C22C2×H(5,𝔽2)11yescharacteristic
401C22C2×2−1+411yescharacteristic
402C22C2×2−1+411yescharacteristic
403C22#58511yescharacteristic
404C22#58511yescharacteristic
405C4—22no—
406C4—22no—
407C4—22no—
408C4—22no—
409C4—22no—
410C4—41no—
411C4—42no—
412C4—42no—
413C4—41no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—42no—
421C4—42no—
422C2—82no—
423C2—82no—
424C2#354411yescharacteristic
425C2#354811yescharacteristic
426C2#355411yescharacteristic
427C2#355811yescharacteristic
428C2#355811yescharacteristic
429C2#356111yescharacteristic
430C2#356411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radical#2733C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2862C2semidirect
2. #2861C2semidirect
3. #2860C2semidirect
4. #2806C2semidirect
5. #2804C2semidirect
6. #2788C2semidirect
7. #2773C2semidirect
8. #2754C2semidirect
10. #2732C2semidirect
11. #2730C2semidirect
12. #2715C2semidirect
13. #2712C2semidirect
14. #2706C2semidirect
15. #2706C2semidirect
16. #2704C2semidirect
17. #2702C2semidirect
18. #2674C2semidirect
19. #2674C2semidirect

Maximal subgroups

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

subgroupindex
#28622
#28612
#28602
#28062
#28042
#27882
#27732
#27542
#27332
#27322
#27302
#27152
#27122
#27062
#27042
#27022
#26742