Finite Group Databasebeta

← #38810

Subgroups — #38810

Order 256.

877 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2864C211yesmaximal, characteristic, complemented
2#2809C211yesmaximal, characteristic, complemented
3#2765C212yesmaximal, complemented
4#2732C211yesmaximal, characteristic, complemented
5#2730C212yesmaximal, complemented
6#2727C211yesmaximal, characteristic, complemented
7#2682C211yesmaximal, characteristic
8#2667C212yesmaximal, complemented
9#2594C212yesmaximal, complemented
10#2576C212yesmaximal, complemented
11#2553C212yesmaximal, complemented
12#2479C212yesmaximal, complemented
13#2457C212yesmaximal, complemented
14#2443C212yesmaximal, complemented
15#2413C212yesmaximal, complemented
16#2360C212yesmaximal, complemented
17#2333C212yesmaximal, complemented
18C4×#124C211yesmaximal, characteristic, complemented
19#2293C211yesmaximal, characteristic, complemented
20C2×#126C2212yes—
21C2×#124C2211yescharacteristic
22C2×#123C2212yes—
23C2×#123C2212yes—
24C2×C22⋊Q8C2212yes—
25C2×#121C2212yes—
26C2×C4×Q8C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29C2×#117C2212yes—
30C22×C4⋊C4C2212yes—
31#401C2212yes—
32#400C2212yes—
33#399C2212yes—
34#399C2212yes—
35#399C2212yes—
36#398C2212yes—
37#398C2212yes—
38#398C2212yes—
39#398C2212yes—
40#398C2212yes—
41#397C2212yes—
42#397C2212yes—
43#396C2212yes—
44#396C2212yes—
45#396C2212yes—
46#396C2212yes—
47#394C2212yes—
48#394C2212yes—
49#394C2212yes—
50#392C2212yes—
51#391C2212yes—
52#391C2211yescharacteristic
53#390C2211yescharacteristic
54#390C2212yes—
55#390C2211yescharacteristic
56#390C2211yescharacteristic
57#389C2212yes—
58#389C2211yescharacteristic
59#389C2211yescharacteristic
60#389C2211yescharacteristic
61#389C2211yescharacteristic
62#389C2212yes—
63#389C2212yes—
64#389C2211yescharacteristic
65#389C2211yescharacteristic
66#388C2212yes—
67#388C2212yes—
68#388C2212yes—
69#388C2212yes—
70#388C2212yes—
71#388C2212yes—
72#388C2212yes—
73#388C2212yes—
74#388C2212yes—
75#387C2212yes—
76#387C2212yes—
77#387C2212yes—
78#387C2212yes—
79#387C2212yes—
80#387C2212yes—
81#387C2212yes—
82#387C2212yes—
83#386C2212yes—
84#386C2212yes—
85#386C2212yes—
86#386C2212yes—
87#386C2212yes—
88#386C2212yes—
89#385C2212yes—
90#385C2212yes—
91#385C2212yes—
92#385C2212yes—
93#382C2211yescharacteristic
94#382C2211yescharacteristic
95#382C2211yescharacteristic
96#381C2212yes—
97#381C2212yes—
98#380C2212yes—
99C4×C4⋊C4C2211yescharacteristic
100C4×C4⋊C4C2212yes—
101C4×C22⋊C4C2212yes—
102C4×C22⋊C4C2212yes—
103C4×C22⋊C4C2212yes—
104C4×C22⋊C4C2212yes—
105C4×C22⋊C4C2212yes—
106C43C2211yescharacteristic
107C22×Q8C2311yescharacteristic
108C22×D4C2311yescharacteristic
109C22×D4C2312yes—
110C22×D4C2312yes—
111C22×D4C2311yescharacteristic
112C23×C4C2312yes—
113C23×C4C2312yes—
114C23×C4C2312yes—
115C23×C4C2312yes—
116#126—44no—
117#124—24no—
118#123—44no—
119#123—44no—
120C22⋊Q8—28no—
121#121—28no—
122C4×Q8—42no—
123C4×D4—24no—
124C4×D4—42no—
125#117—44no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4C2312yes—
128C2×C4⋊C4C2312yes—
129C2×C4⋊C4C2312yes—
130C2×C4⋊C4C2311yescharacteristic
131C2×C4⋊C4C2312yes—
132C2×C4⋊C4C2312yes—
133C2×C4⋊C4C2312yes—
134C2×C4⋊C4C2312yes—
135C2×C4⋊C4C2311yescharacteristic
136C2×C4⋊C4C2311yescharacteristic
137C2×C4⋊C4C2312yes—
138C2×C4⋊C4C2312yes—
139C2×C4⋊C4C2312yes—
140C2×C4⋊C4C2311yescharacteristic
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2312yes—
143C2×C4⋊C4C2312yes—
144C2×C4⋊C4C2312yes—
145C2×C4⋊C4C2311yescharacteristic
146C2×C4⋊C4C2311yescharacteristic
147C2×C4⋊C4C2312yes—
148C2×C4⋊C4C2311yescharacteristic
149C2×C4⋊C4C2311yescharacteristic
150C2×C4⋊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×C22⋊C4C2312yes—
171C2×C42C2311yescharacteristic
172C2×C42C2311yescharacteristic
173C2×C42C2311yescharacteristic
174C2×C42C2311yescharacteristic
175C2×C42C2311yescharacteristic
176C2×C42C2312yes—
177C2×C42C2312yes—
178C2×C42C2312yes—
179C2×C42C2311yescharacteristic
180C2×C42C2311yescharacteristic
181C2×C42C2312yes—
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—41no—
208C2×Q8—41no—
209C2×Q8#4114yes—
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—22no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4#4114yes—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4C2411yescharacteristic
235C22×C4C2412yes—
236C22×C4C2412yes—
237C22×C4C2412yes—
238C22×C4C2412yes—
239C22×C4C2412yes—
240C22×C4C2411yescharacteristic
241C22×C4C2412yes—
242C22×C4C2412yes—
243C22×C4C2412yes—
244C22×C4C2412yes—
245C22×C4C2411yescharacteristic
246C22×C4C2411yescharacteristic
247C22×C4C2412yes—
248C22×C4C2411yescharacteristic
249C22×C4C2411yescharacteristic
250C22×C4C2411yescharacteristic
251C4⋊C4—22no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
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—24no—
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—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—41no—
301C42—41no—
302C42—42no—
303C42—42no—
304C42—42no—
305C42—42no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23C2×#4114yes—
320C23C2511yescharacteristic
321Q8—44no—
322D4—44no—
323D4—44no—
324D4—84no—
325D4—84no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
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—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—48no—
372C2×C4—48no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4C2×#4112yes—
376C2×C4C2×#4112yes—
377C2×C4C2×#4114yes—
378C2×C4H(5,𝔽2)12yes—
379C2×C4H(5,𝔽2)12yes—
380C2×C42−1+412yes—
381C22—24no—
382C22—24no—
383C22—24no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22C22×#4111yescharacteristic
396C22C22×#4111yescharacteristic
397C22C2×H(5,𝔽2)11yescharacteristic
398C22C2×H(5,𝔽2)11yescharacteristic
399C22C2×2−1+411yescharacteristic
400C22#58511yescharacteristic
401C22#58511yescharacteristic
402C4—24no—
403C4—22no—
404C4—22no—
405C4—22no—
406C4—22no—
407C4—22no—
408C4—42no—
409C4—41no—
410C4—42no—
411C4—41no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—42no—
419C2—44no—
420C2—82no—
421C2#345911yescharacteristic
422C2#348211yescharacteristic
423C2#349111yescharacteristic
424C2#351711yescharacteristic
425C2#351911yescharacteristic
426C2#355911yescharacteristic
427C2#356111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radical#2682C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2864C2semidirect
2. #2809C2semidirect
3. #2765C2semidirect
4. #2732C2semidirect
5. #2730C2semidirect
6. #2727C2semidirect
8. #2667C2semidirect
9. #2594C2semidirect
10. #2576C2semidirect
11. #2553C2semidirect
12. #2479C2semidirect
13. #2457C2semidirect
14. #2443C2semidirect
15. #2413C2semidirect
16. #2360C2semidirect
17. #2333C2semidirect
18. C4×#124C2semidirect
19. #2293C2semidirect

Maximal subgroups

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

subgroupindex
#28642
#28092
#27652
#27322
#27302
#27272
#26822
#26672
#25942
#25762
#25532
#24792
#24572
#24432
#24132
#23602
#23332
C4×#1242
#22932