Finite Group Databasebeta

← #49592

Subgroups — #49592

Order 256.

919 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2803C212yesmaximal, complemented
2#2742C212yesmaximal, complemented
3#2710C212yesmaximal, complemented
4#2705C212yesmaximal, complemented
5#2696C212yesmaximal, complemented
6#2672C212yesmaximal, complemented
7#2665C211yesmaximal, characteristic, complemented
8#2615C211yesmaximal, characteristic
9#2610C212yesmaximal, complemented
10#2604C212yesmaximal, complemented
11#2591C211yesmaximal, characteristic, complemented
12#2559C212yesmaximal, complemented
13#2530C212yesmaximal, complemented
14#2471C212yesmaximal, complemented
15#2465C212yesmaximal, complemented
16#2406C211yesmaximal, characteristic, complemented
17#2377C211yesmaximal, characteristic, complemented
18#2341C211yesmaximal, characteristic, complemented
19#2316C211yesmaximal, characteristic, complemented
20C22×#41C2211yescharacteristic
21C2×#126C2212yes—
22C2×#124C2212yes—
23C2×#123C2212yes—
24C2×#123C2212yes—
25C2×C22⋊Q8C2212yes—
26C2×#121C2211yescharacteristic
27C2×#121C2212yes—
28C2×C4×Q8C2212yes—
29C2×#117C2211yescharacteristic
30C2×#117C2211yescharacteristic
31C22×C4⋊C4C2211yescharacteristic
32#400C2211yescharacteristic
33#400C2211yescharacteristic
34#400C2212yes—
35#400C2212yes—
36#399C2212yes—
37#399C2212yes—
38#397C2212yes—
39#397C2212yes—
40#396C2212yes—
41#396C2212yes—
42#395C2211yescharacteristic
43#395C2212yes—
44#395C2211yescharacteristic
45#395C2212yes—
46#394C2211yescharacteristic
47#394C2211yescharacteristic
48#394C2212yes—
49#394C2211yescharacteristic
50#394C2211yescharacteristic
51#394C2212yes—
52#394C2212yes—
53#394C2212yes—
54#393C2212yes—
55#393C2212yes—
56#393C2212yes—
57#392C2212yes—
58#392C2212yes—
59#392C2212yes—
60#391C2212yes—
61#390C2212yes—
62#390C2212yes—
63#390C2212yes—
64#390C2212yes—
65#390C2212yes—
66#389C2212yes—
67#389C2212yes—
68#389C2212yes—
69#389C2212yes—
70#389C2212yes—
71#388C2212yes—
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#386C2212yes—
88#386C2212yes—
89#386C2212yes—
90#386C2212yes—
91#386C2212yes—
92#386C2212yes—
93#385C2212yes—
94#385C2212yes—
95#385C2212yes—
96#381C2211yescharacteristic
97#380C2211yescharacteristic
98#380C2211yescharacteristic
99#380C2211yescharacteristic
100#380C2211yescharacteristic
101#380C2212yes—
102#380C2212yes—
103C4×C22⋊C4C2211yescharacteristic
104C4×C22⋊C4C2212yes—
105C4×C22⋊C4C2212yes—
106#376C2212yes—
107C2×#41—24no—
108C2×#41—44no—
109C22×Q8C2311yescharacteristic
110C22×Q8C2312yes—
111C22×D4C2312yes—
112C22×D4C2311yescharacteristic
113C22×D4C2312yes—
114C22×D4C2311yescharacteristic
115C22×D4C2312yes—
116C22×D4C2312yes—
117C23×C4C2311yescharacteristic
118C23×C4C2312yes—
119C23×C4C2311yescharacteristic
120C23×C4C2312yes—
121C23×C4C2311yescharacteristic
122#126—44no—
123#124—44no—
124#123—44no—
125#123—44no—
126C22⋊Q8—44no—
127#121—24no—
128#121—44no—
129C4×Q8—44no—
130#117—42no—
131#117—42no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4C2311yescharacteristic
134C2×C4⋊C4C2311yescharacteristic
135C2×C4⋊C4C2311yescharacteristic
136C2×C4⋊C4C2311yescharacteristic
137C2×C4⋊C4C2312yes—
138C2×C4⋊C4C2312yes—
139C2×C4⋊C4C2311yescharacteristic
140C2×C4⋊C4C2312yes—
141C2×C4⋊C4C2312yes—
142C2×C4⋊C4C2312yes—
143C2×C4⋊C4C2312yes—
144C2×C4⋊C4C2312yes—
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2312yes—
147C2×C4⋊C4C2312yes—
148C2×C4⋊C4C2312yes—
149C2×C4⋊C4C2312yes—
150C2×C4⋊C4C2312yes—
151C2×C4⋊C4C2312yes—
152C2×C22⋊C4C2312yes—
153C2×C22⋊C4C2312yes—
154C2×C22⋊C4C2312yes—
155C2×C22⋊C4C2312yes—
156C2×C22⋊C4C2312yes—
157C2×C22⋊C4C2312yes—
158C2×C22⋊C4C2312yes—
159C2×C22⋊C4C2311yescharacteristic
160C2×C22⋊C4C2311yescharacteristic
161C2×C22⋊C4C2311yescharacteristic
162C2×C22⋊C4C2311yescharacteristic
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×C22⋊C4C2312yes—
174C2×C22⋊C4C2312yes—
175C2×C22⋊C4C2312yes—
176C2×C22⋊C4C2312yes—
177C2×C22⋊C4C2312yes—
178C2×C42C2312yes—
179C2×C42C2312yes—
180C2×C42C2312yes—
181C2×C42C2312yes—
182C2×C42C2312yes—
183C2×C42C2312yes—
184#95C2312yes—
185#95C2312yes—
186#95C2312yes—
187#95C2312yes—
188#95C2311yescharacteristic
189#95C2312yes—
190#95C2311yescharacteristic
191#95C2312yes—
192#95C2312yes—
193#95C2312yes—
194#95C2312yes—
195#95C2312yes—
196#95C2312yes—
197#95C2312yes—
198#95C2311yescharacteristic
199#95C2311yescharacteristic
200#95C2312yes—
201#95C2312yes—
202#95C2312yes—
203#95C2312yes—
204#95C2312yes—
205#95C2312yes—
206#95C2312yes—
207C24C2412yes—
208C24C2412yes—
209C24C2411yescharacteristic
210#41—416no—
211C2×Q8—22no—
212C2×Q8—24no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×D4—24no—
217C2×D4—24no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—24no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4#4114yes—
245C22×C4C2412yes—
246C22×C4C2412yes—
247C22×C4C2412yes—
248C22×C4C2412yes—
249C22×C4C2412yes—
250C22×C4C2412yes—
251C22×C4C2412yes—
252C22×C4C2412yes—
253C22×C4C2411yescharacteristic
254C22×C4C2411yescharacteristic
255C22×C4C2412yes—
256C22×C4C2411yescharacteristic
257C22×C4C2411yescharacteristic
258C22×C4C2411yescharacteristic
259C22×C4C2412yes—
260C22×C4C2411yescharacteristic
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—24no—
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—41no—
296C22⋊C4—41no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C42—24no—
307C42—24no—
308C42—42no—
309C42—42no—
310C42—42no—
311C42—42no—
312C23—21no—
313C23—22no—
314C23—21no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23C2×#4112yes—
331C23C2511yescharacteristic
332Q8—28no—
333Q8—84no—
334D4—48no—
335D4—44no—
336D4—82no—
337D4—84no—
338D4—84no—
339D4—84no—
340C2×C4—28no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
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—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—44no—
381C2×C4—48no—
382C2×C4—82no—
383C2×C4—84no—
384C2×C4C2×#4112yes—
385C2×C4C2×#4112yes—
386C2×C4H(5,𝔽2)12yes—
387C2×C4H(5,𝔽2)14yes—
388C2×C42−1+412yes—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22C22×#4111yescharacteristic
409C22C2×H(5,𝔽2)12yes—
410C22C2×H(5,𝔽2)11yescharacteristic
411C22C2×2−1+411yescharacteristic
412C22#58512yes—
413C4—22no—
414C4—22no—
415C4—24no—
416C4—22no—
417C4—22no—
418C4—41no—
419C4—41no—
420C4—42no—
421C4—42no—
422C4—42no—
423C4—42no—
424C4—42no—
425C4—42no—
426C4—42no—
427C4—42no—
428C4—42no—
429C2—42no—
430C2—82no—
431C2—82no—
432C2#349011yescharacteristic
433C2#349211yescharacteristic
434C2#351011yescharacteristic
435C2#354612yes—
436C2#354712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radical#2615C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2803C2semidirect
2. #2742C2semidirect
3. #2710C2semidirect
4. #2705C2semidirect
5. #2696C2semidirect
6. #2672C2semidirect
7. #2665C2semidirect
9. #2610C2semidirect
10. #2604C2semidirect
11. #2591C2semidirect
12. #2559C2semidirect
13. #2530C2semidirect
14. #2471C2semidirect
15. #2465C2semidirect
16. #2406C2semidirect
17. #2377C2semidirect
18. #2341C2semidirect
19. #2316C2semidirect

Maximal subgroups

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

subgroupindex
#28032
#27422
#27102
#27052
#26962
#26722
#26652
#26152
#26102
#26042
#25912
#25592
#25302
#24712
#24652
#24062
#23772
#23412
#23162