Finite Group Databasebeta

← #44860

Subgroups — #44860

Order 256.

923 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#2790C212yesmaximal, complemented
2#2728C211yesmaximal, characteristic, complemented
3#2724C211yesmaximal, characteristic, complemented
4#2715C212yesmaximal, complemented
5#2691C212yesmaximal, complemented
6#2586C211yesmaximal, characteristic, complemented
7#2579C212yesmaximal, complemented
8#2555C211yesmaximal, characteristic
9#2545C211yesmaximal, characteristic, complemented
10#2536C211yesmaximal, characteristic, complemented
11#2531C212yesmaximal, complemented
12#2529C212yesmaximal, complemented
13#2461C212yesmaximal, complemented
14#2461C212yesmaximal, complemented
15#2440C212yesmaximal, complemented
16#2399C211yesmaximal, characteristic, complemented
17#2373C212yesmaximal, complemented
18#2361C212yesmaximal, complemented
19#2326C212yesmaximal, complemented
20C2×C4⋊Q8C2211yescharacteristic
21C2×#126C2212yes—
22C2×#124C2212yes—
23C2×#124C2211yescharacteristic
24C2×#124C2211yescharacteristic
25C2×#123C2212yes—
26C2×C22⋊Q8C2212yes—
27C2×C22⋊Q8C2212yes—
28C2×#121C2212yes—
29C2×C4×Q8C2211yescharacteristic
30C2×C4×Q8C2211yescharacteristic
31C2×C4×D4C2212yes—
32C2×C4×D4C2212yes—
33C2×C4×D4C2211yescharacteristic
34C22×C4⋊C4C2212yes—
35#400C2212yes—
36#398C2211yescharacteristic
37#398C2212yes—
38#398C2211yescharacteristic
39#398C2212yes—
40#398C2211yescharacteristic
41#398C2212yes—
42#398C2211yescharacteristic
43#397C2212yes—
44#397C2212yes—
45#397C2212yes—
46#396C2212yes—
47#396C2212yes—
48#396C2212yes—
49#396C2212yes—
50#395C2211yescharacteristic
51#395C2212yes—
52#395C2212yes—
53#395C2211yescharacteristic
54#394C2211yescharacteristic
55#394C2212yes—
56#394C2212yes—
57#394C2211yescharacteristic
58#392C2212yes—
59#391C2212yes—
60#391C2211yescharacteristic
61#390C2212yes—
62#390C2212yes—
63#390C2211yescharacteristic
64#389C2212yes—
65#389C2212yes—
66#389C2212yes—
67#388C2212yes—
68#388C2212yes—
69#388C2212yes—
70#388C2212yes—
71#388C2212yes—
72#388C2212yes—
73#388C2212yes—
74#388C2212yes—
75#388C2212yes—
76#388C2212yes—
77#388C2212yes—
78#387C2212yes—
79#387C2212yes—
80#387C2212yes—
81#387C2212yes—
82#387C2212yes—
83#386C2212yes—
84#386C2212yes—
85#386C2212yes—
86#386C2212yes—
87#385C2212yes—
88#385C2212yes—
89#385C2212yes—
90#385C2212yes—
91#385C2212yes—
92#385C2212yes—
93#385C2212yes—
94#385C2212yes—
95#384C2212yes—
96#382C2211yescharacteristic
97#380C2212yes—
98#380C2212yes—
99C4×C4⋊C4C2212yes—
100C4×C4⋊C4C2212yes—
101C4×C4⋊C4C2211yescharacteristic
102C4×C4⋊C4C2212yes—
103C4×C22⋊C4C2212yes—
104C4×C22⋊C4C2212yes—
105#376C2212yes—
106#376C2211yescharacteristic
107C22×Q8C2311yescharacteristic
108C22×Q8C2311yescharacteristic
109C22×Q8C2311yescharacteristic
110C22×D4C2311yescharacteristic
111C22×D4C2312yes—
112C22×D4C2312yes—
113C22×D4C2311yescharacteristic
114C23×C4C2312yes—
115C23×C4C2312yes—
116C23×C4C2312yes—
117C23×C4C2312yes—
118C4⋊Q8—42no—
119#126—28no—
120#124—28no—
121#124—42no—
122#124—42no—
123#123—44no—
124C22⋊Q8—44no—
125C22⋊Q8—44no—
126#121—28no—
127C4×Q8—42no—
128C4×Q8—42no—
129C4×D4—42no—
130C4×D4—44no—
131C4×D4—44no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4C2312yes—
134C2×C4⋊C4C2312yes—
135C2×C4⋊C4C2312yes—
136C2×C4⋊C4C2312yes—
137C2×C4⋊C4C2312yes—
138C2×C4⋊C4C2312yes—
139C2×C4⋊C4C2312yes—
140C2×C4⋊C4C2312yes—
141C2×C4⋊C4C2312yes—
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⋊C4C2311yescharacteristic
151C2×C4⋊C4C2312yes—
152C2×C4⋊C4C2312yes—
153C2×C4⋊C4C2312yes—
154C2×C4⋊C4C2312yes—
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×C22⋊C4C2312yes—
174C2×C22⋊C4C2312yes—
175C2×C22⋊C4C2312yes—
176C2×C42C2312yes—
177C2×C42C2311yescharacteristic
178C2×C42C2311yescharacteristic
179C2×C42C2311yescharacteristic
180C2×C42C2312yes—
181C2×C42C2312yes—
182C2×C42C2312yes—
183C2×C42C2312yes—
184C2×C42C2311yescharacteristic
185C2×C42C2311yescharacteristic
186#95C2312yes—
187#95C2312yes—
188#95C2312yes—
189#95C2312yes—
190#95C2312yes—
191#95C2311yescharacteristic
192#95C2311yescharacteristic
193#95C2312yes—
194#95C2312yes—
195#95C2312yes—
196#95C2312yes—
197#95C2312yes—
198#95C2312yes—
199#95C2312yes—
200#95C2312yes—
201#95C2312yes—
202#95C2312yes—
203#95C2311yescharacteristic
204#95C2311yescharacteristic
205#95C2312yes—
206#95C2312yes—
207#95C2312yes—
208#95C2312yes—
209C24C2412yes—
210C24C2412yes—
211C2×Q8—24no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—22no—
224C2×D4—24no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—41no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4C2412yes—
243C22×C4C2412yes—
244C22×C4C2412yes—
245C22×C4C2412yes—
246C22×C4C2411yescharacteristic
247C22×C4C2411yescharacteristic
248C22×C4C2411yescharacteristic
249C22×C4C2412yes—
250C22×C4C2412yes—
251C22×C4C2412yes—
252C22×C4C2411yescharacteristic
253C22×C4C2412yes—
254C22×C4C2412yes—
255C22×C4C2411yescharacteristic
256C22×C4C2411yescharacteristic
257C22×C4C2411yescharacteristic
258C22×C4C2412yes—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4#4118yes—
302C42—24no—
303C42—24no—
304C42—24no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—24no—
309C42—42no—
310C42—41no—
311C42—41no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23C2×#4114yes—
326C23C2511yescharacteristic
327Q8—44no—
328Q8—44no—
329Q8—82no—
330D4—48no—
331D4—48no—
332D4—44no—
333D4—82no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—22no—
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—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
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—21no—
365C2×C4—21no—
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—48no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4C2×#4114yes—
380C2×C4C2×#4114yes—
381C2×C4H(5,𝔽2)12yes—
382C2×C4H(5,𝔽2)12yes—
383C2×C4H(5,𝔽2)12yes—
384C2×C42−1+414yes—
385C22—24no—
386C22—24no—
387C22—24no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22C22×#4112yes—
400C22C2×H(5,𝔽2)11yescharacteristic
401C22C2×H(5,𝔽2)11yescharacteristic
402C22C2×H(5,𝔽2)11yescharacteristic
403C22C2×2−1+412yes—
404C4—24no—
405C4—24no—
406C4—24no—
407C4—22no—
408C4—22no—
409C4—22no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—42no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—41no—
421C2—44no—
422C2—82no—
423C2#348111yescharacteristic
424C2#351112yes—
425C2#352012yes—
426C2#353711yescharacteristic
427C2#354411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radical#2555C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2790C2semidirect
2. #2728C2semidirect
3. #2724C2semidirect
4. #2715C2semidirect
5. #2691C2semidirect
6. #2586C2semidirect
7. #2579C2semidirect
9. #2545C2semidirect
10. #2536C2semidirect
11. #2531C2semidirect
12. #2529C2semidirect
13. #2461C2semidirect
14. #2461C2semidirect
15. #2440C2semidirect
16. #2399C2semidirect
17. #2373C2semidirect
18. #2361C2semidirect
19. #2326C2semidirect

Maximal subgroups

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

subgroupindex
#27902
#27282
#27242
#27152
#26912
#25862
#25792
#25552
#25452
#25362
#25312
#25292
#24612
#24402
#23992
#23732
#23612
#23262