Finite Group Databasebeta

← #44377

Subgroups — #44377

Order 256.

940 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2795C211yesmaximal, characteristic, complemented
2#2785C211yesmaximal, characteristic, complemented
3#2731C212yesmaximal, complemented
4#2727C212yesmaximal, complemented
5#2705C212yesmaximal, complemented
6#2612C211yesmaximal, characteristic, complemented
7#2610C212yesmaximal, complemented
8#2566C211yesmaximal, characteristic, complemented
9#2564C211yesmaximal, characteristic, complemented
10#2556C212yesmaximal, complemented
11#2554C212yesmaximal, complemented
12#2551C211yesmaximal, characteristic, complemented
13#2452C212yesmaximal, complemented
14#2439C212yesmaximal, complemented
15#2432C212yesmaximal, complemented
16#2388C211yesmaximal, characteristic
17#2373C212yesmaximal, complemented
18#2359C212yesmaximal, complemented
19#2334C212yesmaximal, complemented
20C22×#41C2211yescharacteristic
21C2×C4⋊Q8C2212yes—
22C2×C4⋊Q8C2212yes—
23C2×#126C2212yes—
24C2×#126C2212yes—
25C2×#125C2211yescharacteristic
26C2×#124C2212yes—
27C2×#123C2212yes—
28C2×C22⋊Q8C2211yescharacteristic
29C2×C22⋊Q8C2212yes—
30C2×C4×D4C2211yescharacteristic
31C2×C4×D4C2212yes—
32C2×#117C2212yes—
33#400C2211yescharacteristic
34#399C2211yescharacteristic
35#399C2212yes—
36#398C2212yes—
37#396C2212yes—
38#396C2212yes—
39#396C2212yes—
40#396C2212yes—
41#396C2212yes—
42#395C2211yescharacteristic
43#395C2212yes—
44#395C2211yescharacteristic
45#394C2212yes—
46#394C2212yes—
47#394C2211yescharacteristic
48#394C2212yes—
49#394C2212yes—
50#394C2212yes—
51#393C2211yescharacteristic
52#393C2212yes—
53#393C2211yescharacteristic
54#393C2212yes—
55#392C2212yes—
56#392C2211yescharacteristic
57#391C2212yes—
58#391C2212yes—
59#390C2211yescharacteristic
60#390C2211yescharacteristic
61#390C2212yes—
62#390C2212yes—
63#390C2212yes—
64#389C2212yes—
65#389C2212yes—
66#389C2212yes—
67#389C2212yes—
68#389C2212yes—
69#389C2212yes—
70#388C2212yes—
71#388C2212yes—
72#388C2212yes—
73#388C2212yes—
74#388C2212yes—
75#388C2212yes—
76#388C2212yes—
77#388C2212yes—
78#388C2212yes—
79#388C2212yes—
80#387C2212yes—
81#387C2212yes—
82#387C2212yes—
83#387C2212yes—
84#387C2212yes—
85#387C2212yes—
86#386C2212yes—
87#386C2212yes—
88#385C2212yes—
89#385C2212yes—
90#385C2212yes—
91#385C2212yes—
92#385C2212yes—
93#384C2211yescharacteristic
94#382C2211yescharacteristic
95#382C2212yes—
96#380C2212yes—
97C4×C4⋊C4C2211yescharacteristic
98C4×C4⋊C4C2212yes—
99C4×C4⋊C4C2212yes—
100C4×C4⋊C4C2212yes—
101C4×C22⋊C4C2211yescharacteristic
102C4×C22⋊C4C2212yes—
103C4×C22⋊C4C2212yes—
104C4×C22⋊C4C2212yes—
105#376C2212yes—
106C2×#95C2211yescharacteristic
107C2×#41—42no—
108C2×#41—44no—
109C22×Q8C2311yescharacteristic
110C22×Q8C2312yes—
111C22×Q8C2311yescharacteristic
112C22×D4C2312yes—
113C22×D4C2311yescharacteristic
114C22×D4C2312yes—
115C22×D4C2311yescharacteristic
116C23×C4C2312yes—
117C23×C4C2312yes—
118C23×C4C2311yescharacteristic
119C23×C4C2311yescharacteristic
120C4⋊Q8—44no—
121C4⋊Q8—44no—
122#126—28no—
123#126—28no—
124#125—42no—
125#124—44no—
126#123—28no—
127C22⋊Q8—42no—
128C22⋊Q8—44no—
129C4×D4—42no—
130C4×D4—44no—
131#117—44no—
132C2×C4⋊C4C2312yes—
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⋊C4C2311yescharacteristic
144C2×C4⋊C4C2312yes—
145C2×C4⋊C4C2311yescharacteristic
146C2×C4⋊C4C2311yescharacteristic
147C2×C4⋊C4C2311yescharacteristic
148C2×C4⋊C4C2312yes—
149C2×C4⋊C4C2312yes—
150C2×C4⋊C4C2312yes—
151C2×C4⋊C4C2312yes—
152C2×C4⋊C4C2312yes—
153C2×C4⋊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⋊C4C2311yescharacteristic
167C2×C22⋊C4C2312yes—
168C2×C22⋊C4C2311yescharacteristic
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×C42C2312yes—
179C2×C42C2312yes—
180C2×C42C2311yescharacteristic
181C2×C42C2311yescharacteristic
182C2×C42C2312yes—
183C2×C42C2312yes—
184C2×C42C2312yes—
185C2×C42C2312yes—
186#95C2312yes—
187#95C2311yescharacteristic
188#95C2311yescharacteristic
189#95C2312yes—
190#95C2312yes—
191#95C2312yes—
192#95C2312yes—
193#95C2312yes—
194#95C2311yescharacteristic
195#95C2312yes—
196#95C2312yes—
197#95C2311yescharacteristic
198#95C2312yes—
199#95C2312yes—
200#95C2312yes—
201#95C2312yes—
202#95C2312yes—
203#95C2312yes—
204#95C2312yes—
205#95C2312yes—
206#95C2312yes—
207#95C2312yes—
208C24C2411yescharacteristic
209C24C2412yes—
210C24C2411yescharacteristic
211#41—416no—
212C2×Q8—24no—
213C2×Q8—24no—
214C2×Q8—41no—
215C2×Q8—42no—
216C2×Q8—41no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×D4—24no—
220C2×D4—22no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—22no—
224C2×D4—41no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—41no—
229C2×D4—42no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4C2412yes—
241C22×C4C2412yes—
242C22×C4C2412yes—
243C22×C4C2411yescharacteristic
244C22×C4C2412yes—
245C22×C4C2412yes—
246C22×C4C2411yescharacteristic
247C22×C4C2412yes—
248C22×C4C2411yescharacteristic
249C22×C4C2412yes—
250C22×C4C2412yes—
251C22×C4C2412yes—
252C22×C4C2412yes—
253C22×C4C2411yescharacteristic
254C22×C4C2412yes—
255C22×C4C2411yescharacteristic
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4#4118yescomplemented
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—24no—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—42no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C42—24no—
301C42—24no—
302C42—24no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—42no—
307C42—42no—
308C42—42no—
309C42—42no—
310C23—22no—
311C23—22no—
312C23—21no—
313C23—21no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—21no—
321C23—22no—
322C23—22no—
323C23—21no—
324C23—22no—
325C23—21no—
326C232−1+412yes—
327C23C2511yescharacteristic
328Q8—44no—
329Q8—84no—
330Q8—82no—
331D4—48no—
332D4—44no—
333D4—48no—
334D4—44no—
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—21no—
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—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—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—48no—
374C2×C4—48no—
375C2×C4—44no—
376C2×C4—82no—
377C2×C4C2×#4114yes—
378C2×C4C2×#4114yes—
379C2×C4C2×#4114yes—
380C2×C4H(5,𝔽2)14yes—
381C2×C4H(5,𝔽2)12yes—
382C2×C42−1+412yes—
383C22—24no—
384C22—22no—
385C22—41no—
386C22—42no—
387C22—41no—
388C22—41no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22—42no—
400C22C22×#4112yes—
401C22C2×H(5,𝔽2)11yescharacteristic
402C22C2×H(5,𝔽2)12yes—
403C22C2×2−1+411yescharacteristic
404C22C2×2−1+411yescharacteristic
405C4—24no—
406C4—24no—
407C4—24no—
408C4—22no—
409C4—24no—
410C4—22no—
411C4—42no—
412C4—42no—
413C4—41no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—41no—
418C4—41no—
419C4—42no—
420C4—42no—
421C2—42no—
422C2—82no—
423C2—81no—
424C2#348311yescharacteristic
425C2#351112yes—
426C2#352012yes—
427C2#353211yescharacteristic
428C2#353811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radical#2388C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2795C2semidirect
2. #2785C2semidirect
3. #2731C2semidirect
4. #2727C2semidirect
5. #2705C2semidirect
6. #2612C2semidirect
7. #2610C2semidirect
8. #2566C2semidirect
9. #2564C2semidirect
10. #2556C2semidirect
11. #2554C2semidirect
12. #2551C2semidirect
13. #2452C2semidirect
14. #2439C2semidirect
15. #2432C2semidirect
17. #2373C2semidirect
18. #2359C2semidirect
19. #2334C2semidirect
277. C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
#27952
#27852
#27312
#27272
#27052
#26122
#26102
#25662
#25642
#25562
#25542
#25512
#24522
#24392
#24322
#23882
#23732
#23592
#23342