Finite Group Databasebeta

← #38083

Subgroups — #38083

Order 256.

1163 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#554C211yesmaximal, characteristic, complemented
2C2×#548C211yesmaximal, characteristic
3#2850C211yesmaximal, characteristic, complemented
4#2700C214yesmaximal, complemented
5#2690C214yesmaximal, complemented
6#2640C211yesmaximal, characteristic, complemented
7#2621C211yesmaximal, characteristic, complemented
8#2568C211yesmaximal, characteristic, complemented
9#2545C211yesmaximal, characteristic, complemented
10#2504C211yesmaximal, characteristic, complemented
11#2503C211yesmaximal, characteristic, complemented
12#2450C214yesmaximal, complemented
13#2444C214yesmaximal, complemented
14#2434C211yesmaximal, characteristic, complemented
15#2432C211yesmaximal, characteristic, complemented
16#2327C211yesmaximal, characteristic, complemented
17#2311C211yesmaximal, characteristic, complemented
18C2×#391C211yesmaximal, characteristic, complemented
19#2290C211yesmaximal, characteristic, complemented
20C22×#41C2211yescharacteristic
21C23×Q8C2211yescharacteristic
22#554—28no—
23#548—28no—
24C2×C4⋊Q8C2211yescharacteristic
25C2×C4⋊Q8C2211yescharacteristic
26C2×C4⋊Q8C2211yescharacteristic
27C2×#125C2212yes—
28C2×#124C2211yescharacteristic
29C2×#124C2212yes—
30C2×#124C2211yescharacteristic
31C2×C22⋊Q8C2211yescharacteristic
32C2×C22⋊Q8C2211yescharacteristic
33C2×C22⋊Q8C2211yescharacteristic
34C2×C22⋊Q8C2212yes—
35C2×C22⋊Q8C2212yes—
36C2×C22⋊Q8C2212yes—
37C2×#121C2211yescharacteristic
38C2×#121C2212yes—
39C2×C4×Q8C2211yescharacteristic
40C2×C4×Q8C2211yescharacteristic
41C2×C4×D4C2212yes—
42C2×C4×D4C2212yes—
43C2×C4×D4C2211yescharacteristic
44C2×#117C2211yescharacteristic
45C22×C4⋊C4C2211yescharacteristic
46C22×C4⋊C4C2211yescharacteristic
47C22×C42C2211yescharacteristic
48#400C2214yes—
49#400C2214yes—
50#398C2214yes—
51#397C2214yes—
52#397C2214yes—
53#397C2214yes—
54#396C2214yes—
55#395C2214yes—
56#394C2214yes—
57#393C2214yes—
58#391C2211yescharacteristic
59#391C2211yescharacteristic
60#391C2211yescharacteristic
61#391C2214yes—
62#391C2211yescharacteristic
63#391C2211yescharacteristic
64#391C2211yescharacteristic
65#391C2214yes—
66#391C2211yescharacteristic
67#390C2211yescharacteristic
68#390C2211yescharacteristic
69#390C2211yescharacteristic
70#388C2214yes—
71#388C2214yes—
72#388C2212yes—
73#387C2214yes—
74#387C2214yes—
75#387C2212yes—
76#386C2214yes—
77#386C2214yes—
78#386C2212yes—
79#386C2214yes—
80#385C2214yes—
81#385C2214yes—
82#385C2212yes—
83#382C2211yescharacteristic
84#382C2211yescharacteristic
85#382C2211yescharacteristic
86#381C2214yes—
87#380C2214yes—
88C4×C4⋊C4C2211yescharacteristic
89C4×C22⋊C4C2211yescharacteristic
90C4×C22⋊C4C2211yescharacteristic
91#376C2211yescharacteristic
92C2×#95C2214yes—
93C2×#41—24no—
94C2×#41—24no—
95C2×#41—42no—
96C22×Q8—24no—
97C22×Q8—24no—
98C22×Q8—42no—
99C22×Q8C2311yescharacteristic
100C22×Q8C2311yescharacteristic
101C22×Q8C2312yes—
102C22×Q8C2311yescharacteristic
103C22×Q8C2312yes—
104C22×D4C2312yes—
105C22×D4C2311yescharacteristic
106C22×D4C2312yes—
107C22×D4C2311yescharacteristic
108C23×C4C2312yes—
109C23×C4C2312yes—
110C23×C4C2311yescharacteristic
111C23×C4C2311yescharacteristic
112C23×C4C2312yes—
113C23×C4C2312yes—
114C23×C4C2312yes—
115C4⋊Q8—24no—
116C4⋊Q8—24no—
117C4⋊Q8—42no—
118#125—28no—
119#124—28no—
120#124—24no—
121#124—42no—
122C22⋊Q8—24no—
123C22⋊Q8—24no—
124C22⋊Q8—24no—
125C22⋊Q8—28no—
126C22⋊Q8—28no—
127C22⋊Q8—28no—
128#121—28no—
129#121—24no—
130C4×Q8—24no—
131C4×Q8—24no—
132C4×D4—28no—
133C4×D4—28no—
134C4×D4D418yes—
135#117—42no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4C2311yescharacteristic
139C2×C4⋊C4C2312yes—
140C2×C4⋊C4C2312yes—
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2311yescharacteristic
143C2×C4⋊C4C2312yes—
144C2×C4⋊C4C2312yes—
145C2×C4⋊C4C2311yescharacteristic
146C2×C4⋊C4C2312yes—
147C2×C4⋊C4C2312yes—
148C2×C4⋊C4C2312yes—
149C2×C4⋊C4C2312yes—
150C2×C4⋊C4C2314yes—
151C2×C4⋊C4C2312yes—
152C2×C4⋊C4C2314yes—
153C2×C4⋊C4C2312yes—
154C2×C4⋊C4C2311yescharacteristic
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⋊C4C2314yes—
164C2×C22⋊C4C2312yes—
165C2×C22⋊C4C2314yes—
166C2×C22⋊C4C2314yes—
167C2×C22⋊C4C2314yes—
168C2×C22⋊C4C2311yescharacteristic
169C2×C22⋊C4C2311yescharacteristic
170C2×C42—42no—
171C2×C42C2311yescharacteristic
172C2×C42C2311yescharacteristic
173C2×C42C2312yes—
174C2×C42C2312yes—
175C2×C42C2311yescharacteristic
176C2×C42C2311yescharacteristic
177C2×C42C2312yes—
178C2×C42C2311yescharacteristic
179#95C2314yes—
180#95C2314yes—
181#95C2314yes—
182#95C2314yes—
183#95C2312yes—
184#95C2312yes—
185#95C2312yes—
186#95C2312yes—
187#95C2312yes—
188#95C2314yes—
189#95C2314yes—
190#95C2314yes—
191#95C2314yes—
192#95C2314yes—
193#95C2314yes—
194C24C2412yes—
195C24C2411yescharacteristic
196C24C2411yescharacteristic
197#41—416no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—24no—
201C2×Q8—22no—
202C2×Q8—24no—
203C2×Q8—22no—
204C2×Q8—24no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—42no—
210C2×Q8—44no—
211C2×Q8—416no—
212C2×Q8#4114yes—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—28no—
217C2×D4—22no—
218C2×D4—24no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—42no—
222C2×D4C2×D414yes—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4C2×D414yes—
241C22×C4C2×D414yes—
242C22×C4C2412yes—
243C22×C4C2412yes—
244C22×C4C2412yes—
245C22×C4C2412yes—
246C22×C4C2412yes—
247C22×C4C2412yes—
248C22×C4C2412yes—
249C22×C4C2412yes—
250C22×C4C2412yes—
251C22×C4C2412yes—
252C22×C4C2411yescharacteristic
253C22×C4C2411yescharacteristic
254C22×C4C2411yescharacteristic
255C22×C4C2411yescharacteristic
256C22×C4C2412yes—
257C22×C4C2411yescharacteristic
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—28no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4C2×D414yes—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—42no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—42no—
287C22⋊C4—44no—
288C22⋊C4C2×D414yes—
289C22⋊C4C2×D414yes—
290C42—24no—
291C42—24no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—42no—
297C42C2×D414yes—
298C23—21no—
299C23—21no—
300C23—22no—
301C23—21no—
302C23—21no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—24no—
309C23—24no—
310C23C22×D412yes—
311C23C22×D412yes—
312C232−1+414yes—
313C23C2511yescharacteristic
314Q8—28no—
315Q8—48no—
316Q8—44no—
317Q8—44no—
318Q8—84no—
319D4—28no—
320D4—48no—
321D4—48no—
322D4—44no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—28no—
332C2×C4—28no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
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—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—48no—
360C2×C4—48no—
361C2×C4—48no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4C22×D412yes—
365C2×C4C22×D412yes—
366C2×C4C22×D412yes—
367C2×C4C22×D412yes—
368C2×C4C22×D412yes—
369C2×C4C2×#4114yes—
370C2×C4C2×#4112yes—
371C2×C4H(5,𝔽2)12yes—
372C22—22no—
373C22—22no—
374C22—24no—
375C22—24no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—44no—
385C22—44no—
386C22#53614yes—
387C22C23×D411yescharacteristic
388C22C22×#4111yescharacteristic
389C22C2×H(5,𝔽2)11yescharacteristic
390C22C2×2−1+412yes—
391C22#58512yes—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—24no—
396C4—22no—
397C4—42no—
398C4—42no—
399C4—42no—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—42no—
404C4—42no—
405C4—42no—
406C4—42no—
407C4#54614yes—
408C2—24no—
409C2—42no—
410C2—82no—
411C2C2×#53611yescharacteristic
412C2C2×#54611yescharacteristic
413C2#350011yescharacteristic
414C2#351612yes—
415C2#354112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radicalC2×#548C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#554C2semidirect
3. #2850C2semidirect
4. #2700C2semidirect
5. #2690C2semidirect
6. #2640C2semidirect
7. #2621C2semidirect
8. #2568C2semidirect
9. #2545C2semidirect
10. #2504C2semidirect
11. #2503C2semidirect
12. #2450C2semidirect
13. #2444C2semidirect
14. #2434C2semidirect
15. #2432C2semidirect
16. #2327C2semidirect
17. #2311C2semidirect
18. C2×#391C2semidirect
19. #2290C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#5482
#28502
#27002
#26902
#26402
#26212
#25682
#25452
#25042
#25032
#24502
#24442
#24342
#24322
#23272
#23112
C2×#3912
#22902