Finite Group Databasebeta

← #35950

Subgroups — #35950

Order 256.

1120 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#554C211yesmaximal, characteristic
2C2×#554C211yesmaximal, characteristic
3#2660C211yesmaximal, characteristic, complemented
4#2659C212yesmaximal, complemented
5#2584C211yesmaximal, characteristic, complemented
6#2583C212yesmaximal, complemented
7#2552C212yesmaximal, complemented
8#2550C212yesmaximal, complemented
9#2549C214yesmaximal, complemented
10#2546C211yesmaximal, characteristic, complemented
11#2502C212yesmaximal, complemented
12#2440C214yesmaximal, complemented
13#2399C211yesmaximal, characteristic, complemented
14C2×#398C212yesmaximal, complemented
15#2349C212yesmaximal
16D4×C4⋊C4C212yesmaximal
17#2275C211yesmaximal, characteristic
18#554—28no—
19#554—28no—
20C2×C4⋊Q8C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic
22C2×C4⋊Q8C2212yes—
23C2×#125C2211yescharacteristic
24C2×#125C2212yes—
25C2×#125C2211yescharacteristic
26C2×C22⋊Q8C2212yes—
27C2×C22⋊Q8C2212yes—
28C2×C22⋊Q8C2212yes—
29C2×C22⋊Q8C2214yes—
30C2×C22⋊Q8C2212yes—
31C2×C4×Q8C2211yescharacteristic
32C2×C4×Q8C2211yescharacteristic
33C2×C4×D4C2211yescharacteristic
34C2×C4×D4C2212yes—
35C2×C4×D4C2211yescharacteristic
36C2×C4×D4C2211yescharacteristic
37C2×C4×D4C2212yes—
38C22×C4⋊C4C2212yes—
39C22×C4⋊C4C2212yes—
40C22×C4⋊C4C2214yes—
41#398C2212yes—
42#398C2212yes—
43#398C2212yes—
44#398C2211yescharacteristic
45#398C2212yes—
46#398C2214yes—
47#398C2214yes—
48#398C2211yescharacteristic
49#398C2212yes—
50#398C2214yes—
51#398C2212yes—
52#395C2211yescharacteristic
53#395C2212yes—
54#395C2211yescharacteristic
55#395C2212yes—
56#395C2214yes—
57#395C2214yes—
58#391C2214yes—
59#391C2212yes—
60#389C2214yes—
61#389C2214yes—
62#389C2212yes—
63#389C2212yes—
64#389C2214yes—
65#389C2212yes—
66#389C2212yes—
67#385C2214yes—
68#385C2214yes—
69#385C2214yes—
70#385C2214yes—
71#384C2212yes—
72#384C2214yes—
73#382C2212yes—
74#380C2214yes—
75#380C2214yes—
76C4×C4⋊C4C2211yescharacteristic
77C4×C4⋊C4C2212yes—
78C4×C4⋊C4C2212yes—
79C4×C4⋊C4C2212yes—
80C4×C22⋊C4C2212yes—
81C4×C22⋊C4C2214yes—
82#376C2211yescharacteristic
83C2×#95C2214yes—
84C2×#95C2212yes—
85C22×Q8C2311yescharacteristic
86C22×Q8C2311yescharacteristic
87C22×Q8C2312yes—
88C22×Q8C2312yes—
89C22×D4C2311yescharacteristic
90C23×C4C2312yes—
91C23×C4C2314yes—
92C23×C4C2312yes—
93C23×C4C2312yes—
94C23×C4C2314yes—
95C4⋊Q8—24no—
96C4⋊Q8—24no—
97C4⋊Q8—44no—
98#125—24no—
99#125—28no—
100#125—24no—
101C22⋊Q8—28no—
102C22⋊Q8—28no—
103C22⋊Q8—216no—
104C22⋊Q8—28no—
105C22⋊Q8—28no—
106C4×Q8—24no—
107C4×Q8—24no—
108C4×D4—28no—
109C4×D4—24no—
110C4×D4—24no—
111C4×D4D418yes—
112C4×D4Q8116yescomplemented
113C2×C4⋊C4—28no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—48no—
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2312yes—
119C2×C4⋊C4C2314yes—
120C2×C4⋊C4C2312yes—
121C2×C4⋊C4C2312yes—
122C2×C4⋊C4C2312yes—
123C2×C4⋊C4C2312yes—
124C2×C4⋊C4C2314yes—
125C2×C4⋊C4C2311yescharacteristic
126C2×C4⋊C4C2311yescharacteristic
127C2×C4⋊C4C2312yes—
128C2×C4⋊C4C2312yes—
129C2×C4⋊C4C2311yescharacteristic
130C2×C4⋊C4C2311yescharacteristic
131C2×C4⋊C4C2312yes—
132C2×C4⋊C4C2312yes—
133C2×C4⋊C4C2314yes—
134C2×C4⋊C4C2314yes—
135C2×C4⋊C4C2314yes—
136C2×C4⋊C4C2314yes—
137C2×C4⋊C4C2312yes—
138C2×C4⋊C4C2312yes—
139C2×C4⋊C4C2312yes—
140C2×C4⋊C4C2314yes—
141C2×C4⋊C4C2314yes—
142C2×C4⋊C4C2311yescharacteristic
143C2×C4⋊C4C2312yes—
144C2×C22⋊C4C2312yes—
145C2×C22⋊C4C2312yes—
146C2×C22⋊C4C2314yes—
147C2×C22⋊C4C2312yes—
148C2×C22⋊C4C2314yes—
149C2×C42C2311yescharacteristic
150C2×C42C2312yes—
151C2×C42C2311yescharacteristic
152C2×C42C2312yes—
153C2×C42C2314yes—
154C2×C42C2311yescharacteristic
155C2×C42C2312yes—
156#95C2311yescharacteristic
157#95C2312yes—
158#95C2312yes—
159#95C2314yes—
160#95C2314yes—
161#95C2314yes—
162#95C2314yes—
163#95C2312yes—
164#95C2311yescharacteristic
165#95C2312yes—
166#95C2312yes—
167#95C2312yes—
168#95C2314yes—
169#95C2314yes—
170#95C2314yes—
171C24C2412yes—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—24no—
175C2×Q8—24no—
176C2×Q8—22no—
177C2×Q8—41no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—44no—
181C2×Q8—42no—
182C2×D4C2×D414yes—
183C2×D4C2×Q818yes—
184C22×C4—24no—
185C22×C4—28no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4C2×D418yes—
196C22×C4C2×Q8116yes—
197C22×C4C2411yescharacteristic
198C22×C4C2411yescharacteristic
199C22×C4C2412yes—
200C22×C4C2412yes—
201C22×C4C2412yes—
202C22×C4C2414yes—
203C22×C4C2411yescharacteristic
204C22×C4C2412yes—
205C22×C4C2411yescharacteristic
206C22×C4C2411yescharacteristic
207C22×C4C2412yes—
208C22×C4C2412yes—
209C22×C4C2412yes—
210C22×C4C2414yes—
211C22×C4C2411yescharacteristic
212C22×C4C2411yescharacteristic
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—24no—
216C4⋊C4—28no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—24no—
223C4⋊C4—22no—
224C4⋊C4—24no—
225C4⋊C4—22no—
226C4⋊C4—24no—
227C4⋊C4—28no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—28no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4—42no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—44no—
239C4⋊C4C2×D414yes—
240C4⋊C4C2×Q818yes—
241C22⋊C4—28no—
242C22⋊C4—24no—
243C22⋊C4—24no—
244C22⋊C4C2×D418yes—
245C22⋊C4C2×Q8116yes—
246C42—24no—
247C42—22no—
248C42—22no—
249C42—44no—
250C42—42no—
251C42C2×D414yes—
252C42C2×Q818yes—
253C23—22no—
254C23—24no—
255C23—22no—
256C23C22×D414yes—
257C23C22×Q818yes—
258C23C2511yescharacteristic
259Q8—44no—
260Q8—44no—
261Q8—84no—
262Q8—84no—
263D4C4⋊Q8116yescomplemented
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—48no—
298C2×C4—416no—
299C2×C4—48no—
300C2×C4—84no—
301C2×C4—88no—
302C2×C4C22×D412yes—
303C2×C4C22×D412yes—
304C2×C4C22×D414yes—
305C2×C4C22×D412yes—
306C2×C4C22×Q814yes—
307C2×C4C22×Q814yes—
308C2×C4C22×Q818yes—
309C2×C4C22×Q814yes—
310C2×C4H(5,𝔽2)12yes—
311C22—24no—
312C22—24no—
313C22—28no—
314C22—28no—
315C22C2×C4⋊Q818yes—
316C22C23×D411yescharacteristic
317C22C23×Q812yes—
318C22C2×H(5,𝔽2)11yescharacteristic
319C22#58512yes—
320C22#58511yescharacteristic
321C4—22no—
322C4—24no—
323C4—22no—
324C4—22no—
325C4—24no—
326C4—28no—
327C4—24no—
328C4—41no—
329C4—42no—
330C4—42no—
331C4—41no—
332C4—42no—
333C4—42no—
334C4—44no—
335C4—41no—
336C4C2×C4⋊Q814yes—
337C2—28no—
338C2C22×C4⋊Q811yescharacteristic
339C2#348511yescharacteristic
340C2#348811yescharacteristic
341C2#350112yes—
342C2#350412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radicalC22×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2660C2semidirect
4. #2659C2semidirect
5. #2584C2semidirect
6. #2583C2semidirect
7. #2552C2semidirect
8. #2550C2semidirect
9. #2549C2semidirect
10. #2546C2semidirect
11. #2502C2semidirect
12. #2440C2semidirect
13. #2399C2semidirect
14. C2×#398C2semidirect
112. C4×D4Q8semidirect
263. D4C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
#26602
#26592
#25842
#25832
#25522
#25502
#25492
#25462
#25022
#24402
#23992
C2×#3982
#23492
D4×C4⋊C42
#22752