Finite Group Databasebeta

← #23932

Subgroups — #23932

Order 256.

778 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#532C211yesmaximal, characteristic, complemented
2C2×#449C212yesmaximal, complemented
3#2297C211yesmaximal, characteristic, complemented
4#2024C211yesmaximal, characteristic, complemented
5#2023C211yesmaximal, characteristic, complemented
6#2014C211yesmaximal, characteristic, complemented
7#2011C211yesmaximal, characteristic, complemented
8#1879C212yesmaximal, complemented
9#1877C212yesmaximal, complemented
10#1875C212yesmaximal, complemented
11#1817C211yesmaximal, characteristic, complemented
12C22×#41—21no—
13C22×#41C2211yescharacteristic
14C22×SD16C2212yes—
15C22×D8C2212yes—
16#532—28no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×#127C2211yescharacteristic
19C2×#124—21no—
20C2×#121—21no—
21C2×#121C2212yes—
22C2×C4×D4—21no—
23C2×C4×D4C2212yes—
24C2×#117C2211yescharacteristic
25C22×C42C2211yescharacteristic
26#449—28no—
27C2×#103C2212yes—
28C2×#102C2212yes—
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×C22⋊C8C2212yes—
32#391C2211yescharacteristic
33#391C2211yescharacteristic
34#390C2211yescharacteristic
35#390C2211yescharacteristic
36#386—21no—
37#386—21no—
38C4×C22⋊C4—21no—
39C4×C22⋊C4—21no—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43#336C2211yescharacteristic
44#336C2211yescharacteristic
45C2×#41—28no—
46C2×#41—22no—
47C2×#41—22no—
48C2×#41—24no—
49C2×#41—42no—
50C2×#41—44no—
51C2×#41—44no—
52C22×Q8—21no—
53C22×Q8C2311yescharacteristic
54C22×D4—22no—
55C22×D4—21no—
56C22×D4—21no—
57C22×D4—21no—
58C22×D4C2312yes—
59C22×D4C2311yescharacteristic
60C23×C4—22no—
61C23×C4—21no—
62C23×C4D411yescharacteristic
63C23×C4D411yescharacteristic
64C23×C4C2311yescharacteristic
65C2×SD16—24no—
66C2×SD16—24no—
67C2×SD16—42no—
68C2×D8—24no—
69C2×D8—24no—
70C2×D8—42no—
71C22×C8C2312yes—
72C22×C8C2312yes—
73C4⋊Q8—24no—
74#127—24no—
75#124—28no—
76#121—28no—
77#121—24no—
78#121D418yes—
79C4×D4—28no—
80C4×D4—24no—
81C4×D4—24no—
82#117—42no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2312yes—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C42—24no—
95C2×C42—21no—
96C2×C42D411yescharacteristic
97C2×C42D411yescharacteristic
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100#103—24no—
101#102—24no—
102#102—42no—
103#102—42no—
104C22⋊C8—24no—
105#95—21no—
106#95—21no—
107#95—21no—
108#95—21no—
109C24—21no—
110C24—22no—
111C24C2×D411yescharacteristic
112#41—48no—
113#41—48no—
114#41—432no—
115C2×Q8—24no—
116C2×Q8—21no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8#4112yescomplemented
122C2×D4—24no—
123C2×D4—24no—
124C2×D4—24no—
125C2×D4—28no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—42no—
135C2×D4—42no—
136C2×D4—42no—
137C2×D4—44no—
138C2×D4—44no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4C2×D414yes—
143C2×D4#4112yescomplemented
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—28no—
151C22×C4—24no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—42no—
160C22×C4—44no—
161C22×C4—42no—
162C22×C4—44no—
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D414yes—
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2411yescharacteristic
170SD16—48no—
171D8—48no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—22no—
178C4⋊C4—24no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4C2×D414yes—
183C22⋊C4—28no—
184C22⋊C4—28no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C42—22no—
191C42—22no—
192C42—22no—
193C42—22no—
194C42—42no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—24no—
202C23—22no—
203C23—41no—
204C23—42no—
205C23—41no—
206C23—42no—
207C23—41no—
208C23—42no—
209C23—41no—
210C23—41no—
211C23—42no—
212C23—42no—
213C23—41no—
214C23—42no—
215C23#12012yes—
216C23C22×D411yescharacteristic
217Q8—24no—
218Q8—48no—
219Q8—42no—
220D4—24no—
221D4—24no—
222D4—24no—
223D4—48no—
224D4—48no—
225D4—48no—
226D4—416no—
227D4—42no—
228D4—42no—
229D4—42no—
230C2×C4—28no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—21no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—44no—
247C2×C4—48no—
248C2×C4—416no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4#12012yes—
260C2×C4#13512yes—
261C2×C4AGL(1,ℤ/8)12yes—
262C2×C4AGL(1,ℤ/8)12yes—
263C2×C4C22×D412yes—
264C2×C4C2×#4111yescharacteristic
265C2×C4C2×#4111yescharacteristic
266C2×C4H(5,𝔽2)12yes—
267C8—42no—
268C8—42no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—44no—
280C22—44no—
281C22—44no—
282C22—82no—
283C22—82no—
284C22—81no—
285C22—82no—
286C22—81no—
287C22—82no—
288C22—81no—
289C22—81no—
290C22C2×#12011yescharacteristic
291C22#52511yescharacteristic
292C22#53511yescharacteristic
293C22C2×#13511yescharacteristic
294C22C2×AGL(1,ℤ/8)11yescharacteristic
295C22C2×AGL(1,ℤ/8)11yescharacteristic
296C22#57511yescharacteristic
297C4—22no—
298C4—22no—
299C4—22no—
300C4—22no—
301C4—42no—
302C4—41no—
303C4—44no—
304C4—81no—
305C4—81no—
306C4#54612yes—
307C4#54612yes—
308C2—42no—
309C2—82no—
310C2—84no—
311C2#240411yescharacteristic
312C2#300211yescharacteristic
313C2#300311yescharacteristic
314C2#310611yescharacteristic
315C2#311511yescharacteristic
316C2#320711yescharacteristic
317C2#321411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#532C2semidirect
2. C2×#449C2semidirect
3. #2297C2semidirect
4. #2024C2semidirect
5. #2023C2semidirect
6. #2014C2semidirect
7. #2011C2semidirect
8. #1879C2semidirect
9. #1877C2semidirect
10. #1875C2semidirect
11. #1817C2semidirect
121. C2×Q8#41semidirect
143. C2×D4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#5322
C2×#4492
#22972
#20242
#20232
#20142
#20112
#18792
#18772
#18752
#18172