Finite Group Databasebeta

← #24242

Subgroups — #24242

Order 256.

745 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 745 conjugacy classes of subgroups — too many to draw. Only its 151 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×#550C211yesmaximal, characteristic
2C2×#460C211yesmaximal, characteristic, complemented
3C2×#452C212yesmaximal, complemented
4C2×C4×Q16C211yesmaximal, characteristic, complemented
5#2538C211yesmaximal, characteristic, complemented
6#2033C212yesmaximal, complemented
7#2000C212yesmaximal, complemented
8#1957C211yesmaximal, characteristic, complemented
9#1954C211yesmaximal, characteristic, complemented
10#1803C211yesmaximal, characteristic, complemented
11#1788C212yesmaximal, complemented
12C22×#41C2212yes—
13C22×Q16C2211yescharacteristic
14C22×SD16C2212yes—
15#550—24no—
16#550—24no—
17C2×#127—21no—
18C2×#127C2211yescharacteristic
19C2×C22⋊Q8C2212yes—
20C2×#121—22no—
21C2×#121C2212yes—
22C2×C4×Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×D4—21no—
25C2×C4×D4C2211yescharacteristic
26#460—24no—
27#452—28no—
28C4×Q16—24no—
29C2×#107C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2212yes—
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×#102C2212yes—
37C2×C22⋊C8C2212yes—
38C2×C22⋊C8C2212yes—
39C2×C4×C8C2211yescharacteristic
40#396—22no—
41#390—21no—
42#389—21no—
43#384C2211yescharacteristic
44#380C2212yes—
45#340C2212yes—
46C2×#41—24no—
47C2×#41—24no—
48C2×#41—24no—
49C2×#41—24no—
50C2×#41—42no—
51C2×#41—42no—
52C22×Q8C2311yescharacteristic
53C22×Q8C2311yescharacteristic
54C22×D4—22no—
55C22×D4—22no—
56C22×D4—21no—
57C22×D4—21no—
58C22×D4D411yescharacteristic
59C22×D4C2312yes—
60C23×C4—22no—
61C23×C4C2312yes—
62C2×Q16—22no—
63C2×Q16—22no—
64C2×Q16D414yescomplemented
65C2×SD16—24no—
66C2×SD16—24no—
67C2×SD16—42no—
68C22×C8C2312yes—
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71#127—28no—
72#127—22no—
73#127D414yes—
74C22⋊Q8—24no—
75C22⋊Q8—42no—
76#121—216no—
77#121—24no—
78#121—24no—
79C4×Q8—22no—
80C4×Q8—22no—
81C4×Q8—22no—
82C4×Q8D414yes—
83C4×D4—28no—
84C4×D4—22no—
85C4×D4—22no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2312yes—
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4D412yes—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42C2311yescharacteristic
102#107—22no—
103C4⋊C8—22no—
104#103—22no—
105#103—24no—
106#103—22no—
107#102—24no—
108#102—24no—
109#102—42no—
110C22⋊C8—24no—
111C22⋊C8—42no—
112C4×C8—22no—
113#95—22no—
114C24—22no—
115C24C2×D412yes—
116#41—48no—
117#41—48no—
118#41—48no—
119#41—48no—
120C2×Q8—21no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8C2×D412yes—
129C2×Q8C2×D412yes—
130C2×Q8C2×D412yes—
131C2×Q8#4112yes—
132C2×D4—28no—
133C2×D4—28no—
134C2×D4—22no—
135C2×D4—24no—
136C2×D4—24no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—44no—
145C2×D4—44no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—81no—
149C2×D4—81no—
150C2×D4—82no—
151C2×D4—82no—
152C2×D4C2×D414yes—
153C22×C4—21no—
154C22×C4—28no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4C2×D412yes—
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2411yescharacteristic
171Q16—28no—
172SD16—48no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—21no—
179C2×C8—22no—
180C2×C8—21no—
181C2×C8C2×D412yes—
182C4⋊C4—24no—
183C4⋊C4—22no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—22no—
187C4⋊C4—21no—
188C4⋊C4—22no—
189C4⋊C4—44no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—81no—
193C4⋊C4C2×D412yes—
194C22⋊C4—28no—
195C22⋊C4—24no—
196C22⋊C4—44no—
197C22⋊C4—82no—
198C42—24no—
199C42—42no—
200C42C2×D412yes—
201C42C2×D412yes—
202C23—22no—
203C23—24no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23C22×D411yescharacteristic
217Q8—22no—
218Q8—22no—
219Q8—22no—
220Q8—22no—
221Q8—22no—
222Q8—41no—
223Q8—41no—
224Q8#12114yes—
225D4—24no—
226D4—24no—
227D4—416no—
228D4—42no—
229D4—42no—
230D4—44no—
231D4—48no—
232D4—84no—
233D4—88no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—416no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4#12012yes—
260C2×C4#12012yes—
261C2×C4#13512yes—
262C2×C4#13712yes—
263C2×C4C22×D411yescharacteristic
264C2×C4C22×D411yescharacteristic
265C2×C4C22×D411yescharacteristic
266C2×C4C2×#4111yescharacteristic
267C2×C4H(5,𝔽2)11yescharacteristic
268C2×C4H(5,𝔽2)11yescharacteristic
269C8—22no—
270C8—41no—
271C8—42no—
272C22—42no—
273C22—44no—
274C22—42no—
275C22—42no—
276C22—44no—
277C22—42no—
278C22—44no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—82no—
283C22—82no—
284C22—82no—
285C22—82no—
286C22C2×#12011yescharacteristic
287C22D4211yescharacteristic
288C22#54611yescharacteristic
289C22C2×#13511yescharacteristic
290C22C2×#13711yescharacteristic
291C22#57611yescharacteristic
292C22#57711yescharacteristic
293C4—24no—
294C4—22no—
295C4—42no—
296C4—42no—
297C4—42no—
298C4—81no—
299C4—82no—
300C4#45214yescomplemented
301C4C2×#12111yescharacteristic
302C4#53511yescharacteristic
303C4D4211yescharacteristic
304C4#54611yescharacteristic
305C2—82no—
306C2—84no—
307C2#240811yescharacteristic
308C2C2×#45211yescharacteristic
309C2#301211yescharacteristic
310C2#327811yescharacteristic
311C2#328711yescharacteristic
312C2#330411yescharacteristic
313C2#333311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#550C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#460C2semidirect
3. C2×#452C2semidirect
4. C2×C4×Q16C2semidirect
5. #2538C2semidirect
6. #2033C2semidirect
7. #2000C2semidirect
8. #1957C2semidirect
9. #1954C2semidirect
10. #1803C2semidirect
11. #1788C2semidirect
64. C2×Q16D4semidirect
300. C4#452semidirect

Maximal subgroups

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

subgroupindex
C2×#5502
C2×#4602
C2×#4522
C2×C4×Q162
#25382
#20332
#20002
#19572
#19542
#18032
#17882