Finite Group Databasebeta

← #24055

Subgroups — #24055

Order 256.

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

normal subgroups only

This group has 669 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×#547C211yesmaximal, characteristic
2C2×#483C211yesmaximal, characteristic, complemented
3C2×#417C211yesmaximal, characteristic, complemented
4#2487C211yesmaximal, characteristic, complemented
5#2043C212yesmaximal, complemented
6#2035C212yesmaximal, complemented
7#1894C211yesmaximal, characteristic, complemented
8#1870C211yesmaximal, characteristic, complemented
9#1864C212yesmaximal, complemented
10#1788C212yesmaximal, complemented
11#1788C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C22×#41C2211yescharacteristic
14C22×M16C2211yescharacteristic
15#547—24no—
16#547—24no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×#123—21no—
19C2×#123—21no—
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121—21no—
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C22×C4⋊C4—21no—
26C22×C4⋊C4C2211yescharacteristic
27#483—24no—
28C2×#106C2212yes—
29#417—24no—
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2212yes—
38#400—21no—
39#399—21no—
40#389C2212yes—
41#386—21no—
42#385—21no—
43#380C2211yescharacteristic
44#380C2212yes—
45#340C2212yes—
46#336C2212yes—
47C2×#41—22no—
48C2×#41—22no—
49C2×#41—22no—
50C2×#41—24no—
51C2×#41—24no—
52C2×#41—42no—
53C2×#41—42no—
54C2×#41D414yes—
55C22×Q8C2311yescharacteristic
56C22×Q8C2311yescharacteristic
57C22×D4—21no—
58C22×D4—21no—
59C22×D4D411yescharacteristic
60C22×D4C2311yescharacteristic
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4C2311yescharacteristic
66C23×C4C2311yescharacteristic
67C2×M16—22no—
68C2×M16—42no—
69C22×C8C2312yes—
70C22×C8C2312yes—
71C4⋊Q8—22no—
72C4⋊Q8—22no—
73#123—28no—
74#123—28no—
75C22⋊Q8—22no—
76C22⋊Q8—22no—
77C22⋊Q8—22no—
78C22⋊Q8D414yes—
79#121—28no—
80C4×D4—22no—
81C4×D4—22no—
82C4×D4—22no—
83C4×D4—22no—
84C2×C4⋊C4—28no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4D411yescharacteristic
94C2×C4⋊C4C2312yes—
95C2×C4⋊C4C2312yes—
96C2×C4⋊C4C2311yescharacteristic
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4D411yescharacteristic
103C2×C22⋊C4D411yescharacteristic
104C2×C42C2311yescharacteristic
105#106—24no—
106#103—24no—
107#103—24no—
108#103—42no—
109#103—42no—
110#102—24no—
111#102—24no—
112C22⋊C8—22no—
113C22⋊C8—42no—
114#95—22no—
115#95—21no—
116#95—22no—
117C24—21no—
118C24C2×D411yescharacteristic
119C24C2×D411yescharacteristic
120#41—28no—
121#41—44no—
122#41—48no—
123#41—48no—
124#41—44no—
125#41—44no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—22no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8C2×D412yes—
133C2×Q8C2×D412yes—
134C2×D4—24no—
135C2×D4—22no—
136C2×D4—24no—
137C2×D4—22no—
138C2×D4—21no—
139C2×D4—22no—
140C2×D4—41no—
141C2×D4—44no—
142C2×D4—44no—
143C2×D4C2×D412yes—
144C2×D4#4114yes—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—24no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—82no—
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D412yes—
169C22×C4C2×D412yes—
170C22×C4C2×D411yescharacteristic
171C22×C4C2411yescharacteristic
172M16—44no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C4⋊C4—24no—
180C4⋊C4—24no—
181C4⋊C4—24no—
182C4⋊C4—22no—
183C4⋊C4—24no—
184C4⋊C4—21no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—42no—
190C4⋊C4—82no—
191C4⋊C4C2×D412yes—
192C22⋊C4—24no—
193C22⋊C4—24no—
194C22⋊C4—24no—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—42no—
198C22⋊C4—82no—
199C42—21no—
200C42—21no—
201C23—22no—
202C23—22no—
203C23—24no—
204C23—21no—
205C23—21no—
206C23—22no—
207C23—21no—
208C23—22no—
209C23—21no—
210C23—21no—
211C23—22no—
212C23—41no—
213C23—41no—
214C23—42no—
215C23#12012yes—
216C23C22×D411yescharacteristic
217Q8—22no—
218Q8—22no—
219Q8—24no—
220Q8—41no—
221Q8—41no—
222Q8—42no—
223D4—22no—
224D4—24no—
225D4—44no—
226D4—48no—
227D4—48no—
228D4#12314yes—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—24no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—48no—
243C2×C4—48no—
244C2×C4—48no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—84no—
259C2×C4#12012yes—
260C2×C4#13712yes—
261C2×C4C22×D411yescharacteristic
262C2×C4C22×D411yescharacteristic
263C2×C4C2×#4112yes—
264C2×C4H(5,𝔽2)12yes—
265C8—42no—
266C8—42no—
267C22—22no—
268C22—24no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—44no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—44no—
277C22—42no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22C2×#12011yescharacteristic
283C22#54612yes—
284C22C2×#13711yescharacteristic
285C22#57511yescharacteristic
286C22#57712yes—
287C4—22no—
288C4—22no—
289C4—41no—
290C4—42no—
291C4—42no—
292C4—42no—
293C4—41no—
294C4—82no—
295C4—82no—
296C4C2×#12311yescharacteristic
297C4#53411yescharacteristic
298C4#54612yes—
299C2—42no—
300C2—44no—
301C2—81no—
302C2#240511yescharacteristic
303C2#300711yescharacteristic
304C2#301011yescharacteristic
305C2#330712yes—
306C2#333112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#547C2
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×#483C2semidirect
3. C2×#417C2semidirect
4. #2487C2semidirect
5. #2043C2semidirect
6. #2035C2semidirect
7. #1894C2semidirect
8. #1870C2semidirect
9. #1864C2semidirect
10. #1788C2semidirect
11. #1788C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
C2×#4832
C2×#4172
#24872
#20432
#20352
#18942
#18702
#18642
#17882