Finite Group Databasebeta

← #25676

Subgroups — #25676

Order 256.

659 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 659 conjugacy classes of subgroups — too many to draw. Only its 135 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
2C2×#501C211yesmaximal, characteristic, complemented
3C2×#490C211yesmaximal, characteristic, complemented
4#2851C211yesmaximal, characteristic, complemented
5#2034C212yesmaximal, complemented
6#2033C212yesmaximal, complemented
7#1993C211yesmaximal, characteristic, complemented
8#1961C211yesmaximal, characteristic, complemented
9#1864C212yesmaximal, complemented
10#1862C212yesmaximal, complemented
11#1749C211yesmaximal, characteristic, complemented
12C22×#41—21no—
13#532—24no—
14#532—24no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×#127C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×#124—21no—
19C2×#121—21no—
20C2×#121C2212yes—
21C2×C4×D4—21no—
22C2×C4×D4C2212yes—
23C22×C42C2211yescharacteristic
24#501—24no—
25#490—24no—
26C2×#107C2212yes—
27C2×#106C2212yes—
28C2×#102C2212yes—
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×#102C2212yes—
32C2×C22⋊C8C2212yes—
33C2×#97C2211yescharacteristic
34C2×#97C2211yescharacteristic
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#387—21no—
38#387—21no—
39#385—21no—
40#385—21no—
41#382C2211yescharacteristic
42#380C2211yescharacteristic
43#380C2211yescharacteristic
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C2×#41—28no—
48C2×#41—44no—
49C2×#41—44no—
50C22×Q8—21no—
51C22×D4—22no—
52C22×D4—21no—
53C22×D4—21no—
54C22×D4C2312yes—
55C23×C4—22no—
56C23×C4D411yescharacteristic
57C23×C4D411yescharacteristic
58C23×C4C2311yescharacteristic
59C22×C8C2312yes—
60C22×C8C2312yes—
61C4⋊Q8—22no—
62C4⋊Q8D414yes—
63#127—22no—
64#127D414yes—
65#125—22no—
66#125—22no—
67#124—28no—
68#121—28no—
69#121—24no—
70#121—24no—
71C4×D4—28no—
72C4×D4—24no—
73C4×D4—24no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2312yes—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C42—22no—
89C2×C42—22no—
90C2×C42D411yescharacteristic
91C2×C42D411yescharacteristic
92C2×C42C2311yescharacteristic
93C2×C42C2311yescharacteristic
94#107—24no—
95#106—24no—
96#102—24no—
97#102—24no—
98#102—24no—
99#102—24no—
100C22⋊C8—24no—
101#97—22no—
102#97—22no—
103#95—21no—
104#95—21no—
105#95—21no—
106#95—21no—
107#95—21no—
108#95—21no—
109C24—22no—
110C24C2×D411yescharacteristic
111#41—432no—
112C2×Q8—24no—
113C2×Q8—42no—
114C2×Q8—42no—
115C2×D4—28no—
116C2×D4—24no—
117C2×D4—24no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—42no—
124C2×D4—42no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4—44no—
128C2×D4—44no—
129C2×D4C2×D414yes—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—28no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2411yescharacteristic
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8#4114yescomplemented
160C4⋊C4—24no—
161C4⋊C4—22no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4C2×D414yes—
173C22⋊C4—28no—
174C22⋊C4—28no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C42—22no—
178C42—22no—
179C42—21no—
180C42—21no—
181C42C2×D412yes—
182C42C2×D412yes—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—24no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23AGL(1,ℤ/8)12yes—
197C23C22×D411yescharacteristic
198Q8—48no—
199D4—24no—
200D4—24no—
201D4—48no—
202D4—48no—
203D4—416no—
204D4—42no—
205D4—42no—
206C2×C4—28no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—24no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—44no—
221C2×C4—416no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4#12012yes—
239C2×C4#12012yes—
240C2×C4AGL(1,ℤ/8)12yes—
241C2×C4C22×D411yescharacteristic
242C2×C4C22×D411yescharacteristic
243C2×C4C2×#4112yes—
244C2×C42−1+412yes—
245C8—24no—
246C8—42no—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—41no—
254C22—44no—
255C22—44no—
256C22—44no—
257C22—82no—
258C22—82no—
259C22—82no—
260C22—82no—
261C22C2×#12011yescharacteristic
262C22#53211yescharacteristic
263C22#53611yescharacteristic
264C22C2×AGL(1,ℤ/8)11yescharacteristic
265C22C2×AGL(1,ℤ/8)11yescharacteristic
266C22#57511yescharacteristic
267C22#57511yescharacteristic
268C4—22no—
269C4—22no—
270C4—22no—
271C4—41no—
272C4—44no—
273C4—41no—
274C4—81no—
275C4—81no—
276C4—81no—
277C4—81no—
278C4#54712yes—
279C4#54812yes—
280C2—42no—
281C2—84no—
282C2#243211yescharacteristic
283C2#300311yescharacteristic
284C2#300511yescharacteristic
285C2#317111yescharacteristic
286C2#318011yescharacteristic
287C2#323411yescharacteristic
288C2#323511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#532C2
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
2. C2×#501C2semidirect
3. C2×#490C2semidirect
4. #2851C2semidirect
5. #2034C2semidirect
6. #2033C2semidirect
7. #1993C2semidirect
8. #1961C2semidirect
9. #1864C2semidirect
10. #1862C2semidirect
11. #1749C2semidirect
159. C2×C8#41semidirect

Maximal subgroups

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

subgroupindex
C2×#5322
C2×#5012
C2×#4902
#28512
#20342
#20332
#19932
#19612
#18642
#18622
#17492