Finite Group Databasebeta

← #25655

Subgroups — #25655

Order 256.

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

normal subgroups only

This group has 662 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×#550C211yesmaximal, characteristic
2C2×#490C211yesmaximal, characteristic, complemented
3C2×#478C211yesmaximal, characteristic, complemented
4#2584C211yesmaximal, characteristic, complemented
5#2041C212yesmaximal, complemented
6#2041C212yesmaximal, complemented
7#1985C211yesmaximal, characteristic, complemented
8#1982C211yesmaximal, characteristic, complemented
9#1864C212yesmaximal, complemented
10#1862C212yesmaximal, complemented
11#1762C211yesmaximal, characteristic, complemented
12C22×#41—21no—
13#550—24no—
14#550—24no—
15C2×#127C2212yes—
16C2×#127C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×#121—22no—
19C2×#121—21no—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4C2212yes—
22C2×C4×D4C2211yescharacteristic
23#490—24no—
24#478—24no—
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2212yes—
31C2×#102C2212yes—
32C2×#102C2212yes—
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×#97C2211yescharacteristic
36#398—21no—
37#395—21no—
38#389—22no—
39#389C2212yes—
40#389C2212yes—
41C4×C4⋊C4C2211yescharacteristic
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C2×#41—28no—
46C2×#41—48no—
47C22×Q8D411yescharacteristic
48C22×D4—21no—
49C22×D4—22no—
50C22×D4—22no—
51C22×D4—21no—
52C22×D4C2312yes—
53C22×D4C2311yescharacteristic
54C23×C4—22no—
55C23×C4—21no—
56C22×C8C2312yes—
57C22×C8C2311yescharacteristic
58C22×C8C2311yescharacteristic
59#127—24no—
60#127—24no—
61#127—22no—
62#127D414yes—
63#125—22no—
64#125—22no—
65#121—28no—
66#121—216no—
67C4×Q8—22no—
68C4×Q8—22no—
69C4×D4—24no—
70C4×D4—24no—
71C4×D4—22no—
72C4×D4D414yes—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4D412yes—
78C2×C4⋊C4D411yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2312yes—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—21no—
85C2×C42—21no—
86C2×C42C2312yes—
87C2×C42C2311yescharacteristic
88#107—22no—
89#106—22no—
90C4⋊C8—22no—
91#102—22no—
92#102—22no—
93#102—24no—
94#102—24no—
95#102—24no—
96#102—24no—
97#102—24no—
98#97—22no—
99#95—21no—
100#95—22no—
101#95—22no—
102C24—22no—
103C24—21no—
104#41—432no—
105C2×Q8—22no—
106C2×Q8—24no—
107C2×D4—28no—
108C2×D4—28no—
109C2×D4—24no—
110C2×D4—24no—
111C2×D4—22no—
112C2×D4—21no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—44no—
120C2×D4—44no—
121C2×D4—44no—
122C2×D4—81no—
123C2×D4—81no—
124C2×D4—82no—
125C2×D4—82no—
126C2×D4C2×D414yes—
127C2×D4C2×D412yes—
128C22×C4—28no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—24no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—44no—
135C22×C4—44no—
136C22×C4—44no—
137C22×C4C2×D412yes—
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D412yes—
141C22×C4C2411yescharacteristic
142C2×C8—22no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C4⋊C4—24no—
150C4⋊C4—22no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—44no—
160C4⋊C4—82no—
161C4⋊C4C2×D412yes—
162C4⋊C4#4114yescomplemented
163C22⋊C4—28no—
164C22⋊C4—24no—
165C42—22no—
166C42—22no—
167C42—81no—
168C42C2×D412yes—
169C42C2×D412yes—
170C23—24no—
171C23—22no—
172C23—42no—
173C23—42no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23—42no—
178C23—42no—
179C23—41no—
180C23—42no—
181C23—41no—
182C23C22×D411yescharacteristic
183Q8—28no—
184D4—22no—
185D4—22no—
186D4—24no—
187D4—24no—
188D4—48no—
189D4—416no—
190D4—42no—
191D4—42no—
192D4—42no—
193D4—84no—
194D4—88no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—48no—
206C2×C4—416no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4#12012yes—
216C2×C4#12012yes—
217C2×C4AGL(1,ℤ/8)14yes—
218C2×C4C22×D411yescharacteristic
219C2×C4C22×D411yescharacteristic
220C2×C4C2×#4112yes—
221C2×C42−1+412yes—
222C8—41no—
223C8—41no—
224C8—42no—
225C22—42no—
226C22—44no—
227C22—42no—
228C22—44no—
229C22—44no—
230C22—42no—
231C22—82no—
232C22—82no—
233C22—82no—
234C22—82no—
235C22—82no—
236C22—82no—
237C22C2×#12011yescharacteristic
238C22#54812yes—
239C22C2×AGL(1,ℤ/8)12yes—
240C22#57611yescharacteristic
241C22#57711yescharacteristic
242C4—22no—
243C4—24no—
244C4—22no—
245C4—44no—
246C4—42no—
247C4—42no—
248C4—81no—
249C4—81no—
250C4#53211yescharacteristic
251C4#53611yescharacteristic
252C4#54712yes—
253C2—82no—
254C2—84no—
255C2#243211yescharacteristic
256C2#300311yescharacteristic
257C2#301211yescharacteristic
258C2#336612yes—
259C2#336812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#550C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#490C2semidirect
3. C2×#478C2semidirect
4. #2584C2semidirect
5. #2041C2semidirect
6. #2041C2semidirect
7. #1985C2semidirect
8. #1982C2semidirect
9. #1864C2semidirect
10. #1862C2semidirect
11. #1762C2semidirect
162. C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#5502
C2×#4902
C2×#4782
#25842
#20412
#19852
#19822
#18642
#18622
#17622