Finite Group Databasebeta

← #25709

Subgroups — #25709

Order 256.

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

normal subgroups only

This group has 638 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×#547C211yesmaximal, characteristic
2C2×#501C211yesmaximal, characteristic, complemented
3C2×#478C211yesmaximal, characteristic, complemented
4#2562C211yesmaximal, characteristic, complemented
5#2034C212yesmaximal, complemented
6#2033C212yesmaximal, complemented
7#1987C211yesmaximal, characteristic, complemented
8#1955C211yesmaximal, characteristic, complemented
9#1853C212yesmaximal, complemented
10#1852C212yesmaximal, complemented
11#1762C211yesmaximal, characteristic, complemented
12C22×#41—21no—
13#547—24no—
14#547—24no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×#125C2211yescharacteristic
17C2×#123—22no—
18C2×C22⋊Q8—21no—
19C2×#121C2212yes—
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C22×C4⋊C4C2212yes—
23#501—24no—
24#478—24no—
25C2×#107C2211yescharacteristic
26C2×#107C2212yes—
27C2×#106C2211yescharacteristic
28C2×#106C2212yes—
29C2×C4⋊C8C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2212yes—
33C2×C22⋊C8C2212yes—
34C2×C22⋊C8C2212yes—
35C2×#97C2211yescharacteristic
36#400—21no—
37#394—21no—
38#388—22no—
39#380C2212yes—
40#380C2212yes—
41C4×C4⋊C4C2211yescharacteristic
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C2×#41—28no—
46C2×#41—48no—
47C22×Q8—21no—
48C22×D4—22no—
49C22×D4D411yescharacteristic
50C22×D4C2311yescharacteristic
51C23×C4—22no—
52C23×C4—21no—
53C23×C4C2312yes—
54C22×C8C2311yescharacteristic
55C22×C8C2311yescharacteristic
56C22×C8C2312yes—
57C4⋊Q8—22no—
58C4⋊Q8D414yes—
59#125—22no—
60#125—22no—
61#123—216no—
62C22⋊Q8—28no—
63#121—24no—
64#121—24no—
65C4×D4—22no—
66C4×D4—22no—
67C4×D4—22no—
68C4×D4D414yes—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4D411yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2312yes—
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2312yes—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4D412yes—
86C2×C42—21no—
87C2×C42C2311yescharacteristic
88#107—22no—
89#107—24no—
90#106—22no—
91#106—24no—
92C4⋊C8—22no—
93#102—22no—
94#102—22no—
95#102—24no—
96C22⋊C8—24no—
97C22⋊C8—24no—
98#97—22no—
99#95—21no—
100#95—22no—
101#95—22no—
102C24—21no—
103C24C2×D412yes—
104#41—432no—
105C2×Q8—24no—
106C2×Q8—44no—
107C2×D4—24no—
108C2×D4—28no—
109C2×D4—22no—
110C2×D4—21no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—44no—
114C2×D4—44no—
115C2×D4C2×D412yes—
116C22×C4—28no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—24no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—44no—
130C22×C4—44no—
131C22×C4—44no—
132C22×C4C2×D412yes—
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2411yescharacteristic
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8#4114yescomplemented
143C4⋊C4—28no—
144C4⋊C4—24no—
145C4⋊C4—22no—
146C4⋊C4—21no—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4C2×D412yes—
157C4⋊C4C2×D414yes—
158C22⋊C4—28no—
159C22⋊C4—24no—
160C22⋊C4—24no—
161C22⋊C4—44no—
162C22⋊C4—82no—
163C42—81no—
164C42C2×D412yes—
165C42C2×D412yes—
166C23—22no—
167C23—22no—
168C23—22no—
169C23—22no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—42no—
174C23—42no—
175C23—41no—
176C23—41no—
177C23AGL(1,ℤ/8)14yes—
178C23C22×D411yescharacteristic
179Q8—48no—
180D4—28no—
181D4—22no—
182D4—22no—
183D4—416no—
184D4—42no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—48no—
194C2×C4—416no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4#12012yes—
211C2×C4#12012yes—
212C2×C4C22×D411yescharacteristic
213C2×C4C22×D411yescharacteristic
214C2×C4C2×#4112yes—
215C2×C42−1+412yes—
216C8—24no—
217C8—41no—
218C8—41no—
219C22—24no—
220C22—24no—
221C22—24no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—82no—
230C22—82no—
231C22C2×#12011yescharacteristic
232C22#54712yes—
233C22C2×AGL(1,ℤ/8)12yes—
234C22#57711yescharacteristic
235C22#57811yescharacteristic
236C4—22no—
237C4—22no—
238C4—42no—
239C4—42no—
240C4—44no—
241C4—82no—
242C4—81no—
243C4—81no—
244C4#53211yescharacteristic
245C4#53611yescharacteristic
246C4#54812yes—
247C2—44no—
248C2—82no—
249C2#243211yescharacteristic
250C2#300311yescharacteristic
251C2#301311yescharacteristic
252C2#334312yes—
253C2#334512yes—
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×#501C2semidirect
3. C2×#478C2semidirect
4. #2562C2semidirect
5. #2034C2semidirect
6. #2033C2semidirect
7. #1987C2semidirect
8. #1955C2semidirect
9. #1853C2semidirect
10. #1852C2semidirect
11. #1762C2semidirect
142. C2×C8#41semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
C2×#5012
C2×#4782
#25622
#20342
#20332
#19872
#19552
#18532
#18522
#17622