Finite Group Databasebeta

← #25658

Subgroups — #25658

Order 256.

538 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 131 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×#559C211yesmaximal, characteristic
2C2×#490C211yesmaximal, characteristic
3C2×#479C211yesmaximal, characteristic, complemented
4#2584C211yesmaximal, characteristic, complemented
5#2042C212yesmaximal, complemented
6#2042C212yesmaximal, complemented
7#1985C211yesmaximal, characteristic, complemented
8#1982C211yesmaximal, characteristic
9#1864C212yesmaximal, complemented
10#1862C212yesmaximal, complemented
11#1763C211yesmaximal, characteristic, complemented
12#559—24no—
13#559—24no—
14C2×#127C2211yescharacteristic
15C2×#125C2211yescharacteristic
16C2×#124—21no—
17C2×#124C2212yes—
18C2×#123—22no—
19C2×C4×Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4—21no—
22C2×C4×D4C2212yes—
23#490—24no—
24#479—24no—
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2212yes—
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—
45C22×Q8D411yescharacteristic
46C22×Q8C2311yescharacteristic
47C22×D4—21no—
48C22×D4—21no—
49C22×D4C2312yes—
50C23×C4—22no—
51C22×C8C2312yes—
52C22×C8C2311yescharacteristic
53C22×C8C2311yescharacteristic
54#127—22no—
55#127D414yes—
56#125—22no—
57#125—22no—
58#124—28no—
59#124—24no—
60#124—24no—
61#123—216no—
62C4×Q8—22no—
63C4×Q8—22no—
64C4×Q8—22no—
65C4×Q8D414yes—
66C4×D4—28no—
67C4×D4—24no—
68C4×D4—24no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4D411yescharacteristic
75C2×C4⋊C4D412yes—
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2312yes—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42C2312yes—
86C2×C42C2311yescharacteristic
87#107—22no—
88#106—22no—
89C4⋊C8—22no—
90#103—22no—
91#103—22no—
92#103—24no—
93#102—24no—
94#102—24no—
95#102—24no—
96#102—24no—
97#97—22no—
98#95—21no—
99#95—22no—
100#95—22no—
101C24—22no—
102C2×Q8—22no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—42no—
107C2×Q8C2×D412yes—
108C2×D4—24no—
109C2×D4—24no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—44no—
116C2×D4—81no—
117C2×D4—81no—
118C2×D4C2×D414yes—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—28no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—82no—
126C22×C4—82no—
127C22×C4C2×D412yes—
128C22×C4C2×D412yes—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2411yescharacteristic
132C2×C8—22no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C4⋊C4—24no—
140C4⋊C4—22no—
141C4⋊C4—24no—
142C4⋊C4—21no—
143C4⋊C4—22no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—44no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—82no—
152C4⋊C4C2×D412yes—
153C4⋊C4#4114yes—
154C22⋊C4—28no—
155C22⋊C4—28no—
156C22⋊C4—28no—
157C42—24no—
158C42—22no—
159C42—22no—
160C42—81no—
161C42C2×D412yes—
162C42C2×D412yes—
163C23—24no—
164C23—42no—
165C23—42no—
166C23—42no—
167C23—42no—
168C23—42no—
169C23—42no—
170C23C22×D411yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—44no—
174Q8—42no—
175D4—24no—
176D4—24no—
177D4—48no—
178D4—42no—
179D4—42no—
180D4—84no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—88no—
204C2×C4#12012yes—
205C2×C4#12012yes—
206C2×C4C22×D411yescharacteristic
207C2×C4C22×D411yescharacteristic
208C2×C4C2×#4112yes—
209C2×C42−1+412yes—
210C8—41no—
211C8—41no—
212C8—42no—
213C22—44no—
214C22—44no—
215C22—44no—
216C22—82no—
217C22—82no—
218C22—82no—
219C22—82no—
220C22C2×#12011yescharacteristic
221C22#54812yes—
222C22#57512yes—
223C22#57611yescharacteristic
224C22#57711yescharacteristic
225C4—22no—
226C4—22no—
227C4—44no—
228C4—42no—
229C4—42no—
230C4—42no—
231C4—42no—
232C4—81no—
233C4—81no—
234C4#53211yescharacteristic
235C4#53611yescharacteristic
236C4#54712yes—
237C2—84no—
238C2#243211yescharacteristic
239C2#300511yescharacteristic
240C2#301211yescharacteristic
241C2#337012yes—
242C2#337112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
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
3. C2×#479C2semidirect
4. #2584C2semidirect
5. #2042C2semidirect
6. #2042C2semidirect
7. #1985C2semidirect
9. #1864C2semidirect
10. #1862C2semidirect
11. #1763C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5592
C2×#4902
C2×#4792
#25842
#20422
#19852
#19822
#18642
#18622
#17632