Finite Group Databasebeta

← #25710

Subgroups — #25710

Order 256.

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

normal subgroups only

This group has 540 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×#555C211yesmaximal, characteristic
2C2×#501C211yesmaximal, characteristic, complemented
3C2×#478C211yesmaximal, characteristic, complemented
4#2562C211yesmaximal, characteristic, complemented
5#2034C212yesmaximal, complemented
6#2033C212yesmaximal, complemented
7#1988C211yesmaximal, characteristic, complemented
8#1955C211yesmaximal, characteristic, complemented
9#1854C212yesmaximal, complemented
10#1854C212yesmaximal, complemented
11#1762C211yesmaximal, characteristic, complemented
12#555—24no—
13#555—24no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#125C2211yescharacteristic
16C2×#124—22no—
17C2×#121—21no—
18C2×#121C2212yes—
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×#117—21no—
22C2×#117C2212yes—
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—
45C22×Q8—21no—
46C22×D4—22no—
47C22×D4D411yescharacteristic
48C22×D4C2311yescharacteristic
49C23×C4—21no—
50C23×C4C2312yes—
51C22×C8C2311yescharacteristic
52C22×C8C2311yescharacteristic
53C22×C8C2312yes—
54C4⋊Q8—22no—
55C4⋊Q8D414yes—
56#125—22no—
57#125—22no—
58#124—216no—
59#121—28no—
60#121—24no—
61#121—24no—
62C4×D4—22no—
63C4×D4—22no—
64C4×D4—22no—
65C4×D4D414yes—
66#117—28no—
67#117—24no—
68#117—24no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4D411yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2312yes—
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2312yes—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4D412yes—
84C2×C42—22no—
85C2×C42—21no—
86C2×C42C2311yescharacteristic
87#107—22no—
88#107—24no—
89#106—22no—
90#106—24no—
91C4⋊C8—22no—
92#102—22no—
93#102—22no—
94#102—24no—
95C22⋊C8—24no—
96C22⋊C8—24no—
97#97—22no—
98#95—21no—
99#95—22no—
100#95—22no—
101C24—21no—
102C24C2×D412yes—
103C2×Q8—24no—
104C2×Q8—81no—
105C2×Q8—81no—
106C2×D4—28no—
107C2×D4—22no—
108C2×D4—21no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—42no—
112C2×D4—82no—
113C2×D4—82no—
114C2×D4C2×D412yes—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—24no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—44no—
128C22×C4C2×D412yes—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2411yescharacteristic
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8#4114yes—
139C4⋊C4—24no—
140C4⋊C4—22no—
141C4⋊C4—21no—
142C4⋊C4—22no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4C2×D412yes—
152C4⋊C4C2×D414yes—
153C22⋊C4—28no—
154C22⋊C4—28no—
155C22⋊C4—24no—
156C22⋊C4—24no—
157C22⋊C4—44no—
158C22⋊C4—82no—
159C42—28no—
160C42—81no—
161C42C2×D412yes—
162C42C2×D412yes—
163C23—22no—
164C23—22no—
165C23—22no—
166C23—22no—
167C23—22no—
168C23—22no—
169C23—22no—
170C23—22no—
171C23—42no—
172C23—42no—
173C23—41no—
174C23—41no—
175C23C22×D411yescharacteristic
176Q8—84no—
177D4—22no—
178D4—22no—
179D4—44no—
180D4—42no—
181D4—88no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—48no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4#12012yes—
207C2×C4#12012yes—
208C2×C4C22×D411yescharacteristic
209C2×C4C22×D411yescharacteristic
210C2×C4C2×#4112yes—
211C2×C42−1+412yes—
212C8—24no—
213C8—41no—
214C8—41no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—82no—
226C22—82no—
227C22C2×#12011yescharacteristic
228C22#54712yes—
229C22#57512yes—
230C22#57611yescharacteristic
231C22#57711yescharacteristic
232C4—22no—
233C4—22no—
234C4—42no—
235C4—42no—
236C4—44no—
237C4—82no—
238C4—81no—
239C4—81no—
240C4#53211yescharacteristic
241C4#53611yescharacteristic
242C4#54812yes—
243C2—82no—
244C2—82no—
245C2#243211yescharacteristic
246C2#300511yescharacteristic
247C2#301211yescharacteristic
248C2#334712yes—
249C2#334812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#555C2
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. #1988C2semidirect
8. #1955C2semidirect
9. #1854C2semidirect
10. #1854C2semidirect
11. #1762C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5552
C2×#5012
C2×#4782
#25622
#20342
#20332
#19882
#19552
#18542
#17622