Finite Group Databasebeta

← #25696

Subgroups — #25696

Order 256.

479 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 479 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#3507C211yesmaximal, characteristic
2#3165C211yesmaximal, characteristic, complemented
3#3141C211yesmaximal, characteristic
4#2568C211yesmaximal, characteristic, complemented
5#2038C214yesmaximal, complemented
6#1986C211yesmaximal, characteristic, complemented
7#1980C211yesmaximal, characteristic
8#1860C214yesmaximal, complemented
9C4×#101C211yesmaximal, characteristic, complemented
10#559—21no—
11#557—21no—
12#556—21no—
13#556—22no—
14#554—21no—
15#552—22no—
16#539—22no—
17#536C2212yes—
18#533C2211yescharacteristic
19#532C2211yescharacteristic
20C2×C4⋊Q8C2211yescharacteristic
21#529—22no—
22#520C2212yes—
23C2×C4×Q8C2211yescharacteristic
24#501—22no—
25#500—21no—
26#498—21no—
27#489—22no—
28#487—21no—
29#486—21no—
30#428C2214yes—
31C2×#104C2214yes—
32#417C2214yes—
33C2×#101C2212yes—
34C4×M16C2211yescharacteristic
35C4×M16C2211yescharacteristic
36#400—22no—
37#391—22no—
38#391C2212yes—
39#382C2212yes—
40C4×C4⋊C4C2211yescharacteristic
41#343C2212yes—
42#339C2214yes—
43C2×#41C2312yes—
44C22×Q8—21no—
45C22×Q8C2311yescharacteristic
46C2×M16C2312yes—
47C2×M16C2312yes—
48C4⋊Q8—22no—
49C4⋊Q8—21no—
50C4⋊Q8—21no—
51C4⋊Q8—21no—
52C4⋊Q8—21no—
53C4⋊Q8D411yescharacteristic
54C4⋊Q8D411yescharacteristic
55C4⋊Q8D411yescharacteristic
56#127D411yescharacteristic
57#126—24no—
58#126—24no—
59#125—22no—
60#125—24no—
61#125D412yes—
62#124—22no—
63#124—22no—
64#124D412yes—
65#123—24no—
66#123—24no—
67C22⋊Q8—24no—
68C22⋊Q8—21no—
69C22⋊Q8—21no—
70C22⋊Q8—22no—
71C22⋊Q8—22no—
72#121—21no—
73#121—21no—
74C4×Q8—22no—
75C4×Q8—22no—
76C4×Q8—21no—
77C4×Q8—21no—
78C4×Q8—21no—
79C4×Q8—21no—
80C4×Q8—21no—
81C4×Q8—21no—
82C4×D4—22no—
83C4×D4—22no—
84C4×D4—24no—
85C4×D4—21no—
86C4×D4—21no—
87#117—24no—
88#117—22no—
89#117C2312yes—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4D412yes—
94C2×C4⋊C4C2312yes—
95C2×C4⋊C4C2311yescharacteristic
96C2×C42D412yes—
97C2×C42C2312yes—
98C2×C42C2311yescharacteristic
99#107—24no—
100#106—24no—
101#104—28no—
102#103—24no—
103#102—24no—
104#101—24no—
105#97—21no—
106#97—21no—
107C4×C8—21no—
108C4×C8—21no—
109#95—21no—
110#95—24no—
111#95—24no—
112#41—24no—
113#41—24no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8—22no—
119C2×Q8—22no—
120C2×Q8—44no—
121C2×Q8—41no—
122C2×Q8—41no—
123C2×Q8C2×D412yes—
124C2×D4—22no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4C2×D412yes—
128C22×C4—24no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4C2×D412yes—
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D412yes—
135C22×C4C2×D411yescharacteristic
136C22×C4C2411yescharacteristic
137M16—22no—
138M16—22no—
139M16—22no—
140M16#4114yes—
141C2×C8—22no—
142C2×C8—22no—
143C4⋊C4—24no—
144C4⋊C4—24no—
145C4⋊C4—24no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—42no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4C2×D412yes—
162C4⋊C4C2×D412yes—
163C22⋊C4—24no—
164C22⋊C4—24no—
165C22⋊C4—24no—
166C22⋊C4—22no—
167C22⋊C4—22no—
168C22⋊C4—22no—
169C42—24no—
170C42—22no—
171C42—22no—
172C42—22no—
173C42—24no—
174C42C2×D411yescharacteristic
175C42C2×D411yescharacteristic
176C42C2×D411yescharacteristic
177C42C2×D411yescharacteristic
178C23—22no—
179C23C22×D411yescharacteristic
180Q8—24no—
181Q8—42no—
182Q8—42no—
183Q8—42no—
184Q8—81no—
185Q8—82no—
186Q8—82no—
187D4—24no—
188D4—42no—
189D4—42no—
190D4—42no—
191D4—81no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—41no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—84no—
209C2×C4#12011yescharacteristic
210C2×C4#12011yescharacteristic
211C2×C4#12011yescharacteristic
212C2×C4#12011yescharacteristic
213C2×C4C22×D411yescharacteristic
214C2×C4C22×D411yescharacteristic
215C2×C4C2×#4112yes—
216C2×C42−1+412yes—
217C8—24no—
218C8—42no—
219C22—21no—
220C22—21no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22C2×#12011yescharacteristic
225C22#53211yescharacteristic
226C22#53611yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—42no—
231C4—42no—
232C4—42no—
233C4—42no—
234C4—42no—
235C4—82no—
236C4—82no—
237C4#45412yes—
238C4#54712yes—
239C4#54812yes—
240C2—21no—
241C2—21no—
242C2—82no—
243C2#243211yescharacteristic
244C2C2×#45411yescharacteristic
245C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3165C2semidirect
4. #2568C2semidirect
5. #2038C2semidirect
6. #1986C2semidirect
8. #1860C2semidirect
9. C4×#101C2semidirect

Maximal subgroups

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

subgroupindex
#35072
#31652
#31412
#25682
#20382
#19862
#19802
#18602
C4×#1012