Finite Group Databasebeta

← #24080

Subgroups — #24080

Order 256.

505 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 123 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#3493C211yesmaximal, characteristic
2#3092C212yesmaximal, complemented
3#2317C211yesmaximal, characteristic, complemented
4#2060C212yesmaximal, complemented
5#2039C212yesmaximal, complemented
6#1897C212yesmaximal, complemented
7#1860C214yesmaximal, complemented
8#1822C211yesmaximal, characteristic, complemented
9#556—22no—
10#555—22no—
11#553—22no—
12#552—22no—
13#544C2211yescharacteristic
14#538—22no—
15#537C2211yescharacteristic
16#529—22no—
17#525C2212yes—
18#520C2212yes—
19C2×#117C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22#484—22no—
23#483—22no—
24#481—22no—
25#480—22no—
26#428C2214yes—
27C2×#104C2214yes—
28#417C2214yes—
29#407C2212yes—
30#389—22no—
31#385—22no—
32#384C2212yes—
33#382C2212yes—
34#343C2212yes—
35#343C2212yes—
36#339C2214yes—
37C2×#41C2312yes—
38C23×C4—21no—
39C23×C4C2311yescharacteristic
40C2×M16C2314yes—
41C4⋊Q8—22no—
42#126—24no—
43#126—24no—
44#125—22no—
45#125—22no—
46#124—22no—
47#124—24no—
48#123—24no—
49#123—22no—
50C22⋊Q8—22no—
51C22⋊Q8—22no—
52C22⋊Q8D412yes—
53C22⋊Q8D412yes—
54#121—24no—
55#121—22no—
56#121—22no—
57#121D412yes—
58#121D412yes—
59#120—22no—
60C4×Q8—22no—
61C4×Q8—22no—
62C4×D4—22no—
63C4×D4—22no—
64C4×D4—24no—
65C4×D4—22no—
66C4×D4—22no—
67#117—24no—
68#117—22no—
69#117—22no—
70#117—22no—
71#117—24no—
72#117C2312yes—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4C2312yes—
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C22⋊C4—21no—
83C2×C22⋊C4D412yes—
84C2×C42D412yes—
85C2×C42C2312yes—
86#107—24no—
87#106—24no—
88#104—28no—
89#103—24no—
90#102—24no—
91C22⋊C8—22no—
92C22⋊C8—22no—
93#95—24no—
94C24C2×D411yescharacteristic
95#41—24no—
96#41—24no—
97C2×Q8—22no—
98C2×Q8C2×D412yes—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4C2×D412yes—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—24no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—42no—
111C22×C4—82no—
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D412yes—
114C22×C4C2×D412yes—
115C22×C4C2×D412yes—
116C22×C4C2×D412yes—
117C22×C4C2411yescharacteristic
118M16—24no—
119M16—24no—
120C2×C8—24no—
121C4⋊C4—24no—
122C4⋊C4—24no—
123C4⋊C4—24no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—24no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4—84no—
134C4⋊C4C2×D412yes—
135C4⋊C4C2×D412yes—
136C22⋊C4—24no—
137C22⋊C4—24no—
138C22⋊C4—24no—
139C22⋊C4—24no—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C42—24no—
146C42—22no—
147C42—22no—
148C42—22no—
149C42—22no—
150C42#4114yes—
151C23—21no—
152C23—21no—
153C23—22no—
154C23—41no—
155C23—41no—
156C23#12012yes—
157C23C22×D411yescharacteristic
158Q8—24no—
159Q8—82no—
160D4—24no—
161D4—42no—
162D4—42no—
163D4—82no—
164D4—82no—
165D4—82no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—24no—
169C2×C4—24no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—44no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—44no—
182C2×C4—42no—
183C2×C4—84no—
184C2×C4—84no—
185C2×C4#12011yescharacteristic
186C2×C4#12011yescharacteristic
187C2×C4C22×D412yes—
188C2×C4C2×#4112yes—
189C2×C42−1+412yes—
190C8—44no—
191C22—21no—
192C22—21no—
193C22—22no—
194C22—22no—
195C22—22no—
196C22—42no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22C2×#12011yescharacteristic
202C22#52511yescharacteristic
203C22#53611yescharacteristic
204C4—21no—
205C4—21no—
206C4—44no—
207C4—42no—
208C4—42no—
209C4—42no—
210C4—42no—
211C4—42no—
212C4—82no—
213C4#54714yes—
214C2—21no—
215C2—21no—
216C2—42no—
217C2—82no—
218C2#240611yescharacteristic
219C2#302111yescharacteristic
220C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3493C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3092C2semidirect
3. #2317C2semidirect
4. #2060C2semidirect
5. #2039C2semidirect
6. #1897C2semidirect
7. #1860C2semidirect
8. #1822C2semidirect

Maximal subgroups

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

subgroupindex
#34932
#30922
#23172
#20602
#20392
#18972
#18602
#18222