Finite Group Databasebeta

← #24581

Subgroups — #24581

Order 256.

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

normal subgroups only

This group has 434 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×#480C211yesmaximal, characteristic
2C22×#106C211yesmaximal, characteristic, complemented
3#2504C211yesmaximal, characteristic
4#2344C211yesmaximal, characteristic, complemented
5#2085C212yesmaximal, complemented
6#2065C212yesmaximal, complemented
7#1949C212yesmaximal, complemented
8#1926C212yesmaximal, complemented
9#1893C211yesmaximal, characteristic
10#1850C211yesmaximal, characteristic, complemented
11#1824C211yesmaximal, characteristic, complemented
12C23×C8C2211yescharacteristic
13C2×#121C2211yescharacteristic
14C2×#117—21no—
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17#480—24no—
18C2×#107C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#106—24no—
21C2×#106C2212yes—
22C2×#106C2212yes—
23C2×C4⋊C8C2212yes—
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2212yes—
27C2×C22⋊C8C2211yescharacteristic
28#397—22no—
29#390C2212yes—
30#389C2212yes—
31#387—22no—
32#386—21no—
33#386—21no—
34#382C2212yes—
35#381—21no—
36#380C2211yescharacteristic
37C4×C4⋊C4C2212yes—
38C4×C22⋊C4C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×D4—21no—
44C22×D4C2311yescharacteristic
45C23×C4—21no—
46C23×C4—21no—
47C23×C4C2311yescharacteristic
48C22×C8—22no—
49C22×C8—24no—
50C22×C8C2311yescharacteristic
51C22×C8C2311yescharacteristic
52C22×C8C2312yes—
53#121—22no—
54#121—22no—
55#117—44no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4D414yes—
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4D411yescharacteristic
73C2×C22⋊C4D411yescharacteristic
74C2×C42—22no—
75C2×C42C2312yes—
76#107—22no—
77#107—22no—
78#106—24no—
79#106—24no—
80C4⋊C8—24no—
81#102—24no—
82#102—22no—
83#102—22no—
84C22⋊C8—22no—
85#95—22no—
86#95—21no—
87#95—22no—
88#95—22no—
89#95—22no—
90#95D412yes—
91C24—21no—
92C24C2×D411yescharacteristic
93C2×D4—21no—
94C2×D4—22no—
95C2×D4—21no—
96C2×D4—22no—
97C2×D4—42no—
98C2×D4—42no—
99C2×D4—42no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—24no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—82no—
112C22×C4C2×D412yes—
113C22×C4C2×D412yes—
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D412yes—
116C22×C4#4112yes—
117C22×C4C2411yescharacteristic
118C2×C8—28no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8#4114yes—
126C4⋊C4—24no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—22no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—21no—
134C4⋊C4—22no—
135C4⋊C4—44no—
136C4⋊C4—42no—
137C4⋊C4C2×D412yes—
138C4⋊C4C2×D412yes—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—42no—
142C22⋊C4—42no—
143C22⋊C4—82no—
144C22⋊C4—82no—
145C42—22no—
146C42—22no—
147C42—44no—
148C23—22no—
149C23—22no—
150C23—41no—
151C23—42no—
152C23—41no—
153C23—41no—
154C23—42no—
155C23#12112yes—
156C23#13512yes—
157C23AGL(1,ℤ/8)12yes—
158C23C22×D411yescharacteristic
159D4—41no—
160D4—41no—
161D4—42no—
162D4—84no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—24no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—48no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—84no—
190C2×C4#12112yes—
191C2×C4C22×D411yescharacteristic
192C2×C4C2×#4112yes—
193C2×C4C2×#4111yescharacteristic
194C2×C42−1+412yes—
195C8—24no—
196C8—41no—
197C8—41no—
198C22—24no—
199C22—24no—
200C22—22no—
201C22—22no—
202C22—82no—
203C22—82no—
204C22—81no—
205C22—81no—
206C22—81no—
207C22#46314yes—
208C22C2×#12111yescharacteristic
209C22#54712yes—
210C22C2×#13511yescharacteristic
211C22C2×AGL(1,ℤ/8)11yescharacteristic
212C22#57712yes—
213C4—22no—
214C4—42no—
215C4—42no—
216C4—42no—
217C4—42no—
218C4—82no—
219C4—81no—
220C4—82no—
221C4#52511yescharacteristic
222C4#53611yescharacteristic
223C4#55212yes—
224C2—24no—
225C2—161no—
226C2#241311yescharacteristic
227C2C2×#46311yescharacteristic
228C2#304311yescharacteristic
229C2#333712yes—
230C2#334512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#106C2semidirect
4. #2344C2semidirect
5. #2085C2semidirect
6. #2065C2semidirect
7. #1949C2semidirect
8. #1926C2semidirect
10. #1850C2semidirect
11. #1824C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4802
C22×#1062
#25042
#23442
#20852
#20652
#19492
#19262
#18932
#18502
#18242