Finite Group Databasebeta

← #29972

Subgroups — #29972

Order 256.

384 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 129 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#2851C211yesmaximal, characteristic
2C2×#389C211yesmaximal, characteristic
3#2101C212yesmaximal, complemented
4#2088C212yesmaximal, complemented
5#2087C212yesmaximal, complemented
6#2059C212yesmaximal, complemented
7#1853C211yesmaximal, characteristic
8#1852C211yesmaximal, characteristic
9#1823C211yesmaximal, characteristic
10#1823C211yesmaximal, characteristic
11#1797C211yesmaximal, characteristic
12C2×#125C2211yescharacteristic
13C22×C4⋊C4—21no—
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C42C2211yescharacteristic
17C2×#107C2212yes—
18C2×#106C2212yes—
19C2×C4⋊C8C2212yes—
20C2×C22⋊C8C2211yescharacteristic
21C2×C22⋊C8C2211yescharacteristic
22#389—21no—
23#389—21no—
24#389C2212yes—
25#389C2211yescharacteristic
26#389C2211yescharacteristic
27#389C2212yes—
28#387—21no—
29#387—21no—
30#385—21no—
31#385—21no—
32#382C2211yescharacteristic
33#380C2211yescharacteristic
34#380C2211yescharacteristic
35C2×#95—21no—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C23×C4—21no—
44C23×C4—21no—
45C23×C4D411yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4C2311yescharacteristic
48C22×C8C2312yes—
49C22×C8C2312yes—
50#125—42no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—42no—
61C2×C4⋊C4—44no—
62C2×C4⋊C4Q814yes—
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C42—42no—
72C2×C42D411yescharacteristic
73C2×C42D411yescharacteristic
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76#107—42no—
77#106—42no—
78C4⋊C8—42no—
79C22⋊C8—22no—
80C22⋊C8—22no—
81#95—22no—
82#95—22no—
83#95—21no—
84#95—21no—
85#95—21no—
86#95—21no—
87#95—21no—
88#95—21no—
89C24C2×D411yescharacteristic
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—24no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×Q812yes—
116C22×C4#4112yes—
117C22×C4C2411yescharacteristic
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C4⋊C4—24no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—44no—
135C4⋊C4—44no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4C2×Q814yes—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C42—22no—
143C42—22no—
144C42—41no—
145C42—41no—
146C23—21no—
147C23—21no—
148C23—21no—
149C23—21no—
150C23C22⋊Q812yes—
151C23#13512yes—
152C23AGL(1,ℤ/8)12yes—
153C23C22×D411yescharacteristic
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—24no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—44no—
171C2×C4—44no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—48no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—84no—
191C2×C4C22⋊Q812yes—
192C2×C4#13712yes—
193C2×C4C22×Q811yescharacteristic
194C2×C4C2×#4111yescharacteristic
195C2×C4H(5,𝔽2)12yes—
196C2×C42−1+412yes—
197C8—42no—
198C8—42no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—22no—
203C22—22no—
204C22—22no—
205C22#47814yes—
206C22C2×C22⋊Q811yescharacteristic
207C22#53411yescharacteristic
208C22#53611yescharacteristic
209C22C2×#13511yescharacteristic
210C22C2×AGL(1,ℤ/8)11yescharacteristic
211C22C2×#13711yescharacteristic
212C22#57511yescharacteristic
213C4—22no—
214C4—22no—
215C4—44no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—81no—
220C4—81no—
221C4—82no—
222C4—81no—
223C4—81no—
224C4#53912yes—
225C4#54412yes—
226C2—24no—
227C2#262711yescharacteristic
228C2C2×#47811yescharacteristic
229C2#307811yescharacteristic
230C2#319011yescharacteristic
231C2#319411yescharacteristic
232C2#323211yescharacteristic
233C2#323811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2101C2semidirect
4. #2088C2semidirect
5. #2087C2semidirect
6. #2059C2semidirect

Maximal subgroups

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

subgroupindex
#28512
C2×#3892
#21012
#20882
#20872
#20592
#18532
#18522
#18232
#17972