Finite Group Databasebeta

← #29971

Subgroups — #29971

Order 256.

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

normal subgroups only

This group has 370 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#2857C211yesmaximal, characteristic
2#2296C211yesmaximal, characteristic
3#2099C211yesmaximal, characteristic, complemented
4#2099C211yesmaximal, characteristic, complemented
5#2088C212yesmaximal, complemented
6#2087C212yesmaximal, complemented
7#2058C211yesmaximal, characteristic, complemented
8#2057C211yesmaximal, characteristic, complemented
9#1854C212yesmaximal
10#1825C212yesmaximal
11#1797C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C22⋊Q8—21no—
14C2×#117C2211yescharacteristic
15C2×#117C2211yescharacteristic
16C22×C42C2211yescharacteristic
17C2×#107C2212yes—
18C2×#106C2212yes—
19C2×C4⋊C8C2212yes—
20C2×C22⋊C8C2212yes—
21#391—21no—
22#389—21no—
23#389C2211yescharacteristic
24#389C2211yescharacteristic
25#389C2211yescharacteristic
26#389C2212yes—
27#389C2211yescharacteristic
28#385—21no—
29#385—21no—
30#384C2211yescharacteristic
31#380—21no—
32#380C2212yes—
33C4×C4⋊C4—21no—
34C4×C4⋊C4—21no—
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43C22×Q8—21no—
44C23×C4D411yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4C2311yescharacteristic
47C22×C8C2312yes—
48C22×C8C2312yes—
49C4⋊Q8—42no—
50C22⋊Q8—44no—
51#117—22no—
52#117—42no—
53#117Q814yescomplemented
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—22no—
71C2×C42—21no—
72C2×C42—21no—
73C2×C42—42no—
74C2×C42D411yescharacteristic
75C2×C42D411yescharacteristic
76C2×C42C2311yescharacteristic
77C2×C42C2311yescharacteristic
78#107—42no—
79#106—42no—
80C4⋊C8—42no—
81C22⋊C8—24no—
82#95—21no—
83#95—21no—
84#95—21no—
85#95—21no—
86#95—22no—
87#95—22no—
88C24C2×D411yescharacteristic
89C2×Q8—42no—
90C2×Q8—42no—
91C2×Q8—42no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×Q812yes—
112C22×C4#4112yes—
113C22×C4C2411yescharacteristic
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—24no—
127C4⋊C4—42no—
128C4⋊C4—44no—
129C4⋊C4—42no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4C2×Q812yes—
136C4⋊C4C2×Q812yes—
137C22⋊C4—24no—
138C22⋊C4—42no—
139C22⋊C4—44no—
140C42—24no—
141C42—22no—
142C42—22no—
143C42—41no—
144C42—41no—
145C42—42no—
146C23—21no—
147C23—21no—
148C23—21no—
149C23—21no—
150C23—21no—
151C23—21no—
152C23C22⋊Q812yes—
153C23C22×D411yescharacteristic
154Q8—84no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—24no—
164C2×C4—24no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—44no—
171C2×C4—44no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4C22⋊Q812yes—
189C2×C4#13512yes—
190C2×C4AGL(1,ℤ/8)12yes—
191C2×C4#13712yes—
192C2×C4C22×Q811yescharacteristic
193C2×C4C2×#4111yescharacteristic
194C2×C4H(5,𝔽2)12yes—
195C2×C42−1+412yes—
196C8—42no—
197C8—42no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22C2×C22⋊Q811yescharacteristic
206C22#53411yescharacteristic
207C22#53611yescharacteristic
208C22C2×#13511yescharacteristic
209C22C2×AGL(1,ℤ/8)11yescharacteristic
210C22C2×#13711yescharacteristic
211C22#57511yescharacteristic
212C4—22no—
213C4—22no—
214C4—22no—
215C4—22no—
216C4—42no—
217C4—42no—
218C4—41no—
219C4—81no—
220C4—81no—
221C4—82no—
222C4—82no—
223C4#53912yes—
224C4#54412yes—
225C2—42no—
226C2#262711yescharacteristic
227C2#307611yescharacteristic
228C2#307911yescharacteristic
229C2#319011yescharacteristic
230C2#319411yescharacteristic
231C2#322811yescharacteristic
232C2#323711yescharacteristic
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. #2099C2semidirect
4. #2099C2semidirect
5. #2088C2semidirect
6. #2087C2semidirect
7. #2058C2semidirect
8. #2057C2semidirect
53. #117Q8semidirect

Maximal subgroups

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

subgroupindex
#28572
#22962
#20992
#20882
#20872
#20582
#20572
#18542
#18252
#17972