Finite Group Databasebeta

← #29159

Subgroups — #29159

Order 256.

414 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 141 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
2C4×C22⋊Q8C211yesmaximal, characteristic
3#2099C211yesmaximal, characteristic, complemented
4#2096C211yesmaximal, characteristic, complemented
5#2058C211yesmaximal, characteristic, complemented
6#2024C211yesmaximal, characteristic, complemented
7#1919C212yesmaximal, complemented
8#1892C212yesmaximal
9#1865C212yesmaximal, complemented
10#1807C212yesmaximal
11C2×#336C211yesmaximal, characteristic
12C23×C8C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C22⋊Q8—21no—
15C2×C22⋊Q8C2211yescharacteristic
16C2×C4×Q8C2211yescharacteristic
17C22×C42C2211yescharacteristic
18C2×#106C2212yes—
19C2×#106C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22#391—21no—
23#391C2211yescharacteristic
24#389—21no—
25#389C2211yescharacteristic
26#389C2212yes—
27#387—21no—
28#385—21no—
29#384C2211yescharacteristic
30#380—21no—
31#380C2211yescharacteristic
32#380C2212yes—
33C4×C4⋊C4—21no—
34C4×C4⋊C4C2211yescharacteristic
35C4×C22⋊C4—21no—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4C2311yescharacteristic
48C22×C8—24no—
49C22×C8—24no—
50C22×C8—42no—
51C22×C8C2312yes—
52C22×C8C2312yes—
53C4⋊Q8—42no—
54C22⋊Q8—22no—
55C22⋊Q8—44no—
56C22⋊Q8Q814yescomplemented
57C4×Q8—42no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C22⋊C4—21no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—21no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42—42no—
76C2×C42D411yescharacteristic
77C2×C42D411yescharacteristic
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80#106—24no—
81#106—42no—
82#103—24no—
83#103—42no—
84#95—21no—
85#95—21no—
86#95—21no—
87#95—21no—
88#95—21no—
89#95—22no—
90#95—22no—
91C24C2×D411yescharacteristic
92C2×Q8—21no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—42no—
96C2×Q8—42no—
97C2×Q8—42no—
98C2×Q8C2×Q812yes—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4SD1614yes—
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×Q812yes—
119C22×C4#4112yes—
120C22×C4C2411yescharacteristic
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—48no—
127C2×C8#4114yes—
128C4⋊C4—24no—
129C4⋊C4—21no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—44no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4C2×Q812yes—
143C22⋊C4—24no—
144C22⋊C4—44no—
145C22⋊C4—42no—
146C42—22no—
147C42—22no—
148C42—41no—
149C42—41no—
150C42—42no—
151C42—42no—
152C23—21no—
153C23—21no—
154C23—21no—
155C23—21no—
156C23C22⋊Q812yes—
157C23C2×SD1612yes—
158C23#13712yes—
159C23C22×D411yescharacteristic
160Q8—24no—
161Q8—42no—
162Q8—84no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—24no—
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—44no—
179C2×C4—44no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4C22⋊Q812yes—
197C2×C4C2×SD1612yes—
198C2×C4C2×SD1612yes—
199C2×C4#13512yes—
200C2×C4C22×Q811yescharacteristic
201C2×C4C2×#4111yescharacteristic
202C2×C4C2×#4112yes—
203C2×C4H(5,𝔽2)12yes—
204C8—24no—
205C8—42no—
206C22—22no—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—22no—
211C22—22no—
212C22#47514yes—
213C22C2×C22⋊Q811yescharacteristic
214C22#53211yescharacteristic
215C22#53411yescharacteristic
216C22C22×SD1611yescharacteristic
217C22C2×#13511yescharacteristic
218C22C2×#13711yescharacteristic
219C22#57511yescharacteristic
220C4—22no—
221C4—22no—
222C4—22no—
223C4—22no—
224C4—42no—
225C4—42no—
226C4—41no—
227C4—82no—
228C4—81no—
229C4—82no—
230C4—81no—
231C4#55112yes—
232C4#55412yes—
233C2—24no—
234C2#256411yescharacteristic
235C2C2×#47511yescharacteristic
236C2#307611yescharacteristic
237C2#316711yescharacteristic
238C2#318111yescharacteristic
239C2#318711yescharacteristic
240C2#319011yescharacteristic
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. #2096C2semidirect
5. #2058C2semidirect
6. #2024C2semidirect
7. #1919C2semidirect
9. #1865C2semidirect
56. C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#28572
C4×C22⋊Q82
#20992
#20962
#20582
#20242
#19192
#18922
#18652
#18072
C2×#3362