Finite Group Databasebeta

← #29240

Subgroups — #29240

Order 256.

421 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C4×C22⋊Q8C211yesmaximal, characteristic
2C2×#389C211yesmaximal, characteristic
3#2088C212yesmaximal, complemented
4#2066C212yesmaximal, complemented
5#1919C212yesmaximal, complemented
6#1872C211yesmaximal, characteristic
7#1872C211yesmaximal, characteristic
8#1865C212yesmaximal, complemented
9#1853C211yesmaximal, characteristic
10#1853C211yesmaximal, characteristic
11#1797C211yesmaximal, characteristic
12C2×C22⋊Q8C2211yescharacteristic
13C2×C4×Q8C2211yescharacteristic
14C22×C4⋊C4—21no—
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C22×C42C2211yescharacteristic
18C2×#106C2212yes—
19C2×#106C2212yes—
20C2×C4⋊C8C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×C22⋊C8C2211yescharacteristic
24C2×C22⋊C8C2211yescharacteristic
25#391C2211yescharacteristic
26#389—21no—
27#389—21no—
28#389C2212yes—
29#389C2211yescharacteristic
30#389C2212yes—
31#387—21no—
32#385—21no—
33#380C2211yescharacteristic
34C4×C4⋊C4—21no—
35C4×C4⋊C4C2211yescharacteristic
36C4×C22⋊C4—21no—
37C2×#95—21no—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C23×C4—21no—
45C23×C4—21no—
46C23×C4D411yescharacteristic
47C23×C4D411yescharacteristic
48C23×C4C2311yescharacteristic
49C22×C8C2312yes—
50C22×C8C2312yes—
51C22⋊Q8—22no—
52C22⋊Q8Q814yes—
53C4×Q8—42no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—42no—
63C2×C4⋊C4—44no—
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C42—21no—
72C2×C42—21no—
73C2×C42—42no—
74C2×C42D411yescharacteristic
75C2×C42D411yescharacteristic
76C2×C42C2311yescharacteristic
77C2×C42C2311yescharacteristic
78#106—24no—
79#106—42no—
80C4⋊C8—42no—
81#103—24no—
82#103—42no—
83C22⋊C8—22no—
84C22⋊C8—22no—
85#95—22no—
86#95—22no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—21no—
91#95—21no—
92C24C2×D411yescharacteristic
93C2×Q8—21no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8C2×Q812yes—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—24no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—41no—
108C22×C4—41no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4SD1614yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×Q812yes—
123C22×C4#4112yes—
124C22×C4C2411yescharacteristic
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C4⋊C4—24no—
132C4⋊C4—24no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—44no—
143C4⋊C4—44no—
144C4⋊C4C2×Q812yes—
145C4⋊C4#4114yes—
146C22⋊C4—24no—
147C22⋊C4—42no—
148C42—22no—
149C42—22no—
150C42—41no—
151C42—41no—
152C42—42no—
153C42—42no—
154C23—21no—
155C23—21no—
156C23—21no—
157C23—21no—
158C23C22⋊Q812yes—
159C23C2×SD1612yes—
160C23#13712yes—
161C23C22×D411yescharacteristic
162Q8—24no—
163Q8—42no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—44no—
182C2×C4—48no—
183C2×C4—44no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—84no—
200C2×C4C22⋊Q812yes—
201C2×C4C2×SD1612yes—
202C2×C4C2×SD1612yes—
203C2×C4C22×Q811yescharacteristic
204C2×C4C2×#4111yescharacteristic
205C2×C4C2×#4112yes—
206C2×C4H(5,𝔽2)12yes—
207C8—42no—
208C8—42no—
209C22—22no—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22#47514yes—
216C22C2×C22⋊Q811yescharacteristic
217C22#52511yescharacteristic
218C22#53411yescharacteristic
219C22C22×SD1611yescharacteristic
220C22C2×#13711yescharacteristic
221C22#57511yescharacteristic
222C22#57511yescharacteristic
223C4—22no—
224C4—22no—
225C4—22no—
226C4—42no—
227C4—42no—
228C4—44no—
229C4—41no—
230C4—41no—
231C4—81no—
232C4—82no—
233C4—81no—
234C4#55312yes—
235C4#55412yes—
236C2—24no—
237C2#257911yescharacteristic
238C2C2×#47511yescharacteristic
239C2#308011yescharacteristic
240C2#310111yescharacteristic
241C2#311711yescharacteristic
242C2#318711yescharacteristic
243C2#319611yescharacteristic
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. #2088C2semidirect
4. #2066C2semidirect
5. #1919C2semidirect
8. #1865C2semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊Q82
C2×#3892
#20882
#20662
#19192
#18722
#18652
#18532
#17972