Finite Group Databasebeta

← #27631

Subgroups — #27631

Order 256.

385 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 127 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
2#2285C211yesmaximal, characteristic
3#2075C212yesmaximal
4#2035C212yesmaximal
5#1948C211yesmaximal, characteristic, complemented
6#1947C211yesmaximal, characteristic, complemented
7#1928C212yesmaximal, complemented
8#1919C212yesmaximal, complemented
9#1803C211yesmaximal, characteristic, complemented
10#1803C211yesmaximal, characteristic, complemented
11#1797C211yesmaximal, characteristic
12C2×C22⋊Q8C2211yescharacteristic
13C2×C4×Q8C2211yescharacteristic
14C2×#117—21no—
15C22×C42C2211yescharacteristic
16C2×#107C2212yes—
17C2×#106C2212yes—
18C2×C4⋊C8C2211yescharacteristic
19C2×C4⋊C8C2211yescharacteristic
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×C22⋊C8C2212yes—
23#391C2211yescharacteristic
24#389C2211yescharacteristic
25#387—21no—
26#385—21no—
27#384C2211yescharacteristic
28#384C2211yescharacteristic
29#382—21no—
30#380—21no—
31#380C2211yescharacteristic
32#380C2212yes—
33C4×C4⋊C4—21no—
34C4×C4⋊C4—21no—
35C4×C4⋊C4C2212yes—
36C4×C4⋊C4C2211yescharacteristic
37C4×C22⋊C4—21no—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C23×C4D411yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4C2311yescharacteristic
47C22×C8C2312yes—
48C22×C8C2312yes—
49C22⋊Q8—22no—
50C22⋊Q8—22no—
51C4×Q8—22no—
52C4×Q8—22no—
53#117—44no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C22⋊C4—21no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—21no—
69C2×C42—22no—
70C2×C42—22no—
71C2×C42—22no—
72C2×C42—21no—
73C2×C42—21no—
74C2×C42D411yescharacteristic
75C2×C42D411yescharacteristic
76C2×C42C2311yescharacteristic
77C2×C42C2311yescharacteristic
78#107—42no—
79#106—24no—
80C4⋊C8—22no—
81C4⋊C8—22no—
82#103—24no—
83#103—42no—
84C22⋊C8—42no—
85#95—21no—
86#95—21no—
87#95—22no—
88#95—21no—
89#95—21no—
90#95—22no—
91#95—21no—
92C24C2×D411yescharacteristic
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8#4112yes—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2411yescharacteristic
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C4⋊C4—24no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—22no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4#4112yes—
146C22⋊C4—24no—
147C22⋊C4—42no—
148C22⋊C4—44no—
149C42—22no—
150C42—22no—
151C42—21no—
152C42—21no—
153C42—44no—
154C42—42no—
155C42—42no—
156C42#4112yes—
157C42#4112yes—
158C23—21no—
159C23—21no—
160C23—21no—
161C23—21no—
162C23—21no—
163C23—21no—
164C23AGL(1,ℤ/8)12yes—
165C23C22×D411yescharacteristic
166Q8—22no—
167Q8—22no—
168Q8—41no—
169Q8—41no—
170C2×C4—28no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—44no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4#12312yes—
205C2×C4#12312yes—
206C2×C4#13512yes—
207C2×C4#13712yes—
208C2×C4#13712yes—
209C2×C4C2×#4111yescharacteristic
210C2×C4C2×#4111yescharacteristic
211C2×C4C2×#4111yescharacteristic
212C2×C4C2×#4111yescharacteristic
213C2×C4H(5,𝔽2)11yescharacteristic
214C2×C42−1+411yescharacteristic
215C8—42no—
216C8—42no—
217C22—22no—
218C22—22no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22C2×#12311yescharacteristic
225C22#52511yescharacteristic
226C22#53711yescharacteristic
227C22C2×#13511yescharacteristic
228C22C2×AGL(1,ℤ/8)11yescharacteristic
229C22C2×#13711yescharacteristic
230C22C2×#13711yescharacteristic
231C4—22no—
232C4—22no—
233C4—41no—
234C4—42no—
235C4—41no—
236C4—42no—
237C4—81no—
238C4—82no—
239C4—81no—
240C4—82no—
241C4#48514yes—
242C4#55211yescharacteristic
243C4#55311yescharacteristic
244C4#55511yescharacteristic
245C4#55611yescharacteristic
246C2—42no—
247C2#248511yescharacteristic
248C2C2×#48511yescharacteristic
249C2#309211yescharacteristic
250C2#310711yescharacteristic
251C2#311211yescharacteristic
252C2#324711yescharacteristic
253C2#325511yescharacteristic
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
5. #1948C2semidirect
6. #1947C2semidirect
7. #1928C2semidirect
8. #1919C2semidirect
9. #1803C2semidirect
10. #1803C2semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊Q82
#22852
#20752
#20352
#19482
#19472
#19282
#19192
#18032
#17972