Finite Group Databasebeta

← #27589

Subgroups — #27589

Order 256.

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

normal subgroups only

This group has 434 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
1C2×#391C211yesmaximal, characteristic
2#2283C211yesmaximal, characteristic
3#2097C212yesmaximal, complemented
4#2074C212yesmaximal, complemented
5#1928C212yesmaximal, complemented
6#1927C212yesmaximal, complemented
7#1853C211yesmaximal, characteristic
8#1852C211yesmaximal, characteristic
9#1817C211yesmaximal, characteristic
10#1787C211yesmaximal, characteristic
11#1787C211yesmaximal, characteristic
12C23×Q8C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C42C2211yescharacteristic
15C2×#107C2212yes—
16C2×#106C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×C22⋊C8C2211yescharacteristic
22C2×C22⋊C8C2211yescharacteristic
23#391—21no—
24#391—21no—
25#391C2212yes—
26#391C2212yes—
27#383—21no—
28#382C2211yescharacteristic
29#382C2211yescharacteristic
30#381—21no—
31#380C2211yescharacteristic
32#380C2211yescharacteristic
33C4×C4⋊C4C2211yescharacteristic
34C4×C4⋊C4C2211yescharacteristic
35C4×C22⋊C4—21no—
36#376—21no—
37C2×#95—21no—
38C2×#95—21no—
39#340C2212yes—
40#340C2212yes—
41#336C2212yes—
42#336C2212yes—
43C22×Q8—24no—
44C22×Q8—21no—
45C22×Q8—22no—
46C22×Q8—22no—
47C22×Q8—42no—
48C22×Q8C2312yes—
49C23×C4—21no—
50C23×C4—21no—
51C23×C4D411yescharacteristic
52C23×C4D411yescharacteristic
53C23×C4C2311yescharacteristic
54C22×C8C2312yes—
55C22×C8C2312yes—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—42no—
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C22⋊C4—21no—
64C2×C22⋊C4—21no—
65C2×C42—22no—
66C2×C42—22no—
67C2×C42—21no—
68C2×C42—21no—
69C2×C42D411yescharacteristic
70C2×C42D411yescharacteristic
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73#107—42no—
74#106—42no—
75#103—24no—
76#103—24no—
77#103—42no—
78#103—42no—
79C22⋊C8—22no—
80C22⋊C8—22no—
81#95—22no—
82#95—22no—
83#95—22no—
84#95—22no—
85#95—21no—
86#95—21no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—21no—
91#95—21no—
92#95—21no—
93C24C2×D411yescharacteristic
94C2×Q8—24no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—48no—
101C2×Q8—48no—
102C2×Q8—42no—
103C2×Q8—42no—
104C2×Q8#4114yes—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—24no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—41no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4#4112yes—
131C22×C4#4112yes—
132C22×C4C2411yescharacteristic
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—42no—
146C22⋊C4—42no—
147C22⋊C4—42no—
148C42—22no—
149C42—22no—
150C42—21no—
151C42—21no—
152C42—21no—
153C42—21no—
154C42—42no—
155C42—42no—
156C23—21no—
157C23—21no—
158C23—21no—
159C23—21no—
160C23#12312yes—
161C23#13512yes—
162C23#13712yes—
163C23C22×D411yescharacteristic
164Q8—24no—
165Q8—24no—
166Q8—48no—
167Q8—42no—
168Q8—42no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—24no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—44no—
186C2×C4—48no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—44no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—84no—
206C2×C4#12312yes—
207C2×C4#13712yes—
208C2×C4C2×#4111yescharacteristic
209C2×C4C2×#4111yescharacteristic
210C2×C4C2×#4112yes—
211C2×C4H(5,𝔽2)12yes—
212C8—42no—
213C8—42no—
214C22—22no—
215C22—22no—
216C22—22no—
217C22—22no—
218C22—22no—
219C22—22no—
220C22#48514yes—
221C22C2×#12311yescharacteristic
222C22#52511yescharacteristic
223C22#53511yescharacteristic
224C22C2×#13511yescharacteristic
225C22C2×#13711yescharacteristic
226C22C2×#13711yescharacteristic
227C22#57511yescharacteristic
228C4—22no—
229C4—22no—
230C4—41no—
231C4—41no—
232C4—41no—
233C4—44no—
234C4—81no—
235C4—81no—
236C4—81no—
237C4—82no—
238C4—81no—
239C4#55112yes—
240C4#55112yes—
241C2—24no—
242C2#248211yescharacteristic
243C2C2×#48511yescharacteristic
244C2#309511yescharacteristic
245C2#310811yescharacteristic
246C2#311711yescharacteristic
247C2#321111yescharacteristic
248C2#321811yescharacteristic
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. #2097C2semidirect
4. #2074C2semidirect
5. #1928C2semidirect
6. #1927C2semidirect

Maximal subgroups

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

subgroupindex
C2×#3912
#22832
#20972
#20742
#19282
#19272
#18532
#18522
#18172
#17872