Finite Group Databasebeta

← #27896

Subgroups — #27896

Order 256.

371 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 119 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#2283C211yesmaximal, characteristic
3#2036C212yesmaximal
4#2035C212yesmaximal
5#1960C211yesmaximal, characteristic, complemented
6#1960C211yesmaximal, characteristic, complemented
7#1928C212yesmaximal, complemented
8#1927C212yesmaximal, complemented
9#1847C211yesmaximal, characteristic, complemented
10#1836C211yesmaximal, characteristic, complemented
11#1749C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C22⋊Q8C2212yes—
14C22×C42C2211yescharacteristic
15C2×#107C2212yes—
16C2×#106C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×C22⋊C8C2212yes—
22C2×#97C2211yescharacteristic
23C2×#97C2211yescharacteristic
24#391C2212yes—
25#389—21no—
26#389—21no—
27#384C2211yescharacteristic
28#383—21no—
29#382C2211yescharacteristic
30#382C2211yescharacteristic
31#381—21no—
32#380—21no—
33#380—21no—
34#380C2211yescharacteristic
35#380C2211yescharacteristic
36C4×C4⋊C4C2211yescharacteristic
37C4×C4⋊C4C2211yescharacteristic
38C4×C22⋊C4—21no—
39#376—21no—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2312yes—
44C23×C4D411yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4C2311yescharacteristic
47C22×C8C2312yes—
48C22×C8C2312yes—
49C4⋊Q8—22no—
50C4⋊Q8—22no—
51C22⋊Q8—42no—
52C22⋊Q8—42no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C22⋊C4—22no—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66C2×C42—22no—
67C2×C42—22no—
68C2×C42—21no—
69C2×C42—21no—
70C2×C42D411yescharacteristic
71C2×C42D411yescharacteristic
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74#107—42no—
75#106—42no—
76#103—24no—
77#103—24no—
78#103—42no—
79#103—42no—
80C22⋊C8—24no—
81#97—22no—
82#97—22no—
83#95—21no—
84#95—21no—
85#95—21no—
86#95—21no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—21no—
91#95—22no—
92#95—22no—
93C24C2×D411yescharacteristic
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2411yescharacteristic
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8#4114yes—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—44no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C22⋊C4—42no—
140C22⋊C4—44no—
141C22⋊C4—42no—
142C42—22no—
143C42—22no—
144C42—21no—
145C42—21no—
146C42—42no—
147C42—42no—
148C42#4112yes—
149C42#4112yes—
150C23—21no—
151C23—21no—
152C23—21no—
153C23—21no—
154C23—21no—
155C23—21no—
156C23#13712yes—
157C23C22×D411yescharacteristic
158Q8—42no—
159Q8—42no—
160Q8—42no—
161Q8—42no—
162C2×C4—21no—
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—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—44no—
175C2×C4—44no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4#12312yes—
197C2×C4#12312yes—
198C2×C4#13712yes—
199C2×C4C2×#4111yescharacteristic
200C2×C4C2×#4112yes—
201C2×C4C2×#4111yescharacteristic
202C2×C4H(5,𝔽2)12yes—
203C8—24no—
204C8—42no—
205C22—22no—
206C22—22no—
207C22—22no—
208C22—41no—
209C22—41no—
210C22—41no—
211C22—41no—
212C22C2×#12311yescharacteristic
213C22#53211yescharacteristic
214C22#53511yescharacteristic
215C22C2×#13711yescharacteristic
216C22C2×#13711yescharacteristic
217C22#57511yescharacteristic
218C22#57511yescharacteristic
219C4—22no—
220C4—22no—
221C4—22no—
222C4—41no—
223C4—41no—
224C4—81no—
225C4—81no—
226C4—81no—
227C4—82no—
228C4—81no—
229C4—82no—
230C4#55312yes—
231C4#55512yes—
232C2—42no—
233C2#250011yescharacteristic
234C2#309511yescharacteristic
235C2#309511yescharacteristic
236C2#317211yescharacteristic
237C2#318111yescharacteristic
238C2#321811yescharacteristic
239C2#321911yescharacteristic
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. #1960C2semidirect
6. #1960C2semidirect
7. #1928C2semidirect
8. #1927C2semidirect
9. #1847C2semidirect
10. #1836C2semidirect

Maximal subgroups

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

subgroupindex
#28572
#22832
#20362
#20352
#19602
#19282
#19272
#18472
#18362
#17492