Finite Group Databasebeta

← #28581

Subgroups — #28581

Order 256.

446 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 129 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×#470C212yesmaximal
2#2857C211yesmaximal, characteristic
3C4×C22⋊Q8C211yesmaximal, characteristic
4#2084C211yesmaximal, characteristic, complemented
5#2072C211yesmaximal, characteristic, complemented
6#2043C211yesmaximal, characteristic, complemented
7#2016C211yesmaximal, characteristic, complemented
8#1927C212yesmaximal, complemented
9#1894C212yesmaximal
10#1878C212yesmaximal, complemented
11#1728C211yesmaximal, characteristic
12C22×Q16C2212yes—
13C22×M16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C22⋊Q8C2212yes—
16C2×C22⋊Q8C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C22×C42C2211yescharacteristic
19#470—28no—
20C2×#106C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25#391C2211yescharacteristic
26#391C2212yes—
27#389—21no—
28#389—21no—
29#389C2211yescharacteristic
30#387—21no—
31#385—21no—
32#384C2211yescharacteristic
33#380—21no—
34#380—21no—
35#380C2211yescharacteristic
36C4×C4⋊C4—21no—
37C4×C4⋊C4C2211yescharacteristic
38C4×C22⋊C4—21no—
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43#336C2212yes—
44C22×Q8C2312yes—
45C22×Q8C2311yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4D411yescharacteristic
48C23×C4C2311yescharacteristic
49C2×Q16—24no—
50C2×Q16—24no—
51C2×Q16—42no—
52C2×M16—24no—
53C2×M16—24no—
54C2×M16—42no—
55C22×C8C2312yes—
56C22×C8C2312yes—
57C4⋊Q8—42no—
58C22⋊Q8—22no—
59C22⋊Q8—24no—
60C22⋊Q8—24no—
61C22⋊Q8D414yes—
62C4×Q8—42no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2312yes—
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C22⋊C4—21no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—21no—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42—42no—
81C2×C42D411yescharacteristic
82C2×C42D411yescharacteristic
83C2×C42C2311yescharacteristic
84C2×C42C2311yescharacteristic
85#106—24no—
86#103—24no—
87#103—24no—
88#103—24no—
89#103—42no—
90#95—21no—
91#95—21no—
92#95—21no—
93#95—21no—
94#95—21no—
95#95—22no—
96#95—22no—
97C24C2×D411yescharacteristic
98C2×Q8—21no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8C2×D412yes—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D412yes—
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4#4112yes—
129C22×C4C2411yescharacteristic
130Q16—48no—
131M16—48no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8#4114yes—
138C4⋊C4—24no—
139C4⋊C4—21no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—44no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4C2×D412yes—
152C22⋊C4—24no—
153C22⋊C4—42no—
154C22⋊C4—44no—
155C42—22no—
156C42—22no—
157C42—41no—
158C42—41no—
159C42—42no—
160C42—42no—
161C23—21no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23#12112yes—
168C23C22×D411yescharacteristic
169Q8—24no—
170Q8—42no—
171Q8—42no—
172Q8—42no—
173Q8—42no—
174Q8—42no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—24no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—44no—
191C2×C4—44no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4#12112yes—
209C2×C4#13712yes—
210C2×C4#13712yes—
211C2×C4#13712yes—
212C2×C4C22×D411yescharacteristic
213C2×C4C2×#4112yes—
214C2×C4C2×#4111yescharacteristic
215C2×C4H(5,𝔽2)12yes—
216C8—24no—
217C8—42no—
218C22—22no—
219C22—22no—
220C22—22no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22C2×#12111yescharacteristic
226C22#53211yescharacteristic
227C22#53411yescharacteristic
228C22C2×#13711yescharacteristic
229C22C2×#13711yescharacteristic
230C22C2×#13711yescharacteristic
231C22#57511yescharacteristic
232C4—22no—
233C4—22no—
234C4—22no—
235C4—22no—
236C4—41no—
237C4—42no—
238C4—42no—
239C4—81no—
240C4—81no—
241C4—82no—
242C4—82no—
243C4#54612yes—
244C4#55312yes—
245C2—42no—
246C2#254311yescharacteristic
247C2#303711yescharacteristic
248C2#303911yescharacteristic
249C2#317211yescharacteristic
250C2#318511yescharacteristic
251C2#319511yescharacteristic
252C2#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
4. #2084C2semidirect
5. #2072C2semidirect
6. #2043C2semidirect
7. #2016C2semidirect
8. #1927C2semidirect
10. #1878C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4702
#28572
C4×C22⋊Q82
#20842
#20722
#20432
#20162
#19272
#18942
#18782
#17282