Finite Group Databasebeta

← #28604

Subgroups — #28604

Order 256.

452 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 123 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×#488C211yesmaximal, characteristic, complemented
2C2×#436C211yesmaximal, characteristic, complemented
3#2562C211yesmaximal, characteristic, complemented
4#2540C211yesmaximal, characteristic
5#2035C212yesmaximal, complemented
6#2034C212yesmaximal, complemented
7#1989C211yesmaximal, characteristic, complemented
8#1983C211yesmaximal, characteristic, complemented
9#1895C212yesmaximal, complemented
10#1894C212yesmaximal, complemented
11C4×#106C211yesmaximal, characteristic, complemented
12C22×M16C2212yes—
13C2×#125C2211yescharacteristic
14C2×#124C2211yescharacteristic
15C2×C22⋊Q8C2212yes—
16C2×#121C2212yes—
17C2×C4×D4C2211yescharacteristic
18#488—24no—
19#436—24no—
20C2×#106C2212yes—
21C2×C4⋊C8C2211yescharacteristic
22C2×#103C2211yescharacteristic
23C2×#103C2211yescharacteristic
24C2×#103C2212yes—
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2212yes—
28C2×C22⋊C8C2212yes—
29C2×C4×C8C2211yescharacteristic
30#400—21no—
31#399—21no—
32#394—21no—
33#392—21no—
34#390—22no—
35#388—22no—
36#380C2212yes—
37#380C2212yes—
38C4×C4⋊C4C2211yescharacteristic
39C4×C4⋊C4C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43C22×Q8C2311yescharacteristic
44C22×D4—22no—
45C22×D4D411yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4C2312yes—
48C2×M16—24no—
49C2×M16—42no—
50C2×M16—42no—
51C22×C8C2312yes—
52C22×C8C2311yescharacteristic
53C22×C8C2311yescharacteristic
54#125—22no—
55#125—22no—
56#124—22no—
57#124—22no—
58C22⋊Q8—24no—
59C22⋊Q8—42no—
60#121—24no—
61#121—42no—
62C4×D4—22no—
63C4×D4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4D411yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2312yes—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4D412yes—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42C2311yescharacteristic
82#106—24no—
83C4⋊C8—22no—
84#103—22no—
85#103—22no—
86#103—42no—
87#102—22no—
88#102—22no—
89#102—42no—
90C22⋊C8—24no—
91C4×C8D414yescomplemented
92#95—21no—
93#95—21no—
94#95—22no—
95#95—22no—
96C24—21no—
97C24C2×D412yes—
98C2×Q8—21no—
99C2×Q8—22no—
100C2×Q8—21no—
101C2×Q8—22no—
102C2×D4—22no—
103C2×D4—21no—
104C2×D4—22no—
105C2×D4—21no—
106C2×D4—22no—
107C2×D4—42no—
108C2×D4—44no—
109C2×D4—44no—
110C2×D4—82no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4C2×D412yes—
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2411yescharacteristic
126M16—48no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8C2×D412yes—
133C2×C8C2×D412yes—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—44no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4#4114yes—
146C22⋊C4—24no—
147C22⋊C4—44no—
148C22⋊C4—44no—
149C22⋊C4—44no—
150C22⋊C4—42no—
151C22⋊C4—42no—
152C22⋊C4—82no—
153C42—81no—
154C42—81no—
155C42C2×D412yes—
156C42#4112yes—
157C23—22no—
158C23—22no—
159C23—22no—
160C23—22no—
161C23—22no—
162C23—22no—
163C23—22no—
164C23—41no—
165C23—42no—
166C23—42no—
167C23—41no—
168C23—41no—
169C23C22×D411yescharacteristic
170Q8—41no—
171Q8—41no—
172Q8—42no—
173D4—44no—
174D4—41no—
175D4—41no—
176D4—42no—
177D4—88no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4#12112yes—
202C2×C4#12112yes—
203C2×C4C22×D411yescharacteristic
204C2×C4C2×#4112yes—
205C2×C4C2×#4111yescharacteristic
206C2×C4H(5,𝔽2)12yes—
207C8—22no—
208C8—22no—
209C8—42no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—42no—
216C22—42no—
217C22—81no—
218C22—81no—
219C22—81no—
220C22—82no—
221C22—82no—
222C22C2×#12111yescharacteristic
223C22#54612yes—
224C22#57512yes—
225C22#57711yescharacteristic
226C22#57711yescharacteristic
227C4—22no—
228C4—22no—
229C4—42no—
230C4—44no—
231C4—81no—
232C4—81no—
233C4—81no—
234C4—82no—
235C4#53211yescharacteristic
236C4#53511yescharacteristic
237C4#55912yes—
238C2—82no—
239C2—161no—
240C2#254411yescharacteristic
241C2#305411yescharacteristic
242C2#306011yescharacteristic
243C2#332312yes—
244C2#332612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2540C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#488C2semidirect
2. C2×#436C2semidirect
3. #2562C2semidirect
5. #2035C2semidirect
6. #2034C2semidirect
7. #1989C2semidirect
8. #1983C2semidirect
9. #1895C2semidirect
10. #1894C2semidirect
11. C4×#106C2semidirect
91. C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4882
C2×#4362
#25622
#25402
#20352
#20342
#19892
#19832
#18952
#18942
C4×#1062