Finite Group Databasebeta

← #28308

Subgroups — #28308

Order 256.

549 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 549 conjugacy classes of subgroups — too many to draw. Only its 139 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×#485C212yesmaximal, complemented
2C2×#452C212yesmaximal
3#2857C211yesmaximal, characteristic, complemented
4#2297C211yesmaximal, characteristic
5#2096C211yesmaximal, characteristic, complemented
6#2093C211yesmaximal, characteristic, complemented
7#2016C211yesmaximal, characteristic, complemented
8#2013C211yesmaximal, characteristic, complemented
9#1969C212yesmaximal, complemented
10#1968C212yesmaximal, complemented
11C4×C22⋊C8C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C22×Q16C2212yes—
14C22×SD16C2212yes—
15C2×C4⋊Q8C2211yescharacteristic
16C2×C22⋊Q8C2212yes—
17C2×#117C2211yescharacteristic
18C22×C42C2211yescharacteristic
19#485—28no—
20#452—28no—
21C2×#107C2212yes—
22C2×#106C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#102C2212yes—
27C2×C22⋊C8C2212yes—
28C2×C4×C8C2212yes—
29#391C2212yes—
30#391C2211yescharacteristic
31#391C2211yescharacteristic
32#390C2211yescharacteristic
33#390C2211yescharacteristic
34#389—22no—
35#386—22no—
36#384C2211yescharacteristic
37#380—22no—
38C4×C22⋊C4—22no—
39#336C2211yescharacteristic
40#336C2211yescharacteristic
41C2×#41—22no—
42C2×#41—22no—
43C2×#41—24no—
44C2×#41—42no—
45C22×Q8C2312yes—
46C22×Q8C2311yescharacteristic
47C22×D4—21no—
48C22×D4C2311yescharacteristic
49C23×C4—21no—
50C23×C4D412yes—
51C23×C4C2311yescharacteristic
52C2×Q16—24no—
53C2×Q16—24no—
54C2×Q16—42no—
55C2×SD16—24no—
56C2×SD16—24no—
57C2×SD16—42no—
58C22×C8C2312yes—
59C22×C8C2312yes—
60C4⋊Q8—42no—
61C22⋊Q8—24no—
62C22⋊Q8—24no—
63#117—24no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C42—21no—
75C2×C42—42no—
76C2×C42D412yes—
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79#107—24no—
80#106—24no—
81#103—24no—
82#103—24no—
83#103—24no—
84#102—24no—
85C22⋊C8D418yes—
86C4×C8—42no—
87#95—22no—
88#95—22no—
89#95—24no—
90C24—21no—
91C24C2×D411yescharacteristic
92#41—48no—
93#41—48no—
94C2×Q8—21no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8#4112yes—
104C2×D4—24no—
105C2×D4—22no—
106C2×D4—21no—
107C2×D4—22no—
108C2×D4—42no—
109C2×D4—42no—
110C2×D4#4112yes—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—24no—
115C22×C4—21no—
116C22×C4—24no—
117C22×C4—24no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D414yes—
127C22×C4C2×D412yes—
128C22×C4C2×D412yes—
129C22×C4C2411yescharacteristic
130Q16—48no—
131SD16—48no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8C2×D414yes—
137C2×C8C2×D414yes—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—44no—
143C4⋊C4—44no—
144C4⋊C4—44no—
145C22⋊C4—42no—
146C22⋊C4—44no—
147C22⋊C4—44no—
148C22⋊C4—44no—
149C42—22no—
150C42—22no—
151C42—42no—
152C42—42no—
153C23—21no—
154C23—21no—
155C23—22no—
156C23—22no—
157C23—22no—
158C23—41no—
159C23—41no—
160C23—42no—
161C23—42no—
162C23#12712yes—
163C23C22×D411yescharacteristic
164Q8—24no—
165Q8—42no—
166Q8—42no—
167Q8—42no—
168Q8—42no—
169Q8—42no—
170D4—24no—
171D4—48no—
172D4—42no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—24no—
181C2×C4—22no—
182C2×C4—48no—
183C2×C4—48no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4#12712yes—
196C2×C4#13514yes—
197C2×C4#13714yes—
198C2×C4C22×D412yes—
199C2×C4C2×#4111yescharacteristic
200C2×C4C2×#4111yescharacteristic
201C2×C4H(5,𝔽2)12yes—
202C8—24no—
203C8—24no—
204C22—22no—
205C22—22no—
206C22—22no—
207C22—42no—
208C22—42no—
209C22—44no—
210C22—42no—
211C22—81no—
212C22—81no—
213C22—82no—
214C22#52511yescharacteristic
215C22C2×#12711yescharacteristic
216C22#53411yescharacteristic
217C22C2×#13512yes—
218C22C2×#13712yes—
219C4—24no—
220C4—24no—
221C4—22no—
222C4—42no—
223C4—81no—
224C4—82no—
225C4—81no—
226C4—82no—
227C4#54612yes—
228C4#54612yes—
229C2—42no—
230C2—82no—
231C2#253911yescharacteristic
232C2#310712yes—
233C2#314711yescharacteristic
234C2#315211yescharacteristic
235C2#319012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
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
1. C2×#485C2semidirect
3. #2857C2semidirect
5. #2096C2semidirect
6. #2093C2semidirect
7. #2016C2semidirect
8. #2013C2semidirect
9. #1969C2semidirect
10. #1968C2semidirect
11. C4×C22⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4852
C2×#4522
#28572
#22972
#20962
#20932
#20162
#20132
#19692
#19682
C4×C22⋊C82