Finite Group Databasebeta

← #28186

Subgroups — #28186

Order 256.

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

normal subgroups only

This group has 417 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×C8×Q8C211yesmaximal, characteristic, complemented
2C2×C4×SD16C211yesmaximal, characteristic
3#2582C211yesmaximal, characteristic, complemented
4#2538C211yesmaximal, characteristic
5#2088C212yesmaximal, complemented
6#2055C212yesmaximal
7#1934C212yesmaximal
8#1933C212yesmaximal, complemented
9#1834C211yesmaximal, characteristic, complemented
10#1803C211yesmaximal, characteristic, complemented
11#1802C211yesmaximal, characteristic
12C22×SD16C2211yescharacteristic
13C2×C22⋊Q8—21no—
14C2×C4×Q8C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16C2×C4×D4C2211yescharacteristic
17C8×Q8—24no—
18C4×SD16—24no—
19C2×#106C2211yescharacteristic
20C2×C4⋊C8C2212yes—
21C2×C4⋊C8C2211yescharacteristic
22C2×#103C2212yes—
23C2×#103C2211yescharacteristic
24C2×#102C2212yes—
25C2×#102C2211yescharacteristic
26C2×C4×C8C2212yes—
27C2×C4×C8C2211yescharacteristic
28#398—22no—
29#396—22no—
30#391—21no—
31#391C2212yes—
32#390C2212yes—
33#389—21no—
34#389C2212yes—
35#389C2212yes—
36#384C2211yescharacteristic
37#384C2211yescharacteristic
38#380—21no—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43C22×Q8D411yescharacteristic
44C22×Q8C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C2×SD16—22no—
48C2×SD16—22no—
49C2×SD16—22no—
50C22×C8C2312yes—
51C22×C8C2311yescharacteristic
52C22×C8C2311yescharacteristic
53C22⋊Q8—44no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×Q8—22no—
58C4×D4—22no—
59C4×D4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4D411yescharacteristic
68C2×C4⋊C4D412yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2312yes—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C42—21no—
77C2×C42C2312yes—
78C2×C42C2311yescharacteristic
79#106—22no—
80C4⋊C8—24no—
81C4⋊C8—22no—
82#103—24no—
83#103—22no—
84#102—24no—
85#102—22no—
86C4×C8—24no—
87C4×C8D414yes—
88#95—22no—
89#95—22no—
90#95—22no—
91#95—22no—
92C24—21no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8SD1614yescomplemented
101C2×Q8#4112yes—
102C2×D4—21no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4#4112yes—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4C2×D412yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D412yes—
119C22×C4C2×D411yescharacteristic
120C22×C4C2411yescharacteristic
121SD16—28no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8C2×D412yes—
130C2×C8C2×D412yes—
131C4⋊C4—24no—
132C4⋊C4—22no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—44no—
144C4⋊C4—42no—
145C4⋊C4—81no—
146C4⋊C4—81no—
147C4⋊C4—82no—
148C22⋊C4—42no—
149C22⋊C4—44no—
150C22⋊C4—82no—
151C42—22no—
152C42—22no—
153C42—42no—
154C42C2×D412yes—
155C42#4112yes—
156C23—22no—
157C23—41no—
158C23—41no—
159C23—41no—
160C23—41no—
161C23—41no—
162C23—41no—
163C23C22×D411yescharacteristic
164Q8—28no—
165Q8—22no—
166Q8—22no—
167Q8—41no—
168Q8—41no—
169D4—22no—
170D4—22no—
171D4—41no—
172D4—41no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—84no—
196C2×C4#12112yes—
197C2×C4#12112yes—
198C2×C4C2×SD1614yes—
199C2×C4C2×SD1612yes—
200C2×C4#13512yes—
201C2×C4C22×D411yescharacteristic
202C2×C4C2×#4111yescharacteristic
203C2×C4C2×#4111yescharacteristic
204C2×C4C2×#4111yescharacteristic
205C2×C42−1+411yescharacteristic
206C2×C42−1+411yescharacteristic
207C8—22no—
208C8—22no—
209C8—42no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—81no—
214C22—81no—
215C22—81no—
216C22—81no—
217C22C2×#12111yescharacteristic
218C22#54811yescharacteristic
219C22#54811yescharacteristic
220C22C22×SD1611yescharacteristic
221C22C2×#13511yescharacteristic
222C22#57611yescharacteristic
223C22#57811yescharacteristic
224C4—24no—
225C4—22no—
226C4—42no—
227C4—42no—
228C4—82no—
229C4—82no—
230C4—81no—
231C4—81no—
232C4#46514yes—
233C4#52511yescharacteristic
234C4#53611yescharacteristic
235C4#55211yescharacteristic
236C4#55211yescharacteristic
237C2—82no—
238C2#253611yescharacteristic
239C2C2×#46511yescharacteristic
240C2#306111yescharacteristic
241C2#335911yescharacteristic
242C2#336111yescharacteristic
243C2#336211yescharacteristic
244C2#336411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C8×Q8C2semidirect
3. #2582C2semidirect
5. #2088C2semidirect
8. #1933C2semidirect
9. #1834C2semidirect
10. #1803C2semidirect
100. C2×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
C2×C8×Q82
C2×C4×SD162
#25822
#25382
#20882
#20552
#19342
#19332
#18342
#18032
#18022