Finite Group Databasebeta

← #24197

Subgroups — #24197

Order 256.

558 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 131 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×#556C211yesmaximal, characteristic, complemented
2C2×#461C211yesmaximal, characteristic, complemented
3C2×#442C211yesmaximal, characteristic
4#2580C211yesmaximal, characteristic, complemented
5#2042C212yesmaximal, complemented
6#2012C212yesmaximal
7#1985C211yesmaximal, characteristic, complemented
8#1960C211yesmaximal, characteristic, complemented
9#1880C212yesmaximal, complemented
10#1865C212yesmaximal, complemented
11#1804C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×D8C2211yescharacteristic
14#556—24no—
15#556—24no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#126—22no—
18C2×#124C2212yes—
19C2×C22⋊Q8—21no—
20C2×C4×Q8C2212yes—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×#117—21no—
25#461—24no—
26#442—24no—
27C2×#106C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2212yes—
35C2×#97C2211yescharacteristic
36#399—21no—
37#393—21no—
38#390C2212yes—
39#389C2212yes—
40#385—22no—
41#382C2211yescharacteristic
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8D411yescharacteristic
46C22×Q8C2312yes—
47C22×D4C2311yescharacteristic
48C22×D4C2311yescharacteristic
49C23×C4—21no—
50C23×C4—21no—
51C2×SD16—24no—
52C2×SD16—24no—
53C2×SD16—42no—
54C2×D8—22no—
55C2×D8—22no—
56C2×D8—22no—
57C22×C8C2312yes—
58C22×C8C2311yescharacteristic
59C22×C8C2311yescharacteristic
60C4⋊Q8—22no—
61C4⋊Q8D414yes—
62#126—216no—
63#124—24no—
64#124—24no—
65C22⋊Q8—28no—
66C4×Q8—24no—
67C4×Q8—24no—
68C4×Q8—22no—
69C4×Q8—22no—
70C4×D4—22no—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4D414yes—
74#117—28no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4D411yescharacteristic
80C2×C4⋊C4D412yes—
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2312yes—
83C2×C4⋊C4C2311yescharacteristic
84C2×C22⋊C4—22no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C42—22no—
91C2×C42C2312yes—
92C2×C42C2311yescharacteristic
93#106—22no—
94C4⋊C8—22no—
95#103—24no—
96#103—24no—
97#103—42no—
98#102—22no—
99#102—22no—
100#102—42no—
101#97—22no—
102#95—21no—
103#95—21no—
104#95—22no—
105C24—21no—
106C24—21no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—42no—
114C2×Q8C2×D414yes—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4C2×D412yes—
126C2×D4#4112yes—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—24no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—44no—
133C22×C4—42no—
134C22×C4—82no—
135C22×C4C2×D412yes—
136C22×C4C2×D412yes—
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2411yescharacteristic
140SD16—48no—
141D8—28no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8#4112yes—
151C4⋊C4—28no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—22no—
155C4⋊C4—24no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—82no—
162C4⋊C4C2×D412yes—
163C22⋊C4—28no—
164C22⋊C4—28no—
165C22⋊C4—24no—
166C22⋊C4—44no—
167C22⋊C4—42no—
168C22⋊C4—82no—
169C42—28no—
170C42—22no—
171C42—22no—
172C42C2×D412yes—
173C42C2×D412yes—
174C23—22no—
175C23—22no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23—41no—
180C23—42no—
181C23—42no—
182C23—42no—
183C23—42no—
184C23C22×D411yescharacteristic
185Q8—24no—
186Q8—24no—
187Q8—44no—
188Q8—42no—
189Q8—42no—
190D4—22no—
191D4—22no—
192D4—22no—
193D4—22no—
194D4—41no—
195D4—41no—
196D4—41no—
197D4—41no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—48no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—84no—
218C2×C4#12012yes—
219C2×C4#12012yes—
220C2×C4C22×D411yescharacteristic
221C2×C4C22×D411yescharacteristic
222C2×C4C2×#4111yescharacteristic
223C2×C4C2×#4111yescharacteristic
224C2×C4H(5,𝔽2)11yescharacteristic
225C2×C42−1+411yescharacteristic
226C8—22no—
227C8—41no—
228C8—42no—
229C22—44no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—82no—
235C22—82no—
236C22—82no—
237C22—81no—
238C22—81no—
239C22C2×#12011yescharacteristic
240C22#54811yescharacteristic
241C22#55011yescharacteristic
242C22#57512yes—
243C22#57712yes—
244C4—22no—
245C4—22no—
246C4—44no—
247C4—42no—
248C4—42no—
249C4—42no—
250C4—82no—
251C4—81no—
252C4#52511yescharacteristic
253C4#53711yescharacteristic
254C4#54611yescharacteristic
255C4#54711yescharacteristic
256C2—82no—
257C2—82no—
258C2#240711yescharacteristic
259C2#300511yescharacteristic
260C2#301011yescharacteristic
261C2#337612yes—
262C2#340112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
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×#556C2semidirect
2. C2×#461C2semidirect
4. #2580C2semidirect
5. #2042C2semidirect
7. #1985C2semidirect
8. #1960C2semidirect
9. #1880C2semidirect
10. #1865C2semidirect
11. #1804C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
C2×#4612
C2×#4422
#25802
#20422
#20122
#19852
#19602
#18802
#18652
#18042