Finite Group Databasebeta

← #24185

Subgroups — #24185

Order 256.

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

normal subgroups only

This group has 575 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×#556C211yesmaximal, characteristic, complemented
2C2×#461C211yesmaximal, characteristic, complemented
3C2×C4×D8C211yesmaximal, characteristic
4#2538C211yesmaximal, characteristic, complemented
5#2042C212yesmaximal, complemented
6#2012C212yesmaximal
7#1984C211yesmaximal, characteristic, complemented
8#1956C211yesmaximal, characteristic, complemented
9#1880C212yesmaximal, complemented
10#1863C212yesmaximal, complemented
11#1802C211yesmaximal, 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×C22⋊Q8—21no—
21C2×C4×Q8C2212yes—
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×#117—21no—
26#461—24no—
27C4×D8—24no—
28C2×#107C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2212yes—
36C2×C4×C8C2211yescharacteristic
37#396—22no—
38#390C2212yes—
39#389C2212yes—
40#384C2211yescharacteristic
41#380—21no—
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—
66C22⋊Q8—44no—
67C4×Q8—24no—
68C4×Q8—24no—
69C4×Q8—22no—
70C4×Q8—22no—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—22no—
74C4×D4D414yes—
75#117—28no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4D412yes—
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2312yes—
86C2×C4⋊C4C2311yescharacteristic
87C2×C22⋊C4—22no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C42—22no—
94C2×C42C2312yes—
95C2×C42C2311yescharacteristic
96#107—22no—
97C4⋊C8—22no—
98#103—24no—
99#103—24no—
100#103—42no—
101#102—22no—
102#102—22no—
103#102—42no—
104C4×C8—22no—
105#95—22no—
106C24—21no—
107C24—21no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8C2×D414yes—
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×D4—21no—
126C2×D4—21no—
127C2×D4C2×D412yes—
128C2×D4#4112yes—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—24no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4C2×D412yes—
140C22×C4C2×D412yes—
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2411yescharacteristic
144SD16—48no—
145D8—28no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8#4112yes—
155C4⋊C4—28no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—22no—
159C4⋊C4—24no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—44no—
166C4⋊C4—42no—
167C4⋊C4—81no—
168C4⋊C4C2×D412yes—
169C22⋊C4—28no—
170C22⋊C4—28no—
171C22⋊C4—24no—
172C22⋊C4—44no—
173C22⋊C4—42no—
174C22⋊C4—82no—
175C42—28no—
176C42—22no—
177C42—22no—
178C42C2×D412yes—
179C42C2×D412yes—
180C23—22no—
181C23—22no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23C22×D411yescharacteristic
195Q8—24no—
196Q8—24no—
197Q8—42no—
198Q8—42no—
199Q8—44no—
200D4—22no—
201D4—22no—
202D4—22no—
203D4—22no—
204D4—41no—
205D4—41no—
206D4—41no—
207D4—41no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—48no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—84no—
230C2×C4#12012yes—
231C2×C4#12012yes—
232C2×C4#13512yes—
233C2×C4AGL(1,ℤ/8)12yes—
234C2×C4C22×D411yescharacteristic
235C2×C4C22×D411yescharacteristic
236C2×C4C2×#4111yescharacteristic
237C2×C4C2×#4111yescharacteristic
238C2×C4H(5,𝔽2)11yescharacteristic
239C2×C42−1+411yescharacteristic
240C8—22no—
241C8—41no—
242C8—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—81no—
250C22—81no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22—81no—
255C22—81no—
256C22—81no—
257C22C2×#12011yescharacteristic
258C22#54811yescharacteristic
259C22#55011yescharacteristic
260C22C2×#13511yescharacteristic
261C22C2×AGL(1,ℤ/8)11yescharacteristic
262C22#57711yescharacteristic
263C22#57811yescharacteristic
264C4—22no—
265C4—44no—
266C4—42no—
267C4—42no—
268C4—42no—
269C4—82no—
270C4—81no—
271C4#44914yes—
272C4#52511yescharacteristic
273C4#53711yescharacteristic
274C4#54611yescharacteristic
275C4#54711yescharacteristic
276C2—82no—
277C2—82no—
278C2#240711yescharacteristic
279C2C2×#44911yescharacteristic
280C2#301311yescharacteristic
281C2#336011yescharacteristic
282C2#336711yescharacteristic
283C2#339311yescharacteristic
284C2#339711yescharacteristic
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. #2538C2semidirect
5. #2042C2semidirect
7. #1984C2semidirect
8. #1956C2semidirect
9. #1880C2semidirect
10. #1863C2semidirect
11. #1802C2semidirect

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×C4×D82
#25382
#20422
#20122
#19842
#19562
#18802
#18632
#18022