Finite Group Databasebeta

← #18776

Subgroups — #18776

Order 256.

596 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 133 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×D4×Q8C211yesmaximal, characteristic, complemented
2#3226C211yesmaximal, characteristic, complemented
3#3100C211yesmaximal, characteristic
4#3037C211yesmaximal, characteristic, complemented
5#1649C214yesmaximal, complemented
6#1627C214yesmaximal, complemented
7#1546C211yesmaximal, characteristic, complemented
8#1486C211yesmaximal, characteristic, complemented
9#1479C211yesmaximal, characteristic, complemented
10C23×Q8—21no—
11C2×#137—21no—
12D4×Q8—22no—
13D4×Q8—22no—
14D4×Q8—22no—
15D4×Q8C2214yes—
16#532C2211yescharacteristic
17C2×C4⋊Q8—21no—
18C2×C4⋊Q8C2211yescharacteristic
19C2×C22⋊Q8—21no—
20C2×C22⋊Q8—22no—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4—21no—
23C2×C4×D4C2211yescharacteristic
24#500C2212yescomplemented
25#486C2212yescomplemented
26#480—21no—
27#480—22no—
28#474—22no—
29#474C2212yes—
30#464—22no—
31#462C2212yescomplemented
32#460C2212yescomplemented
33#459C2212yes—
34#447—21no—
35C4×D8—22no—
36#431C2211yescharacteristic
37#431C2211yescharacteristic
38#423C2211yescharacteristic
39#417—21no—
40#332C2214yescomplemented
41Q8⋊C8C2214yescomplemented
42D4⋊C8C2214yes—
43C2×#41—21no—
44C22×Q8—22no—
45C22×Q8—24no—
46C22×Q8—24no—
47C22×Q8—21no—
48C22×Q8—44no—
49C22×Q8—44no—
50C22×Q8D411yescharacteristic
51C22×D4D411yescharacteristic
52C23×C4—22no—
53C23×C4—21no—
54#137—44no—
55C2×Q16—22no—
56C2×SD16—22no—
57C2×D8—22no—
58C2×M16—21no—
59C4⋊Q8—24no—
60C4⋊Q8—24no—
61C4⋊Q8—21no—
62C4⋊Q8—21no—
63C4⋊Q8D412yes—
64C4⋊Q8C2311yescharacteristic
65C4⋊Q8C2312yes—
66#127C2311yescharacteristic
67#124—21no—
68C22⋊Q8—24no—
69C22⋊Q8—28no—
70C22⋊Q8—24no—
71C22⋊Q8—28no—
72#121—21no—
73#121—21no—
74C4×Q8—21no—
75C4×Q8—21no—
76C4×Q8D412yes—
77C4×Q8C2312yes—
78C4×D4—24no—
79C4×D4—24no—
80C4×D4—21no—
81C4×D4—21no—
82C4×D4—21no—
83C4×D4—21no—
84C4×D4D412yes—
85C4×D4C2312yes—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4D411yescharacteristic
91C2×C4⋊C4D412yes—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—21no—
94C2×C42—21no—
95C2×C42C2311yescharacteristic
96#107—22no—
97#107—24no—
98C4⋊C8—21no—
99C4⋊C8—21no—
100C4⋊C8C2312yes—
101C4⋊C8C2312yes—
102#103—22no—
103#102—22no—
104#102—22no—
105#102—22no—
106#102—24no—
107C22⋊C8—21no—
108C22⋊C8—21no—
109C4×C8—21no—
110C4×C8C2312yes—
111C24—21no—
112#41—42no—
113#41—42no—
114C2×Q8—28no—
115C2×Q8—21no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8—21no—
120C2×Q8—48no—
121C2×Q8—48no—
122C2×Q8—416no—
123C2×Q8—48no—
124C2×Q8—42no—
125C2×Q8—42no—
126C2×Q8D814yescomplemented
127C2×Q8C2×D412yes—
128C2×D4—24no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—41no—
134C2×D4C2×D412yes—
135C22×C4—21no—
136C22×C4—28no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—48no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—41no—
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149Q16—44no—
150SD16—44no—
151D8—44no—
152M16—42no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C4⋊C4—28no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4C2×D412yes—
167C4⋊C4C2×D412yes—
168C4⋊C4C2×D414yes—
169C22⋊C4—28no—
170C22⋊C4—22no—
171C22⋊C4—22no—
172C22⋊C4—41no—
173C22⋊C4—41no—
174C42—22no—
175C42—22no—
176C42C2×D411yescharacteristic
177C42C2×D411yescharacteristic
178C42C2×D411yescharacteristic
179C42C2411yescharacteristic
180C23—24no—
181C23—22no—
182C23—42no—
183C23—42no—
184C23—41no—
185C23#12011yescharacteristic
186Q8—22no—
187Q8—24no—
188Q8—416no—
189Q8—42no—
190Q8—42no—
191Q8—44no—
192Q8—41no—
193Q8—41no—
194Q8C2×D814yes—
195D4—24no—
196D4—22no—
197D4—42no—
198D4—41no—
199D4—41no—
200D4—42no—
201D4—82no—
202D4#13714yescomplemented
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—24no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—416no—
211C2×C4—42no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—44no—
217C2×C4—41no—
218C2×C4—82no—
219C2×C4#12011yescharacteristic
220C2×C4#12011yescharacteristic
221C2×C4#12011yescharacteristic
222C2×C4C2×D812yes—
223C2×C4AGL(1,ℤ/8)12yes—
224C2×C4C22×D411yescharacteristic
225C2×C4C22×D411yescharacteristic
226C2×C4C22×D411yescharacteristic
227C8—42no—
228C8—42no—
229C8—42no—
230C22—21no—
231C22—21no—
232C22—24no—
233C22—48no—
234C22—42no—
235C22—42no—
236C22—81no—
237C22—81no—
238C22—81no—
239C22#44712yes—
240C22C2×#12011yescharacteristic
241C4—24no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—44no—
246C4—42no—
247C4—42no—
248C4—81no—
249C4#44712yes—
250C4C22×D811yescharacteristic
251C4C2×AGL(1,ℤ/8)11yescharacteristic
252C4C2×#13712yes—
253C2—22no—
254C2—44no—
255C2—161no—
256C2C2×#44711yescharacteristic
257C2#300411yescharacteristic
258C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3100C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×D4×Q8C2semidirect
2. #3226C2semidirect
4. #3037C2semidirect
5. #1649C2semidirect
6. #1627C2semidirect
7. #1546C2semidirect
8. #1486C2semidirect
9. #1479C2semidirect
24. #500C22semidirect
25. #486C22semidirect
31. #462C22semidirect
32. #460C22semidirect
40. #332C22semidirect
41. Q8⋊C8C22semidirect
126. C2×Q8D8semidirect
202. D4#137semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Q82
#32262
#31002
#30372
#16492
#16272
#15462
#14862
#14792