Finite Group Databasebeta

← #129556

Subgroups — #129556

Order 512.

650 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3170C211yesmaximal, characteristic
2#12599C211yesmaximal, characteristic
3C2×#1334C214yesmaximal, complemented
4#8595C211yesmaximal, characteristic
5C2×#533—21no—
6#3170—28no—
7C2×#500C411yescharacteristic
8C2×#494C411yescharacteristic
9C2×#487—21no—
10C2×#467—21no—
11C2×#467C412yes—
12C2×C4×Q16—21no—
13C2×C4×Q16C2212yes—
14C22×C4×C8C2211yescharacteristic
15C2×C22⋊C16—21no—
16C2×#346C2212yes—
17#1797—21no—
18C2×#334C2212yes—
19#1374—21no—
20#1334—28no—
21#1277—21no—
22C22×Q16—21no—
23C22×Q16C2×C412yes—
24C23×C8C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26#533—28no—
27#533—44no—
28C2×C4⋊Q8—21no—
29C2×C4⋊Q8—21no—
30C2×#125—41no—
31C2×C22⋊Q8—22no—
32C2×C22⋊Q8—41no—
33C2×C4×Q8—22no—
34C2×C4×Q8—41no—
35C2×#117—41no—
36C22×C42D411yescharacteristic
37C22×C16—22no—
38#500—24no—
39#494—24no—
40#487—28no—
41#467—28no—
42#467—24no—
43#467—24no—
44C4×Q16—28no—
45C4×Q16—24no—
46C4×Q16D418yes—
47C2×#107—21no—
48C2×#107C2×C412yes—
49C2×C4⋊C8—22no—
50C2×#103—22no—
51C2×#103—22no—
52C2×C22⋊C8—41no—
53C2×C4×C8—24no—
54C2×C4×C8C2×C411yescharacteristic
55C2×C4×C8D411yescharacteristic
56C2×C4×C8D411yescharacteristic
57C2×C4×C8C2311yescharacteristic
58C22⋊C16—24no—
59#346—24no—
60#336—41no—
61#334—24no—
62C22×Q8—22no—
63C22×Q8—41no—
64C23×C4C22⋊C411yescharacteristic
65C23×C4D811yescharacteristic
66C23×C4SD1611yescharacteristic
67C2×Q16—24no—
68C2×Q16—22no—
69C2×Q16—24no—
70C2×Q16—24no—
71C2×Q16—42no—
72C2×Q16—42no—
73C2×Q16C22⋊C414yes—
74C22×C8—22no—
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—41no—
80C22×C8—42no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22×C411yescharacteristic
84C22×C8C2×D411yescharacteristic
85C4⋊Q8—24no—
86C4⋊Q8—24no—
87C4⋊Q8—42no—
88C4⋊Q8—42no—
89#125—48no—
90C22⋊Q8—28no—
91C22⋊Q8—44no—
92C22⋊Q8—48no—
93C4×Q8—28no—
94C4×Q8—48no—
95C4×Q8—44no—
96#117—48no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C42—22no—
105C2×C42—22no—
106C2×C42—42no—
107C2×C42C22⋊C411yescharacteristic
108C2×C42D811yescharacteristic
109C2×C42SD1611yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C16—24no—
112C2×C16—42no—
113C2×C16—42no—
114#107—24no—
115#107—22no—
116#107C22⋊C414yes—
117C4⋊C8—42no—
118C4⋊C8—42no—
119#103—28no—
120#103—28no—
121C22⋊C8—82no—
122C4×C8—22no—
123C4×C8—22no—
124C4×C8—22no—
125C4×C8C2×D414yes—
126C24#10211yescharacteristic
127C2×Q8—24no—
128C2×Q8—44no—
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—82no—
135C2×Q8—82no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—41no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4#10211yescharacteristic
151C22×C4#10211yescharacteristic
152C22×C4#10211yescharacteristic
153C22×C4C2×C22⋊C411yescharacteristic
154C22×C4C2×D811yescharacteristic
155C22×C4C2×SD1611yescharacteristic
156Q16—28no—
157Q16—48no—
158Q16—44no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—44no—
167C2×C8—44no—
168C2×C8—82no—
169C2×C8—82no—
170C2×C8—82no—
171C2×C8#10412yes—
172C2×C8C2×C22⋊C412yes—
173C2×C8#12012yes—
174C2×C8#13112yes—
175C2×C8#13112yes—
176C4⋊C4—24no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—48no—
180C4⋊C4—44no—
181C4⋊C4—42no—
182C4⋊C4—48no—
183C22⋊C4—48no—
184C42—21no—
185C42—21no—
186C42—22no—
187C42—22no—
188C42—48no—
189C42#12012yes—
190C42AGL(1,ℤ/8)12yes—
191C16—44no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23#36212yes—
199C23C2×#10211yescharacteristic
200Q8—44no—
201Q8—44no—
202Q8—88no—
203Q8—82no—
204Q8—82no—
205C2×C4—28no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—44no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4#36012yes—
229C2×C4#36212yes—
230C2×C4#37911yescharacteristic
231C2×C4#40811yescharacteristic
232C2×C4#40811yescharacteristic
233C2×C4#41811yescharacteristic
234C2×C4C2×#10411yescharacteristic
235C2×C4#42111yescharacteristic
236C2×C4#44712yes—
237C2×C4#45012yes—
238C8—22no—
239C8—22no—
240C8—41no—
241C8—162no—
242C8#40814yes—
243C22—22no—
244C22—22no—
245C22—22no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22#148011yescharacteristic
251C22#149211yescharacteristic
252C22#178611yescharacteristic
253C22C2×#36011yescharacteristic
254C22#214511yescharacteristic
255C22C2×#36211yescharacteristic
256C22C2×#36211yescharacteristic
257C4—22no—
258C4—22no—
259C4—22no—
260C4—22no—
261C4—41no—
262C4—84no—
263C4—84no—
264C4#148411yescharacteristic
265C4#149611yescharacteristic
266C4#178011yescharacteristic
267C4#178911yescharacteristic
268C2—42no—
269C2#831311yescharacteristic
270C2#1165311yescharacteristic
271C2#1165511yescharacteristic
272C2#1167911yescharacteristic
273C2#1168111yescharacteristic
274C2#1273511yescharacteristic
275C2#1273711yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1334C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31702
#125992
C2×#13342
#85952