Finite Group Databasebeta

← #508119

Subgroups — #508119

Order 512.

762 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 166 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
1C22×#499C211yesmaximal, characteristic
2C2×#2236C211yesmaximal, characteristic
3C2×#2235C211yesmaximal, characteristic
4#33130C211yesmaximal, characteristic
5#13453C214yesmaximal, complemented
6#13366C214yesmaximal, complemented
7#12867C211yesmaximal, characteristic
8#12866C211yesmaximal, characteristic
9#12844C211yesmaximal, characteristic
10C22×#125—21no—
11C2×#499—24no—
12C2×#499—24no—
13C2×#499—22no—
14C2×#499C2214yes—
15C2×#467C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#106—21no—
19C22×C4×C8C2211yescharacteristic
20#2855—21no—
21#2236—24no—
22#2235—24no—
23C2×#367C2212yes—
24C2×#366C2212yes—
25C2×#358C2212yes—
26C2×#358C2212yes—
27C2×C22⋊C16C2211yescharacteristic
28C2×C22⋊C16C2211yescharacteristic
29#1969C2212yes—
30#1942C2212yes—
31#1933—22no—
32#1892—21no—
33#1808C2211yescharacteristic
34#1380C2214yes—
35#1380C2214yes—
36#1374C2214yes—
37C22×Q16C2311yescharacteristic
38C23×C8D411yescharacteristic
39C23×C8C2311yescharacteristic
40C2×#125—28no—
41C2×#125—24no—
42C2×#125—44no—
43C2×#125—42no—
44C2×C22⋊Q8—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—41no—
48C22×C4⋊C4—41no—
49C22×C42D411yescharacteristic
50C22×C16C2312yes—
51C22×C16C2312yes—
52#499—216no—
53#499—28no—
54#499—28no—
55#467—22no—
56#467—22no—
57C2×#107—21no—
58C2×#107—21no—
59C2×#107—22no—
60C2×#107—22no—
61C2×#107D414yes—
62C2×#107D414yes—
63C2×#107C2312yes—
64C2×#107C2311yescharacteristic
65C2×#107C2312yes—
66C2×#107C2312yes—
67C2×#106—28no—
68C2×#106—24no—
69C2×#103—22no—
70C2×#103—21no—
71C2×C4×C8—22no—
72C2×C4×C8—22no—
73C2×C4×C8D412yes—
74C2×C4×C8C2312yes—
75#391—22no—
76#389—22no—
77#387—42no—
78#385—41no—
79#381—41no—
80#380—21no—
81#367—24no—
82#366—24no—
83#358—24no—
84#358—24no—
85C22⋊C16—22no—
86C22⋊C16—22no—
87#340—21no—
88#340—22no—
89C22×Q8—21no—
90C23×C4—41no—
91C23×C4—41no—
92C23×C4D811yescharacteristic
93C23×C4D811yescharacteristic
94C23×C4C2×D411yescharacteristic
95C2×Q16—21no—
96C2×Q16—21no—
97C2×Q16—24no—
98C22×C8—22no—
99C22×C8—24no—
100C22×C8—24no—
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D412yes—
103C22×C8C2×D412yes—
104C22×C8C2×D412yes—
105C22×C8C2411yescharacteristic
106#125—216no—
107#125—48no—
108#125—416no—
109C22⋊Q8—42no—
110C22⋊Q8—42no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—48no—
118C2×C4⋊C4—48no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4—44no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—42no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—41no—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42D812yes—
132C2×C42C2×D412yes—
133C2×C16—22no—
134C2×C16—22no—
135C2×C16—24no—
136C2×C16—24no—
137#107—24no—
138#107—24no—
139#107—22no—
140#107—22no—
141#107—21no—
142#107—21no—
143#107—24no—
144#107C2×D414yes—
145#107C2×D414yes—
146#106—216no—
147#103—42no—
148#103—44no—
149C4×C8—24no—
150C4×C8—22no—
151C4×C8—22no—
152#95—41no—
153#95—42no—
154#95—42no—
155#95—42no—
156C24C2×D811yescharacteristic
157C2×Q8—41no—
158C2×Q8—42no—
159C2×Q8—41no—
160C2×Q8—42no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—42no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—84no—
176C22×C4—82no—
177C22×C4—82no—
178C22×C4#12012yes—
179C22×C4C2×D812yes—
180C22×C4C2×D812yes—
181C22×C4C2×D811yescharacteristic
182C22×C4AGL(1,ℤ/8)12yes—
183C22×C4C22×D411yescharacteristic
184Q16—42no—
185Q16—42no—
186C2×C8—28no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—24no—
191C2×C8—24no—
192C2×C8#12012yes—
193C2×C8#13714yes—
194C2×C8C22×D411yescharacteristic
195C2×C8C22×D411yescharacteristic
196C2×C82−1+414yes—
197C4⋊C4—24no—
198C4⋊C4—24no—
199C4⋊C4—416no—
200C4⋊C4—416no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—84no—
210C4⋊C4—82no—
211C22⋊C4—82no—
212C22⋊C4—82no—
213C42—22no—
214C42—22no—
215C42—24no—
216C16—42no—
217C16—42no—
218C23—22no—
219C23—22no—
220C23#44712yes—
221C23#50812yes—
222C23#51012yes—
223C23C22×D811yescharacteristic
224Q8—81no—
225Q8—82no—
226Q8—81no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—88no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—84no—
245C2×C4—88no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4#44712yes—
254C2×C4C2×#12011yescharacteristic
255C2×C4#53612yes—
256C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
257C2×C4C2×#13712yes—
258C8—22no—
259C8—24no—
260C8#53612yes—
261C8#53612yes—
262C22—24no—
263C22—24no—
264C22—22no—
265C22—22no—
266C22#218914yes—
267C22C2×#44711yescharacteristic
268C22#322612yes—
269C22C2×#50811yescharacteristic
270C22C2×#51011yescharacteristic
271C22#341412yes—
272C4—22no—
273C4—42no—
274C4—42no—
275C4—84no—
276C4—84no—
277C4—162no—
278C4—161no—
279C4—161no—
280C4#261411yescharacteristic
281C4#300411yescharacteristic
282C4#323812yes—
283C2—24no—
284C2#2963811yescharacteristic
285C2C2×#218911yescharacteristic
286C2#3351911yescharacteristic
287C2#3379812yes—
288C2#3380212yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #13453C2semidirect
6. #13366C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4992
C2×#22362
C2×#22352
#331302
#134532
#133662
#128672
#128662
#128442