Finite Group Databasebeta

← #137224

Subgroups — #137224

Order 512.

566 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 110 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×#2075C211yesmaximal, characteristic
2#21940C211yesmaximal, characteristic
3#12866C212yesmaximal, complemented
4C2×#1314C211yesmaximal, characteristic
5#7258C212yesmaximal, complemented
6C22×#107C2211yescharacteristic
7C22×C22⋊C8C2211yescharacteristic
8C2×#380—21no—
9C2×#380—21no—
10C2×C22⋊C16C2212yes—
11#2075—22no—
12#2075C414yescomplemented
13#1852—24no—
14#1808—21no—
15#1808C2212yes—
16C2×#340—21no—
17C2×#340C2211yescharacteristic
18#1380—24no—
19#1314—22no—
20C24×C4D411yescharacteristic
21C23×C8D411yescharacteristic
22C23×C8C2311yescharacteristic
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4—22no—
26C22×C22⋊C4—41no—
27C22×C22⋊C4—41no—
28C22×C16—22no—
29C2×#107—22no—
30C2×#107—22no—
31C2×#107—21no—
32C2×#107—22no—
33C2×#107C2×C412yescomplemented
34C2×#107D414yes—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8C2×C412yes—
38C2×C22⋊C8D412yes—
39#380—22no—
40#380—24no—
41#380—22no—
42#380—41no—
43#380—41no—
44#380—42no—
45C2×#95—41no—
46C2×#95—41no—
47C22⋊C16—22no—
48C22⋊C16—22no—
49#340—22no—
50#340—24no—
51#340—22no—
52#340—21no—
53#340C2×C412yescomplemented
54C25D811yescharacteristic
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—42no—
60C23×C4—42no—
61C23×C4—42no—
62C23×C4—41no—
63C23×C4—41no—
64C23×C4C22⋊C412yes—
65C23×C4D811yescharacteristic
66C23×C4SD1612yes—
67C23×C4C2×D411yescharacteristic
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—42no—
71C22×C8—42no—
72C22×C8C22⋊C412yes—
73C22×C8C22×C411yescharacteristic
74C22×C8C2×D412yes—
75C22×C8C2×D411yescharacteristic
76C22×C8#4112yes—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—82no—
90C2×C4⋊C4—82no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—44no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—82no—
95C2×C22⋊C4—82no—
96C2×C22⋊C4—82no—
97C2×C22⋊C4—82no—
98C2×C16—44no—
99C2×C16—42no—
100#107—24no—
101#107—42no—
102#107—44no—
103#107—82no—
104C22⋊C8—24no—
105C22⋊C8—24no—
106#95—42no—
107#95—44no—
108#95—42no—
109C24—21no—
110C24—21no—
111C24—21no—
112C24—21no—
113C24—42no—
114C24—42no—
115C24#10212yes—
116C24D1612yes—
117C24SD3212yes—
118C24C2×D811yescharacteristic
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—44no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—44no—
136C22×C4—82no—
137C22×C4—81no—
138C22×C4—81no—
139C22×C4—82no—
140C22×C4—84no—
141C22×C4—82no—
142C22×C4—82no—
143C22×C4—84no—
144C22×C4—84no—
145C22×C4—81no—
146C22×C4—81no—
147C22×C4—81no—
148C22×C4—81no—
149C22×C4—81no—
150C22×C4—81no—
151C22×C4—81no—
152C22×C4—81no—
153C22×C4#10212yes—
154C22×C4C2×C22⋊C411yescharacteristic
155C22×C4#12011yescharacteristic
156C22×C4#12311yescharacteristic
157C22×C4C2×SD1611yescharacteristic
158C22×C4AGL(1,ℤ/8)11yescharacteristic
159C22×C4#13711yescharacteristic
160C2×C8—24no—
161C2×C8—22no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—81no—
168C2×C8C4×D412yes—
169C2×C8#12112yes—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—82no—
177C4⋊C4—84no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C22⋊C4—84no—
181C22⋊C4—88no—
182C22⋊C4—82no—
183C22⋊C4—82no—
184C16—82no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—42no—
194C23—42no—
195C23—44no—
196C23—44no—
197C23—42no—
198C23—42no—
199C23—81no—
200C23—81no—
201C23—82no—
202C23—82no—
203C23#35714yes—
204C23C2×#10211yescharacteristic
205C23#44711yescharacteristic
206C23#48011yescharacteristic
207C23C2×D1611yescharacteristic
208C23C2×SD3211yescharacteristic
209C23#50911yescharacteristic
210C23#51011yescharacteristic
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—42no—
220C2×C4—44no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—81no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—84no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—84no—
235C2×C4—84no—
236C2×C4—162no—
237C2×C4—162no—
238C2×C4—168no—
239C2×C4—161no—
240C2×C4—161no—
241C2×C4—161no—
242C2×C4—161no—
243C2×C4#38611yescharacteristic
244C2×C4#41711yescharacteristic
245C2×C4#44811yescharacteristic
246C2×C4#45512yes—
247C2×C4#45812yes—
248C2×C4#48111yescharacteristic
249C8—22no—
250C8—22no—
251C8—42no—
252C22—21no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—22no—
260C22—24no—
261C22—44no—
262C22—42no—
263C22—42no—
264C22—44no—
265C22—44no—
266C22—82no—
267C22—82no—
268C22—82no—
269C22#160712yes—
270C22#187411yescharacteristic
271C22C2×#35711yescharacteristic
272C22#213911yescharacteristic
273C22#218411yescharacteristic
274C22#218511yescharacteristic
275C22#219112yes—
276C22#223211yescharacteristic
277C22#223311yescharacteristic
278C4—21no—
279C4—21no—
280C4—42no—
281C4—162no—
282C4—162no—
283C4—164no—
284C4#160812yes—
285C4#188511yescharacteristic
286C4#189011yescharacteristic
287C2—21no—
288C2—21no—
289C2—21no—
290C2—21no—
291C2—44no—
292C2#902511yescharacteristic
293C2#1227812yes—
294C2#1227912yes—
295C2#1299111yescharacteristic
296C2#1300011yescharacteristic
subgroupG/H
center, socleC23#447
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC25D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #12866C2semidirect
5. #7258C2semidirect
12. #2075C4semidirect
33. C2×#107C2×C4semidirect
53. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20752
#219402
#128662
C2×#13142
#72582