Finite Group Databasebeta

← #136010

Subgroups — #136010

Order 512.

468 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 88 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
1#21939C211yesmaximal, characteristic
2#12928C212yesmaximal, complemented
3#11418C211yesmaximal, characteristic
4#11097C211yesmaximal, characteristic
5#7263C212yesmaximal, complemented
6C22×#108C2211yescharacteristic
7C2×#407—21no—
8C22×C22⋊C8C2211yescharacteristic
9#2280—21no—
10C2×#365—22no—
11#2112C2212yes—
12#1854—22no—
13#1854—22no—
14#1808—21no—
15#1807C2212yes—
16C2×#341—21no—
17C2×#341C2211yescharacteristic
18#1383—22no—
19#1314C414yescomplemented
20C24×C4D411yescharacteristic
21C22×M16—21no—
22C22×M16—21no—
23C23×C8D411yescharacteristic
24C23×C8C2311yescharacteristic
25C2×#117—21no—
26C2×#117—21no—
27C2×M32—22no—
28C2×#108—22no—
29C2×#108—42no—
30C2×#107—21no—
31C2×#107—21no—
32C2×#106D412yes—
33C2×#106D412yes—
34#407—42no—
35#407—42no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—42no—
39C2×C22⋊C8C2×C412yes—
40C2×C22⋊C8Q812yes—
41#381—41no—
42#380—22no—
43#380—22no—
44#379—41no—
45C4×C22⋊C4—41no—
46C4×C22⋊C4—41no—
47#365—44no—
48C22⋊C16—22no—
49C22⋊C16C2×C414yes—
50#341—22no—
51#341—42no—
52#340—22no—
53#340—22no—
54C25SD1611yescharacteristic
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—41no—
60C23×C4—41no—
61C23×C4—41no—
62C23×C4—41no—
63C23×C4C4⋊C412yes—
64C23×C4SD1611yescharacteristic
65C23×C4SD1612yes—
66C23×C4C2×D411yescharacteristic
67C2×M16—22no—
68C2×M16—22no—
69C2×M16—41no—
70C2×M16—41no—
71C2×M16—41no—
72C2×M16—41no—
73C2×M16—41no—
74C2×M16—41no—
75C2×M16—41no—
76C2×M16—41no—
77C2×M16—41no—
78C2×M16—41no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—41no—
82C22×C8—41no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8—41no—
87C22×C8C4⋊C412yes—
88C22×C8C22×C411yescharacteristic
89C22×C8C2×D412yes—
90C22×C8C2×Q811yescharacteristic
91C22×C8#4112yes—
92#117—81no—
93#117—81no—
94#117—81no—
95#117—81no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C42—41no—
103C2×C42—41no—
104M32—42no—
105M32—42no—
106C2×C16—24no—
107#108—24no—
108#108—42no—
109#107—81no—
110#107—81no—
111#106—42no—
112#106—42no—
113C22⋊C8—24no—
114C22⋊C8—24no—
115C22⋊C8—48no—
116#95—42no—
117#95—42no—
118C24—21no—
119C24—21no—
120C24—42no—
121C24—41no—
122C24—41no—
123C24—81no—
124C24#10612yes—
125C24C2×SD1611yescharacteristic
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—81no—
149C22×C4—81no—
150C22×C4—81no—
151C22×C4—81no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4#10612yes—
155C22×C4C2×C4⋊C411yescharacteristic
156C22×C4#12011yescharacteristic
157C22×C4#12311yescharacteristic
158C22×C4C2×SD1611yescharacteristic
159C22×C4AGL(1,ℤ/8)11yescharacteristic
160C22×C4#13711yescharacteristic
161M16—42no—
162M16—42no—
163M16—42no—
164M16—42no—
165M16—81no—
166M16—81no—
167M16—81no—
168M16—81no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—41no—
173C2×C8—41no—
174C2×C8—42no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—81no—
181C2×C8—81no—
182C2×C8—81no—
183C2×C8—81no—
184C2×C8—81no—
185C2×C8C4×D412yes—
186C2×C8C22⋊Q812yes—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C22⋊C4—84no—
192C22⋊C4—84no—
193C22⋊C4—82no—
194C22⋊C4—82no—
195C42—82no—
196C42—82no—
197C16—44no—
198C23—21no—
199C23—21no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—41no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23—81no—
213C23—81no—
214C23—84no—
215C23—84no—
216C23C2×#10611yescharacteristic
217C23#45011yescharacteristic
218C23#48311yescharacteristic
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—81no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4#38511yescharacteristic
247C2×C4#42811yescharacteristic
248C2×C4#44811yescharacteristic
249C2×C4#45311yescharacteristic
250C2×C4#45511yescharacteristic
251C2×C4B(4,𝔽2)11yescharacteristic
252C2×C4#45811yescharacteristic
253C2×C4#48111yescharacteristic
254C8—22no—
255C8—22no—
256C8—42no—
257C8—82no—
258C8—82no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—42no—
267C22—42no—
268C22—44no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—82no—
273C22—82no—
274C22—82no—
275C22—84no—
276C22#160111yescharacteristic
277C22#161311yescharacteristic
278C22#185311yescharacteristic
279C22#215512yes—
280C4—21no—
281C4—21no—
282C4—42no—
283C4—162no—
284C4—162no—
285C4#159611yescharacteristic
286C4#160811yescharacteristic
287C4#185611yescharacteristic
288C4#186711yescharacteristic
289C2—21no—
290C2—21no—
291C2—22no—
292C2—41no—
293C2—82no—
294C2#885411yescharacteristic
295C2#1293011yescharacteristic
296C2#1293211yescharacteristic
subgroupG/H
center, socleC22#1853
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC25SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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
2. #12928C2semidirect
5. #7263C2semidirect
19. #1314C4semidirect

Maximal subgroups

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

subgroupindex
#219392
#129282
#114182
#110972
#72632