Finite Group Databasebeta

← #217060

Subgroups — #217060

Order 512.

811 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35319C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic
3#20815C211yesmaximal, characteristic
4#9907C212yesmaximal, complemented
5#9703C212yesmaximal, complemented
6#9140C212yesmaximal
7#9132C212yesmaximal
8#8589C212yesmaximal, complemented
9#8586C212yesmaximal, complemented
10C2×#541—21no—
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2868C2211yescharacteristic
14#2859C2211yescharacteristic
15#2857C2212yes—
16#2642—21no—
17#2621—21no—
18#2506—22no—
19#2506C414yes—
20#2496—22no—
21#2496C414yes—
22#2327—21no—
23C4×C4⋊Q8C2212yes—
24C4×#117C2211yescharacteristic
25#1924—22no—
26#1917—22no—
27#1842C2212yes—
28#1842C2211yescharacteristic
29#1831C2212yes—
30#1831C2211yescharacteristic
31#1798—21no—
32#1798—21no—
33#1797C2212yes—
34#1768—21no—
35#1768—21no—
36#1767C2212yes—
37C2×#332C2212yes—
38C2×#332C2212yes—
39C2×#332C2212yes—
40C2×#332C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45#541—48no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8C2312yes—
49C2×#126—24no—
50C2×#125—24no—
51C2×#125—21no—
52C2×C22⋊Q8—22no—
53C2×C4×Q8—22no—
54C2×#117D411yescharacteristic
55C2×#117D411yescharacteristic
56C22×C42C2311yescharacteristic
57#432—41no—
58#432—41no—
59#431—41no—
60#431—41no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—22no—
71C2×#97—21no—
72C2×#97—22no—
73C2×C4×C8—21no—
74#400—24no—
75#400—24no—
76#398—24no—
77#397—24no—
78#391—22no—
79#391—22no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389C2×C414yes—
84#389C2×C414yes—
85#389C2×C414yes—
86#387—24no—
87#387—24no—
88#384—22no—
89#384C2×C414yes—
90#384C2312yes—
91#382—21no—
92#380—22no—
93#380C2×C414yes—
94C4×C4⋊C4—22no—
95C4×C4⋊C4C2×C414yes—
96C4×C4⋊C4C2×C412yes—
97C4×C4⋊C4D412yes—
98C4×C22⋊C4C2×C412yes—
99C4×C22⋊C4D412yes—
100#376D411yescharacteristic
101#376D411yescharacteristic
102C43C2311yescharacteristic
103C43C2311yescharacteristic
104#336—22no—
105#336—22no—
106#336—22no—
107#336—22no—
108#336—22no—
109#332—42no—
110#332—42no—
111#332—42no—
112#332—42no—
113C22×Q8—22no—
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—44no—
126C4⋊Q8—44no—
127#126—416no—
128#125—44no—
129#125—416no—
130C22⋊Q8—48no—
131C4×Q8—48no—
132#117—24no—
133#117—42no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22×C414yes—
145C2×C4⋊C4C22×C414yes—
146C2×C22⋊C4—24no—
147C2×C22⋊C4C22⋊C412yes—
148C2×C22⋊C4C22⋊C412yes—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—24no—
154C2×C42C22×C411yescharacteristic
155C2×C42C22×C412yes—
156C2×C42C22×C412yes—
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D412yes—
162C2×C42C2×D412yes—
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2411yescharacteristic
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—44no—
175C4⋊C8—44no—
176C4⋊C8—82no—
177C22⋊C8—42no—
178C22⋊C8—42no—
179C22⋊C8—82no—
180#97—42no—
181#97—82no—
182C4×C8—42no—
183#95—24no—
184#95—24no—
185#95—24no—
186#95—24no—
187#95—24no—
188#95C22⋊C412yes—
189#95C22⋊C412yes—
190#95C22⋊C412yes—
191#95C22⋊C412yes—
192C24C2×C22⋊C411yescharacteristic
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—44no—
196C2×Q8—44no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—42no—
210C22×C4C2×C22⋊C411yescharacteristic
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4C2×C22⋊C412yes—
213C22×C4C2×C22⋊C412yes—
214C22×C4C2×C22⋊C412yes—
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C2×C22⋊C412yes—
217C22×C4C23×C411yescharacteristic
218C22×C4C22×D411yescharacteristic
219C22×C4C22×D411yescharacteristic
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—42no—
236C4⋊C4—44no—
237C4⋊C4—48no—
238C4⋊C4—48no—
239C4⋊C4—48no—
240C4⋊C4—48no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—44no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—48no—
247C22⋊C4—24no—
248C22⋊C4—42no—
249C22⋊C4—48no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—22no—
263C42—42no—
264C42—42no—
265C42—48no—
266C42AGL(1,ℤ/8)14yes—
267C42H(5,𝔽2)11yescharacteristic
268C42H(5,𝔽2)11yescharacteristic
269C422−1+411yescharacteristic
270C422−1+411yescharacteristic
271C23—21no—
272C23—22no—
273C23—21no—
274C23—21no—
275C23—22no—
276C23C22×C22⋊C411yescharacteristic
277Q8—84no—
278Q8—84no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4#41814yes—
312C2×C4#52011yescharacteristic
313C2×C4#52011yescharacteristic
314C2×C4#53511yescharacteristic
315C2×C4#53611yescharacteristic
316C2×C4#53711yescharacteristic
317C2×C4#53711yescharacteristic
318C2×C4C2×AGL(1,ℤ/8)12yes—
319C2×C4#57512yes—
320C2×C4#57712yes—
321C2×C4#57812yes—
322C8—82no—
323C8—82no—
324C8—82no—
325C8—82no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22#231111yescharacteristic
332C22#288912yes—
333C22C2×#41811yescharacteristic
334C22#289611yescharacteristic
335C22#289812yes—
336C4—24no—
337C4—21no—
338C4—21no—
339C4—42no—
340C4—42no—
341C4—42no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—84no—
346C4—84no—
347C4#321412yes—
348C4#323512yes—
349C4#325812yes—
350C4#326011yescharacteristic
351C4#326111yescharacteristic
352C2—81no—
353C2#1775912yes—
354C2#1776712yes—
355C2#2186911yescharacteristic
356C2#2188311yescharacteristic
357C2#2189811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9907C2semidirect
5. #9703C2semidirect
8. #8589C2semidirect
9. #8586C2semidirect

Maximal subgroups

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

subgroupindex
#353192
#219262
#208152
#99072
#97032
#91402
#91322
#85892
#85862