Finite Group Databasebeta

← #327987

Subgroups — #327987

Order 512.

899 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2#33148C211yesmaximal, characteristic
3#33148C211yesmaximal, characteristic
4#27972C211yesmaximal, characteristic
5#10582C214yesmaximal, complemented
6#10355C214yesmaximal, complemented
7#8724C211yesmaximal, characteristic
8#8706C211yesmaximal, characteristic
9#8706C211yesmaximal, characteristic
10C22×#125—21no—
11#2857C2211yescharacteristic
12#2552—22no—
13#2552—22no—
14#2552—22no—
15#2552C2214yes—
16C2×#398—22no—
17C2×#389—21no—
18C2×#389—21no—
19C2×#384—21no—
20C2×#384C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#2072—22no—
24#1987C2212yes—
25#1987C2212yes—
26#1956C2212yes—
27#1948C2212yes—
28#1948C2212yes—
29#1920—22no—
30#1920—22no—
31#1872—22no—
32#1853—21no—
33#1853—21no—
34#1834C2212yes—
35#1823—21no—
36#1823—21no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C4×C22⋊C8C2211yescharacteristic
40#1296C2214yes—
41#1279C2214yes—
42#1279C2214yes—
43C2×C4⋊Q8C2311yescharacteristic
44C2×#125—24no—
45C2×#125—24no—
46C2×#125—44no—
47C2×#125—44no—
48C2×C22⋊Q8—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D412yes—
59C22×C42—21no—
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62C2×#106—22no—
63C2×#106—22no—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×#103—24no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C4×C8C2312yes—
71#398—28no—
72#398—28no—
73#391—21no—
74#389—24no—
75#389—24no—
76#389—24no—
77#389—24no—
78#389—21no—
79#389—21no—
80#384—24no—
81#384—24no—
82#384—21no—
83#384—21no—
84#384D412yes—
85#384C2311yescharacteristic
86#384C2312yes—
87#380—21no—
88#380—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4D412yes—
94C4×C4⋊C4D412yes—
95C4×C4⋊C4C2312yes—
96C4×C4⋊C4C2312yes—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—21no—
100#340—24no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—22no—
107#336—21no—
108#336—21no—
109#336—21no—
110C22×Q8—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C4⋊Q8—24no—
123#125—48no—
124#125—416no—
125#125—48no—
126C22⋊Q8—44no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—48no—
151C2×C4⋊C4—48no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—84no—
157C2×C4⋊C4SD1614yes—
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4C2×D412yes—
160C2×C4⋊C4C2×D412yes—
161C2×C4⋊C4C2×D412yes—
162C2×C4⋊C4C2×D414yes—
163C2×C22⋊C4—41no—
164C2×C42—24no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—42no—
170C2×C42—42no—
171C2×C42—42no—
172C2×C42—42no—
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2411yescharacteristic
177#106—44no—
178#106—44no—
179C4⋊C8—24no—
180C4⋊C8—24no—
181#103—84no—
182C22⋊C8—42no—
183C22⋊C8—42no—
184C22⋊C8—42no—
185C4×C8—24no—
186#95—28no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—41no—
192#95—41no—
193C24#12011yescharacteristic
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—42no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4#12011yescharacteristic
227C22×C4#12011yescharacteristic
228C22×C4#12011yescharacteristic
229C22×C4C2×SD1612yes—
230C22×C4#13512yes—
231C22×C4AGL(1,ℤ/8)12yes—
232C22×C4#13712yes—
233C22×C4C22×D411yescharacteristic
234C22×C4C22×D411yescharacteristic
235C22×C4C22×D411yescharacteristic
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C4⋊C4—24no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—24no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—44no—
254C4⋊C4—42no—
255C4⋊C4—44no—
256C4⋊C4—416no—
257C4⋊C4—416no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C4⋊C4—88no—
271C4⋊C4C2×SD1614yes—
272C4⋊C4#13714yes—
273C22⋊C4—82no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—42no—
278C42—42no—
279C42—42no—
280C42—42no—
281C42—44no—
282C42—44no—
283C422−1+414yes—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23#45012yes—
289C23#45212yes—
290C23#45412yes—
291C23C2×#12011yescharacteristic
292Q8—84no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—21no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—44no—
309C2×C4—48no—
310C2×C4—48no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—84no—
325C2×C4—88no—
326C2×C4—84no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4#45012yes—
334C2×C4#45212yes—
335C2×C4#53612yes—
336C2×C4#53612yes—
337C2×C4#53612yes—
338C2×C4C22×SD1611yescharacteristic
339C2×C4C2×#13511yescharacteristic
340C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
341C2×C4C2×#13712yes—
342C2×C4C2×#13711yescharacteristic
343C2×C4#57512yes—
344C8—82no—
345C8—82no—
346C8—82no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22#163514yes—
354C22#261411yescharacteristic
355C22C2×#45011yescharacteristic
356C22C2×#45211yescharacteristic
357C22#300411yescharacteristic
358C22#300511yescharacteristic
359C22C2×#45411yescharacteristic
360C22#301911yescharacteristic
361C4—22no—
362C4—22no—
363C4—22no—
364C4—44no—
365C4—44no—
366C4—41no—
367C4—82no—
368C4—82no—
369C4—84no—
370C4—161no—
371C4—161no—
372C4#322912yes—
373C4#323212yes—
374C4#323612yes—
375C4#323812yes—
376C2—24no—
377C2C2×#163511yescharacteristic
378C2#1853511yescharacteristic
379C2#1875311yescharacteristic
380C2#1876711yescharacteristic
381C2#2964111yescharacteristic
382C2#2964311yescharacteristic
383C2#2965111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10582C2semidirect
6. #10355C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
#331482
#279722
#105822
#103552
#87242
#87062