Finite Group Databasebeta

← #328001

Subgroups — #328001

Order 512.

1093 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic
2#33146C211yesmaximal, characteristic
3#33108C211yesmaximal, characteristic
4#27960C211yesmaximal, characteristic
5#10575C214yesmaximal, complemented
6#10350C214yesmaximal, complemented
7#8723C211yesmaximal, characteristic
8#8706C211yesmaximal, characteristic
9#8706C211yesmaximal, characteristic
10C22×C4⋊Q8—21no—
11C22×C4⋊Q8—21no—
12C2×#483—21no—
13#2857C2211yescharacteristic
14#2551—22no—
15#2551—22no—
16#2551—22no—
17#2551C2214yes—
18C2×#395—22no—
19C2×#389—21no—
20C2×#389—21no—
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#2099—22no—
25#1990C2212yes—
26#1988C2212yes—
27#1981C2212yes—
28#1948C2212yes—
29#1947C2212yes—
30#1927—22no—
31#1918—22no—
32#1872—21no—
33#1852—21no—
34#1834C2212yes—
35#1823—22no—
36#1823—21no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C4×C22⋊C8C2211yescharacteristic
40#1296C2214yes—
41#1279C2214yes—
42#1279C2214yes—
43C23×Q8—22no—
44C2×C4⋊Q8—24no—
45C2×C4⋊Q8—24no—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—44no—
49C2×C4⋊Q8—44no—
50C2×C4⋊Q8—44no—
51C2×C4⋊Q8—44no—
52C2×C4⋊Q8C2311yescharacteristic
53C2×C22⋊Q8—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D412yes—
63C22×C42—21no—
64C22×C42—21no—
65C22×C42C2311yescharacteristic
66#483—44no—
67C2×#107—22no—
68C2×#106—22no—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2312yes—
71C2×#103—22no—
72C2×#103—22no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C4×C8C2312yes—
77#395—28no—
78#395—28no—
79#391—21no—
80#389—24no—
81#389—24no—
82#389—24no—
83#389—24no—
84#389—21no—
85#389—21no—
86#384—21no—
87#384—21no—
88#384D412yes—
89#384C2311yescharacteristic
90#384C2312yes—
91#380—21no—
92#380—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4D412yes—
98C4×C4⋊C4D412yes—
99C4×C4⋊C4C2312yes—
100C4×C4⋊C4C2312yes—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—21no—
104#340—24no—
105#340—24no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#336—21no—
111#336—21no—
112#336—21no—
113C22×Q8—28no—
114C22×Q8—21no—
115C22×Q8—48no—
116C22×Q8—48no—
117C22×Q8—84no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C4⋊Q8—24no—
130C4⋊Q8—48no—
131C4⋊Q8—48no—
132C4⋊Q8—416no—
133C4⋊Q8—416no—
134C4⋊Q8—48no—
135C4⋊Q8—48no—
136C22⋊Q8—42no—
137C22⋊Q8—42no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—48no—
159C2×C4⋊C4—48no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4D814yes—
167C2×C4⋊C4SD1614yes—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D412yes—
171C2×C4⋊C4C2×D412yes—
172C2×C4⋊C4C2×D414yes—
173C2×C22⋊C4—41no—
174C2×C42—24no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—42no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42—42no—
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2411yescharacteristic
187#107—44no—
188#106—44no—
189C4⋊C8—24no—
190C4⋊C8—24no—
191#103—44no—
192#103—44no—
193C22⋊C8—42no—
194C22⋊C8—42no—
195C22⋊C8—42no—
196C4×C8—24no—
197#95—28no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—41no—
203#95—41no—
204C24#12011yescharacteristic
205C2×Q8—416no—
206C2×Q8—416no—
207C2×Q8—42no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—42no—
211C2×Q8—88no—
212C2×Q8—832no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4#12011yescharacteristic
243C22×C4#12011yescharacteristic
244C22×C4#12011yescharacteristic
245C22×C4C2×D812yes—
246C22×C4C2×SD1612yes—
247C22×C4AGL(1,ℤ/8)12yes—
248C22×C4AGL(1,ℤ/8)12yes—
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C22×C4C22×D411yescharacteristic
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—44no—
270C4⋊C4—416no—
271C4⋊C4—416no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—42no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4C2×D814yes—
287C4⋊C4C2×SD1614yes—
288C4⋊C4#13714yes—
289C4⋊C4#13714yes—
290C22⋊C4—82no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—42no—
295C42—42no—
296C42—42no—
297C42—42no—
298C42—44no—
299C42—44no—
300C422−1+414yes—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23#44712yes—
306C23#45012yes—
307C23#45512yes—
308C23C2×#12011yescharacteristic
309Q8—832no—
310Q8—82no—
311Q8—82no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—21no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—44no—
330C2×C4—48no—
331C2×C4—48no—
332C2×C4—48no—
333C2×C4—48no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—88no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4#44712yes—
353C2×C4#45012yes—
354C2×C4#53612yes—
355C2×C4#53612yes—
356C2×C4#53612yes—
357C2×C4C22×D811yescharacteristic
358C2×C4C22×SD1611yescharacteristic
359C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
360C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
361C2×C4C2×#13712yes—
362C2×C4C2×#13712yes—
363C8—82no—
364C8—82no—
365C8—82no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22#163914yes—
373C22#261411yescharacteristic
374C22C2×#44711yescharacteristic
375C22C2×#45011yescharacteristic
376C22#300411yescharacteristic
377C22#300411yescharacteristic
378C22C2×#45511yescharacteristic
379C22#302011yescharacteristic
380C4—22no—
381C4—22no—
382C4—22no—
383C4—44no—
384C4—44no—
385C4—44no—
386C4—44no—
387C4—41no—
388C4—84no—
389C4—161no—
390C4—161no—
391C4#322612yes—
392C4#322912yes—
393C4#323812yes—
394C4#323812yes—
395C2—24no—
396C2C2×#163911yescharacteristic
397C2#1855111yescharacteristic
398C2#1877611yescharacteristic
399C2#1877711yescharacteristic
400C2#2963811yescharacteristic
401C2#2964111yescharacteristic
402C2#2965211yescharacteristic
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. #10575C2semidirect
6. #10350C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
#331462
#331082
#279602
#105752
#103502
#87232
#87062