Finite Group Databasebeta

← #323853

Subgroups — #323853

Order 512.

756 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic, complemented
2#24751C212yesmaximal
3C8×#121C211yesmaximal, characteristic
4#10651C211yesmaximal, characteristic, complemented
5#10615C211yesmaximal, characteristic, complemented
6#10545C211yesmaximal, characteristic, complemented
7#10264C211yesmaximal, characteristic, complemented
8#9565C212yesmaximal, complemented
9#9217C212yesmaximal, complemented
10#9131C212yesmaximal, complemented
11C2×#1277C211yesmaximal, characteristic, complemented
12C2×#500C2211yescharacteristic
13C2×#465—22no—
14C2×#461C2212yes—
15C2×C8×D4—21no—
16C2×C8×D4C2211yescharacteristic
17C22×C4⋊C8C2212yes—
18C22×C4×C8C2211yescharacteristic
19#2857C2212yes—
20C4×#121C2211yescharacteristic
21#2066—22no—
22#2055—22no—
23#1990C2212yes—
24#1981C2212yes—
25#1954C2211yescharacteristic
26#1948C2212yes—
27#1942—21no—
28#1942—21no—
29#1939C2211yescharacteristic
30#1923—21no—
31#1892—22no—
32#1869—21no—
33#1845C2211yescharacteristic
34#1808—21no—
35#1808—21no—
36#1802C2212yes—
37#1797C2211yescharacteristic
38C8×C4⋊C4C2211yescharacteristic
39C8×C22⋊C4—21no—
40C2×#332C2212yes—
41C2×#332C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2212yes—
45#1277C2211yescharacteristic
46#1277C2211yescharacteristic
47C22×SD16—22no—
48C23×C8—22no—
49C23×C8—21no—
50C23×C8D411yescharacteristic
51C23×C8D411yescharacteristic
52C2×C4⋊Q8C2312yes—
53C2×C22⋊Q8—22no—
54C2×#121D411yescharacteristic
55C2×C4×D4—21no—
56C2×C4×D4C2311yescharacteristic
57C22×C42C2311yescharacteristic
58#500—42no—
59#465—48no—
60#461—24no—
61#461—42no—
62C8×D4—42no—
63C8×D4—44no—
64C2×#107—21no—
65C2×#107—21no—
66C2×#107—21no—
67C2×#107—21no—
68C2×#106—22no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—42no—
73C2×C4⋊C8C2311yescharacteristic
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2312yes—
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—22no—
80C2×#102—22no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8—42no—
87C2×C4×C8D411yescharacteristic
88C2×C4×C8D411yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#391—22no—
92#390D411yescharacteristic
93#389—22no—
94#389—22no—
95#389D411yescharacteristic
96#388—21no—
97#386—21no—
98#384C2312yes—
99#380—22no—
100#380—22no—
101#380D411yescharacteristic
102C4×C4⋊C4C2311yescharacteristic
103C4×C22⋊C4—21no—
104#340—22no—
105#340—22no—
106#340—22no—
107#340—22no—
108#336—21no—
109#336—21no—
110#336—21no—
111#332—24no—
112#332—42no—
113C22×Q8—22no—
114C22×D4—21no—
115C22×D4C2×D411yescharacteristic
116C23×C4—21no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4#4111yescharacteristic
120C2×SD16—44no—
121C2×SD16—44no—
122C2×SD16—82no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—21no—
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—41no—
130C22×C8—41no—
131C22×C8—41no—
132C22×C8—44no—
133C22×C8—42no—
134C22×C8—42no—
135C22×C8—44no—
136C22×C8—42no—
137C22×C8—82no—
138C22×C8SD1614yescomplemented
139C22×C8C2×D411yescharacteristic
140C22×C8C2×D411yescharacteristic
141C22×C8C2×D411yescharacteristic
142C22×C8C2×D411yescharacteristic
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C22⋊Q8—44no—
148C22⋊Q8—44no—
149#121—22no—
150#121D814yes—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—42no—
154C4×D4—42no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4C2×D411yescharacteristic
163C2×C4⋊C4C2×D411yescharacteristic
164C2×C4⋊C4C2×D411yescharacteristic
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—42no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42C2411yescharacteristic
178#107—42no—
179#107—42no—
180#107—42no—
181#107—42no—
182#106—44no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—44no—
189C4⋊C8—42no—
190C4⋊C8#4114yes—
191#103—44no—
192#103—44no—
193#103—82no—
194#102—44no—
195C22⋊C8—42no—
196C22⋊C8—42no—
197C22⋊C8—42no—
198C22⋊C8—42no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—41no—
202C4×C8—41no—
203C4×C8—42no—
204#95—21no—
205#95—21no—
206#95—21no—
207#95—42no—
208#95—42no—
209C24—21no—
210C24#12111yescharacteristic
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×D4—24no—
218C2×D4—22no—
219C2×D4—21no—
220C2×D4—22no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4C2×D812yes—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4C2×D812yes—
248C22×C4C2×SD1612yes—
249C22×C4#13512yes—
250C22×C4#13712yes—
251C22×C4C22×D411yescharacteristic
252C22×C4C22×D411yescharacteristic
253C22×C4C2×#4111yescharacteristic
254SD16—88no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—24no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—44no—
266C2×C8—44no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8—84no—
279C2×C8—88no—
280C2×C8C2×SD1612yes—
281C2×C8C2×SD1612yes—
282C2×C8#13512yes—
283C4⋊C4—21no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—82no—
296C4⋊C4—84no—
297C4⋊C4C2×D812yes—
298C22⋊C4—24no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—84no—
304C42—21no—
305C42—21no—
306C42—22no—
307C42—22no—
308C42—41no—
309C42—41no—
310C42C2×#4112yes—
311C42H(5,𝔽2)12yes—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—22no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23#46112yes—
324C23#46612yes—
325C23#47312yes—
326C23C2×#12111yescharacteristic
327Q8—82no—
328Q8—82no—
329Q8—82no—
330Q8—82no—
331D4—24no—
332D4—48no—
333D4—42no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—44no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4#46112yes—
370C2×C4#46612yes—
371C2×C4#53211yescharacteristic
372C2×C4#53411yescharacteristic
373C2×C4#54612yes—
374C2×C4#55312yes—
375C2×C4C22×D811yescharacteristic
376C2×C4C22×SD1611yescharacteristic
377C2×C4C2×#13511yescharacteristic
378C2×C4C2×#13511yescharacteristic
379C2×C4C2×#13711yescharacteristic
380C2×C4#57511yescharacteristic
381C2×C4#57712yes—
382C8—22no—
383C8—22no—
384C8—22no—
385C8—41no—
386C8—44no—
387C8—81no—
388C8—81no—
389C8—82no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22#167314yes—
404C22#254311yescharacteristic
405C22C2×#46111yescharacteristic
406C22#303511yescharacteristic
407C22C2×#46611yescharacteristic
408C22#306011yescharacteristic
409C22C2×#47311yescharacteristic
410C22#306711yescharacteristic
411C4—22no—
412C4—22no—
413C4—22no—
414C4—42no—
415C4—41no—
416C4—81no—
417C4—81no—
418C4—162no—
419C4—162no—
420C4#316711yescharacteristic
421C4#318111yescharacteristic
422C4#318711yescharacteristic
423C4#319011yescharacteristic
424C4#329412yes—
425C4#330112yes—
426C2—24no—
427C2—82no—
428C2C2×#167311yescharacteristic
429C2#1912511yescharacteristic
430C2#1919411yescharacteristic
431C2#1922811yescharacteristic
432C2#2855111yescharacteristic
433C2#2857111yescharacteristic
434C2#2859111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
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
1. #33209C2semidirect
4. #10651C2semidirect
5. #10615C2semidirect
6. #10545C2semidirect
7. #10264C2semidirect
8. #9565C2semidirect
9. #9217C2semidirect
10. #9131C2semidirect
11. C2×#1277C2semidirect
138. C22×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#332092
#247512
C8×#1212
#106512
#106152
#105452
#102642
#95652
#92172
#91312
C2×#12772