Finite Group Databasebeta

← #242864

Subgroups — #242864

Order 512.

824 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21984C211yesmaximal, characteristic, complemented
2#21786C211yesmaximal, characteristic
3#20738C211yesmaximal, characteristic
4#20732C211yesmaximal, characteristic
5#10625C212yesmaximal, complemented
6#10621C212yesmaximal, complemented
7#10619C212yesmaximal, complemented
8#10616C212yesmaximal, complemented
9#8046C211yesmaximal, characteristic, complemented
10#8038C211yesmaximal, characteristic, complemented
11#7878C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17#2285C2211yescharacteristic
18#2285C2211yescharacteristic
19#1954—22no—
20#1954C2212yes—
21#1954C2212yes—
22#1949—22no—
23#1854—22no—
24#1845C2211yescharacteristic, complemented
25#1844C2211yescharacteristic, complemented
26#1834C2211yescharacteristic, complemented
27#1834C2211yescharacteristic, complemented
28#1824—21no—
29#1823—21no—
30#1802C2212yes—
31#1802C2212yes—
32#1802C2212yes—
33#1802C2212yes—
34#1786—21no—
35#1786—21no—
36#1779—22no—
37#1762—22no—
38C4×#102—22no—
39#1296C2212yescomplemented
40#1296C2212yescomplemented
41#1296C2212yescomplemented
42#1296C2212yescomplemented
43#1277C2212yescomplemented
44#1277C2212yescomplemented
45C23×D4D411yescharacteristic
46C24×C4—21no—
47C2×C4×D4—21no—
48C2×C4×D4—21no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4D412yes—
54C2×C4×D4C2312yes—
55C2×#117—21no—
56C2×#117—21no—
57C22×C4⋊C4D411yescharacteristic
58C22×C22⋊C4—21no—
59C22×C42C2311yescharacteristic
60#432—22no—
61#432—22no—
62#431—22no—
63#431—22no—
64#431—22no—
65#431—22no—
66#431—22no—
67#431—22no—
68C2×#107—22no—
69C2×#106—22no—
70C2×C4⋊C8D412yes—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×#102—22no—
75C2×#102—22no—
76C2×#102—22no—
77C2×#102—22no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8D412yes—
86C2×#97—21no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#384C2311yescharacteristic
92#384C2311yescharacteristic
93#384C2311yescharacteristic
94#384C2311yescharacteristic
95#382—21no—
96#382—21no—
97#380—21no—
98#380—21no—
99#380—21no—
100#380—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105#340—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111C25—21no—
112C22×D4—24no—
113C22×D4—21no—
114C22×D4—22no—
115C22×D4—22no—
116C22×D4—42no—
117C22×D4C2×D412yes—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—82no—
124C23×C4—82no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4#4111yescharacteristic
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8C2×D412yes—
132C22×C8C2×D412yes—
133C4×D4—24no—
134C4×D4—24no—
135C4×D4—22no—
136C4×D4—22no—
137C4×D4—22no—
138C4×D4—22no—
139C4×D4—42no—
140C4×D4—42no—
141C4×D4—42no—
142C4×D4—42no—
143#117—82no—
144#117—82no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4C2×D412yes—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—42no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C42—41no—
168C2×C42—41no—
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42#4112yes—
172C2×C42#4112yes—
173C2×C42#4112yes—
174C2×C42#4112yes—
175C2×C42#4111yescharacteristic
176C2×C42C2411yescharacteristic
177#107—82no—
178#106—82no—
179C4⋊C8—24no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186#102—44no—
187#102—44no—
188#102—44no—
189#102—44no—
190C22⋊C8—24no—
191C22⋊C8—42no—
192C22⋊C8—42no—
193C22⋊C8—42no—
194C22⋊C8—82no—
195C22⋊C8—82no—
196C22⋊C8—82no—
197C22⋊C8—82no—
198#97—42no—
199C4×C8—21no—
200C4×C8—21no—
201C4×C8—21no—
202C4×C8—21no—
203C4×C8—42no—
204C4×C8—42no—
205#95—41no—
206#95—41no—
207#95—41no—
208#95—41no—
209C24—24no—
210C24—22no—
211C24—42no—
212C24—42no—
213C24—81no—
214C24—81no—
215C24—81no—
216C24—81no—
217C24#12111yescharacteristic
218C2×D4—24no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—48no—
225C2×D4—48no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4#13514yes—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—88no—
251C22×C4—88no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12312yes—
258C22×C4#12312yes—
259C22×C4#12312yes—
260C22×C4#12412yes—
261C22×C4#12612yes—
262C22×C4#12612yes—
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C2×#4111yescharacteristic
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—81no—
293C4⋊C4—81no—
294C4⋊C4—81no—
295C4⋊C4—81no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—44no—
299C22⋊C4—82no—
300C22⋊C4—82no—
301C42—82no—
302C42—82no—
303C42#12312yes—
304C42#12312yes—
305C42#12312yes—
306C42#12412yes—
307C42#12612yes—
308C42#12612yes—
309C42C2×#4111yescharacteristic
310C42C2×#4111yescharacteristic
311C42C2×#4111yescharacteristic
312C42C2×#4111yescharacteristic
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—24no—
320C23—48no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—84no—
328C23—84no—
329C23—84no—
330C23—84no—
331C23—84no—
332C23—84no—
333C23#39712yes—
334C23C2×#12111yescharacteristic
335D4—24no—
336D4—24no—
337D4—48no—
338D4—42no—
339D4—42no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—88no—
358C2×C4—84no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4#39712yes—
376C2×C4#39712yes—
377C2×C4#39712yes—
378C2×C4C2×#12311yescharacteristic
379C2×C4C2×#12311yescharacteristic
380C2×C4C2×#12311yescharacteristic
381C2×C4C2×#12411yescharacteristic
382C2×C4C2×#12611yescharacteristic
383C2×C4C2×#12611yescharacteristic
384C2×C4C2×#13512yes—
385C2×C4#57512yes—
386C2×C4#57612yes—
387C2×C4#57712yes—
388C8—42no—
389C8—42no—
390C8—82no—
391C8—82no—
392C8—82no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—44no—
401C22—44no—
402C22—44no—
403C22—816no—
404C22—82no—
405C22—82no—
406C22—82no—
407C22—82no—
408C22C2×#39711yescharacteristic
409C22#304111yescharacteristic
410C22#304211yescharacteristic
411C22#305011yescharacteristic
412C22#305411yescharacteristic
413C22#306811yescharacteristic
414C22#306911yescharacteristic
415C4—22no—
416C4—22no—
417C4—21no—
418C4—21no—
419C4—22no—
420C4—82no—
421C4—161no—
422C4—161no—
423C4—161no—
424C4—161no—
425C4#309112yes—
426C4#309112yes—
427C4#309612yes—
428C4#313612yes—
429C2—42no—
430C2—84no—
431C2#1938811yescharacteristic
432C2#1939411yescharacteristic
433C2#1942511yescharacteristic
434C2#1943111yescharacteristic
435C2#2363211yescharacteristic
436C2#2363811yescharacteristic
437C2#2364711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
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. #21984C2semidirect
5. #10625C2semidirect
6. #10621C2semidirect
7. #10619C2semidirect
8. #10616C2semidirect
9. #8046C2semidirect
10. #8038C2semidirect
11. #7878C2semidirect
24. #1845C22semidirect
25. #1844C22semidirect
26. #1834C22semidirect
27. #1834C22semidirect
39. #1296C22semidirect
40. #1296C22semidirect
41. #1296C22semidirect
42. #1296C22semidirect
43. #1277C22semidirect
44. #1277C22semidirect

Maximal subgroups

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

subgroupindex
#219842
#217862
#207382
#207322
#106252
#106212
#106192
#106162
#80462
#80382
#78782