Finite Group Databasebeta

← #238219

Subgroups — #238219

Order 512.

1258 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic, complemented
2#21803C211yesmaximal, characteristic
3#21803C211yesmaximal, characteristic
4C2×#1982C211yesmaximal, characteristic
5#10354C214yesmaximal, complemented
6#10348C214yesmaximal, complemented
7#8039C211yesmaximal, characteristic, complemented
8#7893C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C2×#544—21no—
11C22×#127C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102—21no—
16C22×#102—21no—
17#2857—22no—
18#2659—22no—
19#2552—21no—
20#2552C2212yescomplemented
21#2551—21no—
22#2551C2212yescomplemented
23#2550—22no—
24#2550—22no—
25#2316—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1982—22no—
30#1982—22no—
31#1982C2214yes—
32#1982C2214yes—
33#1957—21no—
34#1957—21no—
35#1957C2212yes—
36#1957C2212yes—
37#1786—22no—
38#1786—22no—
39#1296C2212yescomplemented
40#1296C2212yescomplemented
41#1279C2214yescomplemented
42#1279C2214yescomplemented
43C23×D4—21no—
44C23×D4—21no—
45C23×D4—21no—
46C23×C8—21no—
47#544—44no—
48#544—44no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×#127—21no—
53C2×#127—21no—
54C2×#127—21no—
55C2×#127—22no—
56C2×#127—22no—
57C2×#127—22no—
58C2×#127Q814yescomplemented
59C2×#127C2312yes—
60C2×#125—22no—
61C2×#125—22no—
62C2×C22⋊Q8—24no—
63C2×#117—21no—
64C2×#117—21no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69#431—22no—
70#431—22no—
71#431—22no—
72#431—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—44no—
86C2×#102—44no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8C2312yes—
91#398—24no—
92#398—24no—
93#395—24no—
94#395—24no—
95#391—24no—
96#389—22no—
97#389—22no—
98#389—24no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#384—22no—
104#384D411yescharacteristic
105#384D411yescharacteristic
106#384C2311yescharacteristic
107#384C2311yescharacteristic
108#382—21no—
109#382—21no—
110#380—24no—
111#380—21no—
112#380—21no—
113#380D412yes—
114#380D412yes—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4D412yes—
120C4×C4⋊C4D412yes—
121C4×C4⋊C4C2312yes—
122C4×C4⋊C4C2312yes—
123C2×#95—21no—
124C25—41no—
125C22×Q8—24no—
126C22×D4—24no—
127C22×D4—24no—
128C22×D4—24no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—41no—
133C22×D4—41no—
134C22×D4—41no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4—82no—
143C22×D4—82no—
144C22×D4—82no—
145C22×D4—82no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—42no—
155C22×C8—42no—
156C22×C8—42no—
157C4⋊Q8—44no—
158C4⋊Q8—48no—
159C4⋊Q8—44no—
160C4⋊Q8—84no—
161#127—24no—
162#127—24no—
163#127—24no—
164#127—22no—
165#127—22no—
166#127—22no—
167#127—42no—
168#127—42no—
169#127—42no—
170#127C2×Q814yes—
171#125—44no—
172#125—48no—
173#125—44no—
174C22⋊Q8—416no—
175#117—42no—
176#117—42no—
177#117—42no—
178#117—42no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4D814yes—
197C2×C4⋊C4SD1614yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—82no—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×Q812yes—
215C2×C42#4112yes—
216C2×C42#4112yes—
217C2×C42#4112yes—
218C2×C42C2411yescharacteristic
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—42no—
224C4⋊C8—42no—
225#102—48no—
226#102—48no—
227#102—84no—
228#102—84no—
229#102—84no—
230#102—84no—
231C22⋊C8—42no—
232C22⋊C8—42no—
233C4×C8—22no—
234C4×C8—22no—
235C4×C8—42no—
236#95—24no—
237#95—24no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244C24—42no—
245C24—44no—
246C24—82no—
247C24—82no—
248C24—82no—
249C24—82no—
250C24—82no—
251C24—82no—
252C24#12011yescharacteristic
253C2×Q8—48no—
254C2×Q8—84no—
255C2×Q8—84no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—44no—
273C2×D4—48no—
274C2×D4—48no—
275C2×D4—88no—
276C2×D4—82no—
277C2×D4—82no—
278C2×D4—82no—
279C2×D4—82no—
280C2×D4—88no—
281C2×D4—88no—
282C2×D4—88no—
283C2×D4—82no—
284C2×D4—82no—
285C2×D4—84no—
286C2×D4—84no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—81no—
300C22×C4—81no—
301C22×C4—81no—
302C22×C4—81no—
303C22×C4#12011yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4#12011yescharacteristic
306C22×C4C22⋊Q812yes—
307C22×C4C22⋊Q812yes—
308C22×C4C22⋊Q812yes—
309C22×C4#12412yes—
310C22×C4#12412yes—
311C22×C4#12412yes—
312C22×C4C2×D812yes—
313C22×C4C2×SD1612yes—
314C22×C4AGL(1,ℤ/8)12yes—
315C22×C4#13712yes—
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—84no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—48no—
338C4⋊C4—48no—
339C4⋊C4—48no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—81no—
349C4⋊C4—81no—
350C4⋊C4—84no—
351C4⋊C4—84no—
352C4⋊C4C2×D814yes—
353C4⋊C4C2×SD1614yes—
354C22⋊C4—44no—
355C22⋊C4—44no—
356C42—42no—
357C42—42no—
358C42—44no—
359C42—44no—
360C42—82no—
361C42C22⋊Q812yes—
362C42C22⋊Q812yes—
363C42C22⋊Q812yes—
364C42#12412yes—
365C42#12412yes—
366C42#12412yes—
367C42C22×Q811yescharacteristic
368C42C2×#4111yescharacteristic
369C42C2×#4111yescharacteristic
370C42C2×#4111yescharacteristic
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—44no—
376C23—88no—
377C23—88no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23—88no—
385C23—164no—
386C23—164no—
387C23—164no—
388C23—164no—
389C23#39312yes—
390C23#44712yes—
391C23#44812yes—
392C23C2×#12011yescharacteristic
393Q8—168no—
394D4—48no—
395D4—48no—
396D4—44no—
397D4—44no—
398D4—84no—
399D4—88no—
400D4—88no—
401D4—82no—
402D4—82no—
403D4—84no—
404D4—88no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—84no—
434C2×C4—84no—
435C2×C4#39312yes—
436C2×C4#39312yes—
437C2×C4#39312yes—
438C2×C4#44712yes—
439C2×C4#44812yes—
440C2×C4#47414yes—
441C2×C4#47614yes—
442C2×C4#48614yes—
443C2×C4#48914yes—
444C2×C4C2×C22⋊Q811yescharacteristic
445C2×C4C2×C22⋊Q811yescharacteristic
446C2×C4C2×C22⋊Q811yescharacteristic
447C2×C4C2×#12411yescharacteristic
448C2×C4C2×#12411yescharacteristic
449C2×C4C2×#12411yescharacteristic
450C2×C4C22×D811yescharacteristic
451C2×C4C22×SD1611yescharacteristic
452C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
453C2×C4C2×#13711yescharacteristic
454C2×C4#57612yes—
455C2×C4#57712yes—
456C8—82no—
457C8—82no—
458C8—82no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—84no—
466C22—84no—
467C22—84no—
468C22—1616no—
469C22—162no—
470C22—162no—
471C22—162no—
472C22—162no—
473C22#202314yes—
474C22C2×#39311yescharacteristic
475C22C2×#44711yescharacteristic
476C22C2×#44811yescharacteristic
477C22#300811yescharacteristic
478C22#301011yescharacteristic
479C22#302111yescharacteristic
480C22#302111yescharacteristic
481C4—22no—
482C4—21no—
483C4—21no—
484C4—22no—
485C4—44no—
486C4—44no—
487C4—82no—
488C4—82no—
489C4—82no—
490C4—82no—
491C4#202314yes—
492C4C2×#47411yescharacteristic
493C4C2×#47611yescharacteristic
494C4#308112yes—
495C4C2×#48611yescharacteristic
496C4C2×#48911yescharacteristic
497C4#313612yes—
498C2—24no—
499C2—164no—
500C2#1889211yescharacteristic
501C2#1889411yescharacteristic
502C2#1891211yescharacteristic
503C2#1891411yescharacteristic
504C2C2×#202311yescharacteristic
505C2#2345611yescharacteristic
506C2#2346311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
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. #35167C2semidirect
5. #10354C2semidirect
6. #10348C2semidirect
7. #8039C2semidirect
8. #7893C2semidirect
9. #7892C2semidirect
20. #2552C22semidirect
22. #2551C22semidirect
39. #1296C22semidirect
40. #1296C22semidirect
41. #1279C22semidirect
42. #1279C22semidirect
58. C2×#127Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
#218032
C2×#19822
#103542
#103482
#80392
#78932
#78922