Finite Group Databasebeta

← #238213

Subgroups — #238213

Order 512.

1114 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic, complemented
2#21884C211yesmaximal, characteristic
3#21859C211yesmaximal, characteristic
4#21257C211yesmaximal, characteristic
5#10356C212yesmaximal, complemented
6#10354C212yesmaximal, complemented
7#10350C212yesmaximal, complemented
8#10348C212yesmaximal, complemented
9C2×#1296C211yesmaximal, characteristic, complemented
10#7893C211yesmaximal, characteristic, complemented
11#7891C211yesmaximal, characteristic, complemented
12C2×#544—21no—
13C2×#532C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C2×#416—21no—
18C2×#416—21no—
19#2868—21no—
20#2859—21no—
21#2660—22no—
22#2552—21no—
23#2552C2212yes—
24#2551—21no—
25#2551C2212yes—
26#2440—22no—
27#2440—22no—
28#2317—21no—
29#2285C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C4×#117C2211yescharacteristic
32#1990C2212yes—
33#1988C2212yes—
34#1983—21no—
35#1983—21no—
36#1982C2211yescharacteristic
37#1982C2211yescharacteristic
38#1982C2212yes—
39#1981C2211yescharacteristic
40#1981C2211yescharacteristic
41#1957C2212yes—
42#1893—22no—
43#1874—22no—
44#1802—22no—
45#1802—21no—
46#1802—21no—
47#1296C2212yes—
48#1296C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53C22×#41—21no—
54C23×C8—21no—
55C23×C8—21no—
56#544—44no—
57#544—44no—
58#532—22no—
59#532—22no—
60#532—22no—
61#532Q814yescomplemented
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8C2311yescharacteristic
67C2×#127C2311yescharacteristic
68C2×#125—22no—
69C2×#125—22no—
70C2×#124—21no—
71C2×C22⋊Q8—24no—
72C2×#121—21no—
73C2×#121—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×#117—21no—
77C2×#117D411yescharacteristic
78C2×#117D411yescharacteristic
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4D412yes—
82C22×C42C2311yescharacteristic
83#431—22no—
84#431—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2312yes—
95#416—44no—
96#416—44no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—24no—
106C2×#102—22no—
107C2×#102—22no—
108C2×C22⋊C8—21no—
109C2×C4×C8C2312yes—
110#398—24no—
111#398—24no—
112#395—24no—
113#395—24no—
114#389—22no—
115#389—22no—
116#389—22no—
117#389—22no—
118#389—22no—
119#389—22no—
120#385—24no—
121#384—22no—
122#384—22no—
123#384—22no—
124#384—21no—
125#384—21no—
126#384D411yescharacteristic
127#384D411yescharacteristic
128#384D412yes—
129#384C2312yes—
130#382—22no—
131#382—21no—
132#380—21no—
133#380D412yes—
134C4×C4⋊C4—21no—
135C4×C4⋊C4D411yescharacteristic
136C4×C4⋊C4D411yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139C4×C4⋊C4C2312yes—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142#376—21no—
143C43—21no—
144#340—24no—
145#340—22no—
146#340—22no—
147C2×#41—24no—
148C2×#41—42no—
149C2×#41—42no—
150C2×#41—42no—
151C2×#41—42no—
152C2×#41—42no—
153C22×Q8—24no—
154C22×Q8—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—41no—
159C23×C4—22no—
160C23×C4—41no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—42no—
169C22×C8—42no—
170C22×C8—42no—
171C22×C8—42no—
172C22×C8—81no—
173C22×C8—81no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—48no—
178C4⋊Q8—44no—
179C4⋊Q8—44no—
180C4⋊Q8—82no—
181C4⋊Q8—82no—
182C4⋊Q8—84no—
183C4⋊Q8C2×Q812yes—
184#127—21no—
185#127—21no—
186#127—21no—
187#127C2×Q812yes—
188#125—48no—
189#125—44no—
190#125—44no—
191#124—24no—
192#124—42no—
193C22⋊Q8—416no—
194#121—24no—
195#121—24no—
196#121—42no—
197#121—42no—
198C4×D4—24no—
199C4×D4—24no—
200C4×D4—42no—
201C4×D4—42no—
202#117—22no—
203#117—22no—
204#117—42no—
205#117—42no—
206#117D814yes—
207#117SD1614yes—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4—81no—
228C2×C4⋊C4—81no—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D414yes—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×Q812yes—
252C2×C42#4112yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255C2×C42C2411yescharacteristic
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—22no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—22no—
262C4⋊C8—42no—
263C4⋊C8—42no—
264C4⋊C8—42no—
265#103—42no—
266#103—42no—
267#103—42no—
268#103—42no—
269#102—42no—
270#102—42no—
271#102—42no—
272#102—42no—
273#102—82no—
274#102—82no—
275#102—84no—
276C22⋊C8—42no—
277C4×C8—22no—
278C4×C8—22no—
279#95—24no—
280#95—24no—
281#95—21no—
282#95—21no—
283#95—22no—
284#95—21no—
285#95—21no—
286#95—22no—
287C24—41no—
288C24#12011yescharacteristic
289#41—48no—
290#41—48no—
291#41—84no—
292#41—84no—
293C2×Q8—22no—
294C2×Q8—48no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—84no—
301C2×Q8—84no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—44no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—82no—
320C2×D4—82no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4#12011yescharacteristic
342C22×C4#12011yescharacteristic
343C22×C4#12011yescharacteristic
344C22×C4C22⋊Q812yes—
345C22×C4C22⋊Q812yes—
346C22×C4C22⋊Q812yes—
347C22×C4#12412yes—
348C22×C4#12412yes—
349C22×C4#12412yes—
350C22×C4C2×D812yes—
351C22×C4C2×SD1612yes—
352C22×C4AGL(1,ℤ/8)12yes—
353C22×C4#13712yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—84no—
370C2×C8—84no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—44no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—48no—
385C4⋊C4—48no—
386C4⋊C4—48no—
387C4⋊C4—42no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—82no—
398C4⋊C4—84no—
399C4⋊C4—84no—
400C4⋊C4C2×D812yes—
401C4⋊C4C2×D812yes—
402C4⋊C4C2×SD1612yes—
403C4⋊C4C2×SD1612yes—
404C22⋊C4—24no—
405C22⋊C4—24no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C42—24no—
410C42—24no—
411C42—42no—
412C42—42no—
413C42—42no—
414C42—42no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42C22⋊Q812yes—
420C42C22⋊Q812yes—
421C42C22⋊Q812yes—
422C42#12412yes—
423C42#12412yes—
424C42#12412yes—
425C42C22×Q811yescharacteristic
426C42C2×#4111yescharacteristic
427C42C2×#4111yescharacteristic
428C42C2×#4111yescharacteristic
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—42no—
434C23—81no—
435C23—81no—
436C23—81no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23#39312yes—
441C23#44712yes—
442C23#44812yes—
443C23C2×#12011yescharacteristic
444Q8—42no—
445Q8—42no—
446Q8—81no—
447Q8—81no—
448Q8—168no—
449D4—44no—
450D4—42no—
451D4—42no—
452D4—42no—
453D4—42no—
454D4—82no—
455D4—88no—
456D4—81no—
457D4—81no—
458D4—81no—
459D4—81no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—88no—
497C2×C4—164no—
498C2×C4#39312yes—
499C2×C4#39312yes—
500C2×C4#39312yes—
501C2×C4#44712yes—
502C2×C4#44812yes—
503C2×C4#47414yes—
504C2×C4#47614yes—
505C2×C4#48614yes—
506C2×C4#48914yes—
507C2×C4C2×C22⋊Q811yescharacteristic
508C2×C4C2×C22⋊Q811yescharacteristic
509C2×C4C2×C22⋊Q811yescharacteristic
510C2×C4C2×#12411yescharacteristic
511C2×C4C2×#12411yescharacteristic
512C2×C4C2×#12411yescharacteristic
513C2×C4C22×D811yescharacteristic
514C2×C4C22×SD1611yescharacteristic
515C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
516C2×C4C2×#13711yescharacteristic
517C2×C4#57611yescharacteristic
518C2×C4#57711yescharacteristic
519C2×C4#57711yescharacteristic
520C2×C4#57811yescharacteristic
521C8—82no—
522C8—81no—
523C8—81no—
524C8—82no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—161no—
535C22—161no—
536C22—161no—
537C22—161no—
538C22#202314yes—
539C22C2×#39311yescharacteristic
540C22C2×#44711yescharacteristic
541C22C2×#44811yescharacteristic
542C22#301211yescharacteristic
543C22#301311yescharacteristic
544C22#301911yescharacteristic
545C22#302011yescharacteristic
546C4—22no—
547C4—21no—
548C4—21no—
549C4—22no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—82no—
555C4—84no—
556C4—82no—
557C4—82no—
558C4#202314yes—
559C4C2×#47411yescharacteristic
560C4C2×#47611yescharacteristic
561C4#308311yescharacteristic
562C4#308411yescharacteristic
563C4C2×#48611yescharacteristic
564C4C2×#48911yescharacteristic
565C4#313611yescharacteristic
566C4#313711yescharacteristic
567C2—24no—
568C2—162no—
569C2#1887011yescharacteristic
570C2#1887211yescharacteristic
571C2#1888211yescharacteristic
572C2#1888411yescharacteristic
573C2C2×#202311yescharacteristic
574C2#2345811yescharacteristic
575C2#2346211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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. #35275C2semidirect
5. #10356C2semidirect
6. #10354C2semidirect
7. #10350C2semidirect
8. #10348C2semidirect
9. C2×#1296C2semidirect
10. #7893C2semidirect
11. #7891C2semidirect
61. #532Q8semidirect

Maximal subgroups

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

subgroupindex
#352752
#218842
#218592
#212572
#103562
#103542
#103502
#103482
C2×#12962
#78932
#78912