Finite Group Databasebeta

← #414735

Subgroups — #414735

Order 512.

1828 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic
2C2×#2538C211yesmaximal, characteristic
3C2×#2584C211yesmaximal, characteristic, complemented
4C2×#1992C211yesmaximal, characteristic, complemented
5C2×#1985C211yesmaximal, characteristic, complemented
6C2×#1762C211yesmaximal, characteristic
7#11328C212yesmaximal, complemented
8#11327C212yesmaximal, complemented
9#11066C212yesmaximal
10#11066C212yesmaximal
11#10849C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×C22⋊Q8—21no—
14C22×C4×Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#478—24no—
17C2×#478—24no—
18C2×#478—24no—
19C22×#107C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×#102C2212yes—
25C22×#97C2211yescharacteristic
26#2584—24no—
27#2584—24no—
28#2538—24no—
29#2538—24no—
30C2×#398—21no—
31C2×#396—22no—
32C2×#395—21no—
33C2×#390C2212yes—
34C2×#389—22no—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#384C2211yescharacteristic
39C2×#380—21no—
40C2×C4×C4⋊C4C2211yescharacteristic
41#1992—24no—
42#1985—24no—
43#1762—24no—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#340C2212yes—
48C2×#340C2212yes—
49C2×#336C2212yes—
50C2×#336C2212yes—
51C23×Q8D411yescharacteristic
52C23×D4C2311yescharacteristic
53C24×C4—21no—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2312yes—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125—22no—
60C2×#125—22no—
61C2×#125—22no—
62C2×#125D414yes—
63C2×C22⋊Q8—28no—
64C2×C22⋊Q8—48no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4D414yes—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4D412yes—
85C22×C4⋊C4D411yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—22no—
93C22×C22⋊C4—22no—
94C22×C42—21no—
95C22×C42C2312yes—
96C22×C42C2311yescharacteristic
97#478—232no—
98C2×#107—22no—
99C2×#107—22no—
100C2×#107—22no—
101C2×#106—22no—
102C2×#106—22no—
103C2×#106—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8D414yes—
107C2×#102—22no—
108C2×#102—22no—
109C2×#102—22no—
110C2×#102—22no—
111C2×#102—22no—
112C2×#102—22no—
113C2×#102—24no—
114C2×#102—24no—
115C2×#102Q818yescomplemented
116C2×#97—22no—
117C2×#97—22no—
118C2×#97—22no—
119#398—28no—
120#396—216no—
121#395—28no—
122#390—24no—
123#390—24no—
124#389—216no—
125#389—24no—
126#389—24no—
127#389—24no—
128#389—24no—
129#389—24no—
130#389—24no—
131#384—22no—
132#384D414yes—
133#380—28no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4D414yes—
136C2×#95—21no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—22no—
140#340—24no—
141#340—24no—
142#340—24no—
143#340—24no—
144#340—24no—
145#336—24no—
146#336—24no—
147C25—21no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—24no—
152C22×Q8—42no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4C2×D412yes—
162C22×D4C2×Q814yes—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—24no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2411yescharacteristic
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8C2×D412yes—
193C22×C8C2×D412yes—
194C22×C8C2×Q814yes—
195#125—28no—
196#125—28no—
197#125—44no—
198#125—44no—
199C22⋊Q8—432no—
200C4×Q8—44no—
201C4×Q8—44no—
202C4×Q8—44no—
203C4×Q8—44no—
204C4×D4—28no—
205C4×D4—28no—
206C4×D4—44no—
207C4×D4—44no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—81no—
254C2×C4⋊C4—81no—
255C2×C4⋊C4—82no—
256C2×C4⋊C4—82no—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4C2×Q814yes—
263C2×C4⋊C4#4114yes—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—82no—
272C2×C22⋊C4—82no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—81no—
281C2×C42—81no—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42#4112yes—
288#107—28no—
289#106—28no—
290C4⋊C8—28no—
291#102—28no—
292#102—28no—
293#102—216no—
294#97—44no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—24no—
299C24—22no—
300C24—24no—
301C24—41no—
302C24—41no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24—41no—
307C24—41no—
308C24—41no—
309C24—41no—
310C24C22×D411yescharacteristic
311C2×Q8—28no—
312C2×Q8—48no—
313C2×Q8—48no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—24no—
319C2×D4—22no—
320C2×D4—24no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4C4⋊Q814yes—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—48no—
387C22×C4—88no—
388C22×C4—88no—
389C22×C4#12012yes—
390C22×C4#12012yes—
391C22×C4#12012yes—
392C22×C4#12012yes—
393C22×C4#12112yes—
394C22×C4#12112yes—
395C22×C4AGL(1,ℤ/8)14yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×Q812yes—
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4112yes—
404C22×C42−1+412yes—
405C22×C42−1+412yes—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—42no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—416no—
435C4⋊C4—48no—
436C4⋊C4—416no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—48no—
446C4⋊C4—44no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—84no—
450C4⋊C4—88no—
451C4⋊C4C22⋊Q818yescomplemented
452C22⋊C4—416no—
453C22⋊C4—48no—
454C22⋊C4—88no—
455C42—42no—
456C42—42no—
457C42—42no—
458C42—42no—
459C42—84no—
460C42#12014yes—
461C42#12114yes—
462C42#12114yes—
463C42#12714yes—
464C23—24no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—42no—
470C23—44no—
471C23—44no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—81no—
483C23—81no—
484C23—82no—
485C23—81no—
486C23—81no—
487C23C2×#12011yescharacteristic
488C23C2×#12011yescharacteristic
489C23C2×#12111yescharacteristic
490C23#54812yes—
491C23D4×Q812yes—
492C23C2×AGL(1,ℤ/8)12yes—
493C23#57512yes—
494C23#57611yescharacteristic
495C23#57711yescharacteristic
496C23#57711yescharacteristic
497C23#57811yescharacteristic
498Q8—416no—
499D4—24no—
500D4—24no—
501D4—42no—
502D4—44no—
503D4—42no—
504D4—44no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—816no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—82no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4#39214yes—
562C2×C4#39214yes—
563C2×C4C2×#12011yescharacteristic
564C2×C4C2×#12111yescharacteristic
565C2×C4C2×#12111yescharacteristic
566C2×C4C2×C22⋊Q812yes—
567C2×C4C2×#12711yescharacteristic
568C2×C4C2×C4⋊Q811yescharacteristic
569C2×C4#53211yescharacteristic
570C2×C4#53312yes—
571C2×C4#53611yescharacteristic
572C2×C4#53611yescharacteristic
573C2×C4#53612yes—
574C2×C4#54112yes—
575C2×C4#54712yes—
576C2×C4#54712yes—
577C2×C4#54812yes—
578C2×C4D4×Q812yes—
579C2×C4D4×Q812yes—
580C2×C4D4×Q812yes—
581C8—42no—
582C8—42no—
583C8—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—84no—
588C22—84no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—84no—
594C22—84no—
595C22C2×#39211yescharacteristic
596C22#240912yes—
597C22#242912yes—
598C22#243211yescharacteristic
599C22#243411yescharacteristic
600C22#300311yescharacteristic
601C22#300511yescharacteristic
602C22#300612yes—
603C22#301211yescharacteristic
604C22#301211yescharacteristic
605C22#301311yescharacteristic
606C22#301311yescharacteristic
607C22#305512yes—
608C22#306011yescharacteristic
609C22#306111yescharacteristic
610C22#336612yes—
611C22#336812yes—
612C22#337612yes—
613C22#337712yes—
614C22#338412yes—
615C22#338612yes—
616C22#338812yes—
617C22#338912yes—
618C4—24no—
619C4—24no—
620C4—48no—
621C4—44no—
622C4—84no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4—82no—
627C4#242912yes—
628C4#243211yescharacteristic
629C4#243411yescharacteristic
630C4#254212yes—
631C4#262111yescharacteristic
632C4#285711yescharacteristic
633C2—84no—
634C2#1517811yescharacteristic
635C2#2339011yescharacteristic
636C2#2339111yescharacteristic
637C2#2434612yes—
638C2#2434712yes—
639C2#2548212yes—
640C2#2548612yes—
641C2#2565511yescharacteristic
642C2#2565611yescharacteristic
643C2#2584411yescharacteristic
644C2#2584511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#2584C2semidirect
4. C2×#1992C2semidirect
5. C2×#1985C2semidirect
7. #11328C2semidirect
8. #11327C2semidirect
11. #10849C2semidirect
115. C2×#102Q8semidirect
451. C4⋊C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4782
C2×#25382
C2×#25842
C2×#19922
C2×#19852
C2×#17622
#113282
#113272
#110662
#108492