Finite Group Databasebeta

← #414875

Subgroups — #414875

Order 512.

1330 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#477C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic
3C2×#1939C211yesmaximal, characteristic
4C2×C4×#103C211yesmaximal, characteristic
5#14161C211yesmaximal, characteristic
6#13926C211yesmaximal, characteristic
7#11256C212yesmaximal
8#11232C212yesmaximal
9#11190C212yesmaximal
10#11077C212yesmaximal
11#10849C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#477—24no—
15C2×#477—24no—
16C2×#477—24no—
17C22×#107C2212yes—
18C22×C4⋊C8C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C2×#391C2212yes—
25C2×#384C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1990—24no—
29#1939—24no—
30C4×#103—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#336C2212yes—
38#1446—22no—
39#1441—22no—
40C4⋊#95—21no—
41C4⋊#95—21no—
42C4⋊#95—21no—
43C4⋊#95—21no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×Q8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8D414yes—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79#477—232no—
80C2×#107—24no—
81C2×#107—24no—
82C2×#107—24no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—24no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—42no—
88C2×C4⋊C8D414yes—
89C2×#103—24no—
90C2×#103—22no—
91C2×#103—22no—
92C2×#103—22no—
93C2×#103—24no—
94C2×#103—22no—
95C2×#103—24no—
96C2×#103—22no—
97C2×#103Q814yescomplemented
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8—22no—
101#391—24no—
102#391—42no—
103#384—22no—
104#384—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95D412yes—
119#340—24no—
120#340—24no—
121#340—22no—
122#340—22no—
123#340—24no—
124#340—42no—
125#336—24no—
126C22×Q8—24no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—81no—
141C22×Q8—81no—
142C22×Q8—81no—
143C22×Q8—81no—
144C22×Q8C2×D412yes—
145C22×Q8C2×Q812yes—
146C22×Q8#4112yes—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8C2×D414yes—
177C22×C8C2×Q812yes—
178C22×C8#4112yes—
179C4⋊Q8—28no—
180C4⋊Q8—28no—
181C4⋊Q8—44no—
182C4⋊Q8—44no—
183C4×Q8—28no—
184C4×Q8—28no—
185C4×Q8—44no—
186C4×Q8—44no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—81no—
229C2×C4⋊C4—81no—
230C2×C4⋊C4Q1614yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×Q812yes—
234C2×C4⋊C4#4112yes—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42#4112yes—
251C2×C42#4112yes—
252C2×C42#4112yes—
253#107—216no—
254C4⋊C8—28no—
255C4⋊C8—48no—
256#103—28no—
257#103—216no—
258#103—28no—
259C4×C8—44no—
260#95—24no—
261#95—44no—
262#95—44no—
263#95—42no—
264#95—82no—
265#95—82no—
266#95—82no—
267#95—82no—
268#95—82no—
269#95—82no—
270C24C22×D411yescharacteristic
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—48no—
299C2×Q8—84no—
300C2×Q8—84no—
301C2×Q8—84no—
302C2×Q8—84no—
303C2×Q8—84no—
304C2×Q8—84no—
305C2×Q8C22⋊Q814yes—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4#12112yes—
366C22×C4#12112yes—
367C22×C4#12112yes—
368C22×C4#12112yes—
369C22×C4#12112yes—
370C22×C4#12112yes—
371C22×C4C2×Q1614yes—
372C22×C4C2×Q1612yes—
373C22×C4#13512yes—
374C22×C4#13712yes—
375C22×C4#13712yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×Q811yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4H(5,𝔽2)11yescharacteristic
387C22×C42−1+411yescharacteristic
388C22×C42−1+411yescharacteristic
389C22×C42−1+411yescharacteristic
390C2×C8—24no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—42no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—42no—
405C2×C8—41no—
406C2×C8—42no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—42no—
411C4⋊C4—28no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—41no—
423C4⋊C4—42no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C42—42no—
438C42—42no—
439C42—42no—
440C42—42no—
441C42—48no—
442C42—84no—
443C42#12014yes—
444C42#12314yes—
445C42#12314yes—
446C42#12414yes—
447C23C2×#12111yescharacteristic
448C23C2×#12111yescharacteristic
449C23C2×#12111yescharacteristic
450C23#54611yescharacteristic
451C23#54711yescharacteristic
452C23#54811yescharacteristic
453C23D4×Q811yescharacteristic
454C23C22×Q1611yescharacteristic
455C23C2×#13511yescharacteristic
456C23C2×#13711yescharacteristic
457C23C2×#13711yescharacteristic
458C23#57611yescharacteristic
459C23#57611yescharacteristic
460C23#57711yescharacteristic
461C23#57711yescharacteristic
462Q8—24no—
463Q8—24no—
464Q8—42no—
465Q8—42no—
466Q8—42no—
467Q8—42no—
468Q8—42no—
469Q8—42no—
470Q8—816no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4#39414yes—
534C2×C4#39414yes—
535C2×C4#46214yes—
536C2×C4#46214yes—
537C2×C4#46414yes—
538C2×C4#46414yes—
539C2×C4#46714yes—
540C2×C4#47014yes—
541C2×C4C2×#12011yescharacteristic
542C2×C4C2×C22⋊Q811yescharacteristic
543C2×C4C2×#12311yescharacteristic
544C2×C4C2×#12311yescharacteristic
545C2×C4#52511yescharacteristic
546C2×C4C2×#12411yescharacteristic
547C2×C4#52911yescharacteristic
548C2×C4#52911yescharacteristic
549C2×C4#53311yescharacteristic
550C2×C4#53311yescharacteristic
551C2×C4#53611yescharacteristic
552C2×C4#53711yescharacteristic
553C2×C4#54111yescharacteristic
554C2×C4#54111yescharacteristic
555C2×C4#54211yescharacteristic
556C2×C4#54211yescharacteristic
557C2×C4#54611yescharacteristic
558C2×C4#54711yescharacteristic
559C2×C4#54811yescharacteristic
560C2×C4#54811yescharacteristic
561C2×C4#54811yescharacteristic
562C2×C4D4×Q811yescharacteristic
563C2×C4D4×Q811yescharacteristic
564C2×C4#55011yescharacteristic
565C2×C4#55211yescharacteristic
566C2×C4#55211yescharacteristic
567C2×C4#55411yescharacteristic
568C2×C4#55611yescharacteristic
569C8—44no—
570C8—42no—
571C8—42no—
572C22C2×#39411yescharacteristic
573C22#241511yescharacteristic
574C22#241711yescharacteristic
575C22#242611yescharacteristic
576C22#242711yescharacteristic
577C22#242911yescharacteristic
578C22#242911yescharacteristic
579C22C2×#46211yescharacteristic
580C22C2×#46211yescharacteristic
581C22C2×#46411yescharacteristic
582C22C2×#46411yescharacteristic
583C22#304111yescharacteristic
584C22#304211yescharacteristic
585C22#304511yescharacteristic
586C22#304511yescharacteristic
587C22C2×#46711yescharacteristic
588C22C2×#47011yescharacteristic
589C22#306211yescharacteristic
590C22#306211yescharacteristic
591C22#329911yescharacteristic
592C22#330511yescharacteristic
593C22#332911yescharacteristic
594C22#333311yescharacteristic
595C22#333811yescharacteristic
596C22#334011yescharacteristic
597C22#335611yescharacteristic
598C22#335711yescharacteristic
599C22#336111yescharacteristic
600C22#336311yescharacteristic
601C22#337211yescharacteristic
602C22#337511yescharacteristic
603C22#338111yescharacteristic
604C22Q8×Q1611yescharacteristic
605C22#338511yescharacteristic
606C22#338711yescharacteristic
607C4—24no—
608C4—44no—
609C4—44no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4—84no—
615C4—162no—
616C4#205618yes—
617C4#240711yescharacteristic
618C4#240911yescharacteristic
619C4#242111yescharacteristic
620C4#242311yescharacteristic
621C4#248911yescharacteristic
622C4#249211yescharacteristic
623C4#262511yescharacteristic
624C4#285511yescharacteristic
625C2#1517911yescharacteristic
626C2C2×#205611yescharacteristic
627C2#2354411yescharacteristic
628C2#2467911yescharacteristic
629C2#2468211yescharacteristic
630C2#2481411yescharacteristic
631C2#2481711yescharacteristic
632C2#2527711yescharacteristic
633C2#2527811yescharacteristic
634C2#2534011yescharacteristic
635C2#2534911yescharacteristic
636C2#2545411yescharacteristic
637C2#2545511yescharacteristic
638C2#2547111yescharacteristic
639C2#2548011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
97. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4772
C2×#19902
C2×#19392
C2×C4×#1032
#141612
#139262
#112562
#112322
#111902
#110772
#108492