Finite Group Databasebeta

← #371939

Subgroups — #371939

Order 512.

1348 conjugacy classes of subgroups.

The subgroup lattice has 765 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1968C211yesmaximal, characteristic
2C2×#1925C211yesmaximal, characteristic
3C2×#1919C211yesmaximal, characteristic, complemented
4C2×#1880C211yesmaximal, characteristic
5C2×#1863C211yesmaximal, characteristic, complemented
6#14260C211yesmaximal, characteristic
7#13786C211yesmaximal, characteristic, complemented
8#11378C211yesmaximal, characteristic, complemented
9#11124C211yesmaximal, characteristic, complemented
10#11123C211yesmaximal, characteristic
11#11023C211yesmaximal, characteristic
12#10928C211yesmaximal, characteristic, complemented
13#10927C211yesmaximal, characteristic
14#10862C211yesmaximal, characteristic, complemented
15#10694C211yesmaximal, characteristic, complemented
16C23×SD16C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#103C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#97C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×#376C2211yescharacteristic
37#1968C418yescomplemented
38#1925—24no—
39#1919—24no—
40#1880—24no—
41#1863—24no—
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#336C2211yescharacteristic
49C2×#336C2211yescharacteristic
50C2×#336C2211yescharacteristic
51C2×#336C2211yescharacteristic
52#1440—22no—
53#1439—22no—
54#1436—22no—
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2211yescharacteristic
59#1434—22no—
60C23×Q8C2311yescharacteristic
61C23×D4C2311yescharacteristic
62C22×SD16—22no—
63C22×SD16—22no—
64C22×SD16—22no—
65C22×SD16—22no—
66C22×SD16—41no—
67C22×SD16—41no—
68C22×SD16C2×C414yes—
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8—41no—
78C2×C4×Q8—41no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C22⋊C4—22no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—22no—
90C22×C22⋊C4—21no—
91C22×C42—21no—
92C22×C42—21no—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C2×#107—22no—
97C2×#107—22no—
98C2×#107—22no—
99C2×#107—22no—
100C2×#107—41no—
101C2×#107—41no—
102C2×#106—22no—
103C2×#106—22no—
104C2×#106C2×C414yes—
105C2×#103—22no—
106C2×#103—22no—
107C2×#103—22no—
108C2×#103—22no—
109C2×#103—22no—
110C2×#103—22no—
111C2×#103—41no—
112C2×#103—41no—
113C2×#103C2×C414yes—
114C2×#102—22no—
115C2×#102—22no—
116C2×#102—22no—
117C2×#102—22no—
118C2×#102—22no—
119C2×#102—22no—
120C2×#102—41no—
121C2×#102—41no—
122C2×#102C2×C414yes—
123C2×#97—22no—
124C2×#97—41no—
125C2×#97—41no—
126C2×C4×C8—41no—
127C2×C4×C8—41no—
128C2×C4×C8C2×C414yes—
129#391—22no—
130#391—22no—
131#391—22no—
132#391C2×C414yes—
133#390—22no—
134#390—22no—
135#390—22no—
136#390—41no—
137#390—41no—
138#390C2×C414yes—
139#389—22no—
140#389—22no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—22no—
143#376—41no—
144#376—41no—
145C2×#95—22no—
146C2×#95—22no—
147C2×#95—21no—
148C2×#95—21no—
149C2×#95—22no—
150C2×#95—22no—
151C2×#95—22no—
152C2×#95—22no—
153C2×#95D412yes—
154C2×#95D412yes—
155#340—22no—
156#340—22no—
157#340—22no—
158#340—22no—
159#340—22no—
160#340—22no—
161#336—22no—
162#336—22no—
163#336D414yes—
164#336D414yes—
165C25—21no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8C22×C412yes—
178C22×Q8#4112yes—
179C22×Q8#4112yes—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4—21no—
189C22×D4—21no—
190C22×D4—21no—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4C22×C412yes—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4C2×D412yes—
202C23×C4C2×D412yes—
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2411yescharacteristic
206C2×SD16—28no—
207C2×SD16—28no—
208C2×SD16—44no—
209C2×SD16—44no—
210C2×SD16—44no—
211C2×SD16—44no—
212C2×SD16—44no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8C22×C412yes—
234C22×C8C22×C412yes—
235C22×C8C2×D412yes—
236C22×C8C2×D412yes—
237C22×C8C2×D412yes—
238C22×C8C2×D412yes—
239C22×C8#4112yes—
240C22×C8#4112yes—
241C4×Q8—44no—
242C4×Q8—44no—
243C4×Q8—44no—
244C4×Q8—44no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—82no—
282C2×C4⋊C4C22×C412yes—
283C2×C4⋊C4C22×C412yes—
284C2×C4⋊C4#4112yes—
285C2×C4⋊C4#4112yes—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—82no—
291C2×C22⋊C4—82no—
292C2×C22⋊C4—82no—
293C2×C22⋊C4—82no—
294C2×C22⋊C4—82no—
295C2×C22⋊C4—82no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—24no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42—82no—
315C2×C42C22×C412yes—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42#4112yes—
319#107—44no—
320#107—44no—
321#106—28no—
322#103—28no—
323#103—44no—
324#103—44no—
325#102—28no—
326#102—44no—
327#102—44no—
328#97—44no—
329C4×C8—44no—
330#95—28no—
331#95—28no—
332#95—42no—
333#95—42no—
334#95—42no—
335#95—42no—
336#95—82no—
337#95—82no—
338#95—82no—
339#95—82no—
340C24—22no—
341C24—41no—
342C24—41no—
343C24—41no—
344C24—41no—
345C24—41no—
346C24—41no—
347C24—42no—
348C24—42no—
349C24—42no—
350C24—42no—
351C24C22×D411yescharacteristic
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—41no—
368C2×Q8—41no—
369C2×Q8—41no—
370C2×Q8—41no—
371C2×Q8—41no—
372C2×Q8—41no—
373C2×Q8—41no—
374C2×Q8—41no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—41no—
378C2×Q8—41no—
379C2×Q8#11714yes—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4C4×D414yes—
467C22×C4#12712yes—
468C22×C4#12712yes—
469C22×C4C23×C411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C2×#4111yescharacteristic
473C22×C4C2×#4111yescharacteristic
474C22×C4C2×#4111yescharacteristic
475C22×C4C2×#4111yescharacteristic
476C22×C4C2×#4111yescharacteristic
477C22×C4C2×#4111yescharacteristic
478C22×C4C2×#4111yescharacteristic
479C22×C4H(5,𝔽2)11yescharacteristic
480C22×C4H(5,𝔽2)11yescharacteristic
481C22×C4H(5,𝔽2)11yescharacteristic
482C22×C42−1+411yescharacteristic
483SD16—416no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—41no—
510C2×C8C4×D414yes—
511C2×C8C4×D414yes—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—48no—
521C4⋊C4—41no—
522C4⋊C4—41no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4—41no—
537C4⋊C4—84no—
538C22⋊C4—88no—
539C22⋊C4—84no—
540C22⋊C4—84no—
541C22⋊C4—88no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—48no—
551C42—41no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—84no—
556C23—42no—
557C23—42no—
558C23—44no—
559C23—44no—
560C23—42no—
561C23—81no—
562C23—81no—
563C23—82no—
564C23—82no—
565C23—82no—
566C23—82no—
567C23—81no—
568C23—81no—
569C23—81no—
570C23—81no—
571C23—82no—
572C23—82no—
573C23—82no—
574C23—82no—
575C23—82no—
576C23—82no—
577C23—81no—
578C23—81no—
579C23C2×C4×D411yescharacteristic
580C23#52511yescharacteristic
581C23#52511yescharacteristic
582C23C2×#12711yescharacteristic
583C23#53211yescharacteristic
584C23#53411yescharacteristic
585C23#53711yescharacteristic
586C23#57612yes—
587C23#57712yes—
588C23#57712yes—
589C23#57812yes—
590Q8—24no—
591Q8—24no—
592Q8—42no—
593Q8—42no—
594Q8—42no—
595Q8—42no—
596Q8—42no—
597Q8—42no—
598D4—42no—
599D4—42no—
600D4—42no—
601D4—42no—
602D4—81no—
603D4—81no—
604D4—81no—
605D4—81no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4—81no—
648C2×C4—81no—
649C2×C4—81no—
650C2×C4—81no—
651C2×C4—81no—
652C2×C4—81no—
653C2×C4—82no—
654C2×C4—82no—
655C2×C4—82no—
656C2×C4—82no—
657C2×C4—82no—
658C2×C4—82no—
659C2×C4—81no—
660C2×C4—81no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4—82no—
664C2×C4—82no—
665C2×C4—81no—
666C2×C4—81no—
667C2×C4C2×#11711yescharacteristic
668C2×C4C2×C4×D411yescharacteristic
669C2×C4C2×C4×D411yescharacteristic
670C2×C4C4×#4111yescharacteristic
671C2×C4#51811yescharacteristic
672C2×C4#52011yescharacteristic
673C2×C4#52911yescharacteristic
674C2×C4#52911yescharacteristic
675C2×C4#53211yescharacteristic
676C2×C4#53211yescharacteristic
677C2×C4#53811yescharacteristic
678C2×C4#53911yescharacteristic
679C2×C4#54611yescharacteristic
680C2×C4#54611yescharacteristic
681C2×C4#54611yescharacteristic
682C2×C4#54611yescharacteristic
683C2×C4#54611yescharacteristic
684C2×C4#54611yescharacteristic
685C2×C4#54811yescharacteristic
686C2×C4#54811yescharacteristic
687C2×C4#55111yescharacteristic
688C2×C4#55111yescharacteristic
689C2×C4#55211yescharacteristic
690C2×C4#55311yescharacteristic
691C2×C4#55311yescharacteristic
692C2×C4#55311yescharacteristic
693C2×C4#55611yescharacteristic
694C2×C4#55911yescharacteristic
695C8—24no—
696C8—24no—
697C8—42no—
698C8—42no—
699C22—82no—
700C22—84no—
701C22—84no—
702C22—82no—
703C22—82no—
704C22—162no—
705C22—162no—
706C22—161no—
707C22—161no—
708C22—161no—
709C22—161no—
710C22#229011yescharacteristic
711C22C4×#12711yescharacteristic
712C22#232511yescharacteristic
713C22#253911yescharacteristic
714C22#254011yescharacteristic
715C22#260111yescharacteristic
716C22#260211yescharacteristic
717C22#294912yes—
718C22#295112yes—
719C22#311912yes—
720C22#311912yes—
721C22#312112yes—
722C22#312112yes—
723C22#315411yescharacteristic
724C22#315411yescharacteristic
725C22#315511yescharacteristic
726C22#315611yescharacteristic
727C22#317412yes—
728C22#317512yes—
729C22#319812yes—
730C22#320112yes—
731C22#325712yes—
732C22#325812yes—
733C4—44no—
734C4—42no—
735C4—42no—
736C4—44no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4—161no—
743C4—161no—
744C4#234211yescharacteristic
745C4#235211yescharacteristic
746C4#236411yescharacteristic
747C4#236711yescharacteristic
748C4#255711yescharacteristic
749C4#255811yescharacteristic
750C4#255911yescharacteristic
751C4#256011yescharacteristic
752C2—162no—
753C2#1408411yescharacteristic
754C2#2113612yes—
755C2#2169511yescharacteristic
756C2#2169611yescharacteristic
757C2#2219412yes—
758C2#2832411yescharacteristic
759C2#2832511yescharacteristic
760C2#2838911yescharacteristic
761C2#2839611yescharacteristic
762C2#2944312yes—
763C2#2947512yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1919C2semidirect
5. C2×#1863C2semidirect
7. #13786C2semidirect
8. #11378C2semidirect
9. #11124C2semidirect
12. #10928C2semidirect
14. #10862C2semidirect
15. #10694C2semidirect
37. #1968C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19682
C2×#19252
C2×#19192
C2×#18802
C2×#18632
#142602
#137862
#113782
#111242
#111232
#110232
#109282
#109272
#108622
#106942