Finite Group Databasebeta

← #374178

Subgroups — #374178

Order 512.

2098 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#480C211yesmaximal, characteristic
2C2×#2344C211yesmaximal, characteristic, complemented
3C2×#1893C211yesmaximal, characteristic
4C22×#340C211yesmaximal, characteristic, complemented
5#14137C211yesmaximal, characteristic
6#11219C212yesmaximal, complemented
7#11066C212yesmaximal, complemented
8#10931C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic, complemented
11#10714C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C22×#117—21no—
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#480—24no—
18C2×#480—24no—
19C2×#480C418yescomplemented
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26#2344—24no—
27#2344—24no—
28C2×#390C2212yes—
29C2×#389C2212yes—
30C2×#387—22no—
31C2×#386—21no—
32C2×#386—21no—
33C2×#381—21no—
34C2×#380C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37#1893—24no—
38C2×#340—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1439—22no—
47C4⋊#95C2212yes—
48C23×D4—21no—
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C23×C8—22no—
54C23×C8—24no—
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—42no—
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C23×C8C2312yes—
61C2×#121—22no—
62C2×#121—22no—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121—22no—
66C2×#121C2×C414yes—
67C2×#117—28no—
68C2×#117—44no—
69C2×#117—44no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—24no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—42no—
80C22×C4⋊C4C2×C414yes—
81C22×C4⋊C4D414yes—
82C22×C4⋊C4D414yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—22no—
92C22×C22⋊C4D411yescharacteristic
93C22×C22⋊C4D411yescharacteristic
94C22×C42—22no—
95C22×C42C2312yes—
96#480—232no—
97C2×#107—22no—
98C2×#107—22no—
99C2×#107—22no—
100C2×#107—22no—
101C2×#107C2×C414yescomplemented
102C2×#107C2×C414yescomplemented
103C2×#102—22no—
104C2×#102—22no—
105C2×#102—22no—
106C2×#102—22no—
107C2×#102—24no—
108C2×#102—24no—
109C2×#102—42no—
110C2×#102C2×C414yescomplemented
111C2×#102C2×C414yescomplemented
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8C2×C414yes—
115#390—24no—
116#390—42no—
117#389—24no—
118#389—24no—
119#387—216no—
120#386—44no—
121#386—44no—
122#381—28no—
123#380—22no—
124#380D414yes—
125C4×C4⋊C4—24no—
126C4×C4⋊C4—24no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C2×#95—22no—
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95D412yes—
135#340—24no—
136#340—24no—
137#340—24no—
138#340—24no—
139#340—42no—
140#336—24no—
141#336—42no—
142C25—21no—
143C25C2×D411yescharacteristic
144C22×D4—24no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—42no—
155C22×D4—42no—
156C22×D4—42no—
157C22×D4—42no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4C22×C412yes—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—21no—
173C23×C4—44no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—82no—
182C23×C4C22×C412yes—
183C23×C4C2×D412yes—
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2×D412yes—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D412yes—
189C23×C4#4112yes—
190C23×C4#4112yes—
191C23×C4C2411yescharacteristic
192C22×C8—28no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—48no—
208C22×C8—48no—
209C22×C8—44no—
210C22×C8—44no—
211C22×C8C22×C412yes—
212C22×C8C22×C412yes—
213C22×C8#4114yes—
214#121—28no—
215#121—28no—
216#121—44no—
217#121—44no—
218#117—432no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—48no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C22⋊C4116yescomplemented
255C2×C4⋊C4C22×C412yes—
256C2×C4⋊C4C22×C412yes—
257C2×C4⋊C4C22×C412yes—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4#4114yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—82no—
280C2×C22⋊C4—82no—
281C2×C22⋊C4D814yes—
282C2×C42—28no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—44no—
290C2×C42—44no—
291#107—28no—
292#107—28no—
293#102—28no—
294#102—28no—
295#102—48no—
296C22⋊C8—28no—
297#95—28no—
298#95—28no—
299#95—24no—
300#95—24no—
301#95—44no—
302#95—44no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—41no—
309C24—41no—
310C24—42no—
311C24—42no—
312C24—41no—
313C24—41no—
314C24—41no—
315C24—41no—
316C24—42no—
317C24—42no—
318C24C4×D412yes—
319C24#12012yes—
320C24#12112yes—
321C24C2×D812yes—
322C24#13512yes—
323C24AGL(1,ℤ/8)12yes—
324C24AGL(1,ℤ/8)12yes—
325C24C22×D411yescharacteristic
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—48no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—42no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—84no—
348C2×D4—84no—
349C2×D4—84no—
350C2×D4—84no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—216no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—88no—
424C22×C4—88no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4C2×C22⋊C414yes—
428C22×C4C4×D414yes—
429C22×C4C4×D412yes—
430C22×C4#12012yes—
431C22×C4#12114yes—
432C22×C4#12112yes—
433C22×C4#12314yes—
434C22×C4C2×D814yes—
435C22×C4C23×C411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4111yescharacteristic
443C22×C4H(5,𝔽2)12yes—
444C22×C42−1+412yes—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—416no—
453C2×C8—42no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—42no—
457C2×C8—41no—
458C2×C8—42no—
459C2×C8—41no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—216no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—48no—
477C4⋊C4—416no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4C2×C22⋊C414yes—
485C4⋊C4C2×C22⋊C414yes—
486C22⋊C4—28no—
487C22⋊C4—48no—
488C22⋊C4—44no—
489C22⋊C4—48no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C42—416no—
493C42—42no—
494C42—42no—
495C42—42no—
496C42—42no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—44no—
517C23—42no—
518C23—82no—
519C23—82no—
520C23—82no—
521C23—82no—
522C23—81no—
523C23—81no—
524C23—82no—
525C23—81no—
526C23—81no—
527C23—82no—
528C23—81no—
529C23—81no—
530C23—82no—
531C23—81no—
532C23—81no—
533C23—82no—
534C23—82no—
535C23—82no—
536C23#38614yes—
537C23C4×D814yes—
538C23#44214yes—
539C23#44714yes—
540C23#44914yes—
541C23#45914yes—
542C23#46314yes—
543C23C2×C4×D411yescharacteristic
544C23C2×#12011yescharacteristic
545C23C2×#12111yescharacteristic
546C23#54612yes—
547C23#54712yes—
548C23C22×D811yescharacteristic
549C23C2×#13511yescharacteristic
550C23C2×AGL(1,ℤ/8)11yescharacteristic
551C23C2×AGL(1,ℤ/8)11yescharacteristic
552C23#57712yes—
553C23#57812yes—
554D4—42no—
555D4—44no—
556D4—42no—
557D4—816no—
558D4—81no—
559D4—81no—
560D4—82no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—42no—
577C2×C4—432no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—816no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—82no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4#38614yes—
623C2×C4C22×C22⋊C411yescharacteristic
624C2×C4C2×C4×D411yescharacteristic
625C2×C4C4×#4112yes—
626C2×C4#52012yes—
627C2×C4C2×#12111yescharacteristic
628C2×C4C2×#12311yescharacteristic
629C2×C4#52511yescharacteristic
630C2×C4#52511yescharacteristic
631C2×C4#53212yes—
632C2×C4#53511yescharacteristic
633C2×C4#53611yescharacteristic
634C2×C4#53712yes—
635C2×C4#54612yes—
636C2×C4#54712yes—
637C2×C4#55112yes—
638C2×C4#55112yes—
639C2×C4#55212yes—
640C2×C4#55212yes—
641C8—42no—
642C8—44no—
643C8—42no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—28no—
648C22—28no—
649C22—24no—
650C22—24no—
651C22—28no—
652C22—24no—
653C22—28no—
654C22—82no—
655C22—84no—
656C22—84no—
657C22—82no—
658C22—82no—
659C22—161no—
660C22—161no—
661C22—161no—
662C22—161no—
663C22—162no—
664C22—162no—
665C22#187918yes—
666C22C2×#38611yescharacteristic
667C22#235912yes—
668C22#241111yescharacteristic
669C22#241311yescharacteristic
670C22#243012yes—
671C22C2×C4×D811yescharacteristic
672C22C2×#44211yescharacteristic
673C22#294712yes—
674C22C2×#44711yescharacteristic
675C22C2×#44911yescharacteristic
676C22#301312yes—
677C22C2×#45911yescharacteristic
678C22C2×#46311yescharacteristic
679C22#304311yescharacteristic
680C22#304411yescharacteristic
681C22#329412yes—
682C22#330012yes—
683C22#330812yes—
684C22#331012yes—
685C22#333412yes—
686C22#333712yes—
687C22#334312yes—
688C22#334512yes—
689C4—24no—
690C4—44no—
691C4—44no—
692C4—44no—
693C4—84no—
694C4—82no—
695C4—84no—
696C4—82no—
697C4#229711yescharacteristic
698C4#231111yescharacteristic
699C4#235912yes—
700C4#241111yescharacteristic
701C4#241311yescharacteristic
702C4#249812yes—
703C2—28no—
704C2—162no—
705C2#1413511yescharacteristic
706C2C2×#187911yescharacteristic
707C2#2093011yescharacteristic
708C2#2291612yes—
709C2#2293812yes—
710C2#2447611yescharacteristic
711C2#2448111yescharacteristic
712C2#2457611yescharacteristic
713C2#2458111yescharacteristic
714C2#2553312yes—
715C2#2553912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2344C2semidirect
4. C22×#340C2semidirect
6. #11219C2semidirect
7. #11066C2semidirect
8. #10931C2semidirect
9. #10907C2semidirect
10. #10730C2semidirect
11. #10714C2semidirect
19. C2×#480C4semidirect
101. C2×#107C2×C4semidirect
102. C2×#107C2×C4semidirect
110. C2×#102C2×C4semidirect
111. C2×#102C2×C4semidirect
254. C2×C4⋊C4C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4802
C2×#23442
C2×#18932
C22×#3402
#141372
#112192
#110662
#109312
#109072
#107302
#107142