Finite Group Databasebeta

← #365270

Subgroups — #365270

Order 512.

1438 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1947C211yesmaximal, characteristic, complemented
2C2×#1925C212yesmaximal
3C2×#1802C211yesmaximal, characteristic
4C2×C4×#102C211yesmaximal, characteristic
5#13906C211yesmaximal, characteristic
6C4⋊C42C211yesmaximal, characteristic, complemented
7#11315C212yesmaximal, complemented
8#11189C212yesmaximal
9#10907C212yesmaximal, complemented
10#10841C211yesmaximal, characteristic, complemented
11#10834C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#107C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C2×#390C2212yes—
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#380—21no—
23C2×#380—21no—
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1947—24no—
29#1925—28no—
30#1802C418yescomplemented
31C4×#102—24no—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1439—22no—
41#1436—22no—
42C4⋊#95—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4C2×C414yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C42—21no—
71C22×C42—22no—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74C2×#107—24no—
75C2×#107—24no—
76C2×#107—24no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8C2×C414yes—
80C2×#102—24no—
81C2×#102—22no—
82C2×#102—24no—
83C2×#102—22no—
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—22no—
87C2×#102C2×C418yes—
88C2×#102D414yes—
89C2×C4×C8—22no—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92#390—24no—
93#390—42no—
94#384—22no—
95#384—22no—
96#384—22no—
97#384C2×C414yes—
98#380—44no—
99#380—44no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—24no—
104C4×C4⋊C4—42no—
105C4×C4⋊C4—44no—
106C4×C4⋊C4D414yes—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95D412yes—
114#340—24no—
115#340—24no—
116#340—22no—
117#340—22no—
118#340—42no—
119#340C2×C418yes—
120#336—24no—
121#336—24no—
122C25—21no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4C22×C412yes—
135C22×D4C2×D412yes—
136C22×D4#4112yes—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—24no—
142C23×C4—22no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8C22×C414yes—
173C22×C8C2×D412yes—
174C22×C8#4112yes—
175C4×D4—28no—
176C4×D4—28no—
177C4×D4—44no—
178C4×D4—44no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—81no—
217C2×C4⋊C4—81no—
218C2×C4⋊C4—81no—
219C2×C4⋊C4—81no—
220C2×C4⋊C4D814yes—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C414yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4#4112yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—82no—
235C2×C22⋊C4—82no—
236C2×C42—24no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42—82no—
248C2×C42C22×C412yes—
249C2×C42C2×D412yes—
250C2×C42#4112yes—
251C2×C42#4112yes—
252C2×C42#4112yes—
253C2×C42#4112yes—
254#107—48no—
255C4⋊C8—28no—
256#102—28no—
257#102—216no—
258#102—28no—
259C4×C8—44no—
260#95—24no—
261#95—24no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—82no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24C22×D411yescharacteristic
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4C4×D414yes—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—84no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4C4×D412yes—
371C22×C4C4×D412yes—
372C22×C4#12112yes—
373C22×C4#12112yes—
374C22×C4#12312yes—
375C22×C4#12312yes—
376C22×C4C2×D814yes—
377C22×C4C2×D812yes—
378C22×C4#13512yes—
379C22×C4AGL(1,ℤ/8)12yes—
380C22×C4#13712yes—
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4H(5,𝔽2)11yescharacteristic
392C22×C42−1+411yescharacteristic
393C22×C42−1+411yescharacteristic
394C22×C42−1+411yescharacteristic
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—28no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—42no—
429C4⋊C4—41no—
430C4⋊C4—42no—
431C4⋊C4—41no—
432C4⋊C4—48no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—84no—
439C4⋊C4—84no—
440C4⋊C4—84no—
441C4⋊C4#12114yes—
442C22⋊C4—48no—
443C22⋊C4—88no—
444C22⋊C4—88no—
445C42—42no—
446C42—42no—
447C42—42no—
448C42—42no—
449C42—48no—
450C42—88no—
451C42#11714yes—
452C42C4×D414yes—
453C42#12414yes—
454C42#12614yes—
455C23—24no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—81no—
475C23—81no—
476C23—81no—
477C23—81no—
478C23—81no—
479C23—81no—
480C23—81no—
481C23—81no—
482C23—81no—
483C23—81no—
484C23—81no—
485C23—81no—
486C23—81no—
487C23—81no—
488C23—81no—
489C23—81no—
490C23C2×C4×D411yescharacteristic
491C23C2×#12111yescharacteristic
492C23C2×#12311yescharacteristic
493C23D4211yescharacteristic
494C23#54711yescharacteristic
495C23#54811yescharacteristic
496C23#54811yescharacteristic
497C23C22×D811yescharacteristic
498C23C2×#13511yescharacteristic
499C23C2×AGL(1,ℤ/8)11yescharacteristic
500C23C2×#13711yescharacteristic
501C23#57611yescharacteristic
502C23#57711yescharacteristic
503C23#57711yescharacteristic
504C23#57811yescharacteristic
505D4—24no—
506D4—24no—
507D4—42no—
508D4—42no—
509D4—42no—
510D4—42no—
511D4—42no—
512D4—42no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—81no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—816no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4#38814yes—
570C2×C4#38814yes—
571C2×C4C4×D814yes—
572C2×C4#44014yes—
573C2×C4#45914yes—
574C2×C4#46414yes—
575C2×C4#48014yes—
576C2×C4#48214yes—
577C2×C4C2×#11711yescharacteristic
578C2×C4C2×C4×D411yescharacteristic
579C2×C4C2×C4×D411yescharacteristic
580C2×C4C4×#4111yescharacteristic
581C2×C4#51911yescharacteristic
582C2×C4#52011yescharacteristic
583C2×C4C2×#12111yescharacteristic
584C2×C4#52511yescharacteristic
585C2×C4C2×#12411yescharacteristic
586C2×C4C2×#12611yescharacteristic
587C2×C4#52911yescharacteristic
588C2×C4#53211yescharacteristic
589C2×C4#53611yescharacteristic
590C2×C4#53711yescharacteristic
591C2×C4#54111yescharacteristic
592C2×C4#54211yescharacteristic
593C2×C4#54711yescharacteristic
594C2×C4#54711yescharacteristic
595C2×C4#54711yescharacteristic
596C2×C4#55211yescharacteristic
597C2×C4#55211yescharacteristic
598C2×C4#55211yescharacteristic
599C2×C4#55311yescharacteristic
600C2×C4#55311yescharacteristic
601C2×C4#55311yescharacteristic
602C2×C4#55611yescharacteristic
603C2×C4#55611yescharacteristic
604C2×C4#55611yescharacteristic
605C8—42no—
606C8—42no—
607C8—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22C2×#38811yescharacteristic
624C22#236111yescharacteristic
625C22#236211yescharacteristic
626C22#241511yescharacteristic
627C22#241611yescharacteristic
628C22#242111yescharacteristic
629C22#242211yescharacteristic
630C22C2×C4×D811yescharacteristic
631C22C2×#44011yescharacteristic
632C22#294611yescharacteristic
633C22#294711yescharacteristic
634C22C2×#45911yescharacteristic
635C22C2×#46411yescharacteristic
636C22#304511yescharacteristic
637C22#304611yescharacteristic
638C22C2×#48011yescharacteristic
639C22C2×#48211yescharacteristic
640C22#309811yescharacteristic
641C22#309911yescharacteristic
642C22#327211yescharacteristic
643C22#327611yescharacteristic
644C22D4×D811yescharacteristic
645C22#328411yescharacteristic
646C22#333411yescharacteristic
647C22#334011yescharacteristic
648C22#335411yescharacteristic
649C22#335711yescharacteristic
650C22#335811yescharacteristic
651C22#335811yescharacteristic
652C22#336111yescharacteristic
653C22#336411yescharacteristic
654C22#336611yescharacteristic
655C22#336711yescharacteristic
656C22#336911yescharacteristic
657C22#337211yescharacteristic
658C4—24no—
659C4—44no—
660C4—44no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—84no—
665C4—162no—
666C4#192518yescomplemented
667C4C4×#12111yescharacteristic
668C4#232311yescharacteristic
669C4#234411yescharacteristic
670C4#234711yescharacteristic
671C4#249411yescharacteristic
672C4#249611yescharacteristic
673C4#254711yescharacteristic
674C4#256211yescharacteristic
675C2—84no—
676C2#1397411yescharacteristic
677C2C2×#192511yescharacteristic
678C2#2106211yescharacteristic
679C2#2299511yescharacteristic
680C2#2300211yescharacteristic
681C2#2306011yescharacteristic
682C2#2306711yescharacteristic
683C2#2467611yescharacteristic
684C2#2467911yescharacteristic
685C2#2472611yescharacteristic
686C2#2472911yescharacteristic
687C2#2499011yescharacteristic
688C2#2499311yescharacteristic
689C2#2501011yescharacteristic
690C2#2501311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×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
1. C2×#1947C2semidirect
6. C4⋊C42C2semidirect
7. #11315C2semidirect
9. #10907C2semidirect
10. #10841C2semidirect
11. #10834C2semidirect
30. #1802C4semidirect
666. C4#1925semidirect

Maximal subgroups

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

subgroupindex
C2×#19472
C2×#19252
C2×#18022
C2×C4×#1022
#139062
C4⋊C422
#113152
#111892
#109072
#108412
#108342