Finite Group Databasebeta

← #406822

Subgroups — #406822

Order 512.

1368 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1854C211yesmaximal, characteristic
2C2×#1824C211yesmaximal, characteristic
3C2×#1809C211yesmaximal, characteristic
4C2×#1736C211yesmaximal, characteristic
5#13904C211yesmaximal, characteristic
6#13879C211yesmaximal, characteristic
7#11378C211yesmaximal, characteristic, complemented
8#11378C211yesmaximal, characteristic, complemented
9#11378C211yesmaximal, characteristic, complemented
10#11377C211yesmaximal, characteristic, complemented
11#11373C211yesmaximal, characteristic, complemented
12#11372C211yesmaximal, characteristic, complemented
13#11325C211yesmaximal, characteristic, complemented
14#11324C211yesmaximal, characteristic, complemented
15#10715C211yesmaximal, characteristic
16C23×M16C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#381—22no—
27C2×#380C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30#1854—24no—
31#1824—24no—
32#1809—24no—
33#1736—24no—
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#336C2211yescharacteristic
47C2×#336C2211yescharacteristic
48C2×#336C2211yescharacteristic
49C2×#336C2211yescharacteristic
50#1445—22no—
51#1439—22no—
52#1436—22no—
53C4⋊#95C2211yescharacteristic
54C4⋊#95C2211yescharacteristic
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2211yescharacteristic
59C24×C4—21no—
60C24×C4C2311yescharacteristic
61C22×M16—22no—
62C22×M16—22no—
63C22×M16—22no—
64C22×M16—22no—
65C22×M16—41no—
66C22×M16—41no—
67C22×M16D414yes—
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C2×#117—22no—
73C2×#117—22no—
74C2×#117—22no—
75C2×#117—22no—
76C2×#117—41no—
77C2×#117—41no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4Q814yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4D412yes—
98C22×C42—21no—
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101C2×#107—22no—
102C2×#107—22no—
103C2×#107—22no—
104C2×#107—41no—
105C2×#107—41no—
106C2×#107—41no—
107C2×#106—22no—
108C2×#106—22no—
109C2×#106—22no—
110C2×#106—22no—
111C2×#106—22no—
112C2×#106—41no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8D414yes—
116#384—22no—
117#384—41no—
118#382—22no—
119#382—41no—
120#381—48no—
121#380—22no—
122#380—22no—
123#380—22no—
124#380—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—21no—
135C2×#95—22no—
136C2×#95D412yes—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#340—22no—
146#340—22no—
147#340—22no—
148#340—22no—
149#336—22no—
150#336—22no—
151#336—41no—
152#336—41no—
153C25C2×D411yescharacteristic
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—24no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×Q812yes—
181C23×C4#4112yes—
182C23×C4#4112yes—
183C23×C4#4112yes—
184C23×C4C2411yescharacteristic
185C2×M16—28no—
186C2×M16—28no—
187C2×M16—44no—
188C2×M16—44no—
189C2×M16—44no—
190C2×M16—44no—
191C2×M16—44no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8C2×D412yes—
215C22×C8C2×D412yes—
216C22×C8C2×D412yes—
217C22×C8C2×D412yes—
218C22×C8#4112yes—
219C22×C8#4112yes—
220#117—28no—
221#117—44no—
222#117—44no—
223#117—44no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
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—21no—
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—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—82no—
278C2×C4⋊C4C2×Q812yes—
279C2×C4⋊C4C2×Q812yes—
280C2×C4⋊C4#4112yes—
281C2×C4⋊C4#4112yes—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—82no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
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—42no—
305C2×C42—42no—
306C2×C42—42no—
307#107—44no—
308#107—44no—
309#106—44no—
310#106—44no—
311C22⋊C8—28no—
312#95—24no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—42no—
318#95—42no—
319#95—44no—
320#95—44no—
321#95—82no—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24C22⋊Q812yes—
331C24#12312yes—
332C24#12712yes—
333C24C22×D411yescharacteristic
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
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—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
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—416no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—48no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—88no—
418C22×C4—84no—
419C22×C4—84no—
420C22×C4C22⋊Q814yes—
421C22×C4C22⋊Q814yes—
422C22×C4C22⋊Q812yes—
423C22×C4#12314yes—
424C22×C4#12314yes—
425C22×C4#12312yes—
426C22×C4#12712yes—
427C22×C4AGL(1,ℤ/8)14yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×Q811yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4H(5,𝔽2)11yescharacteristic
439C22×C4H(5,𝔽2)11yescharacteristic
440C22×C42−1+411yescharacteristic
441C22×C42−1+411yescharacteristic
442M16—416no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8#12314yes—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—48no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—88no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C42—48no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23#39914yes—
541C23C2×C22⋊Q811yescharacteristic
542C23C2×#12311yescharacteristic
543C23C2×#12711yescharacteristic
544C23#54611yescharacteristic
545C23#54611yescharacteristic
546C23#54711yescharacteristic
547C23#54711yescharacteristic
548C23C2×AGL(1,ℤ/8)12yes—
549C23#57512yes—
550C23#57712yes—
551C23#57812yes—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—816no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4#39914yes—
622C2×C4C2×C22⋊Q811yescharacteristic
623C2×C4C2×C22⋊Q811yescharacteristic
624C2×C4C2×#12311yescharacteristic
625C2×C4C2×#12311yescharacteristic
626C2×C4C2×#12311yescharacteristic
627C2×C4#52511yescharacteristic
628C2×C4#52911yescharacteristic
629C2×C4#52911yescharacteristic
630C2×C4#53311yescharacteristic
631C2×C4#53311yescharacteristic
632C2×C4#53411yescharacteristic
633C2×C4#53611yescharacteristic
634C2×C4#53911yescharacteristic
635C2×C4#53911yescharacteristic
636C2×C4#54411yescharacteristic
637C2×C4#54411yescharacteristic
638C2×C4#54611yescharacteristic
639C2×C4#54611yescharacteristic
640C2×C4#54711yescharacteristic
641C2×C4#54711yescharacteristic
642C2×C4#55111yescharacteristic
643C2×C4#55111yescharacteristic
644C2×C4#55111yescharacteristic
645C2×C4#55111yescharacteristic
646C2×C4#55211yescharacteristic
647C2×C4#55211yescharacteristic
648C2×C4#55211yescharacteristic
649C2×C4#55211yescharacteristic
650C8—24no—
651C8—42no—
652C8—42no—
653C8—42no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—42no—
658C22—42no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—42no—
664C22—42no—
665C22C2×#39911yescharacteristic
666C22#241911yescharacteristic
667C22#241911yescharacteristic
668C22#243811yescharacteristic
669C22#243811yescharacteristic
670C22#250111yescharacteristic
671C22#253911yescharacteristic
672C22#307712yes—
673C22#308212yes—
674C22#309412yes—
675C22#309712yes—
676C22#315111yescharacteristic
677C22#315311yescharacteristic
678C22#315411yescharacteristic
679C22#315611yescharacteristic
680C22#330612yes—
681C22#331512yes—
682C22#332712yes—
683C22#333212yes—
684C22#334312yes—
685C22#334812yes—
686C22#335112yes—
687C22#335312yes—
688C4—24no—
689C4—44no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—161no—
696C4—161no—
697C4—161no—
698C4—161no—
699C4#241911yescharacteristic
700C4#241911yescharacteristic
701C4#243811yescharacteristic
702C4#243811yescharacteristic
703C4#248311yescharacteristic
704C4#248411yescharacteristic
705C4#262711yescharacteristic
706C4#285511yescharacteristic
707C2—44no—
708C2#1493611yescharacteristic
709C2#2373111yescharacteristic
710C2#2373611yescharacteristic
711C2#2495412yes—
712C2#2496212yes—
713C2#2600412yes—
714C2#2601212yes—
715C2#2794411yescharacteristic
716C2#2794611yescharacteristic
717C2#2835611yescharacteristic
718C2#2836411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
7. #11378C2semidirect
8. #11378C2semidirect
9. #11378C2semidirect
10. #11377C2semidirect
11. #11373C2semidirect
12. #11372C2semidirect
13. #11325C2semidirect
14. #11324C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18542
C2×#18242
C2×#18092
C2×#17362
#139042
#138792
#113782
#113772
#113732
#113722
#113252
#113242
#107152