Finite Group Databasebeta

← #367454

Subgroups — #367454

Order 512.

1220 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1945C211yesmaximal, characteristic, complemented
2C2×#1855C211yesmaximal, characteristic
3#20843C211yesmaximal, characteristic
4#20836C211yesmaximal, characteristic
5#20628C211yesmaximal, characteristic
6C2×#1750C211yesmaximal, characteristic
7C2×#1733C211yesmaximal, characteristic
8#11329C211yesmaximal, characteristic, complemented
9#11258C211yesmaximal, characteristic, complemented
10#11221C211yesmaximal, characteristic, complemented
11#11221C211yesmaximal, characteristic, complemented
12#10883C211yesmaximal, characteristic, complemented
13#10882C211yesmaximal, characteristic, complemented
14#10860C211yesmaximal, characteristic, complemented
15#10720C211yesmaximal, characteristic
16C23×M16C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#108C2211yescharacteristic
20C22×#108C2211yescharacteristic
21C2×#427C2211yescharacteristic
22C2×#427C2211yescharacteristic
23C2×#424C2211yescharacteristic
24C2×#424C2211yescharacteristic
25C2×#424C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#407C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C22×#97C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32#2280C2211yescharacteristic
33#1945—24no—
34#1855—24no—
35#1854C412yescomplemented
36#1854C412yescomplemented
37#1853C412yescomplemented
38#1852C412yescomplemented
39#1851—22no—
40#1850—22no—
41#1798—22no—
42#1797—22no—
43#1750—24no—
44C2×#341C2211yescharacteristic
45C2×#341C2211yescharacteristic
46#1735C2211yescharacteristic
47#1735C2211yescharacteristic
48#1733—24no—
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59#1729C2211yescharacteristic
60#1729C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C24×C4C2311yescharacteristic
64C22×M16—21no—
65C22×M16—22no—
66C22×M16—22no—
67C22×M16—22no—
68C22×M16—22no—
69C22×M16—41no—
70C22×M16—41no—
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C23×C8—21no—
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C23×C8C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C2×#117—21no—
81C2×#117—22no—
82C2×#117—21no—
83C2×#117—22no—
84C2×#117—22no—
85C2×#117—22no—
86C2×#117—22no—
87C2×#117—41no—
88C2×#117—41no—
89C2×#117—41no—
90C2×#117—41no—
91C2×#117C2×C412yes—
92C2×#117Q814yes—
93C2×#117C2311yescharacteristic
94C2×#117C2311yescharacteristic
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97C22×C4⋊C4—21no—
98C22×C4⋊C4C2×C412yes—
99C22×C22⋊C4—21no—
100C22×C22⋊C4—21no—
101C22×C42—21no—
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104C2×#108—22no—
105C2×#108—22no—
106C2×#108—22no—
107C2×#108—22no—
108C2×#108—41no—
109C2×#108—41no—
110#427—22no—
111#427—22no—
112#427—41no—
113#427—41no—
114C2×#107C2×C412yes—
115C2×#107C2×C412yes—
116C2×#106C2×C412yes—
117C2×#106C2×C412yes—
118#424—22no—
119#424—22no—
120#424—22no—
121#424—22no—
122#424—41no—
123#424—41no—
124#424—41no—
125#424—41no—
126C2×C4⋊C8—22no—
127C2×C4⋊C8—22no—
128C2×C4⋊C8—22no—
129C2×C4⋊C8—22no—
130#407—22no—
131#407—22no—
132C2×C22⋊C8—22no—
133C2×C22⋊C8—22no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8C2×C411yescharacteristic
137C2×C22⋊C8C2×C411yescharacteristic
138C2×C22⋊C8C2×C411yescharacteristic
139C2×C22⋊C8C2×C411yescharacteristic
140C2×C22⋊C8D412yes—
141C2×C22⋊C8D412yes—
142C2×C22⋊C8D414yes—
143C2×C22⋊C8Q814yes—
144C2×#97—22no—
145C2×#97—22no—
146C2×#97—41no—
147C2×C4×C8—22no—
148C2×C4×C8—22no—
149C2×C4×C8—41no—
150#381—21no—
151#380C2×C412yes—
152#380C2×C412yes—
153#379—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156#341—22no—
157#341—22no—
158#340C2×C412yes—
159#340C2×C412yes—
160#340C2×C412yes—
161#340C2×C412yes—
162#339—22no—
163#339—22no—
164#339—22no—
165#339—22no—
166#339—22no—
167#339—41no—
168#339—41no—
169#339D414yes—
170#336—22no—
171#336—22no—
172#336—22no—
173#336—22no—
174#336—22no—
175#336—22no—
176C25#4111yescharacteristic
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—22no—
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×Q812yes—
194C23×C4C2×Q812yes—
195C23×C4#4111yescharacteristic
196C23×C4#4112yes—
197C23×C4C2411yescharacteristic
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—28no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—42no—
222C2×M16—42no—
223C2×M16—44no—
224C2×M16—44no—
225C2×M16—44no—
226C2×M16—42no—
227C2×M16C2×D412yes—
228C2×M16#4112yes—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—42no—
254C22×C8—42no—
255C22×C8—42no—
256C22×C8C22×C411yescharacteristic
257C22×C8C22×C411yescharacteristic
258C22×C8C22×C411yescharacteristic
259C22×C8C22×C411yescharacteristic
260C22×C8C2×D412yes—
261C22×C8C2×D412yes—
262C22×C8C2×D412yes—
263C22×C8C2×D412yes—
264C22×C8C2×Q812yes—
265C22×C8C2×Q812yes—
266#117—21no—
267#117—21no—
268#117—28no—
269#117—21no—
270#117—21no—
271#117—21no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117—21no—
278#117—21no—
279#117—21no—
280#117—21no—
281#117—24no—
282#117—44no—
283#117—44no—
284#117—42no—
285#117—44no—
286#117—42no—
287#117—42no—
288#117—42no—
289#117—42no—
290#117C2×D412yes—
291#117#4112yes—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4C22×C412yes—
299C2×C4⋊C4C22×C412yes—
300C2×C4⋊C4C22×C412yes—
301C2×C4⋊C4C22×C412yes—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—42no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—22no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42C2×D412yes—
330C2×C42C2×Q812yes—
331C2×C42C2×Q812yes—
332C2×C42#4112yes—
333#108—44no—
334#108—44no—
335#107—24no—
336#107—42no—
337#106—24no—
338#106—42no—
339C4⋊C8—44no—
340C4⋊C8—44no—
341C4⋊C8—44no—
342C4⋊C8—44no—
343C22⋊C8—24no—
344C22⋊C8—24no—
345C22⋊C8—42no—
346C22⋊C8—42no—
347C22⋊C8C4⋊C414yes—
348C22⋊C8C4⋊C414yes—
349C22⋊C8C4⋊C414yes—
350C22⋊C8C4⋊C414yes—
351#97—44no—
352C4×C8—44no—
353#95—22no—
354#95—22no—
355C24—21no—
356C24—21no—
357C24—21no—
358C24—21no—
359C24—42no—
360C24—42no—
361C24C4×D411yescharacteristic
362C24C4×D411yescharacteristic
363C24C22⋊Q812yes—
364C24#12512yes—
365C24C2×#4111yescharacteristic
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—44no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—82no—
419C22×C4C2×C4⋊C412yes—
420C22×C4C2×C4⋊C412yes—
421C22×C4C4×D411yescharacteristic
422C22×C4C4×D411yescharacteristic
423C22×C4C4×Q812yes—
424C22×C4C4×Q812yes—
425C22×C4#12112yes—
426C22×C4#12112yes—
427C22×C4C22⋊Q812yes—
428C22×C4#12512yes—
429C22×C4C4⋊Q812yes—
430C22×C4C4⋊Q812yes—
431C22×C4C23×C411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×Q811yescharacteristic
436C22×C4C22×Q811yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4H(5,𝔽2)11yescharacteristic
442C22×C4H(5,𝔽2)11yescharacteristic
443C22×C4H(5,𝔽2)11yescharacteristic
444C22×C42−1+411yescharacteristic
445M16—22no—
446M16—22no—
447M16—22no—
448M16—22no—
449M16—48no—
450M16—41no—
451M16—41no—
452M16—41no—
453M16—41no—
454M16—41no—
455M16—41no—
456M16—41no—
457M16—41no—
458M16—41no—
459M16—41no—
460M16—41no—
461M16—41no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—48no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—41no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8C2×C4⋊C412yes—
506C2×C8C2×C4⋊C412yes—
507C2×C8C2×C4⋊C412yes—
508C2×C8C2×C4⋊C412yes—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—44no—
517C4⋊C4—48no—
518C4⋊C4C4×D414yes—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—44no—
522C22⋊C4—48no—
523C22⋊C4—44no—
524C22⋊C4—41no—
525C22⋊C4—41no—
526C22⋊C4—41no—
527C22⋊C4—41no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C22⋊C4—82no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—44no—
539C42—41no—
540C42—41no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—41no—
546C42—41no—
547C42C22⋊Q814yes—
548C23—22no—
549C23—22no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—82no—
567C23—82no—
568C23—82no—
569C23—82no—
570C23#38912yes—
571C23#38912yes—
572C23C2×C4×D411yescharacteristic
573C23C2×C22⋊Q811yescharacteristic
574C23C2×#12511yescharacteristic
575C23#54611yescharacteristic
576C23#55111yescharacteristic
577C23#55111yescharacteristic
578C23#55211yescharacteristic
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—88no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4#38912yes—
627C2×C4#38912yes—
628C2×C4C22×C4⋊C411yescharacteristic
629C2×C4C2×C4×D411yescharacteristic
630C2×C4C2×C4×Q811yescharacteristic
631C2×C4C4×#4111yescharacteristic
632C2×C4#51811yescharacteristic
633C2×C4#52011yescharacteristic
634C2×C4C2×#12111yescharacteristic
635C2×C4C2×C22⋊Q811yescharacteristic
636C2×C4#52511yescharacteristic
637C2×C4#52911yescharacteristic
638C2×C4C2×C4⋊Q811yescharacteristic
639C2×C4#53311yescharacteristic
640C2×C4#53811yescharacteristic
641C2×C4#53911yescharacteristic
642C2×C4#54311yescharacteristic
643C2×C4#54411yescharacteristic
644C2×C4D4211yescharacteristic
645C2×C4D4211yescharacteristic
646C2×C4#54611yescharacteristic
647C2×C4#54611yescharacteristic
648C2×C4#54611yescharacteristic
649C2×C4#54611yescharacteristic
650C2×C4#54711yescharacteristic
651C2×C4#54711yescharacteristic
652C2×C4D4×Q811yescharacteristic
653C2×C4#55411yescharacteristic
654C2×C4#55411yescharacteristic
655C2×C4#55411yescharacteristic
656C8—22no—
657C8—22no—
658C8—22no—
659C8—22no—
660C8—42no—
661C8—42no—
662C8—42no—
663C8—42no—
664C8—81no—
665C8—81no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—21no—
671C22—21no—
672C22—21no—
673C22—21no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22C2×#38911yescharacteristic
689C22#235811yescharacteristic
690C22#235911yescharacteristic
691C22#243711yescharacteristic
692C22#243811yescharacteristic
693C22#247311yescharacteristic
694C22#250511yescharacteristic
695C22#295512yes—
696C22#295712yes—
697C4—22no—
698C4—22no—
699C4—44no—
700C4—44no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4C4×C22⋊Q811yescharacteristic
706C4#232911yescharacteristic
707C4D4×C4⋊C411yescharacteristic
708C4#234911yescharacteristic
709C4#242411yescharacteristic
710C4#242511yescharacteristic
711C4#244011yescharacteristic
712C4#254911yescharacteristic
713C2—21no—
714C2—21no—
715C2—21no—
716C2—21no—
717C2—42no—
718C2—42no—
719C2#1400411yescharacteristic
720C2#2116311yescharacteristic
721C2#2116411yescharacteristic
722C2#2288912yes—
723C2#2294912yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1945C2semidirect
8. #11329C2semidirect
9. #11258C2semidirect
10. #11221C2semidirect
11. #11221C2semidirect
12. #10883C2semidirect
13. #10882C2semidirect
14. #10860C2semidirect
35. #1854C4semidirect
36. #1854C4semidirect
37. #1853C4semidirect
38. #1852C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19452
C2×#18552
#208432
#208362
#206282
C2×#17502
C2×#17332
#113292
#112582
#112212
#108832
#108822
#108602
#107202