Finite Group Databasebeta

← #4307557

Subgroups — #4307557

Order 512.

3282 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2351C211yesmaximal, characteristic, complemented
2C2×Q8×C22⋊C4C211yesmaximal, characteristic
3C2×Q8×C22⋊C4C211yesmaximal, characteristic
4C2×Q8×C22⋊C4C211yesmaximal, characteristic
5#15241C213yesmaximal, complemented
6#15233C213yesmaximal
7#15192C213yesmaximal, complemented
8#15186C213yesmaximal, complemented
9#15186C213yesmaximal, complemented
10#15183C213yesmaximal
11#15132C213yesmaximal
12C4×#397C213yesmaximal
13Q8×#95C211yesmaximal, characteristic, complemented
14Q8×#95C211yesmaximal, characteristic, complemented
15Q8×#95C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C2×#397C2213yes—
24#2351—28no—
25Q8×C22⋊C4—28no—
26Q8×C22⋊C4—28no—
27Q8×C22⋊C4—28no—
28C2×#391C2213yes—
29C2×#391C2213yes—
30C2×#391C2213yes—
31C2×#391C2213yes—
32C2×#391C2213yes—
33C2×#391C2213yes—
34C2×#389C2213yes—
35C2×#389C2213yes—
36C2×#382C2213yes—
37C2×#380C2213yes—
38C2×#380C2213yes—
39C2×#380C2213yes—
40C2×C4×C4⋊C4C2213yes—
41C2×C4×C22⋊C4C2213yes—
42C2×C4×C22⋊C4C2213yes—
43C2×C4×C22⋊C4C2213yes—
44#1445C2213yes—
45#1445C2216yes—
46#1443C2216yes—
47#1443C2213yes—
48#1443C2213yes—
49#1443C2216yes—
50#1443C2216yes—
51#1443C2216yes—
52#1442C2216yes—
53#1442C2213yes—
54#1440C2216yes—
55#1440C2211yescharacteristic
56#1440C2213yes—
57#1439C2213yes—
58C4⋊#95C2213yes—
59C4⋊#95C2213yes—
60C4⋊#95C2213yes—
61C4⋊#95C2213yes—
62C4⋊#95C2213yes—
63C4⋊#95C2213yes—
64C4⋊#95C2213yes—
65C4⋊#95C2213yes—
66C4⋊#95C2213yes—
67C4×#95C2213yes—
68C4×#95C2213yes—
69C4×#95C2213yes—
70C23×Q8—24no—
71C23×Q8—24no—
72C23×Q8—24no—
73C23×Q8—24no—
74C23×Q8—24no—
75C23×Q8—24no—
76C23×Q8D418yes—
77C23×Q8C2313yes—
78C23×Q8C2311yescharacteristic
79C24×C4C2313yes—
80C2×C4×Q8—24no—
81C2×C4×Q8—24no—
82C2×C4×Q8—24no—
83C2×C4×Q8—24no—
84C2×C4×Q8—24no—
85C2×C4×Q8—24no—
86C2×C4×Q8—24no—
87C2×C4×Q8—24no—
88C2×C4×Q8—24no—
89C2×C4×Q8—24no—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8D418yes—
98C22×C4⋊C4C2313yes—
99C22×C4⋊C4C2313yes—
100C22×C4⋊C4C2313yes—
101C22×C4⋊C4C2313yes—
102C22×C4⋊C4C2313yes—
103C22×C4⋊C4C2313yes—
104C22×C4⋊C4C2313yes—
105C22×C4⋊C4C2313yes—
106C22×C22⋊C4C2313yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2313yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2313yes—
112C22×C42C2313yes—
113C22×C42C2313yes—
114C22×C42C2313yes—
115C22×C42C2313yes—
116C22×C42C2313yes—
117C22×C42C2313yes—
118#397—212no—
119#391—212no—
120#391—212no—
121#391—212no—
122#391—212no—
123#391—212no—
124#391—212no—
125#389—212no—
126#389—212no—
127#382—212no—
128#380—212no—
129#380—212no—
130#380—212no—
131C4×C4⋊C4—212no—
132C4×C22⋊C4—212no—
133C4×C22⋊C4—212no—
134C4×C22⋊C4—212no—
135C2×#95C2313yes—
136C2×#95C2313yes—
137C2×#95C2313yes—
138C2×#95C2311yescharacteristic
139C2×#95C2311yescharacteristic
140C2×#95C2311yescharacteristic
141C2×#95C2313yes—
142C2×#95C2313yes—
143C2×#95C2311yescharacteristic
144C2×#95C2313yes—
145C2×#95C2316yes—
146C2×#95C2313yes—
147C2×#95C2313yes—
148C2×#95C2316yes—
149C2×#95C2313yes—
150C2×#95C2316yes—
151C2×#95C2313yes—
152C2×#95C2313yes—
153C2×#95C2313yes—
154C2×#95C2316yes—
155C2×#95C2313yes—
156C2×#95C2313yes—
157C2×#95C2313yes—
158C2×#95C2313yes—
159C2×#95C2316yes—
160C2×#95C2316yes—
161C2×#95C2316yes—
162C25C2411yescharacteristic
163C22×Q8—232no—
164C22×Q8—232no—
165C22×Q8—232no—
166C22×Q8—26no—
167C22×Q8—26no—
168C22×Q8—26no—
169C22×Q8—26no—
170C22×Q8—26no—
171C22×Q8—26no—
172C22×Q8—416no—
173C22×Q8—416no—
174C22×Q8—416no—
175C22×Q8—416no—
176C22×Q8C2×D4112yes—
177C22×Q8C2×D414yes—
178C22×Q8C2×D414yes—
179C22×Q8#4114yes—
180C22×Q8#4114yes—
181C22×Q8#4114yes—
182C22×Q8#4114yes—
183C22×Q8#4114yes—
184C23×C4—26no—
185C23×C4—26no—
186C23×C4—26no—
187C23×C4—26no—
188C23×C4—26no—
189C23×C4—26no—
190C23×C4C2×D4112yes—
191C23×C4C2413yes—
192C23×C4C2413yes—
193C23×C4C2413yes—
194C23×C4C2413yes—
195C23×C4C2413yes—
196C23×C4C2413yes—
197C23×C4C2411yescharacteristic
198C23×C4C2413yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2413yes—
205C4×Q8—232no—
206C4×Q8—416no—
207C4×Q8—416no—
208C4×Q8—416no—
209C4×Q8—416no—
210C4×Q8—416no—
211C2×C4⋊C4—26no—
212C2×C4⋊C4—26no—
213C2×C4⋊C4—26no—
214C2×C4⋊C4—26no—
215C2×C4⋊C4—26no—
216C2×C4⋊C4—26no—
217C2×C4⋊C4—26no—
218C2×C4⋊C4—26no—
219C2×C4⋊C4—26no—
220C2×C4⋊C4—26no—
221C2×C4⋊C4—26no—
222C2×C4⋊C4—26no—
223C2×C4⋊C4—26no—
224C2×C4⋊C4—26no—
225C2×C4⋊C4—26no—
226C2×C4⋊C4—26no—
227C2×C4⋊C4—26no—
228C2×C4⋊C4—26no—
229C2×C4⋊C4—26no—
230C2×C4⋊C4—26no—
231C2×C4⋊C4—26no—
232C2×C4⋊C4—43no—
233C2×C4⋊C4—43no—
234C2×C4⋊C4C2×D4112yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—26no—
244C2×C22⋊C4—26no—
245C2×C22⋊C4—26no—
246C2×C22⋊C4—26no—
247C2×C22⋊C4—26no—
248C2×C22⋊C4—26no—
249C2×C22⋊C4—43no—
250C2×C22⋊C4—43no—
251C2×C22⋊C4—43no—
252C2×C22⋊C4#4114yes—
253C2×C42—26no—
254C2×C42—26no—
255C2×C42—26no—
256C2×C42—26no—
257C2×C42—26no—
258C2×C42—26no—
259C2×C42—26no—
260C2×C42—26no—
261C2×C42—26no—
262C2×C42—26no—
263C2×C42—26no—
264C2×C42—26no—
265C2×C42—26no—
266C2×C42—26no—
267C2×C42—26no—
268C2×C42—26no—
269C2×C42—26no—
270C2×C42C2×D4112yes—
271#95—26no—
272#95—26no—
273#95—26no—
274#95—26no—
275#95—26no—
276#95—26no—
277#95—22no—
278#95—26no—
279#95—212no—
280#95—212no—
281#95—212no—
282#95—212no—
283#95—26no—
284#95—26no—
285#95—26no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—26no—
290#95—26no—
291#95—43no—
292#95—43no—
293#95—43no—
294#95—43no—
295#95—46no—
296#95—46no—
297#95—46no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—21no—
310C24C22×D412yes—
311C24C2×#4112yes—
312C242−1+412yes—
313C24C2511yescharacteristic
314C2×Q8—224no—
315C2×Q8—224no—
316C2×Q8—224no—
317C2×Q8—4128no—
318C2×Q8—412no—
319C2×Q8—412no—
320C2×Q8—412no—
321C2×Q8—412no—
322C2×Q8#121116yes—
323C2×Q8#123116yes—
324C2×Q8#123116yes—
325C2×Q8#123116yes—
326C2×Q8#124116yes—
327C2×Q8#126116yes—
328C2×Q8#126116yes—
329C22×C4—23no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—23no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—23no—
349C22×C4—23no—
350C22×C4—23no—
351C22×C4—23no—
352C22×C4—23no—
353C22×C4—23no—
354C22×C4—23no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—23no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—23no—
374C22×C4—23no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—224no—
378C22×C4—224no—
379C22×C4—224no—
380C22×C4—23no—
381C22×C4—23no—
382C22×C4—23no—
383C22×C4—23no—
384C22×C4—23no—
385C22×C4—23no—
386C22×C4—23no—
387C22×C4—23no—
388C22×C4—23no—
389C22×C4—23no—
390C22×C4—23no—
391C22×C4—23no—
392C22×C4—23no—
393C22×C4—23no—
394C22×C4—23no—
395C22×C4—23no—
396C22×C4—23no—
397C22×C4—23no—
398C22×C4—23no—
399C22×C4—23no—
400C22×C4—23no—
401C22×C4—23no—
402C22×C4—23no—
403C22×C4—23no—
404C22×C4—23no—
405C22×C4—23no—
406C22×C4—23no—
407C22×C4—23no—
408C22×C4—23no—
409C22×C4—23no—
410C22×C4—23no—
411C22×C4—23no—
412C22×C4—23no—
413C22×C4—23no—
414C22×C4—23no—
415C22×C4—412no—
416C22×C4—412no—
417C22×C4—412no—
418C22×C4—412no—
419C22×C4C22×D416yes—
420C22×C4C22×D412yes—
421C22×C4C22×D416yes—
422C22×C4C22×D416yes—
423C22×C4C22×D416yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4116yes—
427C22×C4C2×#4116yes—
428C22×C4C2×#4116yes—
429C22×C4C2×#4116yes—
430C22×C4C2×#4116yes—
431C22×C42−1+412yes—
432C22×C42−1+412yes—
433C22×C42−1+412yes—
434C4⋊C4—224no—
435C4⋊C4—412no—
436C4⋊C4—412no—
437C4⋊C4—412no—
438C4⋊C4—412no—
439C4⋊C4—412no—
440C4⋊C4—412no—
441C4⋊C4—412no—
442C22⋊C4—28no—
443C22⋊C4—28no—
444C22⋊C4—28no—
445C22⋊C4—412no—
446C22⋊C4—412no—
447C22⋊C4—412no—
448C42—224no—
449C42—412no—
450C42—412no—
451C42—412no—
452C42—412no—
453C42—412no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23#54814yes—
489C23C23×D411yescharacteristic
490C23C23×D411yescharacteristic
491C23C22×#4111yescharacteristic
492C23C22×#4111yescharacteristic
493C23C22×#4111yescharacteristic
494C23C22×#4111yescharacteristic
495C23C22×#4111yescharacteristic
496C23C22×#4111yescharacteristic
497C23C2×2−1+411yescharacteristic
498C23C2×2−1+411yescharacteristic
499C23C2×2−1+411yescharacteristic
500C23C2×2−1+411yescharacteristic
501C23#58511yescharacteristic
502C23#58511yescharacteristic
503C23#58511yescharacteristic
504Q8—448no—
505Q8#397164yescomplemented
506C2×C4—26no—
507C2×C4—22no—
508C2×C4—26no—
509C2×C4—26no—
510C2×C4—26no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—26no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—26no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—43no—
535C2×C4—41no—
536C2×C4—43no—
537C2×C4—43no—
538C2×C4—43no—
539C2×C4—43no—
540C2×C4—43no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—43no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—43no—
562C2×C4—448no—
563C2×C4—43no—
564C2×C4—43no—
565C2×C4—43no—
566C2×C4—43no—
567C2×C4—43no—
568C2×C4—43no—
569C2×C4—43no—
570C2×C4—43no—
571C2×C4—43no—
572C2×C4—43no—
573C2×C4—43no—
574C2×C4—43no—
575C2×C4—43no—
576C2×C4—43no—
577C2×C4—43no—
578C2×C4—43no—
579C2×C4—43no—
580C2×C4—43no—
581C2×C4—43no—
582C2×C4—43no—
583C2×C4—43no—
584C2×C4—43no—
585C2×C4—43no—
586C2×C4—43no—
587C2×C4—43no—
588C2×C4—43no—
589C2×C4—43no—
590C2×C4—43no—
591C2×C4—43no—
592C2×C4—43no—
593C2×C4—43no—
594C2×C4—43no—
595C2×C4—43no—
596C2×C4—43no—
597C2×C4—43no—
598C2×C4C2×#121112yes—
599C2×C4C2×#123112yes—
600C2×C4C2×#123112yes—
601C2×C4C2×#123112yes—
602C2×C4C2×#124112yes—
603C2×C4C2×#126112yes—
604C2×C4C2×#126112yes—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22C22×#12111yescharacteristic
621C22C22×#12311yescharacteristic
622C22C22×#12311yescharacteristic
623C22C22×#12311yescharacteristic
624C22C22×#12411yescharacteristic
625C22C22×#12611yescharacteristic
626C22C22×#12611yescharacteristic
627C22C2×#54811yescharacteristic
628C22D4×#4111yescharacteristic
629C22#348011yescharacteristic
630C22#348011yescharacteristic
631C22#348211yescharacteristic
632C22#348211yescharacteristic
633C22#348211yescharacteristic
634C22#348611yescharacteristic
635C22#348611yescharacteristic
636C22#348711yescharacteristic
637C22#348911yescharacteristic
638C22#349711yescharacteristic
639C22#349711yescharacteristic
640C22#351611yescharacteristic
641C22#351611yescharacteristic
642C22#351611yescharacteristic
643C22#351611yescharacteristic
644C22#351611yescharacteristic
645C22#351611yescharacteristic
646C22#351811yescharacteristic
647C22#351811yescharacteristic
648C22#351811yescharacteristic
649C22#352311yescharacteristic
650C22#352311yescharacteristic
651C22#352311yescharacteristic
652C22#352311yescharacteristic
653C22#352411yescharacteristic
654C22#352411yescharacteristic
655C4—42no—
656C4—46no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—46no—
663C4—83no—
664C4—83no—
665C4—83no—
666C4—83no—
667C4—83no—
668C4C2×#397124yes—
669C2—44no—
670C2C22×#39711yescharacteristic
671C2#3572311yescharacteristic
672C2#3572311yescharacteristic
673C2#3577111yescharacteristic
674C2#3580111yescharacteristic
675C2#3588311yescharacteristic
676C2#3588311yescharacteristic
677C2#3591011yescharacteristic
678C2#3643211yescharacteristic
679C2#3643411yescharacteristic
680C2#3644811yescharacteristic
681C2#3644811yescharacteristic
682C2#3646711yescharacteristic
683C2#3648211yescharacteristic
684C2#3657811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2351C2semidirect
5. #15241C2semidirect
7. #15192C2semidirect
8. #15186C2semidirect
9. #15186C2semidirect
13. Q8×#95C2semidirect
14. Q8×#95C2semidirect
15. Q8×#95C2semidirect
505. Q8#397semidirect

Maximal subgroups

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

subgroupindex
C2×#23512
C2×Q8×C22⋊C42
#152412
#152332
#151922
#151862
#151832
#151322
C4×#3972
Q8×#952