Finite Group Databasebeta

← #386496

Subgroups — #386496

Order 512.

800 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21318C211yesmaximal, characteristic
2#21304C211yesmaximal, characteristic
3#21025C211yesmaximal, characteristic
4#11386C211yesmaximal, characteristic
5#11370C211yesmaximal, characteristic
6#11329C211yesmaximal, characteristic
7#11317C211yesmaximal, characteristic
8#11291C211yesmaximal, characteristic
9#11258C211yesmaximal, characteristic
10#10910C211yesmaximal, characteristic
11#10910C211yesmaximal, characteristic
12#10887C211yesmaximal, characteristic
13#10864C211yesmaximal, characteristic
14#10809C211yesmaximal, characteristic
15#10797C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C22×#108C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C2×#424C2211yescharacteristic
21C2×#405C2211yescharacteristic
22C2×#405C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24#2285C2211yescharacteristic
25#1991C411yescharacteristic, complemented
26#1991C411yescharacteristic, complemented
27#1991C411yescharacteristic, complemented
28#1991C411yescharacteristic, complemented
29#1990C411yescharacteristic, complemented
30#1990C411yescharacteristic, complemented
31#1990C411yescharacteristic, complemented
32#1990C411yescharacteristic, complemented
33#1985—21no—
34#1985—21no—
35#1984—21no—
36#1984—21no—
37#1917—21no—
38#1917—21no—
39#1916—21no—
40#1916—21no—
41C2×#341C2211yescharacteristic
42C2×#341C2211yescharacteristic
43#1735C2211yescharacteristic
44#1735C2211yescharacteristic
45#1735C2211yescharacteristic
46#1735C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55#1729C2211yescharacteristic
56#1729C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59#1728C2211yescharacteristic
60#1728C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C4×#95C2211yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75#533—22no—
76#533—22no—
77#533—41no—
78#533—41no—
79C2×C4⋊Q8C2×C411yescharacteristic
80C2×C4⋊Q8C2×C411yescharacteristic
81C2×#125C2×C411yescharacteristic
82C2×#125C2×C411yescharacteristic
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8—21no—
85C2×C4×Q8—21no—
86C2×C4×Q8—21no—
87C2×#117C2311yescharacteristic
88C2×#117C2311yescharacteristic
89C2×#117C2311yescharacteristic
90C2×#117C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C2×#108—22no—
95C2×#108—41no—
96C2×#108—41no—
97C2×#108—41no—
98C2×#108—41no—
99C2×#108Q814yescomplemented
100#424—22no—
101#424—41no—
102#424—41no—
103#424—41no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108#405—22no—
109#405—41no—
110#405—41no—
111#405—41no—
112C2×#97—21no—
113C2×#97—21no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8—41no—
117C2×C4×C8—41no—
118C2×C4×C8C2×C411yescharacteristic
119C2×C4×C8C2×C411yescharacteristic
120C2×C4×C8C2×C411yescharacteristic
121C2×C4×C8C2×C411yescharacteristic
122#384C2×C411yescharacteristic
123#384C2×C411yescharacteristic
124#382C2×C411yescharacteristic
125#382C2×C411yescharacteristic
126#380—21no—
127#380—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C2×#95—21no—
131C2×#95—21no—
132#341—41no—
133#341—41no—
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#339—22no—
151#339—22no—
152#339—41no—
153#339—41no—
154#339—41no—
155#339—41no—
156#339—41no—
157#339—41no—
158#336—21no—
159#336—21no—
160#336—21no—
161#336—21no—
162#336—21no—
163#336—21no—
164#336—21no—
165#336—21no—
166#336—22no—
167#336—22no—
168#336—21no—
169#336—21no—
170#336—21no—
171#336—21no—
172#336—41no—
173#336—41no—
174#336D411yescharacteristic
175#336D411yescharacteristic
176#336D411yescharacteristic
177#336D411yescharacteristic
178C22×Q8—21no—
179C22×Q8—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4#4111yescharacteristic
183C23×C4#4111yescharacteristic
184C23×C4C2411yescharacteristic
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16C2×Q812yes—
208C2×M16C2×Q812yes—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8C22×C411yescharacteristic
236C22×C8C22×C411yescharacteristic
237C22×C8C22×C411yescharacteristic
238C22×C8C22×C411yescharacteristic
239C22×C8C2×D411yescharacteristic
240C22×C8C2×D411yescharacteristic
241C22×C8C2×D411yescharacteristic
242C22×C8C2×D411yescharacteristic
243C22×C8C2×Q812yes—
244C22×C8#4112yes—
245C4⋊Q8—22no—
246C4⋊Q8—22no—
247C4⋊Q8—41no—
248C4⋊Q8—41no—
249#125—22no—
250#125—22no—
251#125—41no—
252#125—41no—
253C22⋊Q8—42no—
254C22⋊Q8—42no—
255C22⋊Q8—42no—
256C22⋊Q8—42no—
257C4×Q8—42no—
258C4×Q8—42no—
259C4×Q8—42no—
260C4×Q8—42no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—21no—
269#117—21no—
270#117—21no—
271#117—21no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117#4112yes—
276#117#4112yes—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C4⋊C4C22×C411yescharacteristic
289C2×C4⋊C4C22×C411yescharacteristic
290C2×C4⋊C4C22×C411yescharacteristic
291C2×C4⋊C4C22×C411yescharacteristic
292C2×C4⋊C4C22×C411yescharacteristic
293C2×C4⋊C4C22×C411yescharacteristic
294C2×C4⋊C4C22×C411yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42C22×C411yescharacteristic
320C2×C42C22×C411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323#108—44no—
324#108—44no—
325C4⋊C8—42no—
326C4⋊C8—42no—
327C4⋊C8—42no—
328C4⋊C8—42no—
329#97—42no—
330#97—42no—
331C4×C8—41no—
332C4×C8—41no—
333C4×C8—41no—
334C4×C8—41no—
335C4×C8—42no—
336C4×C8—42no—
337#95—24no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—42no—
343C24C2×#4111yescharacteristic
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—81no—
349C2×Q8—81no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4C4×D411yescharacteristic
381C22×C4C4×D411yescharacteristic
382C22×C4C4×D411yescharacteristic
383C22×C4C4×D411yescharacteristic
384C22×C4C23×C411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×Q811yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4H(5,𝔽2)11yescharacteristic
391C22×C4H(5,𝔽2)11yescharacteristic
392C22×C4H(5,𝔽2)11yescharacteristic
393C22×C4H(5,𝔽2)11yescharacteristic
394C22×C42−1+411yescharacteristic
395C22×C42−1+411yescharacteristic
396C22×C42−1+411yescharacteristic
397C22×C42−1+411yescharacteristic
398M16—22no—
399M16—22no—
400M16—22no—
401M16—22no—
402M16—41no—
403M16—41no—
404M16—41no—
405M16—41no—
406M16—41no—
407M16—41no—
408M16—41no—
409M16—41no—
410M16—41no—
411M16—41no—
412M16—41no—
413M16—41no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8C4×Q812yes—
465C2×C8C4×Q812yes—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4#11712yes—
491C4⋊C4#11712yes—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—22no—
505C42—22no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—41no—
523C42—41no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23C2×C4×D411yescharacteristic
529C23#54211yescharacteristic
530C23#54211yescharacteristic
531C23#55111yescharacteristic
532C23#55111yescharacteristic
533C23#55611yescharacteristic
534C23#55711yescharacteristic
535Q8—84no—
536Q8—84no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4#44312yes—
590C2×C4#44412yes—
591C2×C4C2×#11711yescharacteristic
592C2×C4C2×C4×Q811yescharacteristic
593C2×C4#51811yescharacteristic
594C2×C4#51911yescharacteristic
595C2×C4#52011yescharacteristic
596C2×C4#52011yescharacteristic
597C2×C4#53511yescharacteristic
598C2×C4#53611yescharacteristic
599C2×C4#53811yescharacteristic
600C2×C4#53911yescharacteristic
601C2×C4#54111yescharacteristic
602C2×C4#54111yescharacteristic
603C2×C4#54211yescharacteristic
604C2×C4#54211yescharacteristic
605C2×C4#54311yescharacteristic
606C2×C4#54411yescharacteristic
607C2×C4#54611yescharacteristic
608C2×C4#54611yescharacteristic
609C2×C4#54711yescharacteristic
610C2×C4D4×Q811yescharacteristic
611C2×C4#55411yescharacteristic
612C2×C4#55611yescharacteristic
613C2×C4#55711yescharacteristic
614C2×C4#55911yescharacteristic
615C2×C4#56211yescharacteristic
616C2×C4#56211yescharacteristic
617C2×C4#56311yescharacteristic
618C2×C4#56311yescharacteristic
619C8—22no—
620C8—22no—
621C8—41no—
622C8—41no—
623C8—41no—
624C8—41no—
625C8—42no—
626C8—42no—
627C8—41no—
628C8—41no—
629C8—81no—
630C8—81no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22#232611yescharacteristic
644C22#234211yescharacteristic
645C22#236511yescharacteristic
646C22#275511yescharacteristic
647C22#275911yescharacteristic
648C22#279811yescharacteristic
649C22#279911yescharacteristic
650C22C2×#44311yescharacteristic
651C22C2×#44411yescharacteristic
652C22#295711yescharacteristic
653C22#295711yescharacteristic
654C4—22no—
655C4—22no—
656C4—41no—
657C4—42no—
658C4—41no—
659C4—42no—
660C4—82no—
661C4—82no—
662C4—81no—
663C4—81no—
664C4—81no—
665C4—81no—
666C4—81no—
667C4—81no—
668C4#232011yescharacteristic
669C4#232911yescharacteristic
670C4#235511yescharacteristic
671C4#237811yescharacteristic
672C4#268911yescharacteristic
673C4#269211yescharacteristic
674C4#271011yescharacteristic
675C4#271311yescharacteristic
676C2—21no—
677C2—21no—
678C2—21no—
679C2—21no—
680C2#1443011yescharacteristic
681C2#2223411yescharacteristic
682C2#2223611yescharacteristic
683C2#2258611yescharacteristic
684C2#2258711yescharacteristic
685C2#2317011yescharacteristic
686C2#2317111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #1991C4semidirect
26. #1991C4semidirect
27. #1991C4semidirect
28. #1991C4semidirect
29. #1990C4semidirect
30. #1990C4semidirect
31. #1990C4semidirect
32. #1990C4semidirect
99. C2×#108Q8semidirect

Maximal subgroups

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

subgroupindex
#213182
#213042
#210252
#113862
#113702
#113292
#113172
#112912
#112582
#109102
#108872
#108642
#108092
#107972