Finite Group Databasebeta

← #380483

Subgroups — #380483

Order 512.

896 conjugacy classes of subgroups.

The subgroup lattice has 747 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21021C211yesmaximal, characteristic, complemented
2#20872C211yesmaximal, characteristic
3#20375C211yesmaximal, characteristic
4#11376C211yesmaximal, characteristic, complemented
5#11317C211yesmaximal, characteristic, complemented
6#11258C211yesmaximal, characteristic, complemented
7#11221C211yesmaximal, characteristic, complemented
8#11191C211yesmaximal, characteristic
9#11191C211yesmaximal, characteristic
10#11125C211yesmaximal, characteristic
11#11068C211yesmaximal, characteristic
12#10911C211yesmaximal, characteristic, complemented
13#10886C211yesmaximal, characteristic, complemented
14#10798C211yesmaximal, characteristic, complemented
15#10790C211yesmaximal, characteristic
16C2×C4×#41C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C2×#408C2211yescharacteristic
26C2×C4×M16C2211yescharacteristic
27C22×#97C2211yescharacteristic
28#2296C2211yescharacteristic
29#1917—22no—
30#1917—22no—
31#1865C411yescharacteristic, complemented
32#1865C411yescharacteristic, complemented
33#1864C411yescharacteristic, complemented
34#1864C411yescharacteristic, complemented
35#1863C411yescharacteristic, complemented
36#1863C411yescharacteristic, complemented
37#1862C411yescharacteristic, complemented
38#1862C411yescharacteristic, complemented
39#1750—22no—
40#1749—22no—
41C2×#341C2211yescharacteristic
42C2×#341C2211yescharacteristic
43C2×#341C2211yescharacteristic
44#1735C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51#1729C2211yescharacteristic
52#1729C2211yescharacteristic
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55#1729C2211yescharacteristic
56#1728C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C4×#95C2211yescharacteristic
63C22×#41C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C4×#41—41no—
73C4×#41—41no—
74C4×#41—41no—
75C4×#41—41no—
76C2×C4×Q8C2×C411yescharacteristic
77C2×C4×Q8C2×C411yescharacteristic
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C2×C4×D4C2×C411yescharacteristic
81C2×C4×D4C2×C411yescharacteristic
82C2×#117—21no—
83C2×#117C2311yescharacteristic
84C2×#117C2311yescharacteristic
85C2×#117C2311yescharacteristic
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×#108—22no—
91C2×#108—41no—
92C2×#108—41no—
93#427—41no—
94#427—41no—
95C2×#107C2×C411yescharacteristic
96C2×#107C2×C411yescharacteristic
97C2×#106C2×C411yescharacteristic
98C2×#106C2×C411yescharacteristic
99#424—41no—
100#424—41no—
101C2×C4⋊C8—22no—
102C2×#104—22no—
103C2×#104—22no—
104C2×#104—41no—
105C2×#104—41no—
106C2×#104—41no—
107C2×#104—41no—
108#416—22no—
109#416—22no—
110#416—41no—
111#416—41no—
112C2×#103C2×C411yescharacteristic
113C2×#103C2×C411yescharacteristic
114C2×#103C2×C411yescharacteristic
115C2×#103C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120#408—22no—
121#408—22no—
122#408—22no—
123#408—22no—
124C2×C22⋊C8—22no—
125C2×C22⋊C8—22no—
126C4×M16—41no—
127C4×M16—41no—
128C2×#97—21no—
129C2×#97—41no—
130C2×#97—41no—
131C2×#97D412yes—
132C2×#97D412yes—
133C2×C4×C8—21no—
134#389C2×C411yescharacteristic
135#389C2×C411yescharacteristic
136#389C2×C411yescharacteristic
137#389C2×C411yescharacteristic
138#385—21no—
139#385—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C2×#95—21no—
143C2×#95—21no—
144#341—22no—
145#341—41no—
146#341—41no—
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#339—22no—
152#339—22no—
153#339—41no—
154#339—41no—
155#339—41no—
156#339—41no—
157#336—22no—
158#336—22no—
159#336—22no—
160#336—22no—
161#336—22no—
162#336—21no—
163#336—21no—
164#336—41no—
165#336—41no—
166#336—41no—
167#336—41no—
168#336C2×C411yescharacteristic
169#336C2×C411yescharacteristic
170#336C2×C411yescharacteristic
171#336C2×C411yescharacteristic
172C2×#41—21no—
173C2×#41—21no—
174C2×#41—21no—
175C2×#41—21no—
176C2×#41—21no—
177C2×#41—21no—
178C2×#41—21no—
179C2×#41—21no—
180C2×#41—21no—
181C2×#41—21no—
182C2×#41#4112yes—
183C2×#41#4112yes—
184C22×Q8C22×C411yescharacteristic
185C22×D4—21no—
186C22×D4C22×C411yescharacteristic
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4#4111yescharacteristic
193C23×C4#4111yescharacteristic
194C23×C4C2411yescharacteristic
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×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#4112yes—
218C2×M16#4112yes—
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×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×C8C22×C411yescharacteristic
248C22×C8C22×C411yescharacteristic
249C22×C8C22×C411yescharacteristic
250C22×C8C22×C411yescharacteristic
251C22×C8C2×D412yes—
252C22×C8C2×D412yes—
253C4×Q8—41no—
254C4×Q8—41no—
255C4×Q8—41no—
256C4×Q8—41no—
257C4×D4—41no—
258C4×D4—41no—
259C4×D4—41no—
260C4×D4—41no—
261C4×D4—42no—
262C4×D4—42no—
263C4×D4—42no—
264C4×D4—42no—
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—21no—
276#117—21no—
277#117—42no—
278#117—42no—
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⋊C4C22×C411yescharacteristic
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C4⋊C4C22×C411yescharacteristic
289C2×C4⋊C4C22×C411yescharacteristic
290C2×C4⋊C4C22×C411yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—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—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—42no—
318C2×C42—42no—
319C2×C42C22×C411yescharacteristic
320C2×C42C22×C411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323#108—44no—
324#107—41no—
325#107—41no—
326#106—41no—
327#106—41no—
328C4⋊C8—44no—
329#104—44no—
330#104—44no—
331#103—22no—
332#103—22no—
333#103—41no—
334#103—41no—
335#102—22no—
336#102—22no—
337#102—41no—
338#102—41no—
339C22⋊C8—28no—
340C22⋊C8—28no—
341#97—42no—
342#97—42no—
343#97—42no—
344C4×C8—42no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—42no—
350#95—42no—
351C24—21no—
352C24C2×#4111yescharacteristic
353#41—22no—
354#41—22no—
355#41—22no—
356#41—22no—
357#41—22no—
358#41—22no—
359#41—22no—
360#41—22no—
361#41—41no—
362#41—41no—
363#41—41no—
364#41—41no—
365#41—41no—
366#41—41no—
367#41—41no—
368#41—41no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8#11712yes—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4#11712yes—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—81no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4C4×D411yescharacteristic
429C22×C4C4×D411yescharacteristic
430C22×C4C4×D411yescharacteristic
431C22×C4C4×D411yescharacteristic
432C22×C4C23×C411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4H(5,𝔽2)11yescharacteristic
439C22×C4H(5,𝔽2)11yescharacteristic
440C22×C4H(5,𝔽2)11yescharacteristic
441C22×C4H(5,𝔽2)11yescharacteristic
442C22×C4H(5,𝔽2)11yescharacteristic
443C22×C42−1+411yescharacteristic
444C22×C42−1+411yescharacteristic
445C22×C42−1+411yescharacteristic
446M16—22no—
447M16—22no—
448M16—22no—
449M16—22no—
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—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
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—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
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×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—81no—
523C4⋊C4—81no—
524C4⋊C4—81no—
525C4⋊C4—81no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C22⋊C4—41no—
531C22⋊C4—41no—
532C22⋊C4—41no—
533C22⋊C4—41no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—42no—
543C42—42no—
544C42—41no—
545C42—41no—
546C42—41no—
547C42—41no—
548C42—41no—
549C42—41no—
550C42—41no—
551C42—41no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—41no—
557C42—41no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23C2×C4×D411yescharacteristic
570C23#52911yescharacteristic
571C23#52911yescharacteristic
572C23#53811yescharacteristic
573C23#53811yescharacteristic
574C23#54111yescharacteristic
575C23#54211yescharacteristic
576Q8—22no—
577Q8—22no—
578Q8—41no—
579Q8—41no—
580D4—22no—
581D4—22no—
582D4—42no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—41no—
587D4—41no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—81no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4—81no—
638C2×C4—81no—
639C2×C4—81no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4—81no—
644C2×C4—81no—
645C2×C4C2×#11711yescharacteristic
646C2×C4C4×#4111yescharacteristic
647C2×C4#51811yescharacteristic
648C2×C4#51811yescharacteristic
649C2×C4#51911yescharacteristic
650C2×C4#52011yescharacteristic
651C2×C4#52511yescharacteristic
652C2×C4#52511yescharacteristic
653C2×C4#53411yescharacteristic
654C2×C4#53511yescharacteristic
655C2×C4#53611yescharacteristic
656C2×C4#53711yescharacteristic
657C2×C4#53811yescharacteristic
658C2×C4#53811yescharacteristic
659C2×C4#53811yescharacteristic
660C2×C4#53911yescharacteristic
661C2×C4#53911yescharacteristic
662C2×C4#53911yescharacteristic
663C2×C4#53911yescharacteristic
664C2×C4#53911yescharacteristic
665C2×C4#53911yescharacteristic
666C2×C4#53911yescharacteristic
667C2×C4#53911yescharacteristic
668C2×C4#53911yescharacteristic
669C2×C4#54011yescharacteristic
670C2×C4#54211yescharacteristic
671C2×C4#54211yescharacteristic
672C2×C4#54211yescharacteristic
673C8—41no—
674C8—41no—
675C8—41no—
676C8—41no—
677C8—42no—
678C8—42no—
679C8—42no—
680C8—42no—
681C8—81no—
682C8—81no—
683C22—21no—
684C22—21no—
685C22—21no—
686C22—21no—
687C22—21no—
688C22—21no—
689C22—21no—
690C22—21no—
691C22—21no—
692C22—21no—
693C22—21no—
694C22—21no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—81no—
699C22—81no—
700C22—81no—
701C22—81no—
702C22#229011yescharacteristic
703C22#232111yescharacteristic
704C22#232311yescharacteristic
705C22#264911yescharacteristic
706C22#265211yescharacteristic
707C22#265211yescharacteristic
708C22#265311yescharacteristic
709C22#295611yescharacteristic
710C22#295611yescharacteristic
711C22#295711yescharacteristic
712C22#295711yescharacteristic
713C4—41no—
714C4—41no—
715C4—41no—
716C4—41no—
717C4—42no—
718C4—82no—
719C4—82no—
720C4—81no—
721C4—81no—
722C4—81no—
723C4—81no—
724C4—81no—
725C4—81no—
726C4#231311yescharacteristic
727C4#231811yescharacteristic
728C4#233211yescharacteristic
729C4#233511yescharacteristic
730C4#263011yescharacteristic
731C4#263311yescharacteristic
732C4#263611yescharacteristic
733C4#263811yescharacteristic
734C2—21no—
735C2—21no—
736C2—21no—
737C2—21no—
738C2—82no—
739C2#1426911yescharacteristic
740C2#2113911yescharacteristic
741C2#2114011yescharacteristic
742C2#2206211yescharacteristic
743C2#2206211yescharacteristic
744C2#2213211yescharacteristic
745C2#2213211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×#97 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21021C2semidirect
4. #11376C2semidirect
5. #11317C2semidirect
6. #11258C2semidirect
7. #11221C2semidirect
12. #10911C2semidirect
13. #10886C2semidirect
14. #10798C2semidirect
31. #1865C4semidirect
32. #1865C4semidirect
33. #1864C4semidirect
34. #1864C4semidirect
35. #1863C4semidirect
36. #1863C4semidirect
37. #1862C4semidirect
38. #1862C4semidirect

Maximal subgroups

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

subgroupindex
#210212
#208722
#203752
#113762
#113172
#112582
#112212
#111912
#111252
#110682
#109112
#108862
#107982
#107902