Finite Group Databasebeta

← #390636

Subgroups — #390636

Order 512.

871 conjugacy classes of subgroups.

The subgroup lattice has 759 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23730C211yesmaximal, characteristic
2#21051C211yesmaximal, characteristic
3#21025C211yesmaximal, characteristic, complemented
4#11269C211yesmaximal, characteristic, complemented
5#11221C211yesmaximal, characteristic, complemented
6#11191C211yesmaximal, characteristic
7#11191C211yesmaximal, characteristic
8#11068C211yesmaximal, characteristic
9#10934C211yesmaximal, characteristic
10#10923C211yesmaximal, characteristic
11#10909C211yesmaximal, characteristic, complemented
12#10909C211yesmaximal, characteristic, complemented
13#10909C211yesmaximal, characteristic, complemented
14#10839C211yesmaximal, characteristic, complemented
15#10795C211yesmaximal, characteristic, complemented
16C2×#427C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C2×#408C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C2×#405C2211yescharacteristic
28#2297C2211yescharacteristic
29#2285C2211yescharacteristic
30C2×C43C2211yescharacteristic
31#2097C411yescharacteristic, complemented
32#2097C411yescharacteristic, complemented
33#2096C411yescharacteristic, complemented
34#2095C411yescharacteristic, complemented
35#2094C411yescharacteristic, complemented
36#2093C411yescharacteristic, complemented
37#2092C411yescharacteristic, complemented
38#2092C411yescharacteristic, complemented
39#1924—21no—
40#1924—21no—
41#1923—21no—
42#1923—21no—
43#1917—21no—
44#1917—21no—
45#1916—21no—
46#1916—21no—
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49C2×#341C2211yescharacteristic
50C2×#341C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59#1729C2211yescharacteristic
60#1728C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C2×#336C2211yescharacteristic
67C22×#41C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C42—21no—
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#427—41no—
85#427—41no—
86#427—41no—
87#427—41no—
88C2×#107C2×C411yescharacteristic
89C2×#107C2×C411yescharacteristic
90C2×#107C2×C411yescharacteristic
91C2×#107C2×C411yescharacteristic
92C2×#106C2×C411yescharacteristic
93C2×#106C2×C411yescharacteristic
94C2×#106C2×C411yescharacteristic
95C2×#106C2×C411yescharacteristic
96#424—41no—
97#424—41no—
98#423—22no—
99#423—41no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×#104—22no—
105C2×#104—22no—
106C2×#104—22no—
107C2×#104—41no—
108C2×#104—41no—
109C2×#104—41no—
110#416—41no—
111#416—41no—
112#416—41no—
113#416—41no—
114C2×#103C2×C411yescharacteristic
115C2×#103C2×C411yescharacteristic
116C2×#103C2×C411yescharacteristic
117C2×#103C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122#408—22no—
123#408—41no—
124#408—41no—
125#408—41no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8—21no—
130#405—41no—
131#405—41no—
132#405—41no—
133#405—41no—
134C2×#97—21no—
135C2×#97—21no—
136C2×C4×C8—21no—
137C2×C4×C8—21no—
138#391C2×C411yescharacteristic
139#391C2×C411yescharacteristic
140#390C2×C411yescharacteristic
141#390C2×C411yescharacteristic
142#386—21no—
143#386—21no—
144#384C2×C411yescharacteristic
145#384C2×C411yescharacteristic
146#382C2×C411yescharacteristic
147#382C2×C411yescharacteristic
148#380—21no—
149#380—21no—
150C4×C4⋊C4—21no—
151C4×C4⋊C4—21no—
152C4×C22⋊C4—21no—
153C4×C22⋊C4—21no—
154C43—22no—
155C43—41no—
156#341—22no—
157#341—22no—
158#341—22no—
159#341—41no—
160#339—41no—
161#339—41no—
162#339—41no—
163#339—41no—
164#339—41no—
165#339—41no—
166#336—21no—
167#336—21no—
168#336—21no—
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—22no—
175#336—22no—
176#336—22no—
177#336—41no—
178#336C2×C411yescharacteristic
179#336C2×C411yescharacteristic
180#336C2×C411yescharacteristic
181#336C2×C411yescharacteristic
182#336D411yescharacteristic
183#336D411yescharacteristic
184#336D411yescharacteristic
185#336D411yescharacteristic
186C2×#41—21no—
187C2×#41—21no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41—21no—
196C2×#41—21no—
197C2×#41—21no—
198C22×Q8C22×C411yescharacteristic
199C22×D4—21no—
200C22×D4C22×C411yescharacteristic
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4#4111yescharacteristic
205C23×C4#4111yescharacteristic
206C23×C4C2411yescharacteristic
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—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16#4112yes—
230C2×M16#4112yes—
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—21no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8C22×C411yescharacteristic
260C22×C8C22×C411yescharacteristic
261C22×C8C22×C411yescharacteristic
262C22×C8C22×C411yescharacteristic
263C22×C8C2×D411yescharacteristic
264C22×C8C2×D411yescharacteristic
265C22×C8C2×D411yescharacteristic
266C22×C8C2×D411yescharacteristic
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—21no—
278#117—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4C22×C411yescharacteristic
284C2×C4⋊C4C22×C411yescharacteristic
285C2×C4⋊C4C22×C411yescharacteristic
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C4⋊C4C22×C411yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
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—24no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—42no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42C22×C411yescharacteristic
320C2×C42C22×C411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42#4112yes—
324C2×C42#4112yes—
325#107—41no—
326#107—41no—
327#107—41no—
328#107—41no—
329#106—41no—
330#106—41no—
331#106—41no—
332#106—41no—
333C4⋊C8—42no—
334C4⋊C8—42no—
335C4⋊C8—42no—
336C4⋊C8—42no—
337#104—44no—
338#104—44no—
339#103—41no—
340#103—41no—
341#103—41no—
342#103—41no—
343#102—41no—
344#102—41no—
345#102—41no—
346#102—41no—
347C22⋊C8—42no—
348C22⋊C8—42no—
349C22⋊C8—42no—
350C22⋊C8—42no—
351#97—42no—
352#97—42no—
353C4×C8—42no—
354C4×C8—42no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363C24—21no—
364C24C2×#4111yescharacteristic
365#41—41no—
366#41—41no—
367#41—41no—
368#41—41no—
369#41—41no—
370#41—41no—
371#41—41no—
372#41—41no—
373#41—41no—
374#41—41no—
375#41—41no—
376#41—41no—
377#41—41no—
378#41—41no—
379#41—41no—
380#41—41no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4C4×D411yescharacteristic
435C22×C4C4×D411yescharacteristic
436C22×C4C4×D411yescharacteristic
437C22×C4C4×D411yescharacteristic
438C22×C4C23×C411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4H(5,𝔽2)11yescharacteristic
445C22×C4H(5,𝔽2)11yescharacteristic
446C22×C4H(5,𝔽2)11yescharacteristic
447C22×C4H(5,𝔽2)11yescharacteristic
448C22×C4H(5,𝔽2)11yescharacteristic
449C22×C42−1+411yescharacteristic
450C22×C42−1+411yescharacteristic
451C22×C42−1+411yescharacteristic
452M16—22no—
453M16—22no—
454M16—22no—
455M16—22no—
456M16—41no—
457M16—41no—
458M16—41no—
459M16—41no—
460M16—41no—
461M16—41no—
462M16—41no—
463M16—41no—
464M16—41no—
465M16—41no—
466M16—41no—
467M16—41no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—22no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—22no—
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—
508C2×C8—41no—
509C2×C8—41no—
510C2×C8—41no—
511C2×C8—41no—
512C2×C8—41no—
513C2×C8—41no—
514C2×C8—41no—
515C2×C8—41no—
516C2×C8—41no—
517C2×C8—41no—
518C2×C8—41no—
519C2×C8—41no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—81no—
535C4⋊C4—81no—
536C22⋊C4—41no—
537C22⋊C4—41no—
538C22⋊C4—41no—
539C22⋊C4—41no—
540C22⋊C4—41no—
541C22⋊C4—41no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—81no—
545C22⋊C4—81no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—48no—
559C42—48no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—41no—
564C42—41no—
565C42—41no—
566C42—41no—
567C42—41no—
568C42—41no—
569C42—41no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23C2×C4×D411yescharacteristic
582C23#54011yescharacteristic
583C23#54011yescharacteristic
584C23#55111yescharacteristic
585C23#55211yescharacteristic
586C23#55211yescharacteristic
587C23#55611yescharacteristic
588Q8—41no—
589Q8—41no—
590Q8—41no—
591Q8—41no—
592D4—41no—
593D4—41no—
594D4—41no—
595D4—41no—
596D4—81no—
597D4—81no—
598D4—81no—
599D4—81no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
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—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—81no—
650C2×C4—81no—
651C2×C4—81no—
652C2×C4—81no—
653C2×C4#44312yes—
654C2×C4#44412yes—
655C2×C4C4×#4111yescharacteristic
656C2×C4C4×#4111yescharacteristic
657C2×C4#51811yescharacteristic
658C2×C4#51811yescharacteristic
659C2×C4#51911yescharacteristic
660C2×C4#52011yescharacteristic
661C2×C4#53511yescharacteristic
662C2×C4#53511yescharacteristic
663C2×C4#53811yescharacteristic
664C2×C4#53911yescharacteristic
665C2×C4#53911yescharacteristic
666C2×C4#53911yescharacteristic
667C2×C4#54011yescharacteristic
668C2×C4#54211yescharacteristic
669C2×C4#54211yescharacteristic
670C2×C4#54211yescharacteristic
671C2×C4#54611yescharacteristic
672C2×C4#54711yescharacteristic
673C2×C4#54711yescharacteristic
674C2×C4#54811yescharacteristic
675C2×C4#55111yescharacteristic
676C2×C4#55311yescharacteristic
677C2×C4#55311yescharacteristic
678C2×C4#55911yescharacteristic
679C2×C4#56211yescharacteristic
680C2×C4#56211yescharacteristic
681C2×C4#56311yescharacteristic
682C2×C4#56311yescharacteristic
683C8—41no—
684C8—41no—
685C8—41no—
686C8—41no—
687C8—41no—
688C8—41no—
689C8—42no—
690C8—41no—
691C8—41no—
692C8—42no—
693C8—81no—
694C8—81no—
695C22—21no—
696C22—21no—
697C22—21no—
698C22—21no—
699C22—21no—
700C22—21no—
701C22—21no—
702C22—21no—
703C22—21no—
704C22—21no—
705C22—21no—
706C22—21no—
707C22—81no—
708C22—81no—
709C22—81no—
710C22—81no—
711C22—81no—
712C22—81no—
713C22—81no—
714C22#232811yescharacteristic
715C22#235911yescharacteristic
716C22#236211yescharacteristic
717C22#274011yescharacteristic
718C22#274111yescharacteristic
719C22#277311yescharacteristic
720C22#277711yescharacteristic
721C22C2×#44311yescharacteristic
722C22C2×#44411yescharacteristic
723C22#295611yescharacteristic
724C22#295611yescharacteristic
725C4—22no—
726C4—22no—
727C4—44no—
728C4—44no—
729C4—41no—
730C4—41no—
731C4—81no—
732C4—81no—
733C4—81no—
734C4—81no—
735C4—81no—
736C4—81no—
737C4—81no—
738C4#233211yescharacteristic
739C4#233511yescharacteristic
740C4#237411yescharacteristic
741C4#237811yescharacteristic
742C4#271011yescharacteristic
743C4#271111yescharacteristic
744C4#271811yescharacteristic
745C4#272311yescharacteristic
746C2—21no—
747C2—21no—
748C2—21no—
749C2—21no—
750C2—161no—
751C2#1457811yescharacteristic
752C2#2228011yescharacteristic
753C2#2228311yescharacteristic
754C2#2294811yescharacteristic
755C2#2295011yescharacteristic
756C2#2308011yescharacteristic
757C2#2308111yescharacteristic
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 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21025C2semidirect
4. #11269C2semidirect
5. #11221C2semidirect
11. #10909C2semidirect
12. #10909C2semidirect
13. #10909C2semidirect
14. #10839C2semidirect
15. #10795C2semidirect
31. #2097C4semidirect
32. #2097C4semidirect
33. #2096C4semidirect
34. #2095C4semidirect
35. #2094C4semidirect
36. #2093C4semidirect
37. #2092C4semidirect
38. #2092C4semidirect

Maximal subgroups

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

subgroupindex
#237302
#210512
#210252
#112692
#112212
#111912
#110682
#109342
#109232
#109092
#108392
#107952