Finite Group Databasebeta

← #385108

Subgroups — #385108

Order 512.

842 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21058C211yesmaximal, characteristic
2#21052C211yesmaximal, characteristic
3#20755C211yesmaximal, characteristic, complemented
4#11326C211yesmaximal, characteristic, complemented
5#11258C211yesmaximal, characteristic, complemented
6#11191C211yesmaximal, characteristic
7#11125C211yesmaximal, characteristic
8#11125C211yesmaximal, characteristic
9#11053C211yesmaximal, characteristic
10#10929C211yesmaximal, characteristic
11#10909C211yesmaximal, characteristic, complemented
12#10876C211yesmaximal, characteristic, complemented
13#10859C211yesmaximal, characteristic, complemented
14#10809C211yesmaximal, characteristic, complemented
15#10808C211yesmaximal, characteristic, complemented
16C22×#108C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#408C2211yescharacteristic
26C2×#408C2211yescharacteristic
27C2×#405C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2297C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1928C411yescharacteristic, complemented
32#1928C411yescharacteristic, complemented
33#1927C411yescharacteristic, complemented
34#1927C411yescharacteristic, complemented
35#1926C411yescharacteristic, complemented
36#1926C411yescharacteristic, complemented
37#1925C411yescharacteristic, complemented
38#1925C411yescharacteristic, complemented
39#1924—21no—
40#1924—21no—
41#1923—21no—
42#1923—21no—
43#1832—21no—
44#1832—21no—
45#1831—21no—
46#1831—21no—
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49#1735C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56#1729C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59#1729C2211yescharacteristic
60#1729C2211yescharacteristic
61#1728C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C4×#95C2211yescharacteristic
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×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#108—22no—
83C2×#108—41no—
84C2×#108—41no—
85C2×#108—41no—
86C2×#108—41no—
87C2×#108—41no—
88#427—41no—
89#427—41no—
90C2×#107C2×C411yescharacteristic
91C2×#107C2×C411yescharacteristic
92C2×#106C2×C411yescharacteristic
93C2×#106C2×C411yescharacteristic
94#423—41no—
95#423—41no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—41no—
99C2×C4⋊C8—41no—
100C2×C4⋊C8D411yescharacteristic
101C2×C4⋊C8D411yescharacteristic
102C2×C4⋊C8D411yescharacteristic
103C2×C4⋊C8D411yescharacteristic
104C2×#104—22no—
105C2×#104—41no—
106C2×#104—41no—
107C2×#104—41no—
108C2×#104—41no—
109C2×#104—41no—
110#416—22no—
111#416—22no—
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—22no—
131#405—41no—
132C2×#97—21no—
133C2×C4×C8—21no—
134C2×C4×C8—22no—
135C2×C4×C8—41no—
136C2×C4×C8—41no—
137#391C2×C411yescharacteristic
138#391C2×C411yescharacteristic
139#390C2×C411yescharacteristic
140#390C2×C411yescharacteristic
141#386—21no—
142#386—21no—
143C4×C4⋊C4C2×C411yescharacteristic
144C4×C4⋊C4C2×C411yescharacteristic
145C4×C4⋊C4C2×C411yescharacteristic
146C4×C4⋊C4C2×C411yescharacteristic
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
151#376—21no—
152C2×#95—21no—
153C2×#95—21no—
154C43—21no—
155#341—22no—
156#341—41no—
157#340C2×C411yescharacteristic
158#340C2×C411yescharacteristic
159#340C2×C411yescharacteristic
160#340C2×C411yescharacteristic
161#339—22no—
162#339—22no—
163#339—22no—
164#339—41no—
165#339—41no—
166#339—41no—
167#336—21no—
168#336—21no—
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336—22no—
178#336—21no—
179#336—21no—
180#336—41no—
181#336—41no—
182#336C2×C411yescharacteristic
183#336C2×C411yescharacteristic
184#336C2×C411yescharacteristic
185#336C2×C411yescharacteristic
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—21no—
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×C8C22×C411yescharacteristic
258C22×C8C22×C411yescharacteristic
259C22×C8C22×C411yescharacteristic
260C22×C8C22×C411yescharacteristic
261C22×C8C2×D411yescharacteristic
262C22×C8C2×D411yescharacteristic
263C22×C8C2×D411yescharacteristic
264C22×C8C2×D411yescharacteristic
265C22×C8#4112yes—
266C22×C8#4112yes—
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⋊C4C22×C411yescharacteristic
280C2×C4⋊C4C22×C411yescharacteristic
281C2×C4⋊C4C22×C411yescharacteristic
282C2×C4⋊C4C22×C411yescharacteristic
283C2×C4⋊C4C22×C411yescharacteristic
284C2×C4⋊C4C22×C411yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—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×C42C22×C411yescharacteristic
315C2×C42C22×C411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42#4112yes—
319#108—44no—
320#108—44no—
321#107—41no—
322#107—41no—
323#106—41no—
324#106—41no—
325C4⋊C8—41no—
326C4⋊C8—41no—
327C4⋊C8—41no—
328C4⋊C8—41no—
329C4⋊C8—42no—
330C4⋊C8—42no—
331#104—44no—
332#104—44no—
333#103—22no—
334#103—22no—
335#103—41no—
336#103—41no—
337#102—22no—
338#102—22no—
339#102—41no—
340#102—41no—
341C22⋊C8—42no—
342C22⋊C8—42no—
343C22⋊C8—42no—
344C22⋊C8—42no—
345#97—42no—
346C4×C8—42no—
347C4×C8—44no—
348#95—24no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—42no—
358C24—21no—
359C24C2×#4111yescharacteristic
360#41—41no—
361#41—41no—
362#41—41no—
363#41—41no—
364#41—41no—
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—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
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—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4C4×D411yescharacteristic
430C22×C4C4×D411yescharacteristic
431C22×C4C4×D411yescharacteristic
432C22×C4C4×D411yescharacteristic
433C22×C4C23×C411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4H(5,𝔽2)11yescharacteristic
440C22×C4H(5,𝔽2)11yescharacteristic
441C22×C4H(5,𝔽2)11yescharacteristic
442C22×C4H(5,𝔽2)11yescharacteristic
443C22×C4H(5,𝔽2)11yescharacteristic
444C22×C42−1+411yescharacteristic
445C22×C42−1+411yescharacteristic
446C22×C42−1+411yescharacteristic
447M16—22no—
448M16—22no—
449M16—41no—
450M16—41no—
451M16—41no—
452M16—41no—
453M16—41no—
454M16—41no—
455M16—41no—
456M16—41no—
457M16—41no—
458M16—41no—
459M16—41no—
460M16—41no—
461M16—41no—
462M16—41no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
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—22no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
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—
508C2×C8—41no—
509C2×C8—41no—
510C2×C8—41no—
511C2×C8—41no—
512C2×C8—41no—
513C2×C8#11712yes—
514C2×C8#11712yes—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C22⋊C4—42no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C22⋊C4—41no—
531C22⋊C4—41no—
532C22⋊C4—41no—
533C22⋊C4—41no—
534C22⋊C4—42no—
535C22⋊C4—81no—
536C22⋊C4—81no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—22no—
546C42—22no—
547C42—42no—
548C42—41no—
549C42—41no—
550C42—41no—
551C42—41no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—42no—
557C42—41no—
558C42—41no—
559C42—41no—
560C42—41no—
561C42—81no—
562C42—81no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23C2×C4×D411yescharacteristic
575C23#53911yescharacteristic
576C23#53911yescharacteristic
577C23#55311yescharacteristic
578C23#55511yescharacteristic
579C23#55511yescharacteristic
580C23#55611yescharacteristic
581Q8—41no—
582Q8—41no—
583Q8—41no—
584Q8—41no—
585D4—41no—
586D4—41no—
587D4—41no—
588D4—41no—
589D4—81no—
590D4—81no—
591D4—81no—
592D4—81no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—24no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
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—42no—
614C2×C4—42no—
615C2×C4—42no—
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—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
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×C4—81no—
646C2×C4—81no—
647C2×C4#44312yes—
648C2×C4C2×#11711yescharacteristic
649C2×C4C4×#4111yescharacteristic
650C2×C4#51811yescharacteristic
651C2×C4#52011yescharacteristic
652C2×C4#52011yescharacteristic
653C2×C4#52011yescharacteristic
654C2×C4#53411yescharacteristic
655C2×C4#53611yescharacteristic
656C2×C4#53811yescharacteristic
657C2×C4#53811yescharacteristic
658C2×C4#53811yescharacteristic
659C2×C4#53911yescharacteristic
660C2×C4#53911yescharacteristic
661C2×C4#53911yescharacteristic
662C2×C4#54111yescharacteristic
663C2×C4#54211yescharacteristic
664C2×C4#54611yescharacteristic
665C2×C4#54711yescharacteristic
666C2×C4#55011yescharacteristic
667C2×C4#55011yescharacteristic
668C2×C4#55211yescharacteristic
669C2×C4#55211yescharacteristic
670C2×C4#55511yescharacteristic
671C2×C4#55511yescharacteristic
672C2×C4H(3,𝔽4)11yescharacteristic
673C2×C4#56211yescharacteristic
674C2×C4#56211yescharacteristic
675C2×C4#56311yescharacteristic
676C8—22no—
677C8—22no—
678C8—41no—
679C8—41no—
680C8—41no—
681C8—41no—
682C8—41no—
683C8—41no—
684C8—42no—
685C8—81no—
686C8—81no—
687C8—81no—
688C22—21no—
689C22—21no—
690C22—21no—
691C22—21no—
692C22—21no—
693C22—21no—
694C22—21no—
695C22—21no—
696C22—21no—
697C22—21no—
698C22—21no—
699C22—21no—
700C22—81no—
701C22—81no—
702C22—81no—
703C22—81no—
704C22—81no—
705C22—81no—
706C22—81no—
707C22#232211yescharacteristic
708C22#235411yescharacteristic
709C22#236311yescharacteristic
710C22#277511yescharacteristic
711C22#278111yescharacteristic
712C22#278311yescharacteristic
713C22#278511yescharacteristic
714C22C2×#44311yescharacteristic
715C22#295511yescharacteristic
716C22#295611yescharacteristic
717C22#295711yescharacteristic
718C4—22no—
719C4—44no—
720C4—41no—
721C4—41no—
722C4—41no—
723C4—82no—
724C4—81no—
725C4—81no—
726C4—81no—
727C4—81no—
728C4—81no—
729C4—81no—
730C4—81no—
731C4#231811yescharacteristic
732C4#233411yescharacteristic
733C4#234611yescharacteristic
734C4#237911yescharacteristic
735C4#268611yescharacteristic
736C4#269411yescharacteristic
737C4#272111yescharacteristic
738C4#273311yescharacteristic
739C2—21no—
740C2—21no—
741C2—21no—
742C2—21no—
743C2—161no—
744C2#1439911yescharacteristic
745C2#2209511yescharacteristic
746C2#2209711yescharacteristic
747C2#2280211yescharacteristic
748C2#2280411yescharacteristic
749C2#2311311yescharacteristic
750C2#2311511yescharacteristic
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×C4⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20755C2semidirect
4. #11326C2semidirect
5. #11258C2semidirect
11. #10909C2semidirect
12. #10876C2semidirect
13. #10859C2semidirect
14. #10809C2semidirect
15. #10808C2semidirect
31. #1928C4semidirect
32. #1928C4semidirect
33. #1927C4semidirect
34. #1927C4semidirect
35. #1926C4semidirect
36. #1926C4semidirect
37. #1925C4semidirect
38. #1925C4semidirect

Maximal subgroups

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

subgroupindex
#210582
#210522
#207552
#113262
#112582
#111912
#111252
#110532
#109292
#109092
#108762
#108592
#108092
#108082