Finite Group Databasebeta

← #390768

Subgroups — #390768

Order 512.

768 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23656C211yesmaximal, characteristic
2#21027C211yesmaximal, characteristic
3#21026C211yesmaximal, characteristic
4#11376C211yesmaximal, characteristic
5#11376C211yesmaximal, characteristic
6#11317C211yesmaximal, characteristic
7#11269C211yesmaximal, characteristic
8#11221C211yesmaximal, characteristic
9#10909C211yesmaximal, characteristic
10#10887C211yesmaximal, characteristic
11#10878C211yesmaximal, characteristic
12#10839C211yesmaximal, characteristic
13#10809C211yesmaximal, characteristic
14#10809C211yesmaximal, characteristic
15#10795C211yesmaximal, characteristic
16C2×#427C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C2×#405C2211yescharacteristic
21C2×#405C2211yescharacteristic
22#2296C2211yescharacteristic
23#2285C2211yescharacteristic
24C2×C43C2211yescharacteristic
25#2085C411yescharacteristic, complemented
26#2085C411yescharacteristic, complemented
27#2085C411yescharacteristic, complemented
28#2085C411yescharacteristic, complemented
29#2084C411yescharacteristic, complemented
30#2084C411yescharacteristic, complemented
31#2083C411yescharacteristic, complemented
32#2083C411yescharacteristic, complemented
33#1917—21no—
34#1917—21no—
35#1917—21no—
36#1917—21no—
37#1916—21no—
38#1916—21no—
39#1916—21no—
40#1916—21no—
41C2×#341C2211yescharacteristic
42C2×#341C2211yescharacteristic
43C2×#341C2211yescharacteristic
44C2×#341C2211yescharacteristic
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#1729C2211yescharacteristic
60#1728C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C2×#336C2211yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C2×#117C2311yescharacteristic
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—22no—
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—
104#405—22no—
105#405—22no—
106#405—41no—
107#405—41no—
108C2×#97—21no—
109C2×#97—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112#389C2×C411yescharacteristic
113#389C2×C411yescharacteristic
114#389C2×C411yescharacteristic
115#389C2×C411yescharacteristic
116#385—21no—
117#385—21no—
118#384C2×C411yescharacteristic
119#384C2×C411yescharacteristic
120#382C2×C411yescharacteristic
121#382C2×C411yescharacteristic
122#380—21no—
123#380—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C43—22no—
129C43—41no—
130#341—22no—
131#341—41no—
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#339—22no—
143#339—22no—
144#339—22no—
145#339—22no—
146#339—22no—
147#339—41no—
148#339—41no—
149#339—41no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#336—21no—
161#336—21no—
162#336—22no—
163#336—41no—
164#336—41no—
165#336—41no—
166#336C2×C411yescharacteristic
167#336C2×C411yescharacteristic
168#336C2×C411yescharacteristic
169#336C2×C411yescharacteristic
170#336D411yescharacteristic
171#336D411yescharacteristic
172#336D411yescharacteristic
173#336D411yescharacteristic
174C23×C4—21no—
175C23×C4—21no—
176C23×C4#4111yescharacteristic
177C23×C4#4111yescharacteristic
178C23×C4C2411yescharacteristic
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
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#4112yes—
202C2×M16#4112yes—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
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×C8C22×C411yescharacteristic
232C22×C8C22×C411yescharacteristic
233C22×C8C22×C411yescharacteristic
234C22×C8C22×C411yescharacteristic
235C22×C8C2×D411yescharacteristic
236C22×C8C2×D411yescharacteristic
237C22×C8C2×D411yescharacteristic
238C22×C8C2×D411yescharacteristic
239#117—21no—
240#117—21no—
241#117—21no—
242#117—21no—
243#117—21no—
244#117—21no—
245#117—21no—
246#117—21no—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C4⋊C4C22×C411yescharacteristic
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—24no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42C22×C411yescharacteristic
300C2×C42C22×C411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42#4112yes—
304C2×C42#4112yes—
305#107—41no—
306#107—41no—
307#107—41no—
308#107—41no—
309#106—41no—
310#106—41no—
311#106—41no—
312#106—41no—
313C4⋊C8—42no—
314C4⋊C8—42no—
315C4⋊C8—42no—
316C4⋊C8—42no—
317#97—42no—
318#97—42no—
319C4×C8—42no—
320C4×C8—42no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329C24C2×#4111yescharacteristic
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4C4×D411yescharacteristic
361C22×C4C4×D411yescharacteristic
362C22×C4C4×D411yescharacteristic
363C22×C4C4×D411yescharacteristic
364C22×C4C23×C411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4H(5,𝔽2)11yescharacteristic
371C22×C4H(5,𝔽2)11yescharacteristic
372C22×C4H(5,𝔽2)11yescharacteristic
373C22×C42−1+411yescharacteristic
374C22×C42−1+411yescharacteristic
375C22×C42−1+411yescharacteristic
376C22×C42−1+411yescharacteristic
377C22×C42−1+411yescharacteristic
378M16—22no—
379M16—22no—
380M16—22no—
381M16—22no—
382M16—41no—
383M16—41no—
384M16—41no—
385M16—41no—
386M16—41no—
387M16—41no—
388M16—41no—
389M16—41no—
390M16—41no—
391M16—41no—
392M16—41no—
393M16—41no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—48no—
491C42—48no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—41no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23C2×C4×D411yescharacteristic
509C23#54111yescharacteristic
510C23#54111yescharacteristic
511C23#55111yescharacteristic
512C23#55211yescharacteristic
513C23#55311yescharacteristic
514C23#55511yescharacteristic
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
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—41no—
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—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4#44312yes—
568C2×C4#44412yes—
569C2×C4C4×#4111yescharacteristic
570C2×C4C4×#4111yescharacteristic
571C2×C4#51811yescharacteristic
572C2×C4#51911yescharacteristic
573C2×C4#51911yescharacteristic
574C2×C4#52011yescharacteristic
575C2×C4#53611yescharacteristic
576C2×C4#53611yescharacteristic
577C2×C4#53911yescharacteristic
578C2×C4#53911yescharacteristic
579C2×C4#53911yescharacteristic
580C2×C4#53911yescharacteristic
581C2×C4#54111yescharacteristic
582C2×C4#54111yescharacteristic
583C2×C4#54211yescharacteristic
584C2×C4#54211yescharacteristic
585C2×C4#54611yescharacteristic
586C2×C4#54611yescharacteristic
587C2×C4#54711yescharacteristic
588C2×C4#55011yescharacteristic
589C2×C4#55211yescharacteristic
590C2×C4#55211yescharacteristic
591C2×C4#55211yescharacteristic
592C2×C4#55211yescharacteristic
593C2×C4#56211yescharacteristic
594C2×C4#56211yescharacteristic
595C2×C4#56311yescharacteristic
596C2×C4#56311yescharacteristic
597C8—41no—
598C8—41no—
599C8—41no—
600C8—41no—
601C8—41no—
602C8—41no—
603C8—42no—
604C8—42no—
605C8—41no—
606C8—41no—
607C8—81no—
608C8—81no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22#232711yescharacteristic
622C22#235911yescharacteristic
623C22#236311yescharacteristic
624C22#274411yescharacteristic
625C22#274711yescharacteristic
626C22#276611yescharacteristic
627C22#276911yescharacteristic
628C22C2×#44311yescharacteristic
629C22C2×#44411yescharacteristic
630C22#295611yescharacteristic
631C22#295611yescharacteristic
632C4—22no—
633C4—22no—
634C4—44no—
635C4—44no—
636C4—41no—
637C4—41no—
638C4—81no—
639C4—81no—
640C4—81no—
641C4—81no—
642C4—81no—
643C4—81no—
644C4—81no—
645C4—81no—
646C4#233211yescharacteristic
647C4#233611yescharacteristic
648C4#237111yescharacteristic
649C4#237811yescharacteristic
650C4#268711yescharacteristic
651C4#269411yescharacteristic
652C4#269811yescharacteristic
653C4#272111yescharacteristic
654C2—21no—
655C2—21no—
656C2—21no—
657C2—21no—
658C2#1458311yescharacteristic
659C2#2225611yescharacteristic
660C2#2225911yescharacteristic
661C2#2294811yescharacteristic
662C2#2295011yescharacteristic
663C2#2311311yescharacteristic
664C2#2311411yescharacteristic
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. #2085C4semidirect
26. #2085C4semidirect
27. #2085C4semidirect
28. #2085C4semidirect
29. #2084C4semidirect
30. #2084C4semidirect
31. #2083C4semidirect
32. #2083C4semidirect

Maximal subgroups

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

subgroupindex
#236562
#210272
#210262
#113762
#113172
#112692
#112212
#109092
#108872
#108782
#108392
#108092
#107952