Finite Group Databasebeta

← #224979

Subgroups — #224979

Order 512.

785 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22625C211yesmaximal, characteristic, complemented
2#22532C211yesmaximal, characteristic
3C4×#417C211yesmaximal, characteristic
4#9560C211yesmaximal, characteristic, complemented
5#9558C211yesmaximal, characteristic, complemented
6#9542C211yesmaximal, characteristic, complemented
7#9541C211yesmaximal, characteristic, complemented
8#8850C211yesmaximal, characteristic, complemented
9#8848C211yesmaximal, characteristic, complemented
10#8706C211yesmaximal, characteristic, complemented
11#8597C211yesmaximal, characteristic, complemented
12#8312C211yesmaximal, characteristic
13#8213C211yesmaximal, characteristic, complemented
14#8211C211yesmaximal, characteristic, complemented
15#8147C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#446C2211yescharacteristic
18C2×#446C2211yescharacteristic
19C2×#417C411yescharacteristic
20C2×#417C411yescharacteristic
21C2×C4×M16C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1955—21no—
24#1955—21no—
25#1954C2211yescharacteristic
26#1954C2211yescharacteristic
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1916—21no—
30#1916—21no—
31#1797—21no—
32#1797—21no—
33#1797C2211yescharacteristic
34#1797C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36C8×C4⋊C4C2211yescharacteristic
37#1763C411yescharacteristic
38#1763C411yescharacteristic
39#1762C411yescharacteristic
40#1762C411yescharacteristic
41C4×#103C2211yescharacteristic
42C4×#103C2211yescharacteristic
43C4×#102C2211yescharacteristic
44C4×#102C2211yescharacteristic
45#1749—21no—
46C4×C22⋊C8—21no—
47C4×C22⋊C8C2211yescharacteristic
48C4×C22⋊C8C2211yescharacteristic
49#1736C411yescharacteristic
50#1736C411yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×#322C2211yescharacteristic
58C2×#322C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×#41C2×C411yescharacteristic
68C22×M16C2×C411yescharacteristic
69C22×M16C2×C411yescharacteristic
70C4×#41—22no—
71C4×#41—22no—
72C4×#41—41no—
73C4×#41—41no—
74C2×C4×Q8C2×C411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C2×#117—21no—
81C2×#117C2×C411yescharacteristic
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C42—21no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87#446—22no—
88#446—22no—
89#446—41no—
90#446—41no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99#417—24no—
100#417—24no—
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8D411yescharacteristic
117C2×C22⋊C8D411yescharacteristic
118C4×M16—41no—
119C4×M16—41no—
120C2×#97—21no—
121C2×#97—21no—
122C2×#97C2×C411yescharacteristic
123C2×#97C2×C411yescharacteristic
124C2×C4×C8—21no—
125C2×C4×C8—21no—
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138C2×#95—21no—
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#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336D411yescharacteristic
155#336D411yescharacteristic
156#335—41no—
157#334—41no—
158Q8⋊C8—22no—
159Q8⋊C8—22no—
160D4⋊C8—41no—
161D4⋊C8—41no—
162#322—41no—
163#322—41no—
164C2×#41—22no—
165C2×#41—22no—
166C2×#41—22no—
167C2×#41—22no—
168C2×#41—41no—
169C2×#41—41no—
170C22×Q8C22×C411yescharacteristic
171C22×D4—21no—
172C22×D4C22×C411yescharacteristic
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4C22×C411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4#4111yescharacteristic
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—41no—
185C2×M16—41no—
186C2×M16—41no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8C22×C411yescharacteristic
194C22×C8C22×C411yescharacteristic
195C22×C8C22×C411yescharacteristic
196C22×C8C22×C411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8—22no—
204C4×Q8—41no—
205C4×Q8#4112yes—
206C4×Q8#4112yes—
207C4×D4—22no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—41no—
213C4×D4—42no—
214C4×D4—42no—
215C4×D4—42no—
216C4×D4—42no—
217#117—22no—
218#117—42no—
219#117—42no—
220#117—41no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4M1614yes—
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—81no—
252C2×C42—81no—
253C2×C42C22×C411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42#4111yescharacteristic
256C2×C42C2411yescharacteristic
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—42no—
266C4⋊C8—42no—
267C4⋊C8—81no—
268C4⋊C8—81no—
269#103—22no—
270#103—22no—
271#103—22no—
272#103—22no—
273#102—22no—
274#102—22no—
275#102—22no—
276#102M1614yes—
277C22⋊C8—22no—
278C22⋊C8—41no—
279C22⋊C8—42no—
280C22⋊C8—42no—
281C22⋊C8—42no—
282C22⋊C8—42no—
283C22⋊C8—81no—
284C22⋊C8—81no—
285C22⋊C8—81no—
286#97—41no—
287#97—41no—
288#97—42no—
289#97—42no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—81no—
303C4×C8—81no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308C24—21no—
309C24C4×D411yescharacteristic
310#41—28no—
311#41—44no—
312#41—44no—
313#41—44no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8#11712yes—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4C2×M1612yes—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—81no—
365C22×C4—81no—
366C22×C4—81no—
367C22×C4—81no—
368C22×C4C4×D411yescharacteristic
369C22×C4C4×D411yescharacteristic
370C22×C4C4×D411yescharacteristic
371C22×C4C2×M1612yes—
372C22×C4#13112yes—
373C22×C4C23×C411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C2×#4111yescharacteristic
376M16—44no—
377M16—44no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8C2×M1612yes—
419C2×C8#13712yes—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—24no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4#11712yes—
434C4⋊C4C2×M1612yes—
435C4⋊C4C2×M1612yes—
436C4⋊C4C2×M1612yes—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C42—21no—
442C42—21no—
443C42—22no—
444C42—22no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—42no—
450C42—42no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—81no—
456C42—81no—
457C42C2×#4111yescharacteristic
458C42C2×#4111yescharacteristic
459C42H(5,𝔽2)11yescharacteristic
460C42H(5,𝔽2)11yescharacteristic
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—22no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23#43512yes—
475C23C2×C4×D411yescharacteristic
476Q8—22no—
477Q8—22no—
478Q8—41no—
479Q8—41no—
480D4—22no—
481D4—22no—
482D4—48no—
483D4—41no—
484D4—41no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—48no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—84no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4#43512yes—
535C2×C4C2×#11711yescharacteristic
536C2×C4#51811yescharacteristic
537C2×C4#54611yescharacteristic
538C2×C4#54611yescharacteristic
539C2×C4#55111yescharacteristic
540C2×C4#55111yescharacteristic
541C2×C4C22×M1611yescharacteristic
542C2×C4C22×M1611yescharacteristic
543C2×C4C2×#13111yescharacteristic
544C2×C4#56811yescharacteristic
545C2×C4C2×#13711yescharacteristic
546C2×C4#57511yescharacteristic
547C2×C4#57611yescharacteristic
548C2×C4#57711yescharacteristic
549C8—22no—
550C8—22no—
551C8—42no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—41no—
556C8—41no—
557C8—41no—
558C8—81no—
559C8—81no—
560C8—81no—
561C8—81no—
562C8—81no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22#234211yescharacteristic
578C22C2×#43511yescharacteristic
579C22D4×M1611yescharacteristic
580C22C4×#13711yescharacteristic
581C22#294611yescharacteristic
582C22#295511yescharacteristic
583C22#295611yescharacteristic
584C4—22no—
585C4—21no—
586C4—21no—
587C4—42no—
588C4—41no—
589C4—41no—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—81no—
595C4—81no—
596C4#292011yescharacteristic
597C4#292411yescharacteristic
598C4#297411yescharacteristic
599C4#297511yescharacteristic
600C4#330911yescharacteristic
601C4#331611yescharacteristic
602C4#332611yescharacteristic
603C4#333311yescharacteristic
604C2—42no—
605C2—82no—
606C2#1787011yescharacteristic
607C2#1787311yescharacteristic
608C2#1804711yescharacteristic
609C2#1805011yescharacteristic
610C2#2255311yescharacteristic
611C2#2257911yescharacteristic
612C2#2258811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22625C2semidirect
4. #9560C2semidirect
5. #9558C2semidirect
6. #9542C2semidirect
7. #9541C2semidirect
8. #8850C2semidirect
9. #8848C2semidirect
10. #8706C2semidirect
11. #8597C2semidirect
13. #8213C2semidirect
14. #8211C2semidirect
15. #8147C2semidirect

Maximal subgroups

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

subgroupindex
#226252
#225322
C4×#4172
#95602
#95582
#95422
#95412
#88502
#88482
#87062
#85972
#83122
#82132
#82112
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