Finite Group Databasebeta

← #360900

Subgroups — #360900

Order 512.

1152 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20628C211yesmaximal, characteristic
2#20379C211yesmaximal, characteristic
3C2×#1735C211yesmaximal, characteristic
4#11317C212yesmaximal, complemented
5#11291C212yesmaximal, complemented
6#10887C212yesmaximal, complemented
7#10809C212yesmaximal, complemented
8#10733C211yesmaximal, characteristic
9#10733C211yesmaximal, characteristic
10#10723C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C2×#424C2212yes—
16C2×#407C2211yescharacteristic
17C2×#407C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C2×#405C2212yes—
20C2×C4×C22⋊C4C2211yescharacteristic
21#1798—22no—
22#1797—21no—
23#1797—21no—
24#1750—22no—
25#1749—21no—
26C4×C22⋊C8—21no—
27C2×#341C2212yes—
28C2×#341C2212yes—
29#1735C411yescharacteristic, complemented
30#1735C411yescharacteristic, complemented
31#1735C411yescharacteristic, complemented
32#1735C411yescharacteristic, complemented
33#1735C2211yescharacteristic
34#1735C2211yescharacteristic
35#1735C2211yescharacteristic
36#1735C2211yescharacteristic
37C2×#340C411yescharacteristic, complemented
38C2×#340C411yescharacteristic, complemented
39C2×#340C411yescharacteristic, complemented
40C2×#340C411yescharacteristic, complemented
41C2×#339C2212yes—
42C2×#339C2212yes—
43C2×#339C2212yes—
44C2×#339C2212yes—
45#1729C2212yes—
46#1729C2212yes—
47C2×#336C2212yes—
48C2×#336C2212yes—
49C24×C4C2311yescharacteristic
50C22×M16C2312yes—
51C22×M16C2312yes—
52C23×C8—22no—
53C23×C8—22no—
54C23×C8—42no—
55C23×C8—42no—
56C23×C8C2×C411yescharacteristic
57C23×C8C2×C411yescharacteristic
58C23×C8C2×C411yescharacteristic
59C23×C8C2×C411yescharacteristic
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2312yes—
63C2×#117—21no—
64C2×#117—21no—
65C2×#117—21no—
66C2×#117—21no—
67C2×#117—24no—
68C2×#117—24no—
69C2×#117—42no—
70C2×#117—42no—
71C2×#117C2×C411yescharacteristic
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117C2×C411yescharacteristic
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C4⋊C4C2×C411yescharacteristic
80C22×C4⋊C4C2×C411yescharacteristic
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C42—21no—
87C22×C42—21no—
88C22×C42C2312yes—
89#424—24no—
90#424—42no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93#407—22no—
94#407—22no—
95#407—22no—
96#407Q814yes—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8D411yescharacteristic
104C2×C22⋊C8D411yescharacteristic
105C2×C22⋊C8D412yes—
106#405—24no—
107#405—42no—
108C2×#97—22no—
109C2×C4×C8—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C2×#95—21no—
113#341—42no—
114#341—42no—
115#340C2×C411yescharacteristic
116#340C2×C411yescharacteristic
117#340C2×C411yescharacteristic
118#340C2×C411yescharacteristic
119#340C2×C411yescharacteristic
120#340C2×C411yescharacteristic
121#340C2×C411yescharacteristic
122#340C2×C411yescharacteristic
123#340C2×C411yescharacteristic
124#340C2×C411yescharacteristic
125#340C2×C411yescharacteristic
126#340C2×C411yescharacteristic
127#340C2×C411yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#339—24no—
132#339—24no—
133#339—42no—
134#339—42no—
135#336—22no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—42no—
140#336—42no—
141C25#4111yescharacteristic
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×Q812yes—
156C23×C4#4112yes—
157C23×C4#4112yes—
158C23×C4#4111yescharacteristic
159C23×C4#4112yes—
160C23×C4C2411yescharacteristic
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16C2×Q814yes—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—41no—
194C22×C8—41no—
195C22×C8—41no—
196C22×C8—41no—
197C22×C8—41no—
198C22×C8—41no—
199C22×C8—41no—
200C22×C8—41no—
201C22×C8—48no—
202C22×C8—48no—
203C22×C8C22×C411yescharacteristic
204C22×C8C22×C411yescharacteristic
205C22×C8C22×C411yescharacteristic
206C22×C8C22×C411yescharacteristic
207C22×C8C2×D412yes—
208C22×C8C2×D412yes—
209#117—22no—
210#117—22no—
211#117—22no—
212#117—22no—
213#117—22no—
214#117—22no—
215#117—22no—
216#117—22no—
217#117—22no—
218#117—22no—
219#117—44no—
220#117—44no—
221#117—44no—
222#117—44no—
223#117—41no—
224#117—41no—
225#117—41no—
226#117—41no—
227#117#4114yes—
228#117#4114yes—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—44no—
277C2×C42—44no—
278C4⋊C8—44no—
279C4⋊C8—44no—
280C22⋊C8—24no—
281C22⋊C8—24no—
282C22⋊C8—22no—
283C22⋊C8—22no—
284C22⋊C8—24no—
285C22⋊C8—42no—
286C22⋊C8—42no—
287#97—44no—
288C4×C8—44no—
289#95—42no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—41no—
295C24—41no—
296C24C4×D411yescharacteristic
297C24C4×D411yescharacteristic
298C24#12512yes—
299C24#12612yes—
300C24C2×#4111yescharacteristic
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—24no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—81no—
347C22×C4—81no—
348C22×C4—81no—
349C22×C4—81no—
350C22×C4#11712yes—
351C22×C4#11712yes—
352C22×C4C4×D411yescharacteristic
353C22×C4C4×D411yescharacteristic
354C22×C4C4×Q812yes—
355C22×C4C4×Q812yes—
356C22×C4C22⋊Q812yes—
357C22×C4C22⋊Q812yes—
358C22×C4#12312yes—
359C22×C4#12312yes—
360C22×C4#12512yes—
361C22×C4#12612yes—
362C22×C4C23×C411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×Q811yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4112yes—
368C22×C4C2×#4112yes—
369C22×C4C2×#4111yescharacteristic
370C22×C4H(5,𝔽2)12yes—
371C22×C4H(5,𝔽2)12yes—
372M16—24no—
373M16—24no—
374M16—42no—
375M16—42no—
376M16—42no—
377M16—42no—
378M16—42no—
379M16—42no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
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—24no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—44no—
400C2×C8—44no—
401C2×C8—44no—
402C2×C8—44no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4#11712yes—
433C4⋊C4#11712yes—
434C4⋊C4#11712yes—
435C4⋊C4#11712yes—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—48no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C42—24no—
446C42—24no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—42no—
451C42—42no—
452C42—42no—
453C42—42no—
454C42—42no—
455C42—42no—
456C42—42no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—24no—
468C23—24no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—24no—
474C23—24no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23#38712yes—
482C23#38712yes—
483C23#44312yes—
484C23#44412yes—
485C23C2×C4×D411yescharacteristic
486C23C2×#12511yescharacteristic
487C23C2×#12611yescharacteristic
488C23#55112yes—
489C23#55112yes—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
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—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—84no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4#38712yes—
544C2×C4#38712yes—
545C2×C4C2×#11711yescharacteristic
546C2×C4C2×#11711yescharacteristic
547C2×C4C2×#11711yescharacteristic
548C2×C4C2×C4×Q811yescharacteristic
549C2×C4#51811yescharacteristic
550C2×C4#51811yescharacteristic
551C2×C4C2×C22⋊Q811yescharacteristic
552C2×C4C2×#12311yescharacteristic
553C2×C4#52912yes—
554C2×C4#52912yes—
555C2×C4#53812yes—
556C2×C4#53812yes—
557C2×C4#54612yes—
558C2×C4#54612yes—
559C2×C4#55312yes—
560C2×C4#55312yes—
561C2×C4#55412yes—
562C2×C4#55412yes—
563C8—41no—
564C8—41no—
565C8—41no—
566C8—41no—
567C8—42no—
568C8—42no—
569C8—44no—
570C8—82no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—48no—
589C22—48no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22#192214yes—
595C22C2×#38711yescharacteristic
596C22#234211yescharacteristic
597C22#234211yescharacteristic
598C22#250512yes—
599C22#251012yes—
600C22C2×#44311yescharacteristic
601C22C2×#44411yescharacteristic
602C22#295711yescharacteristic
603C22#295711yescharacteristic
604C4—22no—
605C4—22no—
606C4—42no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—84no—
611C4—81no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4#228311yescharacteristic
616C4#231211yescharacteristic
617C4#234511yescharacteristic
618C4#234511yescharacteristic
619C4#244412yes—
620C4#245012yes—
621C2—22no—
622C2—24no—
623C2—22no—
624C2—42no—
625C2#1388311yescharacteristic
626C2C2×#192211yescharacteristic
627C2#2104311yescharacteristic
628C2#2258611yescharacteristic
629C2#2258611yescharacteristic
630C2#2258711yescharacteristic
631C2#2258711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11317C2semidirect
5. #11291C2semidirect
6. #10887C2semidirect
7. #10809C2semidirect
29. #1735C4semidirect
30. #1735C4semidirect
31. #1735C4semidirect
32. #1735C4semidirect
37. C2×#340C4semidirect
38. C2×#340C4semidirect
39. C2×#340C4semidirect
40. C2×#340C4semidirect

Maximal subgroups

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

subgroupindex
#206282
#203792
C2×#17352
#113172
#112912
#108872
#108092
#107332
#107232
#107142