Finite Group Databasebeta

← #228148

Subgroups — #228148

Order 512.

658 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22983C211yesmaximal, characteristic
2Q8×#104C211yesmaximal, characteristic
3#18519C211yesmaximal, characteristic
4#17998C211yesmaximal, characteristic, complemented
5#17509C211yesmaximal, characteristic
6#9848C211yesmaximal, characteristic, complemented
7#9589C211yesmaximal, characteristic, complemented
8#9584C211yesmaximal, characteristic, complemented
9#9232C211yesmaximal, characteristic, complemented
10#9207C211yesmaximal, characteristic
11#9056C211yesmaximal, characteristic, complemented
12#8907C211yesmaximal, characteristic, complemented
13#8902C211yesmaximal, characteristic
14#8191C211yesmaximal, characteristic
15#8175C211yesmaximal, characteristic, complemented
16Q8×#41C2211yescharacteristic
17#3170C2211yescharacteristic
18#3104C2211yescharacteristic
19#3076C2211yescharacteristic
20C8×#41C2211yescharacteristic
21Q8×M16C2211yescharacteristic
22#2962C2211yescharacteristic
23#2906C2211yescharacteristic
24C42×Q8C2211yescharacteristic
25#1994—21no—
26#1994C2211yescharacteristic
27#1994C2211yescharacteristic
28#1986—21no—
29#1986C2211yescharacteristic
30#1950C2211yescharacteristic
31#1946—21no—
32#1922—21no—
33#1866—21no—
34#1806—21no—
35#1806C2211yescharacteristic
36#1767C2211yescharacteristic
37#1765—21no—
38C4×#104—21no—
39C4×#104C2211yescharacteristic
40#1744C2211yescharacteristic
41#1649C411yescharacteristic, complemented
42#1648C411yescharacteristic, complemented
43#1645C411yescharacteristic, complemented
44#1644C411yescharacteristic, complemented
45#1637C411yescharacteristic, complemented
46#1636C411yescharacteristic, complemented
47#1633C411yescharacteristic, complemented
48#1632C411yescharacteristic, complemented
49#1591—21no—
50#1584—21no—
51#1584—21no—
52C8×Q16—21no—
53#1571C2211yescharacteristic
54#1567C2211yescharacteristic
55#1542C2211yescharacteristic
56#1503C2211yescharacteristic
57#1493—21no—
58#1493C2211yescharacteristic
59#1493C2211yescharacteristic
60#1487—21no—
61#1487C2211yescharacteristic
62#1481—21no—
63#1481C2211yescharacteristic
64#1476C2211yescharacteristic
65C2×Q8⋊C8—21no—
66C2×Q8⋊C8C2211yescharacteristic
67C2×Q8⋊C8C2211yescharacteristic
68C8×M16C2211yescharacteristic
69#1303C2211yescharacteristic
70#1301C2211yescharacteristic
71#1279C2211yescharacteristic
72#1279C2211yescharacteristic
73#1278C2211yescharacteristic
74#1276C2211yescharacteristic
75C2×#131—21no—
76Q82C2×C411yescharacteristic
77#557—21no—
78#557C2×C411yescharacteristic
79#554—21no—
80#554—21no—
81#554—41no—
82#554C2×C411yescharacteristic
83D4×Q8—21no—
84D4×Q8C2×C411yescharacteristic
85#533—21no—
86#533—21no—
87#533—41no—
88#533C2311yescharacteristic
89#533C2311yescharacteristic
90C4×#41—21no—
91C4×#41C2311yescharacteristic
92C2×C4×Q8—21no—
93C2×C4×Q8—21no—
94C2×C4×Q8C2311yescharacteristic
95#500C2×C411yescharacteristic
96#494C2×C411yescharacteristic
97#487C2×C411yescharacteristic
98#487C2×C411yescharacteristic
99#484—21no—
100#479—21no—
101#478—21no—
102#477C2×C411yescharacteristic
103#477C2×C411yescharacteristic
104#477C2×C411yescharacteristic
105#476C2×C411yescharacteristic
106#475C2×C411yescharacteristic
107#474C2×C411yescharacteristic
108#467—41no—
109#462C2×C411yescharacteristic
110#462C2×C411yescharacteristic
111#451—21no—
112#446—21no—
113#446—21no—
114#446—41no—
115#446—41no—
116C8×Q8—21no—
117C8×Q8—21no—
118C8×Q8—21no—
119C4×Q16—21no—
120C4×Q16—21no—
121C4×Q16—41no—
122C8×D4—21no—
123C8×D4—41no—
124#431—21no—
125#431C2311yescharacteristic
126#431C2311yescharacteristic
127C2×#108—21no—
128#427—21no—
129#424—21no—
130#424—21no—
131#424—41no—
132#423—21no—
133#423—21no—
134#423C2311yescharacteristic
135#423C2311yescharacteristic
136C2×C4⋊C8—21no—
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2311yescharacteristic
139C2×#104—21no—
140C2×#104D411yescharacteristic
141C2×#104D411yescharacteristic
142#416—21no—
143#405—21no—
144C4×M16—21no—
145C4×M16C2311yescharacteristic
146C2×#97—21no—
147C2×C4×C8—21no—
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150C4×C4⋊C4—21no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4—41no—
153C4×C4⋊C4C2311yescharacteristic
154C43—21no—
155C43C2311yescharacteristic
156#341—21no—
157#339—21no—
158#339—21no—
159#339—21no—
160#339—21no—
161#339—21no—
162#339—21no—
163#339—21no—
164#339D411yescharacteristic
165#339D411yescharacteristic
166#336—21no—
167#336—41no—
168#335—21no—
169#335—21no—
170#335—41no—
171#333C2×C411yescharacteristic
172#333C2×C411yescharacteristic
173#332C2×C411yescharacteristic
174#332C2×C411yescharacteristic
175#328—41no—
176Q8⋊C8—21no—
177Q8⋊C8—21no—
178Q8⋊C8—21no—
179Q8⋊C8—21no—
180Q8⋊C8—21no—
181Q8⋊C8—21no—
182Q8⋊C8—21no—
183Q8⋊C8—21no—
184Q8⋊C8C2×C411yescharacteristic
185Q8⋊C8C2×C411yescharacteristic
186Q8⋊C8C2×C411yescharacteristic
187Q8⋊C8C2×C411yescharacteristic
188Q8⋊C8C2×C411yescharacteristic
189Q8⋊C8C2×C411yescharacteristic
190Q8⋊C8D411yescharacteristic
191Q8⋊C8D411yescharacteristic
192Q8⋊C8D411yescharacteristic
193Q8⋊C8D411yescharacteristic
194D4⋊C8C2×C411yescharacteristic
195D4⋊C8C2×C411yescharacteristic
196#322—21no—
197C82—21no—
198C2×#41C2×D411yescharacteristic
199C22×Q8—21no—
200C22×Q8#4111yescharacteristic
201C2×Q16—21no—
202C2×Q16—21no—
203C2×Q16—41no—
204#131—42no—
205#131—81no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—41no—
209C2×M16C2×D411yescharacteristic
210C2×M16C2×D411yescharacteristic
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—41no—
216C22×C8—41no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—41no—
222C4⋊Q8—41no—
223C4⋊Q8C22×C411yescharacteristic
224C4⋊Q8C22×C411yescharacteristic
225C4⋊Q8C22×C411yescharacteristic
226C4⋊Q8C22×C411yescharacteristic
227#125—21no—
228#125—21no—
229#125—21no—
230#125—21no—
231#125—41no—
232#125—41no—
233#125—41no—
234C22⋊Q8—21no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C22⋊Q8—41no—
241C22⋊Q8—41no—
242C22⋊Q8—41no—
243C22⋊Q8—41no—
244C22⋊Q8—41no—
245C22⋊Q8—41no—
246C22⋊Q8—41no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—41no—
258C4×Q8—41no—
259C4×Q8—41no—
260C4×Q8—42no—
261C4×Q8—42no—
262C4×Q8C22×C411yescharacteristic
263C4×Q8C22×C411yescharacteristic
264C4×Q8C22×C411yescharacteristic
265C4×Q8C2×D411yescharacteristic
266C4×Q8C2×D411yescharacteristic
267C4×Q8#4111yescharacteristic
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—21no—
272C4×D4—41no—
273C4×D4—41no—
274C4×D4C22×C411yescharacteristic
275C4×D4#4111yescharacteristic
276#117—21no—
277#117—21no—
278#117—21no—
279#117—41no—
280#117C2×D411yescharacteristic
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—41no—
285C2×C4⋊C4—41no—
286C2×C4⋊C4#4111yescharacteristic
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—41no—
293C2×C42—41no—
294C2×C42—41no—
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2411yescharacteristic
298#108—81no—
299#107—21no—
300#107—21no—
301#107—21no—
302#107—21no—
303#107—41no—
304#106—21no—
305#106—21no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—41no—
314C4⋊C8—41no—
315C4⋊C8—41no—
316C4⋊C8—41no—
317C4⋊C8—41no—
318C4⋊C8—41no—
319C4⋊C8—41no—
320C4⋊C8C22×C411yescharacteristic
321C4⋊C8C22×C411yescharacteristic
322C4⋊C8C22×C411yescharacteristic
323C4⋊C8C22×C411yescharacteristic
324C4⋊C8C2×D411yescharacteristic
325C4⋊C8C2×D411yescharacteristic
326#104—22no—
327#104—42no—
328#104Q1614yescomplemented
329#103—21no—
330#103—21no—
331#103—21no—
332#103—21no—
333#103—21no—
334#103—21no—
335#103—41no—
336#103—41no—
337#102—21no—
338#102—21no—
339C22⋊C8—21no—
340C22⋊C8—21no—
341C22⋊C8—41no—
342#97—21no—
343#97—41no—
344#97—41no—
345#97—81no—
346C4×C8—21no—
347C4×C8—21no—
348C4×C8—21no—
349C4×C8—21no—
350C4×C8—21no—
351C4×C8—41no—
352C4×C8—41no—
353C4×C8—41no—
354C4×C8C22×C411yescharacteristic
355C4×C8C22×C411yescharacteristic
356C4×C8C2×D411yescharacteristic
357C4×C8C2×D411yescharacteristic
358#95—21no—
359#95—41no—
360#41—21no—
361#41—21no—
362#41C2×Q1612yes—
363#41#13512yes—
364C2×Q8—21no—
365C2×Q8—22no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—41no—
371C2×Q8—41no—
372C2×Q8—41no—
373C2×Q8—42no—
374C2×Q8—81no—
375C2×Q8C4×D411yescharacteristic
376C2×Q8C4×D411yescharacteristic
377C2×Q8C4×D411yescharacteristic
378C2×D4—21no—
379C2×D4C4×D411yescharacteristic
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4C22×D411yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4H(5,𝔽2)11yescharacteristic
393Q16—42no—
394Q16—81no—
395Q16—82no—
396M16—21no—
397M16—21no—
398M16—21no—
399M16—41no—
400M16—41no—
401M16—41no—
402M16—41no—
403M16—82no—
404M16C2×Q1612yes—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—81no—
425C2×C8—81no—
426C2×C8—82no—
427C2×C8—82no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—44no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4C4×D411yescharacteristic
459C4⋊C4C4×D411yescharacteristic
460C4⋊C4C4×D411yescharacteristic
461C4⋊C4C4×D411yescharacteristic
462C22⋊C4—21no—
463C22⋊C4—21no—
464C22⋊C4—21no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—41no—
480C42—41no—
481C42—41no—
482C42—41no—
483C42—44no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42—41no—
488C42—81no—
489C42—82no—
490C42C2×Q1612yes—
491C42C23×C411yescharacteristic
492C42C22×D411yescharacteristic
493C42C2×#4111yescharacteristic
494C42H(5,𝔽2)11yescharacteristic
495C23—21no—
496C23#54611yescharacteristic
497Q8—21no—
498Q8—21no—
499Q8—21no—
500Q8—21no—
501Q8—42no—
502Q8—41no—
503Q8—41no—
504Q8—41no—
505Q8—41no—
506Q8—82no—
507Q8—81no—
508Q8—81no—
509Q8—161no—
510Q8C4×Q1612yes—
511Q8#44412yes—
512D4—22no—
513D4—22no—
514D4—21no—
515D4C4×Q1612yes—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—81no—
543C2×C4—82no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—161no—
548C2×C4C2×C4×D411yescharacteristic
549C2×C4C2×C4×D411yescharacteristic
550C2×C4#51811yescharacteristic
551C2×C4D4211yescharacteristic
552C2×C4#54611yescharacteristic
553C2×C4#55111yescharacteristic
554C2×C4C22×Q1611yescharacteristic
555C2×C4C2×#13511yescharacteristic
556C2×C4#57711yescharacteristic
557C2×C4#57811yescharacteristic
558C8—22no—
559C8—41no—
560C8—41no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—41no—
565C8—41no—
566C8—41no—
567C8—41no—
568C8—41no—
569C8—41no—
570C8—81no—
571C8—81no—
572C8—81no—
573C22—21no—
574C22—21no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22#235811yescharacteristic
579C22#329911yescharacteristic
580C22#330011yescharacteristic
581C4—22no—
582C4—21no—
583C4—22no—
584C4—21no—
585C4—21no—
586C4—42no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—81no—
593C4—82no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4—81no—
598C4C2×C4×Q1611yescharacteristic
599C4#294711yescharacteristic
600C4C2×#44411yescharacteristic
601C4#295611yescharacteristic
602C4#328211yescharacteristic
603C4D4×Q1611yescharacteristic
604C2—21no—
605C2—21no—
606C2—42no—
607C2#2287111yescharacteristic
608C2#2288611yescharacteristic
609C2#2289811yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2358
prime radicalC2×#41C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17998C2semidirect
6. #9848C2semidirect
7. #9589C2semidirect
8. #9584C2semidirect
9. #9232C2semidirect
11. #9056C2semidirect
12. #8907C2semidirect
15. #8175C2semidirect
41. #1649C4semidirect
42. #1648C4semidirect
43. #1645C4semidirect
44. #1644C4semidirect
45. #1637C4semidirect
46. #1636C4semidirect
47. #1633C4semidirect
48. #1632C4semidirect
328. #104Q16semidirect

Maximal subgroups

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

subgroupindex
#229832
Q8×#1042
#185192
#179982
#175092
#98482
#95892
#95842
#92322
#92072
#90562
#89072
#89022
#81912
#81752