Finite Group Databasebeta

← #228156

Subgroups — #228156

Order 512.

649 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22983C211yesmaximal, characteristic
2#22529C211yesmaximal, characteristic
3#18519C211yesmaximal, characteristic
4#17998C211yesmaximal, characteristic, complemented
5#17509C211yesmaximal, characteristic
6#9845C211yesmaximal, characteristic, complemented
7#9589C211yesmaximal, characteristic, complemented
8#9584C211yesmaximal, characteristic, complemented
9#9236C211yesmaximal, characteristic, complemented
10#9207C211yesmaximal, characteristic
11#9056C211yesmaximal, characteristic, complemented
12#8907C211yesmaximal, characteristic, complemented
13#8902C211yesmaximal, characteristic
14C8×#104C211yesmaximal, characteristic
15#8176C211yesmaximal, characteristic, complemented
16Q8×#41C2211yescharacteristic
17#3170C2211yescharacteristic
18#3104C2211yescharacteristic
19#3076C2211yescharacteristic
20C8×#41C2211yescharacteristic
21Q8×M16C2211yescharacteristic
22#2962C2211yescharacteristic
23#2906C2211yescharacteristic
24#2295C2211yescharacteristic
25#1995—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—
38#1759—21no—
39C4×#104C2211yescharacteristic
40C42×C8C2211yescharacteristic
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×Q8C2311yescharacteristic
94#500C2×C411yescharacteristic
95#494C2×C411yescharacteristic
96#487C2×C411yescharacteristic
97#487C2×C411yescharacteristic
98#484—21no—
99#479—21no—
100#478—21no—
101#477C2×C411yescharacteristic
102#477C2×C411yescharacteristic
103#477C2×C411yescharacteristic
104#476C2×C411yescharacteristic
105#475C2×C411yescharacteristic
106#474C2×C411yescharacteristic
107#467—41no—
108#462C2×C411yescharacteristic
109#462C2×C411yescharacteristic
110#451—21no—
111#446—21no—
112#446—21no—
113#446—41no—
114#446—41no—
115C8×Q8—21no—
116C8×Q8—21no—
117C8×Q8—21no—
118C4×Q16—21no—
119C4×Q16—21no—
120C4×Q16—41no—
121C8×D4—21no—
122C8×D4—41no—
123#431—21no—
124#431C2311yescharacteristic
125#431C2311yescharacteristic
126C2×#108—21no—
127#427—21no—
128#424—21no—
129#424—21no—
130#424—41no—
131#423—21no—
132#423—21no—
133#423C2311yescharacteristic
134#423C2311yescharacteristic
135C2×C4⋊C8—21no—
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2311yescharacteristic
138C2×#104—21no—
139C2×#104D411yescharacteristic
140C2×#104D411yescharacteristic
141#416—21no—
142#405—21no—
143C4×M16—21no—
144C4×M16C2311yescharacteristic
145C2×#97—21no—
146C2×C4×C8—21no—
147C2×C4×C8—41no—
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150#391—21no—
151#389—21no—
152#387—41no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4C2311yescharacteristic
155#376—21no—
156C43C2311yescharacteristic
157#341—21no—
158#339—21no—
159#339—21no—
160#339—21no—
161#339—21no—
162#339—21no—
163#339—21no—
164#339—21no—
165#339D411yescharacteristic
166#339D411yescharacteristic
167#336—21no—
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×Q8C22×C411yescharacteristic
261C4×Q8C22×C411yescharacteristic
262C4×Q8C22×C411yescharacteristic
263C4×Q8C2×D411yescharacteristic
264C4×Q8C2×D411yescharacteristic
265C4×Q8#4111yescharacteristic
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—41no—
271C4×D4—41no—
272C4×D4C22×C411yescharacteristic
273C4×D4#4111yescharacteristic
274#117—21no—
275#117—21no—
276#117—21no—
277#117—41no—
278#117C2×D411yescharacteristic
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—41no—
283C2×C4⋊C4—41no—
284C2×C4⋊C4#4111yescharacteristic
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—41no—
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2411yescharacteristic
294#108—81no—
295#107—21no—
296#107—21no—
297#107—21no—
298#107—21no—
299#107—41no—
300#106—21no—
301#106—21no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8—41no—
310C4⋊C8—41no—
311C4⋊C8—41no—
312C4⋊C8—41no—
313C4⋊C8—41no—
314C4⋊C8—41no—
315C4⋊C8—41no—
316C4⋊C8C22×C411yescharacteristic
317C4⋊C8C22×C411yescharacteristic
318C4⋊C8C22×C411yescharacteristic
319C4⋊C8C22×C411yescharacteristic
320C4⋊C8C2×D411yescharacteristic
321C4⋊C8C2×D411yescharacteristic
322#104—22no—
323#104—42no—
324#104Q1614yescomplemented
325#103—21no—
326#103—21no—
327#103—21no—
328#103—21no—
329#103—21no—
330#103—21no—
331#103—41no—
332#103—41no—
333#102—21no—
334#102—21no—
335C22⋊C8—21no—
336C22⋊C8—21no—
337C22⋊C8—41no—
338#97—21no—
339#97—41no—
340#97—41no—
341#97—81no—
342C4×C8—21no—
343C4×C8—21no—
344C4×C8—21no—
345C4×C8—21no—
346C4×C8—21no—
347C4×C8—41no—
348C4×C8—41no—
349C4×C8—41no—
350C4×C8—44no—
351C4×C8C22×C411yescharacteristic
352C4×C8C22×C411yescharacteristic
353C4×C8C2×D411yescharacteristic
354C4×C8C2×D411yescharacteristic
355#95—21no—
356#95—41no—
357#95—41no—
358#95—41no—
359#41—21no—
360#41—21no—
361#41C2×Q1612yes—
362#41#13512yes—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—41no—
369C2×Q8—41no—
370C2×Q8—41no—
371C2×Q8—41no—
372C2×Q8—42no—
373C2×Q8—81no—
374C2×Q8C4×D411yescharacteristic
375C2×Q8C4×D411yescharacteristic
376C2×Q8C4×D411yescharacteristic
377C2×D4—21no—
378C2×D4C4×D411yescharacteristic
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4C22×D411yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4H(5,𝔽2)11yescharacteristic
392Q16—42no—
393Q16—82no—
394Q16—81no—
395M16—21no—
396M16—21no—
397M16—21no—
398M16—41no—
399M16—41no—
400M16—41no—
401M16—41no—
402M16—82no—
403M16C2×Q1612yes—
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—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—82no—
426C2×C8—82no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—41no—
443C4⋊C4—41no—
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—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—82no—
457C4⋊C4C4×D411yescharacteristic
458C4⋊C4C4×D411yescharacteristic
459C4⋊C4C4×D411yescharacteristic
460C4⋊C4C4×D411yescharacteristic
461C22⋊C4—21no—
462C22⋊C4—21no—
463C22⋊C4—21no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—41no—
479C42—41no—
480C42—42no—
481C42—41no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42C2×Q1612yes—
488C42C23×C411yescharacteristic
489C42C22×D411yescharacteristic
490C42C2×#4111yescharacteristic
491C42H(5,𝔽2)11yescharacteristic
492C23—21no—
493C23#54611yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—21no—
498Q8—21no—
499Q8—41no—
500Q8—41no—
501Q8—41no—
502Q8—41no—
503Q8—81no—
504Q8—81no—
505Q8—81no—
506Q8—82no—
507Q8—161no—
508Q8C4×Q1612yes—
509D4—22no—
510D4—22no—
511D4—21no—
512D4C4×Q1612yes—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
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—41no—
536C2×C4—41no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—161no—
545C2×C4C2×C4×D411yescharacteristic
546C2×C4C2×C4×D411yescharacteristic
547C2×C4#51811yescharacteristic
548C2×C4D4211yescharacteristic
549C2×C4#54611yescharacteristic
550C2×C4#55111yescharacteristic
551C2×C4C22×Q1611yescharacteristic
552C2×C4C2×#13511yescharacteristic
553C2×C4#57711yescharacteristic
554C2×C4#57811yescharacteristic
555C8—22no—
556C8—22no—
557C8—22no—
558C8—41no—
559C8—41no—
560C8—41no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—41no—
565C8—41no—
566C8—41no—
567C8—81no—
568C8—81no—
569C8—81no—
570C22—21no—
571C22—21no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22#235811yescharacteristic
576C22#329911yescharacteristic
577C22#330011yescharacteristic
578C4—22no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—42no—
583C4—42no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—161no—
595C4C2×C4×Q1611yescharacteristic
596C4#294711yescharacteristic
597C4#295511yescharacteristic
598C4#295711yescharacteristic
599C4#328211yescharacteristic
600C4D4×Q1611yescharacteristic
601C2—21no—
602C2—21no—
603C2—42no—
604C2#2287111yescharacteristic
605C2#2288711yescharacteristic
606C2#2289911yescharacteristic
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. #9845C2semidirect
7. #9589C2semidirect
8. #9584C2semidirect
9. #9236C2semidirect
11. #9056C2semidirect
12. #8907C2semidirect
15. #8176C2semidirect
41. #1649C4semidirect
42. #1648C4semidirect
43. #1645C4semidirect
44. #1644C4semidirect
45. #1637C4semidirect
46. #1636C4semidirect
47. #1633C4semidirect
48. #1632C4semidirect
324. #104Q16semidirect

Maximal subgroups

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

subgroupindex
#229832
#225292
#185192
#179982
#175092
#98452
#95892
#95842
#92362
#92072
#90562
#89072
#89022
C8×#1042
#81762