Finite Group Databasebeta

← #228269

Subgroups — #228269

Order 512.

617 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23080C211yesmaximal, characteristic
2#22529C211yesmaximal, characteristic
3#18519C211yesmaximal, characteristic
4#18003C211yesmaximal, characteristic, complemented
5#17514C211yesmaximal, characteristic
6#9848C211yesmaximal, characteristic, complemented
7#9591C211yesmaximal, characteristic, complemented
8#9579C211yesmaximal, characteristic, complemented
9#9236C211yesmaximal, characteristic, complemented
10#9209C211yesmaximal, characteristic
11#9056C211yesmaximal, characteristic, complemented
12#8907C211yesmaximal, characteristic, complemented
13#8903C211yesmaximal, characteristic
14#8195C211yesmaximal, characteristic
15#8182C211yesmaximal, characteristic, complemented
16Q8×#41C2211yescharacteristic
17#3170C2211yescharacteristic
18#3104C2211yescharacteristic
19#3076C2211yescharacteristic
20#2965C2211yescharacteristic
21Q8×M16C2211yescharacteristic
22#2962C2211yescharacteristic
23#2925C2211yescharacteristic
24#2295C2211yescharacteristic
25#1995—21no—
26#1994C2211yescharacteristic
27#1994C2211yescharacteristic
28#1986—21no—
29#1986C2211yescharacteristic
30#1950C2211yescharacteristic
31#1937—21no—
32#1929—21no—
33#1922—21no—
34#1806—21no—
35#1806C2211yescharacteristic
36#1772—21no—
37C4×C4⋊C8C2211yescharacteristic
38#1759—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#1593—21no—
50#1587—21no—
51#1587—21no—
52#1579—21no—
53#1570C2211yescharacteristic
54#1567C2211yescharacteristic
55#1542C2211yescharacteristic
56#1503C2211yescharacteristic
57#1497—21no—
58#1493C2211yescharacteristic
59#1493C2211yescharacteristic
60#1491—21no—
61#1487C2211yescharacteristic
62#1485—21no—
63#1481C2211yescharacteristic
64#1479—21no—
65#1476C2211yescharacteristic
66C2×Q8⋊C8C2211yescharacteristic
67C2×Q8⋊C8C2211yescharacteristic
68#1451C2211yescharacteristic
69#1303C2211yescharacteristic
70#1301C2211yescharacteristic
71#1279C2211yescharacteristic
72#1279C2211yescharacteristic
73#1278C2211yescharacteristic
74#1276C2211yescharacteristic
75Q82C2×C411yescharacteristic
76#557—21no—
77#557C2×C411yescharacteristic
78#554—21no—
79#554—21no—
80#554—41no—
81#554C2×C411yescharacteristic
82D4×Q8—21no—
83D4×Q8C2×C411yescharacteristic
84#533—21no—
85#533—21no—
86#533—41no—
87#533C2311yescharacteristic
88#533C2311yescharacteristic
89C4×#41—21no—
90C4×#41C2311yescharacteristic
91C2×C4×Q8—21no—
92C2×C4×Q8C2311yescharacteristic
93#500C2×C411yescharacteristic
94#494C2×C411yescharacteristic
95#487C2×C411yescharacteristic
96#487C2×C411yescharacteristic
97#484—21no—
98#479—21no—
99#478—21no—
100#477C2×C411yescharacteristic
101#477C2×C411yescharacteristic
102#477C2×C411yescharacteristic
103#476C2×C411yescharacteristic
104#475C2×C411yescharacteristic
105#474C2×C411yescharacteristic
106#467—41no—
107#462C2×C411yescharacteristic
108#462C2×C411yescharacteristic
109#451—21no—
110#446—21no—
111#446—21no—
112#446—21no—
113#446—41no—
114#446—41no—
115C8×Q8—21no—
116C8×Q8—21no—
117C4×Q16—21no—
118C4×Q16—21no—
119C4×Q16—41no—
120#436—21no—
121#435—41no—
122#433—41no—
123#432—21no—
124#431—21no—
125#431—21no—
126#431C2311yescharacteristic
127#431C2311yescharacteristic
128C2×#108—21no—
129#427—21no—
130#424—21no—
131#423—21no—
132#423C2311yescharacteristic
133#423C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×#104—21no—
137C2×#104D411yescharacteristic
138C2×#104D411yescharacteristic
139#416—21no—
140#408—21no—
141#405—21no—
142#405—21no—
143C4×M16—21no—
144C4×M16—21no—
145C4×M16C2311yescharacteristic
146C2×C4×C8—21no—
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149#391—21no—
150#389—21no—
151#387—41no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4C2311yescharacteristic
154#376—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—41no—
169#334—21no—
170#334—21no—
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—
197#322—21no—
198C2×#41C2×D411yescharacteristic
199C22×Q8—21no—
200C22×Q8#4111yescharacteristic
201C2×Q16—21no—
202C2×Q16—21no—
203C2×Q16—41no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—41no—
207C2×M16C2×D411yescharacteristic
208C2×M16C2×D411yescharacteristic
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—41no—
218C4⋊Q8—41no—
219C4⋊Q8C22×C411yescharacteristic
220C4⋊Q8C22×C411yescharacteristic
221C4⋊Q8C22×C411yescharacteristic
222C4⋊Q8C22×C411yescharacteristic
223#125—21no—
224#125—21no—
225#125—21no—
226#125—21no—
227#125—41no—
228#125—41no—
229#125—41no—
230C22⋊Q8—21no—
231C22⋊Q8—21no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234C22⋊Q8—21no—
235C22⋊Q8—21no—
236C22⋊Q8—41no—
237C22⋊Q8—41no—
238C22⋊Q8—41no—
239C22⋊Q8—41no—
240C22⋊Q8—41no—
241C22⋊Q8—41no—
242C22⋊Q8—41no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8—41no—
254C4×Q8—41no—
255C4×Q8—41no—
256C4×Q8C22×C411yescharacteristic
257C4×Q8C22×C411yescharacteristic
258C4×Q8C22×C411yescharacteristic
259C4×Q8C2×D411yescharacteristic
260C4×Q8C2×D411yescharacteristic
261C4×Q8#4111yescharacteristic
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—41no—
267C4×D4—41no—
268C4×D4C22×C411yescharacteristic
269C4×D4#4111yescharacteristic
270#117—21no—
271#117—21no—
272#117—21no—
273#117—41no—
274#117C2×D411yescharacteristic
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—41no—
279C2×C4⋊C4—41no—
280C2×C4⋊C4#4111yescharacteristic
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—41no—
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2411yescharacteristic
290#108—42no—
291#107—21no—
292#107—21no—
293#107—21no—
294#107—21no—
295#107—41no—
296#106—21no—
297#106—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—21no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8—41no—
306C4⋊C8—41no—
307C4⋊C8—41no—
308C4⋊C8—41no—
309C4⋊C8—41no—
310C4⋊C8C22×C411yescharacteristic
311C4⋊C8C22×C411yescharacteristic
312C4⋊C8C22×C411yescharacteristic
313C4⋊C8C22×C411yescharacteristic
314C4⋊C8C2×D411yescharacteristic
315C4⋊C8C2×D411yescharacteristic
316#104—22no—
317#104—41no—
318#104—42no—
319#103—21no—
320#103—21no—
321#103—21no—
322#103—21no—
323#103—21no—
324#103—21no—
325#103—41no—
326#103—41no—
327#102—21no—
328#102—21no—
329C22⋊C8—21no—
330C22⋊C8—21no—
331C22⋊C8—41no—
332C22⋊C8—41no—
333C22⋊C8—41no—
334#97—21no—
335#97—41no—
336#97—41no—
337#97—41no—
338#97—41no—
339C4×C8—21no—
340C4×C8—21no—
341C4×C8—21no—
342C4×C8—21no—
343C4×C8—21no—
344C4×C8—41no—
345C4×C8—41no—
346C4×C8—41no—
347C4×C8—42no—
348C4×C8C22×C411yescharacteristic
349C4×C8C22×C411yescharacteristic
350C4×C8C2×D411yescharacteristic
351C4×C8C2×D411yescharacteristic
352#95—21no—
353#95—41no—
354#95—41no—
355#95—41no—
356#41—21no—
357#41—21no—
358#41—21no—
359#41—21no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—41no—
366C2×Q8—41no—
367C2×Q8—41no—
368C2×Q8—41no—
369C2×Q8—42no—
370C2×Q8—81no—
371C2×Q8C4×D411yescharacteristic
372C2×Q8C4×D411yescharacteristic
373C2×Q8C4×D411yescharacteristic
374C2×D4—21no—
375C2×D4C4×D411yescharacteristic
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4C22×D411yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4H(5,𝔽2)11yescharacteristic
389Q16—42no—
390Q16—81no—
391Q16—82no—
392M16—21no—
393M16—21no—
394M16—21no—
395M16—21no—
396M16—41no—
397M16—41no—
398M16—41no—
399M16—41no—
400M16—82no—
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—21no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—81no—
421C2×C8—81no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
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—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
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—82no—
452C4⋊C4C4×D411yescharacteristic
453C4⋊C4C4×D411yescharacteristic
454C4⋊C4C4×D411yescharacteristic
455C4⋊C4C4×D411yescharacteristic
456C22⋊C4—21no—
457C22⋊C4—21no—
458C22⋊C4—21no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42—41no—
480C42—41no—
481C42—81no—
482C42#13712yes—
483C42C23×C411yescharacteristic
484C42C22×D411yescharacteristic
485C42C2×#4111yescharacteristic
486C42H(5,𝔽2)11yescharacteristic
487C23—21no—
488C23#54611yescharacteristic
489Q8—21no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—41no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499Q8—81no—
500Q8—81no—
501Q8—81no—
502Q8—82no—
503Q8—161no—
504D4—21no—
505D4—21no—
506D4—41no—
507D4—41no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
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—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—82no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—161no—
540C2×C4C2×C4×D411yescharacteristic
541C2×C4C2×C4×D411yescharacteristic
542C2×C4#51811yescharacteristic
543C2×C4D4211yescharacteristic
544C2×C4#54611yescharacteristic
545C2×C4#55111yescharacteristic
546C2×C4C2×#13711yescharacteristic
547C2×C4#57511yescharacteristic
548C2×C4#57811yescharacteristic
549C2×C4#57811yescharacteristic
550C8—41no—
551C8—41no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—41no—
556C8—41no—
557C8—41no—
558C8—41no—
559C8—41no—
560C8—41no—
561C8—41no—
562C8—81no—
563C8—81no—
564C8—81no—
565C22—21no—
566C22—21no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22#235811yescharacteristic
571C22#331211yescharacteristic
572C22#331311yescharacteristic
573C4—21no—
574C4—21no—
575C4—21no—
576C4—42no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—161no—
590C4C4×#13711yescharacteristic
591C4#295011yescharacteristic
592C4#295511yescharacteristic
593C4#295711yescharacteristic
594C4#327611yescharacteristic
595C4#327711yescharacteristic
596C2—21no—
597C2—21no—
598C2—81no—
599C2#2286911yescharacteristic
600C2#2288911yescharacteristic
601C2#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. #18003C2semidirect
6. #9848C2semidirect
7. #9591C2semidirect
8. #9579C2semidirect
9. #9236C2semidirect
11. #9056C2semidirect
12. #8907C2semidirect
15. #8182C2semidirect
41. #1649C4semidirect
42. #1648C4semidirect
43. #1645C4semidirect
44. #1644C4semidirect
45. #1637C4semidirect
46. #1636C4semidirect
47. #1633C4semidirect
48. #1632C4semidirect

Maximal subgroups

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

subgroupindex
#230802
#225292
#185192
#180032
#175142
#98482
#95912
#95792
#92362
#92092
#90562
#89072
#89032
#81952
#81822