Finite Group Databasebeta

← #228141

Subgroups — #228141

Order 512.

743 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23568C211yesmaximal, characteristic
2#23088C211yesmaximal, characteristic
3#22509C211yesmaximal, characteristic, complemented
4C2×#1590C211yesmaximal, characteristic, complemented
5C2×#1497C211yesmaximal, characteristic, complemented
6#9660C211yesmaximal, characteristic, complemented
7#9553C211yesmaximal, characteristic, complemented
8#9543C211yesmaximal, characteristic, complemented
9#9074C211yesmaximal, characteristic
10#8949C211yesmaximal, characteristic, complemented
11#8948C211yesmaximal, characteristic, complemented
12#8848C211yesmaximal, characteristic, complemented
13#8834C211yesmaximal, characteristic, complemented
14#8201C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#446C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20C2×C4×SD16C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×#432C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C4×#121C2211yescharacteristic
25#2058C411yescharacteristic
26#2043C411yescharacteristic
27#2035C411yescharacteristic
28#2035C411yescharacteristic
29#2023C411yescharacteristic
30#2013C411yescharacteristic
31#2000C411yescharacteristic
32#2000C411yescharacteristic
33#1985—21no—
34#1984—21no—
35#1984C2211yescharacteristic
36#1984C2211yescharacteristic
37#1954—21no—
38#1954C2211yescharacteristic
39#1939C2211yescharacteristic
40#1924—21no—
41#1870—21no—
42#1798—21no—
43#1797C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#103C2211yescharacteristic
47#1753—21no—
48#1750—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51#1590—24no—
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54#1497—24no—
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C22×SD16C2×C411yescharacteristic
71C22×M16—21no—
72#533—22no—
73#533—22no—
74#533—41no—
75#533—41no—
76C2×C4⋊Q8C2×C411yescharacteristic
77C2×C4⋊Q8C2×C411yescharacteristic
78C2×#125—21no—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8C2×C411yescharacteristic
81C2×C22⋊Q8C2×C411yescharacteristic
82C2×#121C2×C411yescharacteristic
83C2×C4×Q8—21no—
84C2×C4×Q8C2311yescharacteristic
85C2×C4×Q8C2311yescharacteristic
86C2×C4×D4—21no—
87C2×C4×D4C2311yescharacteristic
88C2×#117—21no—
89C22×C42C2311yescharacteristic
90#446—22no—
91#446—41no—
92C8×Q8—22no—
93C8×Q8—41no—
94C4×SD16—22no—
95C4×SD16—22no—
96C4×SD16—41no—
97C4×SD16—41no—
98#436—22no—
99#436—41no—
100#432—22no—
101#432—41no—
102#431—22no—
103#431—41no—
104C2×#106C2×C411yescharacteristic
105C2×#106C2×C411yescharacteristic
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×#103C2×C411yescharacteristic
115C2×#103C2×C411yescharacteristic
116C2×#103C2×C411yescharacteristic
117C2×#103C2×C411yescharacteristic
118C2×#103C2×C411yescharacteristic
119C2×#103C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8C2×C411yescharacteristic
129C2×C22⋊C8C2×C411yescharacteristic
130C2×C22⋊C8D411yescharacteristic
131C2×C22⋊C8D411yescharacteristic
132C2×#97—21no—
133C2×#97—21no—
134C2×C4×C8—21no—
135C2×C4×C8—21no—
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142#390C2×C411yescharacteristic
143#389C2×C411yescharacteristic
144#388—21no—
145#386—21no—
146#380C2×C411yescharacteristic
147C4×C4⋊C4C2311yescharacteristic
148C4×C22⋊C4—21no—
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#340C2×C411yescharacteristic
152#340C2×C411yescharacteristic
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#336C2×C411yescharacteristic
161#336C2×C411yescharacteristic
162#336D411yescharacteristic
163#336D411yescharacteristic
164#335—41no—
165#334—22no—
166Q8⋊C8—22no—
167Q8⋊C8—22no—
168Q8⋊C8—22no—
169Q8⋊C8D414yescomplemented
170D4⋊C8—22no—
171D4⋊C8—41no—
172#322—41no—
173C82—22no—
174C22×Q8—21no—
175C22×Q8C22×C411yescharacteristic
176C22×Q8C22×C411yescharacteristic
177C22×D4—21no—
178C22×D4C22×C411yescharacteristic
179C23×C4—21no—
180C23×C4C22×C411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4#4111yescharacteristic
183C2×SD16—22no—
184C2×SD16—22no—
185C2×SD16—22no—
186C2×SD16—41no—
187C2×SD16—41no—
188C2×SD16—41no—
189C2×M16—42no—
190C2×M16—42no—
191C2×M16—81no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8C22×C411yescharacteristic
199C22×C8C22×C411yescharacteristic
200C22×C8C22×C411yescharacteristic
201C22×C8C22×C411yescharacteristic
202C22×C8C2×D411yescharacteristic
203C22×C8C2×D411yescharacteristic
204C22×C8C2×D411yescharacteristic
205C22×C8C2×D411yescharacteristic
206C4⋊Q8—22no—
207C4⋊Q8—22no—
208C4⋊Q8—41no—
209C4⋊Q8—41no—
210#125—42no—
211#125—42no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214C22⋊Q8—41no—
215C22⋊Q8—41no—
216C22⋊Q8—42no—
217C22⋊Q8—42no—
218#121—22no—
219#121—41no—
220C4×Q8—21no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—42no—
227C4×Q8—42no—
228C4×Q8C2×D412yes—
229C4×Q8#4112yes—
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—81no—
235C4×D4—81no—
236#117—42no—
237#117—42no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C22⋊C4—21no—
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×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42C22×C411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42#4111yescharacteristic
268C2×C42C2411yescharacteristic
269#106—22no—
270#106—41no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—42no—
279C4⋊C8—42no—
280C4⋊C8—42no—
281C4⋊C8—81no—
282C4⋊C8C2×D412yes—
283#103—22no—
284#103—22no—
285#103—22no—
286#103—22no—
287#103—22no—
288#103—41no—
289#102—22no—
290#102—41no—
291C22⋊C8—22no—
292C22⋊C8—22no—
293C22⋊C8—22no—
294C22⋊C8—41no—
295C22⋊C8—42no—
296C22⋊C8—42no—
297C22⋊C8—42no—
298C22⋊C8—42no—
299C22⋊C8—42no—
300C22⋊C8—81no—
301#97—42no—
302#97—42no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—42no—
315C4×C8—81no—
316C4×C8C2×D412yes—
317#95—21no—
318#95—21no—
319#95—21no—
320C24—21no—
321C24C4×D411yescharacteristic
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—81no—
336C2×Q8C4×D412yes—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—81no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4C4×D411yescharacteristic
376C22×C4C4×D411yescharacteristic
377C22×C4C4×D411yescharacteristic
378C22×C4C23×C411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C2×#4111yescharacteristic
381SD16—44no—
382SD16—44no—
383M16—84no—
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—22no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—22no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
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×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8#13112yes—
424C2×C8#13112yes—
425C2×C8#13712yes—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—81no—
444C4⋊C4C4×D412yes—
445C4⋊C4#13112yes—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—81no—
452C22⋊C4—81no—
453C22⋊C4—81no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—22no—
458C42—22no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42C22×D411yescharacteristic
467C42C2×#4111yescharacteristic
468C42H(5,𝔽2)11yescharacteristic
469C42H(5,𝔽2)11yescharacteristic
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23C2×C4×D411yescharacteristic
485Q8—22no—
486Q8—22no—
487Q8—41no—
488Q8—41no—
489Q8—41no—
490Q8—41no—
491Q8—41no—
492Q8—41no—
493Q8—84no—
494D4—41no—
495D4—41no—
496D4—41no—
497D4—41no—
498D4—84no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
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—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—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4C2×C4×D411yescharacteristic
546C2×C4#51811yescharacteristic
547C2×C4D4211yescharacteristic
548C2×C4#54611yescharacteristic
549C2×C4#54611yescharacteristic
550C2×C4#55111yescharacteristic
551C2×C4C2×#13111yescharacteristic
552C2×C4C2×#13111yescharacteristic
553C2×C4C2×#13111yescharacteristic
554C2×C4#56811yescharacteristic
555C2×C4C2×#13711yescharacteristic
556C2×C4#57511yescharacteristic
557C2×C4#57711yescharacteristic
558C2×C4#57711yescharacteristic
559C8—22no—
560C8—22no—
561C8—22no—
562C8—42no—
563C8—41no—
564C8—41no—
565C8—41no—
566C8—42no—
567C8—41no—
568C8—41no—
569C8—81no—
570C8—81no—
571C8—81no—
572C8—81no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22#235811yescharacteristic
588C22#293111yescharacteristic
589C22#293311yescharacteristic
590C22C4×#13711yescharacteristic
591C22#294911yescharacteristic
592C22#295511yescharacteristic
593C22#295711yescharacteristic
594C4—21no—
595C4—21no—
596C4—41no—
597C4—42no—
598C4—41no—
599C4—41no—
600C4—42no—
601C4—42no—
602C4—81no—
603C4—81no—
604C4—81no—
605C4—81no—
606C4#293111yescharacteristic
607C4#293311yescharacteristic
608C4#297611yescharacteristic
609C4#297911yescharacteristic
610C4#326611yescharacteristic
611C4#327111yescharacteristic
612C4#332611yescharacteristic
613C4#332911yescharacteristic
614C2—81no—
615C2—161no—
616C2#1796711yescharacteristic
617C2#1797111yescharacteristic
618C2#1801311yescharacteristic
619C2#1802811yescharacteristic
620C2#2285311yescharacteristic
621C2#2288511yescharacteristic
622C2#2288911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×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
3. #22509C2semidirect
4. C2×#1590C2semidirect
5. C2×#1497C2semidirect
6. #9660C2semidirect
7. #9553C2semidirect
8. #9543C2semidirect
10. #8949C2semidirect
11. #8948C2semidirect
12. #8848C2semidirect
13. #8834C2semidirect
14. #8201C2semidirect
15. #8148C2semidirect
169. Q8⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
#235682
#230882
#225092
C2×#15902
C2×#14972
#96602
#95532
#95432
#90742
#89492
#89482
#88482
#88342
#82012
#81482