Finite Group Databasebeta

← #191971

Subgroups — #191971

Order 512.

922 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2994C211yesmaximal, characteristic
2#21952C211yesmaximal, characteristic, complemented
3#21403C211yesmaximal, characteristic
4C2×#1565C211yesmaximal, characteristic, complemented
5C2×#1482C211yesmaximal, characteristic
6#8850C211yesmaximal, characteristic, complemented
7#8849C211yesmaximal, characteristic, complemented
8#8848C211yesmaximal, characteristic, complemented
9#8847C211yesmaximal, characteristic, complemented
10#8846C211yesmaximal, characteristic, complemented
11#8845C211yesmaximal, characteristic, complemented
12#8844C211yesmaximal, characteristic, complemented
13#8843C211yesmaximal, characteristic, complemented
14#7893C211yesmaximal, characteristic, complemented
15#7850C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17#2994—24no—
18#2994—24no—
19C2×#446C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×#435—21no—
23C2×C8×D4—21no—
24C2×C8×D4C2211yescharacteristic
25C2×#433—21no—
26C2×#431C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C2×#424—21no—
30C22×C4⋊C8C2211yescharacteristic
31C2×C4×M16C2211yescharacteristic
32#2285C2211yescharacteristic
33#1992C411yescharacteristic
34#1992C411yescharacteristic
35#1959C411yescharacteristic
36#1959C411yescharacteristic
37#1939—21no—
38#1939—21no—
39#1939C2211yescharacteristic
40#1939C2211yescharacteristic
41#1939C2211yescharacteristic
42#1939C2211yescharacteristic
43#1895C411yescharacteristic
44#1895C411yescharacteristic
45#1894C411yescharacteristic
46#1894C411yescharacteristic
47#1798—21no—
48#1798—21no—
49C4×#103C2211yescharacteristic
50C4×#103C2211yescharacteristic
51C4×#102C2211yescharacteristic
52C4×#102C2211yescharacteristic
53#1565—24no—
54C2×#335C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58#1482—24no—
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69#1273C2211yescharacteristic
70#1273C2211yescharacteristic
71C22×M16—21no—
72C22×M16C2×C411yescharacteristic
73C22×M16C2×C411yescharacteristic
74C23×C8—21no—
75C23×C8—21no—
76#529—22no—
77#529—22no—
78#529D414yes—
79#529Q814yes—
80C2×#126—21no—
81C2×#125C2×C411yescharacteristic
82C2×#124C2×C411yescharacteristic
83C2×#123—21no—
84C2×C22⋊Q8C2×C411yescharacteristic
85C2×#121C2×C411yescharacteristic
86C2×C4×Q8C2311yescharacteristic
87C2×C4×D4—21no—
88C2×C4×D4C2311yescharacteristic
89C2×#117—21no—
90C2×#117—21no—
91C22×C42C2311yescharacteristic
92#446—22no—
93#446—22no—
94C8×Q8—22no—
95C8×Q8—22no—
96#436—22no—
97#436—22no—
98#435—28no—
99C8×D4—28no—
100C8×D4—22no—
101C8×D4—22no—
102#433—28no—
103#431—22no—
104#431—22no—
105#431—22no—
106#431—22no—
107#431—22no—
108#431—22no—
109#424—28no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—22no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8D411yescharacteristic
120C2×C4⋊C8D411yescharacteristic
121C2×C4⋊C8D414yes—
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×#103C2×C411yescharacteristic
127C2×#103C2×C411yescharacteristic
128C2×#103C2×C411yescharacteristic
129C2×#103C2×C411yescharacteristic
130C2×#102C2×C411yescharacteristic
131C2×#102C2×C411yescharacteristic
132C2×#102C2×C411yescharacteristic
133C2×#102C2×C411yescharacteristic
134C2×C22⋊C8—21no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8—21no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8D411yescharacteristic
139C2×C22⋊C8D411yescharacteristic
140C4×M16—22no—
141C4×M16—22no—
142C2×#97—21no—
143C2×#97C2×C411yescharacteristic
144C2×#97C2×C411yescharacteristic
145C2×C4×C8—21no—
146C2×C4×C8—21no—
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153#384C2×C411yescharacteristic
154#384C2×C411yescharacteristic
155#382—21no—
156#380—21no—
157#380C2×C411yescharacteristic
158#380C2×C411yescharacteristic
159C4×C4⋊C4—21no—
160C4×C4⋊C4C2311yescharacteristic
161C4×C4⋊C4C2311yescharacteristic
162#340C2×C411yescharacteristic
163#340C2×C411yescharacteristic
164#340C2×C411yescharacteristic
165#340C2×C411yescharacteristic
166#340C2×C411yescharacteristic
167#340C2×C411yescharacteristic
168#340C2×C411yescharacteristic
169#340C2×C411yescharacteristic
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#335—22no—
175#335—22no—
176#334—22no—
177#334—22no—
178Q8⋊C8—22no—
179Q8⋊C8—22no—
180D4⋊C8—22no—
181D4⋊C8—22no—
182C22×Q8C22×C411yescharacteristic
183C22×D4—21no—
184C22×D4C22×C411yescharacteristic
185C23×C4—21no—
186C23×C4C22×C411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4#4111yescharacteristic
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—24no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—41no—
195C2×M16—41no—
196C2×M16—81no—
197C2×M16—81no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—24no—
201C22×C8—24no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—81no—
207C22×C8—81no—
208C22×C8—81no—
209C22×C8—81no—
210C22×C8C22×C411yescharacteristic
211C22×C8C22×C411yescharacteristic
212C22×C8C22×C411yescharacteristic
213C22×C8C22×C411yescharacteristic
214C22×C8C2×D411yescharacteristic
215C22×C8C2×D411yescharacteristic
216C22×C8C2×D411yescharacteristic
217C22×C8C2×D411yescharacteristic
218#126—24no—
219#126—24no—
220#125—22no—
221#125C4⋊C414yes—
222#124—22no—
223#124C4⋊C414yes—
224#123—24no—
225#123—24no—
226C22⋊Q8—22no—
227C22⋊Q8C4⋊C414yes—
228#121—22no—
229#121C4⋊C414yes—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8C2×D412yes—
233C4×Q8C2×Q812yes—
234C4×D4—24no—
235C4×D4—24no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4C2×D412yes—
239C4×D4C2×Q812yes—
240#117—24no—
241#117—24no—
242#117—81no—
243#117—81no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42C22×C411yescharacteristic
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D411yescharacteristic
272C2×C42C2×Q812yes—
273C2×C42#4112yes—
274C2×C42#4111yescharacteristic
275C2×C42C2411yescharacteristic
276C4⋊C8—24no—
277C4⋊C8—24no—
278C4⋊C8—24no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—42no—
288C4⋊C8—81no—
289C4⋊C8—81no—
290C4⋊C8—81no—
291C4⋊C8—81no—
292C4⋊C8C2×D412yes—
293C4⋊C8C2×D412yes—
294#103—22no—
295#103—22no—
296#103—41no—
297#103—41no—
298#102—22no—
299#102—22no—
300#102—41no—
301#102—41no—
302C22⋊C8—24no—
303C22⋊C8—24no—
304C22⋊C8—22no—
305C22⋊C8—22no—
306C22⋊C8—42no—
307C22⋊C8—42no—
308#97—24no—
309#97—22no—
310#97—22no—
311C4×C8—24no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—42no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329C24—21no—
330C24C4×D411yescharacteristic
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8C2×C4⋊C412yes—
337C2×D4—24no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4C2×C4⋊C412yes—
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—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4C2×C4⋊C412yes—
370C22×C4C4×D412yes—
371C22×C4C4×D411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4C4×D411yescharacteristic
374C22×C4#12012yes—
375C22×C4C22⋊Q812yes—
376C22×C4C22⋊Q812yes—
377C22×C4#12312yes—
378C22×C4C23×C411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C2×#4111yescharacteristic
381M16—28no—
382M16—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—21no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—21no—
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—21no—
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—41no—
424C2×C8—84no—
425C2×C8—84no—
426C2×C8#13112yes—
427C2×C8#13112yes—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—81no—
446C4⋊C4—81no—
447C4⋊C4C2×C4⋊C412yes—
448C4⋊C4C2×C4⋊C412yes—
449C22⋊C4—24no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—81no—
456C22⋊C4—81no—
457C42—22no—
458C42—22no—
459C42—41no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42C2×C4⋊C412yes—
464C42C4×D412yes—
465C42#12012yes—
466C42C22⋊Q812yes—
467C42C22⋊Q812yes—
468C42#12312yes—
469C42C22×D411yescharacteristic
470C42C22×D411yescharacteristic
471C42C22×Q811yescharacteristic
472C42C2×#4111yescharacteristic
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—22no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23#38512yes—
487C23C2×C4×D411yescharacteristic
488Q8—22no—
489Q8—22no—
490Q8—41no—
491Q8—41no—
492D4—22no—
493D4—22no—
494D4—48no—
495D4—41no—
496D4—41no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
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—41no—
540C2×C4#38512yes—
541C2×C4#38512yes—
542C2×C4#38512yes—
543C2×C4C22×C4⋊C411yescharacteristic
544C2×C4C2×C4×D411yescharacteristic
545C2×C4C2×#12011yescharacteristic
546C2×C4C2×C22⋊Q811yescharacteristic
547C2×C4C2×C22⋊Q811yescharacteristic
548C2×C4C2×#12311yescharacteristic
549C2×C4C2×#13111yescharacteristic
550C2×C4C2×#13111yescharacteristic
551C2×C4#56811yescharacteristic
552C2×C4#56811yescharacteristic
553C2×C4#57611yescharacteristic
554C2×C4#57711yescharacteristic
555C2×C4#57711yescharacteristic
556C2×C4#57811yescharacteristic
557C8—22no—
558C8—22no—
559C8—41no—
560C8—41no—
561C8—41no—
562C8—41no—
563C8—42no—
564C8—42no—
565C8—41no—
566C8—41no—
567C8—81no—
568C8—81no—
569C8—81no—
570C8—81no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C2×#38511yescharacteristic
586C22#293511yescharacteristic
587C22#293511yescharacteristic
588C22#294611yescharacteristic
589C22#294711yescharacteristic
590C22#295511yescharacteristic
591C22#295511yescharacteristic
592C4—22no—
593C4—22no—
594C4—21no—
595C4—21no—
596C4—22no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4—81no—
604C4#290611yescharacteristic
605C4#290611yescharacteristic
606C4#292911yescharacteristic
607C4#293011yescharacteristic
608C4#301211yescharacteristic
609C4#301311yescharacteristic
610C4#308311yescharacteristic
611C4#308411yescharacteristic
612C2—42no—
613C2—82no—
614C2#1808011yescharacteristic
615C2#1808111yescharacteristic
616C2#1808211yescharacteristic
617C2#1808311yescharacteristic
618C2#2083811yescharacteristic
619C2#2087811yescharacteristic
620C2#2088411yescharacteristic
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 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21952C2semidirect
4. C2×#1565C2semidirect
6. #8850C2semidirect
7. #8849C2semidirect
8. #8848C2semidirect
9. #8847C2semidirect
10. #8846C2semidirect
11. #8845C2semidirect
12. #8844C2semidirect
13. #8843C2semidirect
14. #7893C2semidirect
15. #7850C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29942
#219522
#214032
C2×#15652
C2×#14822
#88502
#88492
#88482
#88472
#88462
#88452
#88442
#88432
#78932
#78502