Finite Group Databasebeta

← #217691

Subgroups — #217691

Order 512.

769 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22098C211yesmaximal, characteristic, complemented
2#21847C211yesmaximal, characteristic
3C4×#452C211yesmaximal, characteristic
4#9658C211yesmaximal, characteristic, complemented
5#9655C211yesmaximal, characteristic, complemented
6#9556C211yesmaximal, characteristic, complemented
7#9553C211yesmaximal, characteristic, complemented
8#9543C211yesmaximal, characteristic, complemented
9#9542C211yesmaximal, characteristic, complemented
10#8948C211yesmaximal, characteristic, complemented
11#8946C211yesmaximal, characteristic, complemented
12#8937C211yesmaximal, characteristic, complemented
13#8830C211yesmaximal, characteristic, complemented
14#8312C211yesmaximal, characteristic
15#8148C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#452C411yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×SD16C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C4×C22⋊Q8C2211yescharacteristic
24#1984C2211yescharacteristic
25#1984C2211yescharacteristic
26#1984C2211yescharacteristic
27#1954C2211yescharacteristic
28#1954C2211yescharacteristic
29#1939—21no—
30#1939C2211yescharacteristic
31#1923—21no—
32#1923—21no—
33#1916—21no—
34#1878C411yescharacteristic
35#1877C411yescharacteristic
36#1873C411yescharacteristic
37#1869—21no—
38#1869—21no—
39#1865C411yescharacteristic
40#1862C411yescharacteristic
41#1854C411yescharacteristic
42#1797—21no—
43#1797—21no—
44#1797C2211yescharacteristic
45#1788C411yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#334C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×#41C2×C411yescharacteristic
68C22×Q16C2×C411yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C23×C8—21no—
71C2×C22⋊Q8C2×C411yescharacteristic
72C4×#41—41no—
73C4×#41—41no—
74C4×#41—41no—
75C4×#41—41no—
76C2×C4×Q8C2×C411yescharacteristic
77C2×C4×Q8C2311yescharacteristic
78C2×C4×Q8C2311yescharacteristic
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4C2×C411yescharacteristic
82C2×C4×D4C2311yescharacteristic
83C2×#117—21no—
84C2×#117C2×C411yescharacteristic
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87#452C818yescomplemented
88C8×Q8—41no—
89C8×Q8—41no—
90C4×Q16—41no—
91C4×Q16—41no—
92C4×SD16—41no—
93C4×SD16—41no—
94#431—41no—
95#431—41no—
96C2×#107C2×C411yescharacteristic
97C2×#106C2×C411yescharacteristic
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—41no—
102C2×C4⋊C8—41no—
103C2×C4⋊C8D411yescharacteristic
104C2×C4⋊C8D411yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×#103C2×C411yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#103C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8C2×C411yescharacteristic
124C2×C22⋊C8C2×C411yescharacteristic
125C2×C22⋊C8D411yescharacteristic
126C2×C22⋊C8D411yescharacteristic
127C2×C4×C8—21no—
128C2×C4×C8—21no—
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135#391C2×C411yescharacteristic
136#389C2×C411yescharacteristic
137#387—21no—
138#385—21no—
139#380C2×C411yescharacteristic
140C4×C4⋊C4—21no—
141C4×C4⋊C4C2311yescharacteristic
142C4×C22⋊C4—21no—
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336C2×C411yescharacteristic
158#336C2×C411yescharacteristic
159#334—41no—
160#334—41no—
161Q8⋊C8—41no—
162Q8⋊C8—41no—
163Q8⋊C8—41no—
164Q8⋊C8—41no—
165D4⋊C8—41no—
166D4⋊C8—41no—
167#322—41no—
168C82—41no—
169C2×#41—22no—
170C2×#41—41no—
171C2×#41—41no—
172C2×#41—41no—
173C2×#41—41no—
174C2×#41C2×C814yes—
175C22×Q8C22×C411yescharacteristic
176C22×Q8C22×C411yescharacteristic
177C22×D4—21no—
178C22×D4C22×C411yescharacteristic
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4C22×C411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4#4111yescharacteristic
185C2×Q16—41no—
186C2×Q16—41no—
187C2×Q16C2×C814yescomplemented
188C2×SD16—41no—
189C2×SD16—41no—
190C2×SD16C2×C814yescomplemented
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—42no—
198C22×C8—81no—
199C22×C8—81no—
200C22×C8C22×C411yescharacteristic
201C22×C8C22×C411yescharacteristic
202C22×C8C22×C411yescharacteristic
203C22×C8C22×C411yescharacteristic
204C22×C8C2×D411yescharacteristic
205C22×C8C2×D411yescharacteristic
206C22×C8C2×D411yescharacteristic
207C22×C8C2×D411yescharacteristic
208C22⋊Q8—22no—
209C22⋊Q8C2×C814yes—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×Q8—21no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×Q8—21no—
217C4×Q8—21no—
218C4×Q8—41no—
219C4×Q8—41no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4—41no—
225C4×D4—41no—
226C4×D4—42no—
227C4×D4—42no—
228C4×D4—42no—
229C4×D4—42no—
230#117—42no—
231#117—42no—
232#117—41no—
233#117—41no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42C22×C411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42#4111yescharacteristic
263C2×C42C2411yescharacteristic
264#107—41no—
265#106—41no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—41no—
275C4⋊C8—41no—
276C4⋊C8—42no—
277C4⋊C8—42no—
278C4⋊C8—81no—
279#103—41no—
280#103—41no—
281#103—41no—
282#103—41no—
283#103C2×C814yescomplemented
284#102—41no—
285#102—41no—
286#102C2×C814yescomplemented
287C22⋊C8—22no—
288C22⋊C8—41no—
289C22⋊C8—41no—
290C22⋊C8—42no—
291C22⋊C8—81no—
292C22⋊C8—81no—
293C22⋊C8—81no—
294C22⋊C8—81no—
295C22⋊C8—81no—
296C22⋊C8C2×C814yes—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—42no—
310C4×C8—81no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316C24—21no—
317C24C4×D411yescharacteristic
318#41—44no—
319#41—44no—
320#41—44no—
321#41—44no—
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×Q8C22×C812yes—
333C2×Q8C22×C812yes—
334C2×D4—24no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4C22×C812yes—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
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—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—81no—
375C22×C4—81no—
376C22×C4C4×D411yescharacteristic
377C22×C4C4×D411yescharacteristic
378C22×C4C4×D411yescharacteristic
379C22×C4C22×C812yes—
380C22×C4#13112yes—
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C2×#4111yescharacteristic
384Q16—44no—
385SD16—44no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—22no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
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—21no—
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—41no—
425C2×C8—41no—
426C2×C8—84no—
427C2×C8C22×C812yes—
428C2×C8C22×C812yes—
429C4⋊C4—21no—
430C4⋊C4—24no—
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—42no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—81no—
445C4⋊C4C22×C812yes—
446C22⋊C4—24no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—81no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—42no—
459C42—42no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42H(5,𝔽2)11yescharacteristic
467C42H(5,𝔽2)11yescharacteristic
468C42H(5,𝔽2)11yescharacteristic
469C42H(5,𝔽2)11yescharacteristic
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—22no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23C8×D412yes—
484C23C2×C4×D411yescharacteristic
485Q8—22no—
486Q8—22no—
487Q8—22no—
488Q8—22no—
489Q8—41no—
490Q8—41no—
491Q8—41no—
492Q8—41no—
493D4—22no—
494D4—22no—
495D4—48no—
496D4—41no—
497D4—41no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—48no—
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—81no—
539C2×C4—81no—
540C2×C4—84no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4C8×D412yes—
546C2×C4#51811yescharacteristic
547C2×C4#51811yescharacteristic
548C2×C4#53411yescharacteristic
549C2×C4#53511yescharacteristic
550C2×C4#53811yescharacteristic
551C2×C4#53811yescharacteristic
552C2×C4C23×C811yescharacteristic
553C2×C4C2×#13111yescharacteristic
554C2×C4#56811yescharacteristic
555C2×C4#56811yescharacteristic
556C2×C4#57711yescharacteristic
557C2×C4#57711yescharacteristic
558C2×C4#57711yescharacteristic
559C2×C4#57811yescharacteristic
560C8—22no—
561C8—22no—
562C8—42no—
563C8—41no—
564C8—41no—
565C8—41no—
566C8—41no—
567C8—41no—
568C8—41no—
569C8—81no—
570C8—81no—
571C8—81no—
572C8—81no—
573C8—81no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22#232111yescharacteristic
589C22C2×C8×D411yescharacteristic
590C22#293011yescharacteristic
591C22#294711yescharacteristic
592C22#295211yescharacteristic
593C22#295611yescharacteristic
594C22#295711yescharacteristic
595C4—22no—
596C4—21no—
597C4—21no—
598C4—42no—
599C4—42no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—81no—
604C4—81no—
605C4—81no—
606C4—81no—
607C4#297211yescharacteristic
608C4#297211yescharacteristic
609C4#297611yescharacteristic
610C4#297611yescharacteristic
611C4#320111yescharacteristic
612C4#320411yescharacteristic
613C4#322211yescharacteristic
614C4#322511yescharacteristic
615C2—42no—
616C2—82no—
617C2#1785311yescharacteristic
618C2#1785811yescharacteristic
619C2#1794911yescharacteristic
620C2#1795511yescharacteristic
621C2#2202811yescharacteristic
622C2#2205911yescharacteristic
623C2#2206211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×#41C2×C4
LCSG ▷ C2×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
1. #22098C2semidirect
4. #9658C2semidirect
5. #9655C2semidirect
6. #9556C2semidirect
7. #9553C2semidirect
8. #9543C2semidirect
9. #9542C2semidirect
10. #8948C2semidirect
11. #8946C2semidirect
12. #8937C2semidirect
13. #8830C2semidirect
15. #8148C2semidirect
87. #452C8semidirect
187. C2×Q16C2×C8semidirect
190. C2×SD16C2×C8semidirect
283. #103C2×C8semidirect
286. #102C2×C8semidirect

Maximal subgroups

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

subgroupindex
#220982
#218472
C4×#4522
#96582
#96552
#95562
#95532
#95432
#95422
#89482
#89462
#89372
#88302
#83122
#81482