Finite Group Databasebeta

← #189635

Subgroups — #189635

Order 512.

827 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2993C211yesmaximal, characteristic
2#21993C211yesmaximal, characteristic, complemented
3#17806C211yesmaximal, characteristic, complemented
4#17492C211yesmaximal, characteristic
5#17487C211yesmaximal, characteristic
6#8665C211yesmaximal, characteristic, complemented
7#8664C211yesmaximal, characteristic, complemented
8#8663C211yesmaximal, characteristic, complemented
9#8662C211yesmaximal, characteristic, complemented
10#8661C211yesmaximal, characteristic, complemented
11#8660C211yesmaximal, characteristic, complemented
12#8659C211yesmaximal, characteristic, complemented
13#8658C211yesmaximal, characteristic, complemented
14#8079C211yesmaximal, characteristic, complemented
15#8078C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17#2993—21no—
18#2993—21no—
19#2993—21no—
20#2993—21no—
21#2993C412yes—
22#2993C412yes—
23#2993C412yes—
24#2993C412yes—
25C2×#446C2211yescharacteristic
26C2×C8×Q8C2211yescharacteristic
27C2×#436—21no—
28C2×#436C2211yescharacteristic
29C2×#435—21no—
30C2×C8×D4C2211yescharacteristic
31#2920C2211yescharacteristic
32#2920C2211yescharacteristic
33C2×#433—21no—
34C2×#431C2211yescharacteristic
35C2×#431C2211yescharacteristic
36C2×#424—21no—
37C2×#423C2211yescharacteristic
38C2×#423C2211yescharacteristic
39#1950—21no—
40#1950—21no—
41#1855—21no—
42#1855—21no—
43#1848C2211yescharacteristic
44#1848C2211yescharacteristic
45#1837C2211yescharacteristic
46#1837C2211yescharacteristic
47#1574—21no—
48#1574—21no—
49#1573—21no—
50#1573—21no—
51#1564C2211yescharacteristic
52#1564C2211yescharacteristic
53#1564C2211yescharacteristic
54#1564C2211yescharacteristic
55#1495—22no—
56#1494—22no—
57#1487C2211yescharacteristic
58#1487C2211yescharacteristic
59#1487C2211yescharacteristic
60#1487C2211yescharacteristic
61#1486C2211yescharacteristic
62#1486C2211yescharacteristic
63#1486C2211yescharacteristic
64#1486C2211yescharacteristic
65#1483—22no—
66#1482—22no—
67#1301C2211yescharacteristic
68#1301C2211yescharacteristic
69#1301C2211yescharacteristic
70#1301C2211yescharacteristic
71#1301C2211yescharacteristic
72#1301C2211yescharacteristic
73#1301C2211yescharacteristic
74#1301C2211yescharacteristic
75C22×M16—21no—
76C22×M16—21no—
77C22×M16—21no—
78C23×C8—21no—
79#529—21no—
80#529—21no—
81#529—21no—
82#529—21no—
83#529C2×C412yes—
84#529C2×C412yes—
85#529D412yes—
86#529D412yes—
87C2×#126—21no—
88C2×#125Q811yescharacteristic
89C2×#124Q811yescharacteristic
90C2×#123—21no—
91C2×C22⋊Q8Q811yescharacteristic
92C2×#121Q811yescharacteristic
93C2×C4×Q8C2311yescharacteristic
94C2×C4×D4—21no—
95C2×C4×D4C2311yescharacteristic
96C2×#117—21no—
97C22×C42C2311yescharacteristic
98#446—21no—
99#446—21no—
100#446C2×C412yes—
101#446C2×C412yes—
102C8×Q8—21no—
103C8×Q8—21no—
104C8×Q8C2×C412yes—
105C8×Q8C2×C412yes—
106#436—22no—
107#436—22no—
108#436—22no—
109#436—22no—
110#436—21no—
111#436—21no—
112#436C2×C412yes—
113#436C2×C412yes—
114#435—22no—
115#435—22no—
116#435—22no—
117#435—22no—
118C8×D4—21no—
119C8×D4—21no—
120C8×D4C2×C412yes—
121C8×D4C2×C412yes—
122#433—22no—
123#433—22no—
124#433—22no—
125#433—22no—
126#432—21no—
127#432—21no—
128#432—21no—
129#432—21no—
130#431—21no—
131#431—21no—
132#431—21no—
133#431—21no—
134#431—21no—
135#431—21no—
136#431C2×C411yescharacteristic
137#431C2×C411yescharacteristic
138#431C2×C411yescharacteristic
139#431C2×C411yescharacteristic
140#431D411yescharacteristic
141#431D411yescharacteristic
142#431D411yescharacteristic
143#431D411yescharacteristic
144#431C2311yescharacteristic
145#431C2311yescharacteristic
146#431C2311yescharacteristic
147#431C2311yescharacteristic
148C2×#108—21no—
149C2×#108—21no—
150C2×#108—21no—
151C2×#108—21no—
152#424—22no—
153#424—22no—
154#424—22no—
155#424—22no—
156#423—21no—
157#423—21no—
158#423—21no—
159#423—21no—
160#423—21no—
161#423—21no—
162#423C2×C412yes—
163#423C2×C412yes—
164#423C2311yescharacteristic
165#423C2311yescharacteristic
166#423C2311yescharacteristic
167#423C2311yescharacteristic
168C2×C4⋊C8—21no—
169C2×C4⋊C8—21no—
170C2×C4⋊C8—21no—
171C2×C4⋊C8—21no—
172C2×C4⋊C8—21no—
173C2×C4⋊C8—21no—
174C2×C4⋊C8D411yescharacteristic
175C2×C4⋊C8D411yescharacteristic
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2311yescharacteristic
178#407—21no—
179#407—21no—
180C2×C22⋊C8—21no—
181C2×C22⋊C8—21no—
182C2×C22⋊C8—21no—
183C2×C22⋊C8D411yescharacteristic
184C2×C22⋊C8D411yescharacteristic
185C4×M16—21no—
186C4×M16—21no—
187C2×#97—21no—
188C2×C4×C8—21no—
189C2×C4×C8—21no—
190C2×C4×C8C2311yescharacteristic
191C2×C4×C8C2311yescharacteristic
192#341—21no—
193#341—21no—
194#341—21no—
195#341—21no—
196#335—21no—
197#335—21no—
198#335—21no—
199#335—21no—
200#334—21no—
201#334—21no—
202#334—21no—
203#334—21no—
204Q8⋊C8—22no—
205Q8⋊C8—22no—
206Q8⋊C8—22no—
207Q8⋊C8—22no—
208D4⋊C8—22no—
209D4⋊C8—22no—
210D4⋊C8—22no—
211D4⋊C8—22no—
212C22×Q8C2×Q811yescharacteristic
213C22×D4—21no—
214C22×D4C2×Q811yescharacteristic
215C23×C4—21no—
216C23×C4C2×D411yescharacteristic
217C23×C4C2×Q811yescharacteristic
218C23×C4#4111yescharacteristic
219C2×M16—22no—
220C2×M16—22no—
221C2×M16—22no—
222C2×M16—22no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—41no—
228C2×M16—41no—
229C2×M16—81no—
230C2×M16—81no—
231C2×M16—81no—
232C2×M16—81no—
233C2×M16—81no—
234C2×M16—81no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—41no—
240C22×C8—41no—
241C22×C8—81no—
242C22×C8—81no—
243C22×C8C2×D411yescharacteristic
244C22×C8C2×D411yescharacteristic
245C22×C8C2×D411yescharacteristic
246C22×C8C2×D411yescharacteristic
247#126—22no—
248#126—22no—
249#126—22no—
250#126—22no—
251#125—21no—
252#125—21no—
253#125C4⋊C412yes—
254#125C4⋊C412yes—
255#124—21no—
256#124—21no—
257#124C4⋊C412yes—
258#124C4⋊C412yes—
259#123—22no—
260#123—22no—
261#123—22no—
262#123—22no—
263C22⋊Q8—21no—
264C22⋊Q8—21no—
265C22⋊Q8C4⋊C412yes—
266C22⋊Q8C4⋊C412yes—
267#121—21no—
268#121—21no—
269#121C4⋊C412yes—
270#121C4⋊C412yes—
271C4×Q8—21no—
272C4×Q8—21no—
273C4×Q8C22×C412yes—
274C4×Q8C2×D412yes—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4C22×C412yes—
282C4×D4C2×D412yes—
283#117—22no—
284#117—22no—
285#117—22no—
286#117—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4C2×Q811yescharacteristic
292C2×C4⋊C4C2×Q811yescharacteristic
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C42—21no—
299C2×C42C22×C411yescharacteristic
300C2×C42C22×C411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×Q811yescharacteristic
307C2×C42#4111yescharacteristic
308C2×C42#4111yescharacteristic
309C2×C42#4111yescharacteristic
310C2×C42C2411yescharacteristic
311#108—81no—
312#108—81no—
313#108—81no—
314#108—81no—
315C4⋊C8—22no—
316C4⋊C8—22no—
317C4⋊C8—22no—
318C4⋊C8—22no—
319C4⋊C8—22no—
320C4⋊C8—22no—
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8—21no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8—21no—
333C4⋊C8—21no—
334C4⋊C8—21no—
335C4⋊C8—41no—
336C4⋊C8—41no—
337C4⋊C8—41no—
338C4⋊C8—41no—
339C4⋊C8C22⋊C411yescharacteristic
340C4⋊C8C22⋊C411yescharacteristic
341C4⋊C8C22⋊C411yescharacteristic
342C4⋊C8C22⋊C411yescharacteristic
343C4⋊C8C22⋊C411yescharacteristic
344C4⋊C8C22⋊C411yescharacteristic
345C4⋊C8C22⋊C411yescharacteristic
346C4⋊C8C22⋊C411yescharacteristic
347C4⋊C8C22×C412yes—
348C4⋊C8C22×C412yes—
349C22⋊C8—22no—
350C22⋊C8—22no—
351C22⋊C8—22no—
352C22⋊C8—22no—
353C22⋊C8—21no—
354C22⋊C8—21no—
355C22⋊C8—21no—
356C22⋊C8—21no—
357C22⋊C8—41no—
358C22⋊C8—41no—
359C22⋊C8—41no—
360C22⋊C8—41no—
361C22⋊C8C22⋊C411yescharacteristic
362C22⋊C8C22⋊C411yescharacteristic
363C22⋊C8C22⋊C411yescharacteristic
364C22⋊C8C22⋊C411yescharacteristic
365C22⋊C8C22⋊C411yescharacteristic
366C22⋊C8C22⋊C411yescharacteristic
367C22⋊C8C22⋊C411yescharacteristic
368C22⋊C8C22⋊C411yescharacteristic
369#97—22no—
370#97—22no—
371#97—41no—
372#97—41no—
373C4×C8—22no—
374C4×C8—22no—
375C4×C8—21no—
376C4×C8—21no—
377C4×C8—21no—
378C4×C8—21no—
379C4×C8—41no—
380C4×C8—41no—
381C4×C8C22×C411yescharacteristic
382C4×C8C22×C411yescharacteristic
383C4×C8C22×C411yescharacteristic
384C4×C8C22×C411yescharacteristic
385C4×C8C2×D411yescharacteristic
386C4×C8C2×D411yescharacteristic
387C4×C8C2×D411yescharacteristic
388C4×C8C2×D411yescharacteristic
389C24—21no—
390C24C22⋊Q811yescharacteristic
391C2×Q8—21no—
392C2×Q8—42no—
393C2×Q8C2×C4⋊C412yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—42no—
398C2×D4—82no—
399C2×D4C2×C4⋊C412yes—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—42no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—81no—
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C4⋊C411yescharacteristic
418C22×C4C2×C4⋊C411yescharacteristic
419C22×C4#11711yescharacteristic
420C22×C4#11711yescharacteristic
421C22×C4#12111yescharacteristic
422C22×C4#12111yescharacteristic
423C22×C4#12111yescharacteristic
424C22×C4#12111yescharacteristic
425C22×C4C22⋊Q811yescharacteristic
426C22×C4C22⋊Q811yescharacteristic
427C22×C4C22⋊Q811yescharacteristic
428C22×C4C22⋊Q811yescharacteristic
429C22×C4C22⋊Q811yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×Q811yescharacteristic
432C22×C4C2×#4111yescharacteristic
433M16—41no—
434M16—41no—
435M16—41no—
436M16—41no—
437M16—41no—
438M16—41no—
439M16—41no—
440M16—41no—
441M16—82no—
442M16—82no—
443M16—82no—
444M16—82no—
445M16—82no—
446M16—82no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—82no—
470C2×C8—82no—
471C2×C8—82no—
472C2×C8—82no—
473C2×C8C2×C22⋊C411yescharacteristic
474C2×C8C2×C22⋊C411yescharacteristic
475C2×C8C2×C22⋊C411yescharacteristic
476C2×C8C2×C22⋊C411yescharacteristic
477C2×C8C2×C22⋊C411yescharacteristic
478C2×C8C2×C22⋊C411yescharacteristic
479C2×C8C2×C22⋊C411yescharacteristic
480C2×C8C2×C22⋊C411yescharacteristic
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4C2×C4⋊C412yes—
492C4⋊C4C2×C4⋊C412yes—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C42—22no—
504C42—22no—
505C42C2×C22⋊C411yescharacteristic
506C42C2×C22⋊C411yescharacteristic
507C42C2×C4⋊C411yescharacteristic
508C42C2×C4⋊C411yescharacteristic
509C42#11711yescharacteristic
510C42#11711yescharacteristic
511C42#12111yescharacteristic
512C42#12111yescharacteristic
513C42#12111yescharacteristic
514C42#12111yescharacteristic
515C42C22⋊Q811yescharacteristic
516C42C22⋊Q811yescharacteristic
517C42C23×C411yescharacteristic
518C42C22×D411yescharacteristic
519C42C22×D411yescharacteristic
520C42C2×#4111yescharacteristic
521C23—21no—
522C23—21no—
523C23—22no—
524C23—41no—
525C23—41no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23#38011yescharacteristic
531C23#38011yescharacteristic
532C23C2×C22⋊Q811yescharacteristic
533Q8—24no—
534Q8—42no—
535D4—24no—
536D4—42no—
537D4—82no—
538D4—82no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4#38011yescharacteristic
573C2×C4#38011yescharacteristic
574C2×C4#38011yescharacteristic
575C2×C4#38011yescharacteristic
576C2×C4#38011yescharacteristic
577C2×C4#38011yescharacteristic
578C2×C4C22×C22⋊C411yescharacteristic
579C2×C4C22×C4⋊C411yescharacteristic
580C2×C4C2×#11711yescharacteristic
581C2×C4C2×#12111yescharacteristic
582C2×C4C2×#12111yescharacteristic
583C2×C4C2×C22⋊Q811yescharacteristic
584C2×C4#57611yescharacteristic
585C2×C4#57711yescharacteristic
586C2×C4#57711yescharacteristic
587C2×C4#57811yescharacteristic
588C8—21no—
589C8—21no—
590C8—21no—
591C8—21no—
592C8—21no—
593C8—21no—
594C8—21no—
595C8—21no—
596C8—42no—
597C8—42no—
598C8—81no—
599C8—81no—
600C8—81no—
601C8—81no—
602C8—81no—
603C8—81no—
604C8—81no—
605C8—81no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—81no—
618C22—81no—
619C22—82no—
620C22—82no—
621C22C2×#38011yescharacteristic
622C22#308311yescharacteristic
623C22#308411yescharacteristic
624C4—21no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—21no—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4#289711yescharacteristic
637C4#289811yescharacteristic
638C4#291612yes—
639C4#304511yescharacteristic
640C4#304611yescharacteristic
641C2—21no—
642C2—21no—
643C2—41no—
644C2—41no—
645C2—82no—
646C2#2065211yescharacteristic
647C2#2066811yescharacteristic
648C2#2067511yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#380
prime radicalC2×#121Q8
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21993C2semidirect
3. #17806C2semidirect
6. #8665C2semidirect
7. #8664C2semidirect
8. #8663C2semidirect
9. #8662C2semidirect
10. #8661C2semidirect
11. #8660C2semidirect
12. #8659C2semidirect
13. #8658C2semidirect
14. #8079C2semidirect
15. #8078C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29932
#219932
#178062
#174922
#174872
#86652
#86642
#86632
#86622
#86612
#86602
#86592
#86582
#80792
#80782