Finite Group Databasebeta

← #6529079

Subgroups — #6529079

Order 512.

3230 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3503C211yesmaximal, characteristic, complemented
2C2×#3110C211yesmaximal, characteristic, complemented
3C22×#474C211yesmaximal, characteristic, complemented
4#34963C211yesmaximal, characteristic
5#24341C214yesmaximal, complemented
6#24298C214yesmaximal, complemented
7#24153C214yesmaximal, complemented
8#24130C214yesmaximal, complemented
9#24098C214yesmaximal, complemented
10#23969C214yesmaximal, complemented
11#21889C211yesmaximal, characteristic, complemented
12#21851C211yesmaximal, characteristic, complemented
13#21841C211yesmaximal, characteristic, complemented
14#3503—28no—
15#3503—28no—
16C2×Q82C2212yescomplemented
17C2×#557—21no—
18C2×#554C2214yes—
19C2×#554C2212yescomplemented
20C2×D4×Q8—22no—
21C2×D4×Q8—21no—
22C2×#544—21no—
23C2×#544—22no—
24C2×#533C2211yescharacteristic
25C2×#533C2212yescomplemented
26C22×C4⋊Q8C2211yescharacteristic
27C22×C4⋊Q8C2212yescomplemented
28C2×#520—21no—
29C2×C4×#41C2211yescharacteristic
30C22×C4×D4C2211yescharacteristic
31#3110—28no—
32C2×#484C2212yes—
33C2×#474—28no—
34C2×#474C2214yes—
35C2×#474C2214yes—
36C2×#474C2212yes—
37C2×#461C2212yes—
38C2×#450C2212yes—
39C2×#440C2214yes—
40C2×#432C2211yescharacteristic
41C22×#107C2212yes—
42C22×C4⋊C8C2211yescharacteristic
43C22×#102C2212yes—
44#2598—22no—
45#2580C2212yes—
46#2535C2212yes—
47#2487C2212yes—
48#2482—21no—
49#2408C2212yes—
50#2404—21no—
51#2398—21no—
52#2397—21no—
53C4×#121C2214yes—
54C4×#120—22no—
55#2283C2211yescharacteristic
56#2057C2218yescomplemented
57#2036C2218yescomplemented
58#2023C2218yes—
59#2000C2218yes—
60#1993C2214yescomplemented
61#1988C2214yescomplemented
62#1988C2212yescomplemented
63#1985C2212yescomplemented
64#1981C2212yescomplemented
65#1955C2212yescomplemented
66#1892C2214yes—
67#1880C2218yes—
68#1874C2218yes—
69#1869C2214yes—
70#1863C2218yescomplemented
71#1852C2218yescomplemented
72#1824C2214yes—
73#1805C2214yescomplemented
74#1797C2214yescomplemented
75C22×#41D411yescharacteristic
76C23×Q8—21no—
77C23×Q8—22no—
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C22×SD16C2314yes—
81C23×C8C2312yes—
82Q82—28no—
83Q82—28no—
84#557—216no—
85#554—28no—
86#554—216no—
87#554—28no—
88#554—216no—
89D4×Q8—232no—
90D4×Q8—216no—
91#544—232no—
92#544—216no—
93#533—28no—
94#533—28no—
95#533—24no—
96#533D418yescomplemented
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—28no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—24no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8D418yescomplemented
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2314yes—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2311yescharacteristic
112C2×#125—24no—
113C2×#125—22no—
114C2×#125—21no—
115C2×#125—22no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8—24no—
120C2×C22⋊Q8—24no—
121C2×C22⋊Q8C2314yes—
122C2×C22⋊Q8C2312yes—
123C2×#121C2314yes—
124C2×#120—22no—
125#520—216no—
126C4×#41—24no—
127C4×#41—24no—
128C2×C4×Q8—22no—
129C2×C4×Q8—21no—
130C2×C4×Q8—22no—
131C2×C4×Q8D411yescharacteristic
132C2×C4×Q8C2314yes—
133C2×C4×Q8C2312yes—
134C2×C4×Q8C2311yescharacteristic
135C2×C4×D4—24no—
136C2×C4×D4—22no—
137C2×C4×D4—24no—
138C2×C4×D4—21no—
139C2×C4×D4—21no—
140C2×C4×D4—24no—
141C2×C4×D4—24no—
142C2×C4×D4D411yescharacteristic
143C2×C4×D4D412yes—
144C2×C4×D4C2312yes—
145C2×C4×D4C2312yes—
146C2×C4×D4C2312yes—
147C2×C4×D4C2311yescharacteristic
148C2×#117—21no—
149C2×#117—22no—
150C2×#117D412yes—
151C2×#117D411yescharacteristic
152C22×C4⋊C4—21no—
153C22×C4⋊C4—22no—
154C22×C4⋊C4C2311yescharacteristic
155C22×C4⋊C4C2314yes—
156C22×C4⋊C4C2312yes—
157C22×C22⋊C4—21no—
158C22×C42C2312yes—
159C22×C42C2311yescharacteristic
160#484—28no—
161#474—28no—
162#474—216no—
163#474—216no—
164#461—28no—
165#450—28no—
166#440—216no—
167#432—24no—
168C2×#107—28no—
169C2×#107C2314yes—
170C2×#107C2318yes—
171C2×C4⋊C8—24no—
172C2×C4⋊C8C2312yes—
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2312yes—
175C2×#103C2314yes—
176C2×#103C2318yes—
177C2×#102—28no—
178C2×#102C2318yes—
179C2×#102C2314yes—
180C2×C22⋊C8C2312yes—
181C2×C22⋊C8C2318yes—
182C2×#97C2312yes—
183#400—22no—
184#399—22no—
185#399—22no—
186#397—22no—
187#394—22no—
188#393—22no—
189#392—22no—
190#392—22no—
191#390C2314yes—
192#389C2314yes—
193#388—24no—
194#386—22no—
195#386—24no—
196#386—24no—
197#385—24no—
198#385—22no—
199#383—21no—
200#382C2312yes—
201#381—21no—
202#380C2312yes—
203#380C2314yes—
204#379—22no—
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4—21no—
207C4×C22⋊C4—24no—
208#376—21no—
209#340C2318yes—
210#340C2318yes—
211#340C2318yes—
212#336C2318yes—
213C25—21no—
214C2×#41—28no—
215C2×#41—24no—
216C22×Q8—216no—
217C22×Q8—28no—
218C22×Q8—21no—
219C22×Q8—22no—
220C22×Q8—24no—
221C22×Q8—22no—
222C22×Q8—84no—
223C22×Q8—88no—
224C22×Q8C2×D411yescharacteristic
225C22×Q8C2412yes—
226C22×D4—24no—
227C22×D4—24no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—42no—
232C22×D4C2×D411yescharacteristic
233C22×D4C2×D412yes—
234C22×D4C2412yes—
235C23×C4—24no—
236C23×C4—22no—
237C23×C4—21no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—44no—
241C23×C4—84no—
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248C2×SD16—28no—
249C2×SD16—28no—
250C2×SD16—44no—
251C22×C8—24no—
252C22×C8—44no—
253C22×C8C2414yes—
254C22×C8C2414yes—
255C4⋊Q8—216no—
256C4⋊Q8—216no—
257C4⋊Q8—232no—
258C4⋊Q8—216no—
259C4⋊Q8—28no—
260C4⋊Q8—28no—
261C4⋊Q8—28no—
262C4⋊Q8—28no—
263C4⋊Q8—22no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—22no—
267C4⋊Q8C2×D414yes—
268C4⋊Q8C2×D414yes—
269C4⋊Q8C2×D418yes—
270C4⋊Q8C2×D418yes—
271#125—216no—
272#125—232no—
273#125—216no—
274#125—28no—
275C22⋊Q8—216no—
276C22⋊Q8—232no—
277C22⋊Q8—232no—
278C22⋊Q8—232no—
279C22⋊Q8—28no—
280C22⋊Q8—28no—
281C22⋊Q8—24no—
282C22⋊Q8—28no—
283C22⋊Q8C2×D418yes—
284#121—28no—
285#121—44no—
286#120—48no—
287C4×Q8—216no—
288C4×Q8—216no—
289C4×Q8—28no—
290C4×Q8—28no—
291C4×Q8—28no—
292C4×Q8—24no—
293C4×Q8—22no—
294C4×Q8—22no—
295C4×Q8—24no—
296C4×Q8C2×D418yes—
297C4×D4—232no—
298C4×D4—216no—
299C4×D4—28no—
300C4×D4—24no—
301C4×D4—24no—
302C4×D4—24no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4—24no—
306C4×D4—24no—
307C4×D4—24no—
308C4×D4—24no—
309C4×D4—44no—
310C4×D4—44no—
311C4×D4—88no—
312#117—216no—
313#117—28no—
314#117—24no—
315#117—28no—
316C2×C4⋊C4—216no—
317C2×C4⋊C4—28no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—28no—
332C2×C4⋊C4—28no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D418yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2414yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2414yes—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—44no—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C42—24no—
360C2×C42—22no—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—22no—
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D414yes—
369C2×C42C2×D414yes—
370C2×C42C2412yes—
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373#107—216no—
374#107—28no—
375C4⋊C8—24no—
376C4⋊C8—24no—
377C4⋊C8—24no—
378#103—28no—
379#103—48no—
380#102—216no—
381#102—28no—
382C22⋊C8—24no—
383C22⋊C8—48no—
384#97—24no—
385#95—22no—
386#95—22no—
387#95—24no—
388#95—22no—
389#95—24no—
390#95—22no—
391C24—22no—
392C24—44no—
393C24—42no—
394C24—44no—
395C24—82no—
396C24C22×D412yes—
397C24C22×D411yescharacteristic
398#41—232no—
399C2×Q8—216no—
400C2×Q8—22no—
401C2×Q8—28no—
402C2×Q8—24no—
403C2×Q8—28no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—48no—
409C2×Q8—44no—
410C2×Q8—84no—
411C2×Q8—88no—
412C2×Q8—816no—
413C2×Q8—832no—
414C2×Q8C22×D414yes—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—28no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—48no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—48no—
429C2×D4—48no—
430C2×D4—42no—
431C2×D4—84no—
432C2×D4#13718yes—
433C2×D42−1+414yes—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—216no—
445C22×C4—28no—
446C22×C4—24no—
447C22×C4—28no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—84no—
459C22×C4—816no—
460C22×C4—82no—
461C22×C4—84no—
462C22×C4—88no—
463C22×C4#12014yes—
464C22×C4#12014yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C2511yescharacteristic
476SD16—416no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—28no—
482C2×C8—48no—
483C4⋊C4—216no—
484C4⋊C4—216no—
485C4⋊C4—216no—
486C4⋊C4—216no—
487C4⋊C4—24no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—28no—
494C4⋊C4—28no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—42no—
506C4⋊C4—84no—
507C4⋊C4C22×D414yes—
508C4⋊C4C22×D418yes—
509C22⋊C4—216no—
510C22⋊C4—28no—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—48no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C42—216no—
522C42—24no—
523C42—24no—
524C42—28no—
525C42—22no—
526C42—22no—
527C42—44no—
528C42#12014yes—
529C42#12014yes—
530C42C22×D412yes—
531C42C22×D412yes—
532C42C22×D412yes—
533C42C22×D412yes—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—22no—
539C23—24no—
540C23—24no—
541C23—48no—
542C23—42no—
543C23—42no—
544C23—44no—
545C23—44no—
546C23—816no—
547C23—84no—
548C23—84no—
549C23C2×#12012yes—
550C23C2×#13714yes—
551C23C2×#13714yes—
552C23C23×D411yescharacteristic
553Q8—28no—
554Q8—28no—
555Q8—42no—
556Q8—42no—
557Q8—416no—
558Q8—48no—
559Q8—88no—
560Q8—816no—
561D4—28no—
562D4—216no—
563D4—24no—
564D4—24no—
565D4—48no—
566D4—42no—
567D4—42no—
568D4—816no—
569C2×C4—216no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—44no—
589C2×C4—48no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—88no—
601C2×C4—88no—
602C2×C4—816no—
603C2×C4—84no—
604C2×C4—88no—
605C2×C4C2×#12012yes—
606C2×C4C2×#12012yes—
607C2×C4C2×#12012yes—
608C2×C4C2×#12012yes—
609C2×C4C2×#12012yes—
610C2×C4C2×#13714yes—
611C2×C4#57614yes—
612C2×C4C23×D411yescharacteristic
613C2×C4C23×D411yescharacteristic
614C2×C4C2×2−1+412yes—
615C2×C4#58512yes—
616C8—44no—
617C8—44no—
618C22—22no—
619C22—28no—
620C22—22no—
621C22—28no—
622C22—24no—
623C22—42no—
624C22—44no—
625C22—42no—
626C22—44no—
627C22—48no—
628C22—816no—
629C22—84no—
630C22—82no—
631C22—82no—
632C22#300414yes—
633C22C22×#12011yescharacteristic
634C22#348912yes—
635C22C22×#13712yes—
636C22C2×#57612yes—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—24no—
641C4—22no—
642C4—42no—
643C4—44no—
644C4—42no—
645C4—48no—
646C4—44no—
647C4—84no—
648C4—82no—
649C4#300814yes—
650C4#348611yescharacteristic
651C4#348711yescharacteristic
652C4#348912yes—
653C2—24no—
654C2—44no—
655C2—84no—
656C2#3565911yescharacteristic
657C2C2×#300411yescharacteristic
658C2C2×#300811yescharacteristic
659C2#6212714yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#34963C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3503C2semidirect
2. C2×#3110C2semidirect
3. C22×#474C2semidirect
5. #24341C2semidirect
6. #24298C2semidirect
7. #24153C2semidirect
8. #24130C2semidirect
9. #24098C2semidirect
10. #23969C2semidirect
11. #21889C2semidirect
12. #21851C2semidirect
13. #21841C2semidirect
16. C2×Q82C22semidirect
19. C2×#554C22semidirect
25. C2×#533C22semidirect
27. C22×C4⋊Q8C22semidirect
56. #2057C22semidirect
57. #2036C22semidirect
60. #1993C22semidirect
61. #1988C22semidirect
62. #1988C22semidirect
63. #1985C22semidirect
64. #1981C22semidirect
65. #1955C22semidirect
70. #1863C22semidirect
71. #1852C22semidirect
73. #1805C22semidirect
74. #1797C22semidirect
96. #533D4semidirect
106. C2×C4⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#35032
C2×#31102
C22×#4742
#349632
#243412
#242982
#241532
#241302
#240982
#239692
#218892
#218512
#218412