Finite Group Databasebeta

← #6489664

Subgroups — #6489664

Order 512.

3190 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2C2×#3236C211yesmaximal, characteristic, complemented
3C2×#3101C211yesmaximal, characteristic
4#35248C211yesmaximal, characteristic, complemented
5#33009C212yesmaximal
6#33007C212yesmaximal
7#29643C212yesmaximal, complemented
8#29641C212yesmaximal, complemented
9#25945C214yesmaximal, complemented
10#25851C214yesmaximal, complemented
11#24337C218yesmaximal, complemented
12#21836C211yesmaximal, characteristic, complemented
13#21820C211yesmaximal, characteristic, complemented
14#20739C211yesmaximal, characteristic
15C2×#554—28no—
16C2×#554—22no—
17C2×#554—22no—
18C2×#554—28no—
19C2×#554C2214yes—
20C2×#554C2214yes—
21C2×#533C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C22×#125C2211yescharacteristic
24C22×#125C2211yescharacteristic
25C22×C22⋊Q8—21no—
26C22×C22⋊Q8—21no—
27C22×C22⋊Q8—21no—
28C22×C4×Q8C2211yescharacteristic
29C22×C4×D4C2211yescharacteristic
30C22×C4×D4C2211yescharacteristic
31C22×C4×D4C2211yescharacteristic
32C23×C4⋊C4—21no—
33#3236—28no—
34C2×#501C2212yescomplemented
35C2×#491C2212yescomplemented
36#3101—28no—
37C2×#484C2212yes—
38C2×#483C2212yes—
39C2×#483C2212yes—
40C2×#479C2214yescomplemented
41C2×#476C2212yes—
42C2×#461C2212yes—
43C2×#450C2212yes—
44C2×C4×SD16C2214yes—
45C2×#432C2211yescharacteristic
46C2×#431C2211yescharacteristic
47#2855C2212yes—
48#2855C2212yes—
49#2634—22no—
50#2581C2214yescomplemented
51#2568C2212yescomplemented
52#2546C2212yescomplemented
53#2503—21no—
54#2484—21no—
55#2484—21no—
56#2434—21no—
57#2327—22no—
58#2283C2211yescharacteristic
59#2059C2218yescomplemented
60#2058C2218yescomplemented
61#2026C2218yescomplemented
62#2024C2218yescomplemented
63#1991C2214yescomplemented
64#1989C2214yescomplemented
65#1985C2214yescomplemented
66#1984C2214yescomplemented
67#1877C2218yes—
68#1877C2218yes—
69#1864C2218yes—
70#1862C2218yes—
71#1824C2212yes—
72#1823C2212yes—
73#1823C2212yes—
74#1817C2212yes—
75#1817C2212yes—
76#1804C2214yes—
77#1786C2212yes—
78#1762C2214yes—
79C23×Q8—21no—
80C23×Q8D411yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4—21no—
83C24×C4—21no—
84C24×C4—21no—
85C22×SD16C2314yes—
86#554—232no—
87#554—232no—
88#554—216no—
89#554—216no—
90#554—216no—
91#554—216no—
92#533—24no—
93#533—24no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2314yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×#125—24no—
102C2×#125—24no—
103C2×#125—22no—
104C2×#125—21no—
105C2×#125—21no—
106C2×#125—24no—
107C2×#125—21no—
108C2×#125D418yes—
109C2×#125C2314yes—
110C2×#125C2312yes—
111C2×#125C2312yes—
112C2×C22⋊Q8—216no—
113C2×C22⋊Q8—216no—
114C2×C22⋊Q8—216no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8—24no—
120C2×C22⋊Q8—24no—
121C2×C22⋊Q8—24no—
122C2×C22⋊Q8C2312yes—
123C2×C4×Q8—24no—
124C2×C4×Q8—24no—
125C2×C4×Q8—21no—
126C2×C4×Q8D412yes—
127C2×C4×Q8D412yes—
128C2×C4×Q8C2312yes—
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2312yes—
131C2×C4×D4—24no—
132C2×C4×D4—24no—
133C2×C4×D4—24no—
134C2×C4×D4—24no—
135C2×C4×D4—22no—
136C2×C4×D4—22no—
137C2×C4×D4—21no—
138C2×C4×D4—21no—
139C2×C4×D4—24no—
140C2×C4×D4D418yes—
141C2×C4×D4C2314yes—
142C2×C4×D4C2314yes—
143C2×C4×D4C2312yes—
144C2×C4×D4C2312yes—
145C2×#117—21no—
146C22×C4⋊C4—216no—
147C22×C4⋊C4—21no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—24no—
153C22×C4⋊C4—24no—
154C22×C4⋊C4D411yescharacteristic
155C22×C4⋊C4D411yescharacteristic
156C22×C4⋊C4D411yescharacteristic
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2311yescharacteristic
159C22×C4⋊C4C2311yescharacteristic
160C22×C22⋊C4—21no—
161C22×C22⋊C4—21no—
162C22×C22⋊C4—21no—
163C22×C42C2311yescharacteristic
164C22×C42C2311yescharacteristic
165C22×C42C2311yescharacteristic
166#501—28no—
167#491—28no—
168#484—28no—
169#483—28no—
170#483—28no—
171#479—216no—
172#476—28no—
173#461—28no—
174#450—28no—
175C4×SD16—216no—
176#432—24no—
177#431—24no—
178C2×#107C2314yes—
179C2×#106C2314yes—
180C2×#106C2314yes—
181C2×C4⋊C8C2312yes—
182C2×C4⋊C8C2312yes—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#102C2318yes—
187C2×#102C2314yes—
188C2×C22⋊C8C2312yes—
189C2×C22⋊C8C2312yes—
190C2×#97C2312yes—
191C2×C4×C8C2312yes—
192#400—22no—
193#400—22no—
194#400—24no—
195#398—22no—
196#398—22no—
197#395—24no—
198#391—22no—
199#391—22no—
200#391C2314yes—
201#391C2314yes—
202#389C2314yes—
203#389C2314yes—
204#387—24no—
205#387—24no—
206#387—24no—
207#387—24no—
208#385—22no—
209#385—22no—
210#383—21no—
211#382—22no—
212#382C2312yes—
213#381—22no—
214#381—22no—
215#381—21no—
216#380C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—21no—
219#376—21no—
220#340C2318yes—
221#340C2318yes—
222#340C2318yes—
223#336C2318yes—
224#336C2318yes—
225C25—21no—
226C22×Q8—28no—
227C22×Q8—24no—
228C22×Q8—24no—
229C22×Q8—21no—
230C22×Q8—44no—
231C22×Q8—84no—
232C22×Q8C2×D412yes—
233C22×Q8C2×D412yes—
234C22×Q8C2412yes—
235C22×D4—28no—
236C22×D4—21no—
237C22×D4—22no—
238C22×D4C2×D414yes—
239C22×D4C2412yes—
240C23×C4—28no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—28no—
245C23×C4—24no—
246C23×C4—28no—
247C23×C4—24no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—48no—
251C23×C4—84no—
252C23×C4—84no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2411yescharacteristic
260C2×SD16—28no—
261C2×SD16—216no—
262C22×C8C2414yes—
263C22×C8C2414yes—
264C4⋊Q8—216no—
265C4⋊Q8—28no—
266C4⋊Q8—28no—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270C4⋊Q8—22no—
271C4⋊Q8—48no—
272#125—216no—
273#125—28no—
274#125—28no—
275#125—28no—
276#125—28no—
277#125—24no—
278#125—24no—
279#125—44no—
280#125C2×D418yes—
281#125C2×D418yes—
282C22⋊Q8—232no—
283C22⋊Q8—232no—
284C22⋊Q8—232no—
285C22⋊Q8—232no—
286C22⋊Q8—232no—
287C22⋊Q8—232no—
288C22⋊Q8—24no—
289C22⋊Q8—24no—
290C22⋊Q8—44no—
291C4×Q8—28no—
292C4×Q8—28no—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—44no—
300C4×D4—216no—
301C4×D4—216no—
302C4×D4—28no—
303C4×D4—28no—
304C4×D4—28no—
305C4×D4—28no—
306C4×D4—28no—
307C4×D4—28no—
308C4×D4—24no—
309C4×D4—24no—
310C4×D4C2×D418yes—
311C4×D4C2×D418yes—
312#117—44no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—232no—
316C2×C4⋊C4—232no—
317C2×C4⋊C4—28no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—28no—
320C2×C4⋊C4—28no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2×D412yes—
346C2×C4⋊C4C2×D412yes—
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2×D412yes—
349C2×C4⋊C4C2×D414yes—
350C2×C4⋊C4C2×D414yes—
351C2×C4⋊C4C2×D414yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2414yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C22⋊C4—28no—
358C2×C22⋊C4—28no—
359C2×C22⋊C4—28no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42C2×D412yes—
373C2×C42C2×D412yes—
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D414yes—
377C2×C42C2×D414yes—
378C2×C42C2412yes—
379C2×C42C2412yes—
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382#107—28no—
383#106—28no—
384#106—28no—
385C4⋊C8—24no—
386C4⋊C8—24no—
387#103—28no—
388#103—28no—
389#103—28no—
390#102—28no—
391#102—216no—
392C22⋊C8—24no—
393C22⋊C8—24no—
394#97—24no—
395C4×C8—24no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406C24—28no—
407C24—24no—
408C24—22no—
409C24—44no—
410C24—44no—
411C24C22×D411yescharacteristic
412C2×Q8—216no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—28no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—416no—
423C2×Q8—44no—
424C2×Q8—42no—
425C2×Q8—88no—
426C2×Q8—816no—
427C2×D4—28no—
428C2×D4—216no—
429C2×D4—216no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—24no—
433C2×D4C22×D414yes—
434C2×D42−1+418yes—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—216no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—216no—
452C22×C4—28no—
453C22×C4—28no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—432no—
460C22×C4—48no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—48no—
465C22×C4—816no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4—88no—
469C22×C4—816no—
470C22×C4#12014yes—
471C22×C4#12014yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C2511yescharacteristic
484SD16—416no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—28no—
488C2×C8—28no—
489C4⋊C4—216no—
490C4⋊C4—216no—
491C4⋊C4—216no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—28no—
511C4⋊C4—48no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—48no—
518C4⋊C4C22×D414yes—
519C4⋊C4C22×D414yes—
520C4⋊C4C22×D414yes—
521C22⋊C4—216no—
522C22⋊C4—216no—
523C22⋊C4—28no—
524C22⋊C4—28no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C42—24no—
528C42—24no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—44no—
534C42—44no—
535C42#12014yes—
536C42#12014yes—
537C42C22×D412yes—
538C42C22×D412yes—
539C42C22×D412yes—
540C42C22×D412yes—
541C23—22no—
542C23—22no—
543C23—216no—
544C23—24no—
545C23—28no—
546C23—48no—
547C23—432no—
548C23—48no—
549C23—44no—
550C23—44no—
551C23C2×#12012yes—
552C23#57714yes—
553C23C23×D411yescharacteristic
554Q8—48no—
555Q8—48no—
556Q8—44no—
557Q8—44no—
558Q8—84no—
559Q8—816no—
560D4—216no—
561D4—24no—
562D4—28no—
563D4#53618yes—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—44no—
586C2×C4—432no—
587C2×C4—48no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—88no—
604C2×C4—88no—
605C2×C4—816no—
606C2×C4C2×#12012yes—
607C2×C4C2×#12012yes—
608C2×C4C2×#12012yes—
609C2×C4C2×#12012yes—
610C2×C4C2×#12012yes—
611C2×C4#57514yes—
612C2×C4C23×D411yescharacteristic
613C2×C4C23×D411yescharacteristic
614C2×C4C2×2−1+414yes—
615C8—44no—
616C8—44no—
617C22—28no—
618C22—22no—
619C22—22no—
620C22—28no—
621C22—432no—
622C22—416no—
623C22—44no—
624C22—44no—
625C22#300914yes—
626C22C22×#12011yescharacteristic
627C22#348612yes—
628C22C2×#57512yes—
629C22C2×#57712yes—
630C4—22no—
631C4—24no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—42no—
637C4—44no—
638C4—42no—
639C4—42no—
640C4—44no—
641C4—44no—
642C4—84no—
643C4—82no—
644C4—82no—
645C4C2×#53612yes—
646C4#348612yes—
647C2—24no—
648C2—48no—
649C2#3524211yescharacteristic
650C2#6035911yescharacteristic
651C2C2×#300911yescharacteristic
652C2#6184114yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3236C2semidirect
4. #35248C2semidirect
7. #29643C2semidirect
8. #29641C2semidirect
9. #25945C2semidirect
10. #25851C2semidirect
11. #24337C2semidirect
12. #21836C2semidirect
13. #21820C2semidirect
34. C2×#501C22semidirect
35. C2×#491C22semidirect
40. C2×#479C22semidirect
50. #2581C22semidirect
51. #2568C22semidirect
52. #2546C22semidirect
59. #2059C22semidirect
60. #2058C22semidirect
61. #2026C22semidirect
62. #2024C22semidirect
63. #1991C22semidirect
64. #1989C22semidirect
65. #1985C22semidirect
66. #1984C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#32362
C2×#31012
#352482
#330092
#330072
#296432
#296412
#259452
#258512
#243372
#218362
#218202
#207392