Finite Group Databasebeta

← #6436556

Subgroups — #6436556

Order 512.

2898 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×#541C211yesmaximal, characteristic
2C2×#3113C211yesmaximal, characteristic, complemented
3C2×#3104C211yesmaximal, characteristic, complemented
4#35325C211yesmaximal, characteristic
5#29699C214yesmaximal, complemented
6#29674C214yesmaximal, complemented
7#29643C212yesmaximal, complemented
8#29627C212yesmaximal, complemented
9#29599C212yesmaximal
10#29597C212yesmaximal
11#23586C218yesmaximal, complemented
12#20741C211yesmaximal, characteristic, complemented
13#20735C211yesmaximal, characteristic, complemented
14#20717C211yesmaximal, characteristic
15C2×#541—28no—
16C2×#541—28no—
17C2×#541—22no—
18C2×#541—22no—
19C2×#541C2214yes—
20C2×#541C2214yes—
21C2×#533C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C22×#126—21no—
24C22×#126—21no—
25C22×#125—21no—
26C22×#125—21no—
27C22×#125C2211yescharacteristic
28C22×C22⋊Q8C2211yescharacteristic
29C22×C22⋊Q8C2211yescharacteristic
30C22×C4×Q8C2211yescharacteristic
31C22×C4×Q8C2211yescharacteristic
32C22×#117C2211yescharacteristic
33#3113—28no—
34#3104—28no—
35C2×#484C2212yes—
36C2×#483C2212yes—
37C2×#477C2212yescomplemented
38C2×#475C2212yescomplemented
39C2×#462C2212yescomplemented
40C2×#462C2212yescomplemented
41C2×#451C2212yes—
42C2×#451C2212yes—
43C2×#441C2214yescomplemented
44C2×C4×Q16C2214yescomplemented
45C2×#432C2211yescharacteristic
46C2×#431C2211yescharacteristic
47#2855C2212yes—
48#2848C2212yes—
49#2628C2212yes—
50#2621C2212yes—
51#2611—22no—
52#2599—22no—
53#2583—21no—
54#2561—21no—
55#2484—21no—
56#2482—21no—
57#2323C2214yes—
58#2283C2211yescharacteristic
59#2059C2218yescomplemented
60#2043C2218yescomplemented
61#2036C2218yes—
62#2035C2218yes—
63#2026C2218yescomplemented
64#2016C2218yescomplemented
65#2001C2218yes—
66#2001C2218yes—
67#1824C2212yes—
68#1823C2212yes—
69#1817C2212yes—
70#1805C2214yescomplemented
71#1805C2214yescomplemented
72#1798C2214yes—
73#1787C2212yes—
74#1787C2212yes—
75#1779C2212yes—
76#1763C2214yescomplemented
77C4×#103C2214yescomplemented
78#1750C2214yes—
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C22×Q16C2314yes—
83C22×Q16C2314yes—
84#541—232no—
85#541—232no—
86#541—216no—
87#541—216no—
88#541—216no—
89#541—216no—
90#533—24no—
91#533—24no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2314yes—
97C2×C4⋊Q8C2311yescharacteristic
98C2×#126—216no—
99C2×#126—216no—
100C2×#126—24no—
101C2×#126—24no—
102C2×#126—24no—
103C2×#126—24no—
104C2×#125—216no—
105C2×#125—216no—
106C2×#125—24no—
107C2×#125—24no—
108C2×#125—24no—
109C2×#125—24no—
110C2×#125—24no—
111C2×#125—24no—
112C2×#125—22no—
113C2×#125—21no—
114C2×#125C2314yes—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—24no—
121C2×C22⋊Q8—24no—
122C2×C22⋊Q8C2312yes—
123C2×C22⋊Q8C2314yes—
124C2×C22⋊Q8C2314yes—
125C2×C22⋊Q8C2314yes—
126C2×C4×Q8—24no—
127C2×C4×Q8—24no—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8—21no—
132C2×C4×Q8—21no—
133C2×C4×Q8D418yes—
134C2×C4×Q8D418yes—
135C2×C4×Q8C2312yes—
136C2×C4×Q8C2312yes—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2312yes—
140C2×#117—21no—
141C2×#117—24no—
142C2×#117—24no—
143C2×#117D412yes—
144C2×#117D412yes—
145C2×#117C2312yes—
146C2×#117C2312yes—
147C22×C4⋊C4—21no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—21no—
153C22×C4⋊C4—21no—
154C22×C4⋊C4—21no—
155C22×C4⋊C4D411yescharacteristic
156C22×C4⋊C4D411yescharacteristic
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2311yescharacteristic
159C22×C22⋊C4—21no—
160C22×C22⋊C4—21no—
161C22×C22⋊C4D411yescharacteristic
162C22×C22⋊C4D411yescharacteristic
163C22×C42—21no—
164C22×C42—21no—
165C22×C42C2311yescharacteristic
166C22×C42C2311yescharacteristic
167#484—28no—
168#483—28no—
169#477—28no—
170#475—28no—
171#462—28no—
172#462—28no—
173#451—28no—
174#451—28no—
175#441—216no—
176C4×Q16—216no—
177#432—24no—
178#431—24no—
179C2×#107C2314yes—
180C2×#106C2314yes—
181C2×C4⋊C8C2312yes—
182C2×C4⋊C8C2312yes—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#103C2314yes—
187C2×#103C2318yes—
188C2×#103C2318yes—
189C2×C22⋊C8C2312yes—
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2318yes—
192C2×#97C2312yes—
193C2×C4×C8C2312yes—
194#400—22no—
195#399—22no—
196#398—22no—
197#398—22no—
198#397—22no—
199#397—22no—
200#395—22no—
201#393—22no—
202#391C2314yes—
203#389—24no—
204#389C2314yes—
205#388—24no—
206#388—24no—
207#387—24no—
208#387—24no—
209#385—22no—
210#385—22no—
211#385—24no—
212#383—21no—
213#382C2312yes—
214#381—22no—
215#381—21no—
216#380C2314yes—
217#380C2312yes—
218#379—22no—
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—21no—
221#376—21no—
222#340C2318yes—
223#340C2318yes—
224#336C2318yes—
225C25C2×D411yescharacteristic
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—21no—
229C22×Q8—21no—
230C22×Q8—21no—
231C22×Q8—44no—
232C22×Q8—44no—
233C22×Q8C2×D414yes—
234C22×Q8C2×D414yes—
235C22×Q8C2412yes—
236C22×Q8C2412yes—
237C22×Q8C2412yes—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—44no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C2×Q16—28no—
254C2×Q16—28no—
255C2×Q16—48no—
256C2×Q16—48no—
257C22×C8C2414yes—
258C22×C8C2414yes—
259C4⋊Q8—216no—
260C4⋊Q8—28no—
261C4⋊Q8—28no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266#126—232no—
267#126—232no—
268#126—232no—
269#126—232no—
270#125—232no—
271#125—232no—
272#125—232no—
273#125—232no—
274#125—216no—
275#125—28no—
276#125—28no—
277#125—44no—
278C22⋊Q8—216no—
279C22⋊Q8—216no—
280C22⋊Q8—28no—
281C22⋊Q8—28no—
282C22⋊Q8—28no—
283C22⋊Q8—28no—
284C22⋊Q8—24no—
285C22⋊Q8—24no—
286C22⋊Q8—48no—
287C22⋊Q8—44no—
288C4×Q8—28no—
289C4×Q8—28no—
290C4×Q8—28no—
291C4×Q8—28no—
292C4×Q8—24no—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—44no—
299C4×Q8C2×D418yes—
300C4×Q8C2×D418yes—
301C4×Q8C2×D418yes—
302C4×Q8C2×D418yes—
303#117—28no—
304#117—28no—
305#117—24no—
306#117—24no—
307#117—24no—
308#117—24no—
309#117—44no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—28no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2×D414yes—
346C2×C4⋊C4C2×D414yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2414yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C22⋊C4—28no—
352C2×C22⋊C4—28no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C22⋊C4C2×D412yes—
365C2×C42—28no—
366C2×C42—28no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42C2×D412yes—
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D414yes—
379C2×C42C2×D414yes—
380C2×C42C2412yes—
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383#107—28no—
384#106—28no—
385C4⋊C8—24no—
386C4⋊C8—24no—
387#103—28no—
388#103—28no—
389#103—28no—
390#103—28no—
391#103—48no—
392#103—48no—
393C22⋊C8—24no—
394C22⋊C8—24no—
395C22⋊C8—48no—
396#97—24no—
397C4×C8—24no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406C24—22no—
407C24—24no—
408C24—42no—
409C24—42no—
410C24C22×D412yes—
411C24C22×D411yescharacteristic
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—24no—
419C2×Q8—24no—
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8—28no—
423C2×Q8—48no—
424C2×Q8—48no—
425C2×Q8—44no—
426C2×Q8—44no—
427C2×Q8—48no—
428C2×Q8—48no—
429C2×Q8—42no—
430C2×Q8—82no—
431C2×Q8C22×D414yes—
432C2×Q8C22×D414yes—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4#12014yes—
467C22×C4#12014yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C2511yescharacteristic
479Q16—416no—
480Q16—416no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—28no—
484C2×C8—28no—
485C4⋊C4—216no—
486C4⋊C4—216no—
487C4⋊C4—216no—
488C4⋊C4—216no—
489C4⋊C4—216no—
490C4⋊C4—216no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—28no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—28no—
506C4⋊C4—48no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—84no—
513C4⋊C4C22×D414yes—
514C4⋊C4C22×D414yes—
515C22⋊C4—216no—
516C22⋊C4—216no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—84no—
525C42—28no—
526C42—28no—
527C42—28no—
528C42—28no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—44no—
533C42—82no—
534C42#12014yes—
535C42#12014yes—
536C42C22×D412yes—
537C42C22×D412yes—
538C42C22×D412yes—
539C42C22×D412yes—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—24no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—88no—
551C23C2×#12012yes—
552C23#57814yes—
553C23C23×D411yescharacteristic
554Q8—24no—
555Q8—24no—
556Q8—24no—
557Q8—24no—
558Q8—48no—
559Q8—48no—
560Q8—44no—
561Q8—44no—
562Q8—44no—
563Q8—44no—
564Q8—84no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—44no—
585C2×C4—48no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—44no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4C2×#12012yes—
607C2×C4C2×#12012yes—
608C2×C4C2×#12012yes—
609C2×C4C2×#12012yes—
610C2×C4C2×#12012yes—
611C2×C4C23×D411yescharacteristic
612C2×C4C23×D411yescharacteristic
613C2×C4#58514yes—
614C8—44no—
615C8—44no—
616C22—28no—
617C22—22no—
618C22—22no—
619C22—44no—
620C22—44no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—88no—
625C22#301114yes—
626C22C22×#12011yescharacteristic
627C22#344912yes—
628C22C2×#57812yes—
629C22#358612yes—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—42no—
636C4—44no—
637C4—44no—
638C4—42no—
639C4—44no—
640C4—42no—
641C4—44no—
642C4—84no—
643C4—82no—
644C4—82no—
645C4#344912yes—
646C4#344912yes—
647C2—24no—
648C2—82no—
649C2#3463211yescharacteristic
650C2C2×#301111yescharacteristic
651C2#6036811yescharacteristic
652C2#6098314yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3113C2semidirect
3. C2×#3104C2semidirect
5. #29699C2semidirect
6. #29674C2semidirect
7. #29643C2semidirect
8. #29627C2semidirect
11. #23586C2semidirect
12. #20741C2semidirect
13. #20735C2semidirect
37. C2×#477C22semidirect
38. C2×#475C22semidirect
39. C2×#462C22semidirect
40. C2×#462C22semidirect
43. C2×#441C22semidirect
44. C2×C4×Q16C22semidirect
59. #2059C22semidirect
60. #2043C22semidirect
63. #2026C22semidirect
64. #2016C22semidirect
70. #1805C22semidirect
71. #1805C22semidirect
76. #1763C22semidirect
77. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5412
C2×#31132
C2×#31042
#353252
#296992
#296742
#296432
#296272
#295992
#295972
#235862
#207412
#207352
#207172