Finite Group Databasebeta

← #6436555

Subgroups — #6436555

Order 512.

3012 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#541C211yesmaximal, characteristic, complemented
2C2×#3112C211yesmaximal, characteristic, complemented
3C2×#3103C211yesmaximal, characteristic, complemented
4#35301C211yesmaximal, characteristic
5#29699C214yesmaximal, complemented
6#29671C214yesmaximal, complemented
7#29640C212yesmaximal, complemented
8#29624C212yesmaximal, complemented
9#29595C212yesmaximal, complemented
10#29593C212yesmaximal, complemented
11#23584C218yesmaximal, complemented
12#20741C211yesmaximal, characteristic, complemented
13#20735C211yesmaximal, characteristic, complemented
14#20717C211yesmaximal, characteristic, complemented
15C2×#541—28no—
16C2×#541—28no—
17C2×#541—22no—
18C2×#541—22no—
19C2×#541C2214yes—
20C2×#541C2214yes—
21C2×#532C2211yescharacteristic
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#3112—28no—
34#3103—28no—
35C2×#481C2212yes—
36C2×#480C2212yes—
37C2×#477C2212yescomplemented
38C2×#475C2212yescomplemented
39C2×#460C2212yescomplemented
40C2×#460C2212yescomplemented
41C2×#448C2212yes—
42C2×#448C2212yes—
43C2×#440C2214yescomplemented
44C2×C4×SD16C2214yescomplemented
45C2×#432C2211yescharacteristic
46C2×#431C2211yescharacteristic
47#2854C2212yes—
48#2847C2212yes—
49#2628C2212yes—
50#2619C2212yes—
51#2610—22no—
52#2598—22no—
53#2591—21no—
54#2539—21no—
55#2406—21no—
56#2404—21no—
57#2324C2214yes—
58#2283C2211yescharacteristic
59#2059C2218yescomplemented
60#2043C2218yescomplemented
61#2036C2218yes—
62#2035C2218yes—
63#2025C2218yescomplemented
64#2013C2218yescomplemented
65#2000C2218yes—
66#2000C2218yes—
67#1824C2212yes—
68#1823C2212yes—
69#1817C2212yes—
70#1805C2214yescomplemented
71#1805C2214yescomplemented
72#1798C2214yes—
73#1787C2212yes—
74#1787C2212yes—
75#1779C2212yes—
76#1763C2214yescomplemented
77C4×#103C2214yescomplemented
78#1750C2214yes—
79C22×#41—21no—
80C23×Q8C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C22×SD16C2314yes—
84C22×SD16C2314yes—
85#541—232no—
86#541—232no—
87#541—216no—
88#541—216no—
89#541—216no—
90#541—216no—
91#532—24no—
92#532—24no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8C2314yes—
97C2×C4⋊Q8C2311yescharacteristic
98C2×#127C2311yescharacteristic
99C2×#126—216no—
100C2×#126—216no—
101C2×#126—24no—
102C2×#126—24no—
103C2×#126—24no—
104C2×#126—24no—
105C2×#125—216no—
106C2×#125—216no—
107C2×#125—24no—
108C2×#125—24no—
109C2×#125—24no—
110C2×#125—24no—
111C2×#125—24no—
112C2×#125—24no—
113C2×#125—22no—
114C2×#125C2314yes—
115C2×#124—21no—
116C2×#123—22no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8—22no—
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8—24no—
122C2×C22⋊Q8—24no—
123C2×C22⋊Q8C2314yes—
124C2×C22⋊Q8C2314yes—
125C2×#121—21no—
126C2×#121C2312yes—
127C2×#121C2314yes—
128C2×#120—22no—
129C2×C4×Q8—24no—
130C2×C4×Q8—24no—
131C2×C4×Q8—21no—
132C2×C4×Q8—21no—
133C2×C4×Q8—21no—
134C2×C4×Q8—21no—
135C2×C4×Q8D418yescomplemented
136C2×C4×Q8D418yescomplemented
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2312yes—
140C2×C4×Q8C2312yes—
141C2×C4×D4—21no—
142C2×C4×D4C2312yes—
143C2×#117—24no—
144C2×#117—24no—
145C2×#117D412yes—
146C2×#117D412yes—
147C2×#117C2312yes—
148C2×#117C2312yes—
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⋊C4—21no—
156C22×C4⋊C4—21no—
157C22×C4⋊C4D411yescharacteristic
158C22×C4⋊C4D411yescharacteristic
159C22×C4⋊C4C2311yescharacteristic
160C22×C4⋊C4C2311yescharacteristic
161C22×C22⋊C4—21no—
162C22×C22⋊C4—21no—
163C22×C22⋊C4D411yescharacteristic
164C22×C22⋊C4D411yescharacteristic
165C22×C42—21no—
166C22×C42—21no—
167C22×C42C2311yescharacteristic
168C22×C42C2311yescharacteristic
169#481—28no—
170#480—28no—
171#477—28no—
172#475—28no—
173#460—28no—
174#460—28no—
175#448—28no—
176#448—28no—
177#440—216no—
178C4×SD16—216no—
179#432—24no—
180#431—24no—
181C2×#107C2314yes—
182C2×#106C2314yes—
183C2×C4⋊C8C2312yes—
184C2×C4⋊C8C2312yes—
185C2×#103C2314yes—
186C2×#103C2314yes—
187C2×#103C2318yes—
188C2×#103C2318yes—
189C2×#102C2314yes—
190C2×#102C2314yes—
191C2×C22⋊C8C2312yes—
192C2×C22⋊C8C2312yes—
193C2×C22⋊C8C2318yes—
194C2×#97C2312yes—
195C2×C4×C8C2312yes—
196#400—22no—
197#399—22no—
198#395—22no—
199#394—22no—
200#394—22no—
201#393—22no—
202#392—22no—
203#392—22no—
204#390—24no—
205#390C2314yes—
206#389C2314yes—
207#388—24no—
208#388—24no—
209#386—22no—
210#386—22no—
211#386—24no—
212#386—24no—
213#385—24no—
214#383—21no—
215#382C2312yes—
216#381—21no—
217#380C2314yes—
218#380C2312yes—
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—21no—
221#376—21no—
222#340C2318yes—
223#340C2318yes—
224#336C2318yes—
225C25C2×D411yescharacteristic
226C2×#41—44no—
227C2×#41—84no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—21no—
231C22×Q8—21no—
232C22×Q8—21no—
233C22×Q8—44no—
234C22×Q8—44no—
235C22×Q8C2×D414yes—
236C22×Q8C2×D414yes—
237C22×Q8C2412yes—
238C22×Q8C2412yes—
239C22×D4—22no—
240C22×D4—24no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2412yes—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—44no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2412yes—
259C23×C4C2411yescharacteristic
260C2×SD16—28no—
261C2×SD16—28no—
262C2×SD16—48no—
263C2×SD16—48no—
264C22×C8C2414yes—
265C22×C8C2414yes—
266C4⋊Q8—216no—
267C4⋊Q8—28no—
268C4⋊Q8—28no—
269C4⋊Q8—22no—
270C4⋊Q8—22no—
271#127—22no—
272#127—22no—
273#126—232no—
274#126—232no—
275#126—232no—
276#126—232no—
277#125—232no—
278#125—232no—
279#125—232no—
280#125—232no—
281#125—216no—
282#125—28no—
283#125—28no—
284#124—44no—
285#123—84no—
286C22⋊Q8—216no—
287C22⋊Q8—216no—
288C22⋊Q8—28no—
289C22⋊Q8—28no—
290C22⋊Q8—28no—
291C22⋊Q8—28no—
292#121—24no—
293#121—24no—
294#121—48no—
295#121—44no—
296#120—84no—
297C4×Q8—28no—
298C4×Q8—28no—
299C4×Q8—28no—
300C4×Q8—28no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×Q8C2×D418yes—
306C4×Q8C2×D418yes—
307C4×Q8C2×D418yes—
308C4×Q8C2×D418yes—
309C4×D4—24no—
310C4×D4—24no—
311C4×D4—44no—
312#117—28no—
313#117—28no—
314#117—24no—
315#117—24no—
316#117—24no—
317#117—24no—
318C2×C4⋊C4—28no—
319C2×C4⋊C4—28no—
320C2×C4⋊C4—28no—
321C2×C4⋊C4—28no—
322C2×C4⋊C4—28no—
323C2×C4⋊C4—28no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—28no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2×D412yes—
346C2×C4⋊C4C2×D412yes—
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2×D414yes—
349C2×C4⋊C4C2×D414yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2414yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C22⋊C4—28no—
355C2×C22⋊C4—28no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4C2×D412yes—
368C2×C22⋊C4C2×D412yes—
369C2×C22⋊C4C2×D412yes—
370C2×C22⋊C4C2×D412yes—
371C2×C42—28no—
372C2×C42—28no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42C2×D412yes—
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D414yes—
384C2×C42C2×D414yes—
385C2×C42C2412yes—
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388#107—28no—
389#106—28no—
390C4⋊C8—24no—
391C4⋊C8—24no—
392#103—28no—
393#103—28no—
394#103—48no—
395#103—48no—
396#102—28no—
397#102—28no—
398C22⋊C8—24no—
399C22⋊C8—24no—
400C22⋊C8—48no—
401#97—24no—
402C4×C8—24no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—24no—
408#95—24no—
409C24—22no—
410C24—24no—
411C24—22no—
412C24—42no—
413C24—42no—
414C24C22×D412yes—
415C24C22×D411yescharacteristic
416#41—816no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8—48no—
426C2×Q8—48no—
427C2×Q8—44no—
428C2×Q8—44no—
429C2×Q8—48no—
430C2×Q8—48no—
431C2×Q8—42no—
432C2×Q8—82no—
433C2×Q8C22×D414yes—
434C2×Q8C22×D414yes—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—28no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—48no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—82no—
445C2×D4—82no—
446C2×D4—84no—
447C2×D4—84no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4#12014yes—
484C22×C4#12014yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C2511yescharacteristic
496SD16—416no—
497SD16—416no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—28no—
501C2×C8—28no—
502C4⋊C4—216no—
503C4⋊C4—216no—
504C4⋊C4—216no—
505C4⋊C4—216no—
506C4⋊C4—216no—
507C4⋊C4—216no—
508C4⋊C4—28no—
509C4⋊C4—28no—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—28no—
523C4⋊C4—42no—
524C4⋊C4—84no—
525C4⋊C4C22×D414yes—
526C4⋊C4C22×D414yes—
527C22⋊C4—216no—
528C22⋊C4—216no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—48no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—84no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C42—28no—
541C42—28no—
542C42—28no—
543C42—28no—
544C42—24no—
545C42—22no—
546C42—22no—
547C42—82no—
548C42#12014yes—
549C42#12014yes—
550C42C22×D412yes—
551C42C22×D412yes—
552C42C22×D412yes—
553C42C22×D412yes—
554C23—22no—
555C23—22no—
556C23—24no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—24no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—42no—
566C23—44no—
567C23—42no—
568C23—42no—
569C23—88no—
570C23C2×#12012yes—
571C23#57714yes—
572C23C23×D411yescharacteristic
573Q8—24no—
574Q8—24no—
575Q8—24no—
576Q8—24no—
577Q8—48no—
578Q8—48no—
579Q8—44no—
580Q8—44no—
581Q8—84no—
582D4—44no—
583D4—44no—
584D4—84no—
585D4—84no—
586D4—88no—
587D4—168no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—28no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—44no—
608C2×C4—48no—
609C2×C4—44no—
610C2×C4—48no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—48no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—88no—
629C2×C4C2×#12012yes—
630C2×C4C2×#12012yes—
631C2×C4C2×#12012yes—
632C2×C4C2×#12012yes—
633C2×C4C2×#12012yes—
634C2×C4C23×D411yescharacteristic
635C2×C4C23×D411yescharacteristic
636C2×C4#58514yes—
637C8—44no—
638C8—44no—
639C22—28no—
640C22—22no—
641C22—22no—
642C22—44no—
643C22—44no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—88no—
648C22—84no—
649C22—84no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22#301014yes—
654C22C22×#12011yescharacteristic
655C22#344912yes—
656C22C2×#57712yes—
657C22#358612yes—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—42no—
664C4—44no—
665C4—44no—
666C4—42no—
667C4—44no—
668C4—42no—
669C4—44no—
670C4—84no—
671C4—82no—
672C4#344912yes—
673C4#344912yes—
674C2—24no—
675C2—82no—
676C2—162no—
677C2#3463211yescharacteristic
678C2C2×#301011yescharacteristic
679C2#6036811yescharacteristic
680C2#6098214yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35301C2
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
1. C22×#541C2semidirect
2. C2×#3112C2semidirect
3. C2×#3103C2semidirect
5. #29699C2semidirect
6. #29671C2semidirect
7. #29640C2semidirect
8. #29624C2semidirect
9. #29595C2semidirect
10. #29593C2semidirect
11. #23584C2semidirect
12. #20741C2semidirect
13. #20735C2semidirect
14. #20717C2semidirect
37. C2×#477C22semidirect
38. C2×#475C22semidirect
39. C2×#460C22semidirect
40. C2×#460C22semidirect
43. C2×#440C22semidirect
44. C2×C4×SD16C22semidirect
59. #2059C22semidirect
60. #2043C22semidirect
63. #2025C22semidirect
64. #2013C22semidirect
70. #1805C22semidirect
71. #1805C22semidirect
76. #1763C22semidirect
77. C4×#103C22semidirect
135. C2×C4×Q8D4semidirect
136. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5412
C2×#31122
C2×#31032
#353012
#296992
#296712
#296402
#296242
#295952
#295932
#235842
#207412
#207352
#207172