Finite Group Databasebeta

← #6491251

Subgroups — #6491251

Order 512.

3025 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#541C211yesmaximal, characteristic, complemented
2C2×#3260C212yesmaximal, complemented
3#35261C211yesmaximal, characteristic
4#32960C212yesmaximal, complemented
5#32903C212yesmaximal, complemented
6#29701C212yesmaximal, complemented
7#29671C212yesmaximal, complemented
8#25511C214yesmaximal, complemented
9#24387C212yesmaximal, complemented
10#24381C212yesmaximal, complemented
11#24305C212yesmaximal, complemented
12#24209C212yesmaximal, complemented
13#24163C214yesmaximal, complemented
14#21898C212yesmaximal, complemented
15#20628C211yesmaximal, characteristic, complemented
16C2×#541—216no—
17C2×#541—22no—
18C2×#541—22no—
19C2×#541C2212yes—
20C2×#541C2212yes—
21C2×#541C2212yes—
22C2×#541C2212yes—
23C2×#540C2211yescharacteristic
24C22×C4⋊Q8C2211yescharacteristic
25C22×#126—22no—
26C22×#125—22no—
27C22×#125C2211yescharacteristic
28C22×C22⋊Q8C2212yes—
29C22×C4×Q8C2212yes—
30C22×#117C2211yescharacteristic
31#3260—216no—
32C2×#479C2214yescomplemented
33C2×#478C2214yescomplemented
34C2×#468C2214yes—
35C2×#465C2214yes—
36C2×#460C2214yescomplemented
37C2×#461C2214yescomplemented
38C2×#448C2212yes—
39C2×#448C2212yes—
40C2×#424C2212yes—
41C2×#407C2211yescharacteristic
42C22×C22⋊C8C2211yescharacteristic
43#2856C2211yescharacteristic
44#2851C2211yescharacteristic
45#2847C2212yes—
46#2629C2211yescharacteristic
47#2624C2212yes—
48#2619C2211yescharacteristic
49#2476—22no—
50#2471—22no—
51#2443—22no—
52#2424—22no—
53#2324C2212yes—
54#2314C2211yescharacteristic
55C4×#121C2212yes—
56#2059C2214yescomplemented
57#2043C2214yescomplemented
58#2036C2214yes—
59#2035C2214yes—
60#2025C2214yescomplemented
61#2013C2214yescomplemented
62#2000C2214yes—
63#2000C2214yes—
64#1989C2214yescomplemented
65#1981C2214yescomplemented
66#1894C2214yes—
67#1892C2214yes—
68#1880C2214yescomplemented
69#1880C2214yescomplemented
70#1874C2212yes—
71#1874C2212yes—
72#1865C2214yescomplemented
73#1863C2214yescomplemented
74#1853C2212yes—
75#1852C2212yes—
76#1805C2214yescomplemented
77#1803C2214yescomplemented
78#1798C2212yes—
79#1798C2212yes—
80#1797C2212yes—
81#1797C2212yes—
82#1787C2212yes—
83#1787C2212yes—
84#1729C2212yes—
85C23×Q8C2312yes—
86C24×C4C2311yescharacteristic
87C22×SD16C2314yes—
88C22×SD16C2314yes—
89C22×M16C2312yes—
90C23×C8C2312yes—
91#541—232no—
92#541—232no—
93#541—28no—
94#541—28no—
95#541—216no—
96#541—216no—
97#541—28no—
98#541D4116yescomplemented
99#540—28no—
100C2×C4⋊Q8—28no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×#127C2311yescharacteristic
105C2×#126—232no—
106C2×#126—24no—
107C2×#126—24no—
108C2×#126—24no—
109C2×#126—24no—
110C2×#125—232no—
111C2×#125—28no—
112C2×#125—24no—
113C2×#125—24no—
114C2×#125—24no—
115C2×#125—24no—
116C2×#125—22no—
117C2×#125C2311yescharacteristic
118C2×#125C2312yes—
119C2×#125C2312yes—
120C2×#123—22no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8—216no—
124C2×C22⋊Q8C2314yes—
125C2×C22⋊Q8C2314yes—
126C2×#121—22no—
127C2×#121—22no—
128C2×#121C2312yes—
129C2×#121C2312yes—
130C2×#121C2312yes—
131C2×#120—22no—
132C2×C4×Q8—216no—
133C2×C4×Q8—22no—
134C2×C4×Q8—22no—
135C2×C4×Q8C2314yes—
136C2×C4×Q8C2314yes—
137C2×C4×D4—22no—
138C2×C4×D4C2312yes—
139C2×#117—28no—
140C2×#117D412yes—
141C2×#117D412yes—
142C2×#117C2311yescharacteristic
143C2×#117C2311yescharacteristic
144C2×#117C2311yescharacteristic
145C2×#117C2311yescharacteristic
146C22×C4⋊C4—22no—
147C22×C4⋊C4—22no—
148C22×C4⋊C4—22no—
149C22×C4⋊C4—22no—
150C22×C4⋊C4D412yes—
151C22×C4⋊C4C2312yes—
152C22×C22⋊C4—22no—
153C22×C22⋊C4D412yes—
154C22×C42—22no—
155C22×C42C2311yescharacteristic
156C22×C42C2311yescharacteristic
157#479—216no—
158#478—216no—
159#468—216no—
160#465—216no—
161#461—216no—
162#460—216no—
163#448—44no—
164#448—44no—
165C2×#107C2314yes—
166C2×#106C2314yes—
167#424—28no—
168C2×C4⋊C8C2314yes—
169C2×C4⋊C8C2314yes—
170C2×#103C2314yes—
171C2×#103C2314yes—
172C2×#103C2314yes—
173C2×#103C2314yes—
174C2×#103C2314yes—
175C2×#103C2314yes—
176C2×#102C2314yes—
177C2×#102C2314yes—
178#407—42no—
179C2×C22⋊C8—42no—
180C2×C22⋊C8C2312yes—
181C2×C22⋊C8C2312yes—
182C2×C22⋊C8C2312yes—
183C2×C22⋊C8C2312yes—
184#399—22no—
185#398—22no—
186#397—22no—
187#396—22no—
188#394—22no—
189#394—22no—
190#394—22no—
191#392—22no—
192#390—22no—
193#390—22no—
194#390C2312yes—
195#390C2312yes—
196#389C2312yes—
197#389C2312yes—
198#388—22no—
199#388—22no—
200#388—22no—
201#387—22no—
202#387—22no—
203#386—22no—
204#386—22no—
205#386—22no—
206#386—22no—
207#386—22no—
208#385—22no—
209#385—22no—
210#384C2311yescharacteristic
211#382C2311yescharacteristic
212#380C2312yes—
213#380C2312yes—
214#380C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—22no—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#336C2314yes—
222#336C2314yes—
223C25C2×D411yescharacteristic
224C22×Q8—28no—
225C22×Q8—22no—
226C22×Q8—24no—
227C22×Q8—24no—
228C22×Q8—44no—
229C22×Q8C2414yes—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4C2412yes—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—24no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—42no—
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×SD16—28no—
251C2×SD16—28no—
252C2×SD16—44no—
253C2×SD16—44no—
254C2×SD16—44no—
255C2×SD16—44no—
256C2×M16—24no—
257C2×M16—42no—
258C2×M16—42no—
259C22×C8—24no—
260C22×C8—42no—
261C22×C8—42no—
262C22×C8C2414yes—
263C22×C8C2414yes—
264C4⋊Q8—216no—
265C4⋊Q8—28no—
266C4⋊Q8—24no—
267C4⋊Q8C2×D418yes—
268#127—24no—
269#126—232no—
270#126—232no—
271#126—232no—
272#126—232no—
273#125—232no—
274#125—232no—
275#125—232no—
276#125—216no—
277#125—232no—
278#125—28no—
279#125—24no—
280#125—24no—
281#125C2×D418yes—
282#123—48no—
283C22⋊Q8—216no—
284C22⋊Q8—216no—
285C22⋊Q8—28no—
286C22⋊Q8—28no—
287C22⋊Q8—28no—
288C22⋊Q8C2×D4116yes—
289#121—28no—
290#121—24no—
291#121—24no—
292#121—44no—
293#121—48no—
294#121—84no—
295#120—84no—
296C4×Q8—216no—
297C4×Q8—216no—
298C4×Q8—28no—
299C4×Q8—28no—
300C4×Q8—28no—
301C4×Q8C2×D4116yes—
302C4×D4—28no—
303C4×D4—84no—
304#117—24no—
305#117—28no—
306#117—28no—
307#117—24no—
308#117—24no—
309#117C2×D418yes—
310C2×C4⋊C4—216no—
311C2×C4⋊C4—216no—
312C2×C4⋊C4—216no—
313C2×C4⋊C4—216no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—28no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2414yes—
337C2×C22⋊C4—216no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—28no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—216no—
357C2×C42—24no—
358C2×C42—24no—
359C2×C42—24no—
360C2×C42—24no—
361C2×C42—22no—
362C2×C42C2×D412yes—
363C2×C42C2×D412yes—
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368#107—28no—
369#106—28no—
370C4⋊C8—28no—
371C4⋊C8—28no—
372#103—28no—
373#103—28no—
374#103—44no—
375#103—44no—
376#103—44no—
377#103—44no—
378#102—28no—
379#102—28no—
380C22⋊C8—24no—
381C22⋊C8—24no—
382C22⋊C8—42no—
383C22⋊C8—42no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391C24—22no—
392C24—22no—
393C24—24no—
394C24—22no—
395C24—41no—
396C24—42no—
397C24C22×D411yescharacteristic
398C24C22×D411yescharacteristic
399C24C22×D411yescharacteristic
400C2×Q8—28no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—416no—
407C2×Q8—416no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8C22×D418yes—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—24no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4—82no—
426C2×D4—82no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D414yes—
474C22×C4C22×D411yescharacteristic
475C22×C4H(5,𝔽2)12yes—
476C22×C42−1+412yes—
477C22×C4C2511yescharacteristic
478SD16—88no—
479SD16—88no—
480M16—84no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—84no—
488C4⋊C4—28no—
489C4⋊C4—216no—
490C4⋊C4—216no—
491C4⋊C4—216no—
492C4⋊C4—216no—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—216no—
496C4⋊C4—216no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—44no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4#12018yes—
510C4⋊C4C22×D418yes—
511C22⋊C4—216no—
512C22⋊C4—216no—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4#12018yes—
528C42—216no—
529C42—216no—
530C42—24no—
531C42—24no—
532C42—22no—
533C42—22no—
534C42—82no—
535C42C22×D414yes—
536C42C22×D414yes—
537C23—22no—
538C23—21no—
539C23—21no—
540C23—22no—
541C23—21no—
542C23—21no—
543C23—22no—
544C23—22no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—22no—
549C23—44no—
550C23—48no—
551C23—44no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—84no—
560C23—82no—
561C23C2×#12012yes—
562C23#53712yes—
563C23#57714yes—
564C23C23×D411yescharacteristic
565Q8—28no—
566Q8—28no—
567Q8—416no—
568Q8—44no—
569Q8—44no—
570D4—44no—
571D4—44no—
572D4—88no—
573D4—164no—
574D4—164no—
575D4—164no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—416no—
605C2×C4—416no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—164no—
622C2×C4C2×#12014yes—
623C2×C4C2×#12014yes—
624C2×C4C2×#12012yes—
625C2×C4#53712yes—
626C2×C4C23×D412yes—
627C2×C4C2×H(5,𝔽2)11yescharacteristic
628C2×C4C2×2−1+411yescharacteristic
629C2×C4#58512yes—
630C8—44no—
631C8—44no—
632C22—24no—
633C22—22no—
634C22—24no—
635C22—22no—
636C22—22no—
637C22—24no—
638C22—42no—
639C22—41no—
640C22—41no—
641C22—42no—
642C22—41no—
643C22—42no—
644C22—41no—
645C22—41no—
646C22—88no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22#326014yescomplemented
655C22C22×#12011yescharacteristic
656C22C2×#53711yescharacteristic
657C22#344911yescharacteristic
658C22C2×#57712yes—
659C22#358612yes—
660C4—22no—
661C4—22no—
662C4—24no—
663C4—24no—
664C4—41no—
665C4—48no—
666C4—42no—
667C4—42no—
668C4—48no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4#348812yes—
673C4#348912yes—
674C2—24no—
675C2—42no—
676C2—81no—
677C2—162no—
678C2#3525111yescharacteristic
679C2#6037412yes—
680C2C2×#326011yescharacteristic
681C2#6092111yescharacteristic
682C2#6098212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35261C2
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×#3260C2semidirect
4. #32960C2semidirect
5. #32903C2semidirect
6. #29701C2semidirect
7. #29671C2semidirect
8. #25511C2semidirect
9. #24387C2semidirect
10. #24381C2semidirect
11. #24305C2semidirect
12. #24209C2semidirect
13. #24163C2semidirect
14. #21898C2semidirect
15. #20628C2semidirect
32. C2×#479C22semidirect
33. C2×#478C22semidirect
36. C2×#460C22semidirect
37. C2×#461C22semidirect
56. #2059C22semidirect
57. #2043C22semidirect
60. #2025C22semidirect
61. #2013C22semidirect
64. #1989C22semidirect
65. #1981C22semidirect
68. #1880C22semidirect
69. #1880C22semidirect
72. #1865C22semidirect
73. #1863C22semidirect
76. #1805C22semidirect
77. #1803C22semidirect
98. #541D4semidirect
654. C22#3260semidirect

Maximal subgroups

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

subgroupindex
C22×#5412
C2×#32602
#352612
#329602
#329032
#297012
#296712
#255112
#243872
#243812
#243052
#242092
#241632
#218982
#206282