Finite Group Databasebeta

← #7505065

Subgroups — #7505065

Order 512.

1720 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35662C211yesmaximal, characteristic, complemented
2#35620C211yesmaximal, characteristic
3#31549C214yesmaximal, complemented
4#31548C214yesmaximal, complemented
5#31533C214yesmaximal, complemented
6#31532C214yesmaximal, complemented
7C2×#2097C211yesmaximal, characteristic, complemented
8#23661C211yesmaximal, characteristic, complemented
9#23611C211yesmaximal, characteristic
10#23555C211yesmaximal, characteristic, complemented
11#23457C211yesmaximal, characteristic, complemented
12#23451C211yesmaximal, characteristic
13#21323C211yesmaximal, characteristic, complemented
14#21237C211yesmaximal, characteristic
15#21022C211yesmaximal, characteristic, complemented
16#20931C211yesmaximal, characteristic
17#20930C211yesmaximal, characteristic
18#20882C211yesmaximal, characteristic
19#20651C211yesmaximal, characteristic, complemented
20C22×#137C2211yescharacteristic
21C2×#544C2211yescharacteristic
22C2×#533C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#417C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C2×C4×M16C2211yescharacteristic
36#2582—22no—
37#2581—22no—
38#2568C2212yes—
39#2566C2212yes—
40#2565C2212yes—
41#2563C2212yes—
42#2453C2214yes—
43#2451C2214yes—
44#2450—24no—
45#2442—22no—
46#2439—22no—
47#2438—24no—
48#2317C2211yescharacteristic
49#2317C2211yescharacteristic
50#2316C2211yescharacteristic
51#2311C2211yescharacteristic
52#2310C2211yescharacteristic
53#2297C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#391C2211yescharacteristic
56C2×#382C2211yescharacteristic
57#2097C2214yes—
58#2097C2214yes—
59#2085C2214yes—
60#2084C2214yes—
61#2068C2214yes—
62#2067C2214yes—
63#2059C2214yes—
64#2057C2214yes—
65#2026C2214yes—
66#2026C2212yes—
67#2025C2212yes—
68#2024C2214yes—
69#2024C2212yes—
70#2023C2212yes—
71#1993C2214yes—
72#1991C2214yes—
73#1973C2212yes—
74#1971C2212yes—
75#1969C2212yes—
76#1968C2212yes—
77#1917C2214yes—
78#1916C2214yes—
79#1880C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1877C2214yes—
83#1865C2214yes—
84#1864C2214yes—
85#1805C2214yes—
86#1803C2214yes—
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C2×#137—42no—
100C2×#137—42no—
101C2×#137Q818yescomplemented
102C22×Q16C2312yes—
103C22×SD16C2312yes—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#544—42no—
109#544—42no—
110#533—24no—
111#533—42no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2312yes—
115C2×#125—21no—
116C2×#125C2312yes—
117C2×#125C2312yes—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—22no—
120#520—42no—
121#520—42no—
122#519—24no—
123#519—42no—
124C2×C4×Q8—22no—
125C2×C4×Q8—22no—
126C2×C4×Q8C2314yes—
127C2×C4×Q8C2312yes—
128C2×C4×D4—22no—
129C2×C4×D4C2312yes—
130C2×#117—21no—
131C2×#117—22no—
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C22×C4⋊C4—21no—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#428—42no—
143C2×#107—42no—
144C2×#107C2312yes—
145C2×#107C2314yes—
146C2×#106—42no—
147C2×#106C2312yes—
148C2×#106C2314yes—
149#424—42no—
150C2×C4⋊C8C2314yes—
151#419—42no—
152#417—42no—
153#416—24no—
154#416—42no—
155C2×#103—42no—
156C2×#103—42no—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2314yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2314yes—
164C2×#102C2312yes—
165C2×#102C2312yes—
166C2×#102C2312yes—
167C4×M16—42no—
168C2×#97C2312yes—
169C2×C4×C8C2312yes—
170#400—24no—
171#400—24no—
172#398—24no—
173#398—24no—
174#398—24no—
175#396—24no—
176#395—24no—
177#393—24no—
178#391—21no—
179#391—21no—
180#391—24no—
181#391—22no—
182#391—24no—
183#391C2311yescharacteristic
184#391C2312yes—
185#391C2314yes—
186#391C2311yescharacteristic
187#391C2312yes—
188#391C2312yes—
189#390C2311yescharacteristic
190#390C2312yes—
191#390C2311yescharacteristic
192#389—22no—
193#389—22no—
194#389—24no—
195#389D412yes—
196#389D412yes—
197#389C2314yes—
198#389C2314yes—
199#387—24no—
200#387—24no—
201#387—24no—
202#386—22no—
203#386—22no—
204#385—22no—
205#385—22no—
206#385—24no—
207#385D414yes—
208#384—21no—
209#384C2312yes—
210#382—21no—
211#382—22no—
212#382C2312yes—
213#382C2312yes—
214#382C2312yes—
215#381—22no—
216#380—24no—
217#380—22no—
218#380—22no—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—22no—
222C2×#95—22no—
223C2×#95—22no—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#336C2314yes—
233#336C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41C2×Q814yes—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8C2×Q814yes—
252C22×Q8C2412yes—
253C22×Q8C2411yescharacteristic
254C22×D4—21no—
255C22×D4C2411yescharacteristic
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—22no—
262C23×C4C2×D412yes—
263C23×C4C2411yescharacteristic
264#137—416no—
265C2×Q16—42no—
266C2×Q16—42no—
267C2×Q16C2×Q818yes—
268C2×SD16—42no—
269C2×SD16—42no—
270C2×SD16C2×Q818yes—
271C2×M16—22no—
272C2×M16—22no—
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—22no—
276C2×M16C2×Q814yes—
277C22×C8—22no—
278C22×C8—22no—
279C22×C8—22no—
280C22×C8—22no—
281C22×C8—22no—
282C22×C8—22no—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C4⋊Q8—24no—
288C4⋊Q8—42no—
289C4⋊Q8—42no—
290C4⋊Q8—42no—
291C4⋊Q8—44no—
292#125—24no—
293#125—42no—
294#125—42no—
295#125—42no—
296#125—44no—
297C22⋊Q8—48no—
298C22⋊Q8—48no—
299C4×Q8—28no—
300C4×Q8—48no—
301C4×Q8—44no—
302C4×Q8—42no—
303C4×Q8—42no—
304C4×Q8—48no—
305C4×D4—42no—
306C4×D4—42no—
307C4×D4—48no—
308#117—22no—
309#117—22no—
310#117—22no—
311#117—22no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—48no—
318#117—44no—
319#117#4114yes—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—44no—
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D414yes—
345C2×C4⋊C4C2×D414yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2411yescharacteristic
352C2×C4⋊C4C2412yes—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D412yes—
362C2×C42—21no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—24no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42C2×D412yes—
370C2×C42C2412yes—
371C2×C42C2412yes—
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374#107—44no—
375#107—42no—
376#106—44no—
377#106—42no—
378C4⋊C8—44no—
379#103—24no—
380#103—44no—
381#103—44no—
382#103—42no—
383#103—42no—
384#103—42no—
385#103—42no—
386#102—24no—
387#102—42no—
388#102—42no—
389#97—42no—
390C4×C8—42no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—22no—
396#95—22no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95C2×D414yes—
404#95C2×D414yes—
405C24—21no—
406C24C22×D411yescharacteristic
407#41—28no—
408#41—28no—
409#41—44no—
410#41—44no—
411C2×Q8—24no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—28no—
415C2×Q8—28no—
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8—21no—
422C2×Q8—22no—
423C2×Q8—21no—
424C2×Q8—44no—
425C2×Q8—44no—
426C2×Q8—44no—
427C2×Q8—42no—
428C2×Q8—44no—
429C2×Q8—42no—
430C2×Q8—44no—
431C2×Q8—82no—
432C2×Q8—84no—
433C2×Q8C22×Q814yes—
434C2×Q8C22×Q812yes—
435C2×D4—24no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4C22×Q812yes—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—84no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D414yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×Q812yes—
477C22×C4C2×#4112yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C42−1+412yes—
482C22×C4C2511yescharacteristic
483Q16—48no—
484SD16—48no—
485M16—28no—
486M16—44no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—44no—
499C2×C8—44no—
500C2×C8C22×Q814yes—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—48no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C4⋊C4—84no—
533C4⋊C4C2×#4114yes—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—44no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—84no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C42—24no—
544C42—24no—
545C42—21no—
546C42—21no—
547C42—22no—
548C42—21no—
549C42—21no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—48no—
554C42—44no—
555C42—42no—
556C42—42no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—42no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23#53512yes—
565C23#54612yes—
566C23D4×Q812yes—
567C23C23×D411yescharacteristic
568Q8—24no—
569Q8—22no—
570Q8—24no—
571Q8—22no—
572Q8—42no—
573Q8—41no—
574Q8—48no—
575Q8—42no—
576Q8—41no—
577Q8—88no—
578Q8—164no—
579D4—22no—
580D4—22no—
581D4—48no—
582D4—41no—
583D4—41no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—48no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—48no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—84no—
622C2×C4—164no—
623C2×C4—162no—
624C2×C4—162no—
625C2×C4—162no—
626C2×C4#53512yes—
627C2×C4#54214yes—
628C2×C4#54314yes—
629C2×C4#54612yes—
630C2×C4D4×Q812yes—
631C2×C4#55414yes—
632C2×C4#56214yes—
633C2×C4C23×Q811yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C2×C4C2×H(5,𝔽2)11yescharacteristic
638C2×C4C2×2−1+411yescharacteristic
639C8—24no—
640C8—42no—
641C8—42no—
642C8—42no—
643C22—24no—
644C22—22no—
645C22—22no—
646C22—24no—
647C22—42no—
648C22—44no—
649C22—82no—
650C22—81no—
651C22—81no—
652C22#269214yes—
653C22C2×#53511yescharacteristic
654C22C2×#54611yescharacteristic
655C22C2×D4×Q811yescharacteristic
656C22#358412yes—
657C22#358612yes—
658C4—44no—
659C4—42no—
660C4—42no—
661C4—44no—
662C4—42no—
663C4—44no—
664C4—42no—
665C4—42no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—81no—
670C4—81no—
671C4—82no—
672C4#269214yes—
673C4C2×#54211yescharacteristic
674C4C2×#54311yescharacteristic
675C4C2×#55411yescharacteristic
676C4C2×#56211yescharacteristic
677C2—24no—
678C2—82no—
679C2C2×#269211yescharacteristic
680C2#6081911yescharacteristic
681C2#6082611yescharacteristic
682C2#6119712yes—
683C2#6138612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35662C2semidirect
3. #31549C2semidirect
4. #31548C2semidirect
5. #31533C2semidirect
6. #31532C2semidirect
7. C2×#2097C2semidirect
8. #23661C2semidirect
10. #23555C2semidirect
11. #23457C2semidirect
13. #21323C2semidirect
15. #21022C2semidirect
19. #20651C2semidirect
101. C2×#137Q8semidirect

Maximal subgroups

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

subgroupindex
#356622
#356202
#315492
#315482
#315332
#315322
C2×#20972
#236612
#236112
#235552
#234572
#234512
#213232
#212372
#210222
#209312
#209302
#208822
#206512