Finite Group Databasebeta

← #7509280

Subgroups — #7509280

Order 512.

1670 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35620C211yesmaximal, characteristic, complemented
2#35558C211yesmaximal, characteristic, complemented
3#32453C214yesmaximal, complemented
4#32452C214yesmaximal, complemented
5#32437C214yesmaximal, complemented
6#32436C214yesmaximal, complemented
7C2×#2092C211yesmaximal, characteristic, complemented
8#23632C211yesmaximal, characteristic, complemented
9#23627C211yesmaximal, characteristic, complemented
10#23555C211yesmaximal, characteristic, complemented
11#23544C211yesmaximal, characteristic
12#23456C211yesmaximal, characteristic, complemented
13#21236C211yesmaximal, characteristic
14#21174C211yesmaximal, characteristic, complemented
15#21067C211yesmaximal, characteristic, complemented
16#21022C211yesmaximal, characteristic, complemented
17#20932C211yesmaximal, characteristic
18#20882C211yesmaximal, characteristic, complemented
19#20765C211yesmaximal, characteristic, complemented
20C22×AGL(1,ℤ/8)C2211yescharacteristic
21C2×#544C2211yescharacteristic
22C2×#520C2211yescharacteristic
23C2×#428C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#417C2211yescharacteristic
28C2×#418C2211yescharacteristic
29C2×#416C2211yescharacteristic
30C2×#416C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C2×C4×M16C2211yescharacteristic
34#2568C2212yescomplemented
35#2566C2212yescomplemented
36#2565C2212yescomplemented
37#2563C2212yescomplemented
38#2513—22no—
39#2513C2214yes—
40#2512—22no—
41#2512C2214yes—
42#2480—24no—
43#2450—24no—
44#2442—22no—
45#2439—22no—
46#2317C2211yescharacteristic
47#2317C2211yescharacteristic
48#2311C2211yescharacteristic
49#2309C2211yescharacteristic
50#2309C2211yescharacteristic
51#2297C2211yescharacteristic
52#2296C2211yescharacteristic
53C2×#390C2211yescharacteristic
54#2285C2211yescharacteristic
55C2×#382C2211yescharacteristic
56#2274C2211yescharacteristic
57#2092C2214yes—
58#2092C2214yes—
59#2074C2212yescomplemented
60#2073C2214yes—
61#2073C2212yescomplemented
62#2072C2212yescomplemented
63#2071C2214yes—
64#2071C2212yescomplemented
65#2059C2214yescomplemented
66#2057C2214yescomplemented
67#2055C2214yes—
68#2054C2214yes—
69#2025C2214yes—
70#2023C2214yes—
71#1973C2212yes—
72#1972C2212yes—
73#1970C2212yes—
74#1968C2212yes—
75#1949C2214yescomplemented
76#1948C2214yescomplemented
77#1928C2214yescomplemented
78#1925C2214yescomplemented
79#1917C2214yescomplemented
80#1916C2214yescomplemented
81#1880C2214yes—
82#1879C2214yes—
83#1865C2214yescomplemented
84#1864C2214yescomplemented
85#1836C2214yescomplemented
86#1835C2214yescomplemented
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C2×AGL(1,ℤ/8)—24no—
100C2×AGL(1,ℤ/8)—24no—
101C2×AGL(1,ℤ/8)—42no—
102C22×SD16C2312yes—
103C22×D8C2312yes—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#544—42no—
109#544—42no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8C2312yes—
112C2×#126—24no—
113C2×#125—21no—
114C2×#125C2312yes—
115C2×C22⋊Q8—22no—
116#520—42no—
117#520—42no—
118C2×C4×Q8C2312yes—
119C2×C4×D4—22no—
120C2×C4×D4C2312yes—
121C2×#117—21no—
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—22no—
132C22×C42C2311yescharacteristic
133#428—42no—
134C2×#107—42no—
135C2×#107C2312yes—
136C2×#107C2314yes—
137C2×#106—42no—
138C2×#106C2312yes—
139C2×#106C2314yes—
140#424—24no—
141C2×C4⋊C8C2314yes—
142#418—42no—
143#417—42no—
144#416—24no—
145#416—24no—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#102—42no—
150C2×#102—42no—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2312yes—
155C2×#102C2314yes—
156C2×#102C2312yes—
157C2×#102C2312yes—
158C4×M16—24no—
159C2×#97C2312yes—
160C2×C4×C8C2312yes—
161#400—24no—
162#398—24no—
163#397—24no—
164#397—24no—
165#397—24no—
166#396—24no—
167#396—24no—
168#393—24no—
169#391—24no—
170#391C2311yescharacteristic
171#391C2312yes—
172#391C2311yescharacteristic
173#390—21no—
174#390—22no—
175#390—21no—
176#390—22no—
177#390C2311yescharacteristic
178#390C2312yes—
179#390C2314yes—
180#390C2314yes—
181#390C2311yescharacteristic
182#390C2312yes—
183#390C2312yes—
184#389—22no—
185#389D412yes—
186#389D412yes—
187#389C2314yes—
188#388—24no—
189#388—24no—
190#388—24no—
191#387—24no—
192#386—22no—
193#386—22no—
194#385—22no—
195#385—22no—
196#385—24no—
197#385D414yes—
198#384—21no—
199#384C2312yes—
200#384C2312yes—
201#382—21no—
202#382—22no—
203#382C2312yes—
204#382C2312yes—
205#382C2312yes—
206#382C2312yes—
207#380—22no—
208#380—24no—
209#380—22no—
210#379—22no—
211#379—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C2×#95—22no—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#336C2314yes—
229#336C2314yes—
230#336C2312yes—
231#336C2312yes—
232#336C2314yes—
233C25—21no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41#4114yes—
240C22×Q8—22no—
241C22×Q8C2411yescharacteristic
242C22×D4—21no—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4C2412yes—
251C22×D4C2411yescharacteristic
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4C2×D412yes—
258C23×C4C2411yescharacteristic
259AGL(1,ℤ/8)—416no—
260C2×SD16—24no—
261C2×SD16—24no—
262C2×SD16—42no—
263C2×D8—24no—
264C2×D8—24no—
265C2×D8—42no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—22no—
270C2×M16—22no—
271C2×M16#4114yes—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8—22no—
277C22×C8—22no—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C4⋊Q8—42no—
283C4⋊Q8—42no—
284C4⋊Q8—44no—
285#126—88no—
286#125—42no—
287#125—42no—
288#125—44no—
289C22⋊Q8—48no—
290C4×Q8—42no—
291C4×Q8—42no—
292C4×D4—42no—
293C4×D4—42no—
294C4×D4—48no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117—22no—
302#117—22no—
303#117—22no—
304#117—22no—
305#117—44no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D414yes—
328C2×C4⋊C4C2×D414yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—84no—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42—24no—
359C2×C42C2×D412yes—
360C2×C42C2412yes—
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—44no—
365#107—42no—
366#106—44no—
367#106—42no—
368C4⋊C8—28no—
369#103—24no—
370#103—24no—
371#103—42no—
372#102—24no—
373#102—24no—
374#102—44no—
375#102—44no—
376#102—42no—
377#102—42no—
378#102—42no—
379#97—24no—
380C4×C8—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95C2×D414yes—
392#95C2×D414yes—
393C24—21no—
394C24—22no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—82no—
399C24—82no—
400C24C22×D411yescharacteristic
401#41—28no—
402#41—28no—
403#41—44no—
404#41—44no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—44no—
411C2×Q8—84no—
412C2×Q8C2×#4112yes—
413C2×D4—24no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4C2×#4112yes—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—84no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4C22×D412yes—
458C22×C4C22×D414yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4H(5,𝔽2)12yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C42−1+412yes—
468C22×C42−1+412yes—
469C22×C4C2511yescharacteristic
470SD16—48no—
471D8—48no—
472M16—28no—
473M16—44no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—44no—
486C2×C8—44no—
487C2×C8C2×#4114yes—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—84no—
516C4⋊C4—82no—
517C4⋊C4—84no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—82no—
528C22⋊C4—84no—
529C22⋊C4—82no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C42—24no—
534C42—21no—
535C42—21no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—21no—
540C42—21no—
541C42—22no—
542C42—44no—
543C42—42no—
544C42—42no—
545C42—42no—
546C42—84no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—42no—
551C23—42no—
552C23—41no—
553C23—41no—
554C23—44no—
555C23—44no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—88no—
560C23—84no—
561C23—164no—
562C23—162no—
563C23—162no—
564C23#53512yes—
565C23#54712yes—
566C23#54812yes—
567C23C23×D411yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572Q8—164no—
573D4—22no—
574D4—22no—
575D4—48no—
576D4—42no—
577D4—41no—
578D4—42no—
579D4—42no—
580D4—41no—
581D4—42no—
582D4—84no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—48no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—84no—
617C2×C4—164no—
618C2×C4—162no—
619C2×C4—162no—
620C2×C4—162no—
621C2×C4#53512yes—
622C2×C4#53914yes—
623C2×C4#54214yes—
624C2×C4#54712yes—
625C2×C4#54812yes—
626C2×C4#55114yes—
627C2×C4#56314yes—
628C2×C4C22×#4111yescharacteristic
629C2×C4C22×#4111yescharacteristic
630C2×C4C2×H(5,𝔽2)11yescharacteristic
631C2×C4C2×H(5,𝔽2)11yescharacteristic
632C2×C4C2×2−1+411yescharacteristic
633C2×C4C2×2−1+411yescharacteristic
634C8—24no—
635C8—42no—
636C8—42no—
637C8—42no—
638C22—24no—
639C22—22no—
640C22—22no—
641C22—24no—
642C22—44no—
643C22—42no—
644C22—82no—
645C22—81no—
646C22—81no—
647C22—84no—
648C22—82no—
649C22—82no—
650C22—84no—
651C22—82no—
652C22—168no—
653C22#272314yes—
654C22C2×#53511yescharacteristic
655C22C2×#54711yescharacteristic
656C22C2×#54811yescharacteristic
657C22#358412yes—
658C22#358512yes—
659C4—44no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—81no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—81no—
670C4—82no—
671C4—82no—
672C4#272314yes—
673C4C2×#53911yescharacteristic
674C4C2×#54211yescharacteristic
675C4C2×#55111yescharacteristic
676C4C2×#56311yescharacteristic
677C2—24no—
678C2—82no—
679C2—162no—
680C2C2×#272311yescharacteristic
681C2#6081911yescharacteristic
682C2#6082511yescharacteristic
683C2#6127412yes—
684C2#6134912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×AGL(1,ℤ/8)C22
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. #35620C2semidirect
2. #35558C2semidirect
3. #32453C2semidirect
4. #32452C2semidirect
5. #32437C2semidirect
6. #32436C2semidirect
7. C2×#2092C2semidirect
8. #23632C2semidirect
9. #23627C2semidirect
10. #23555C2semidirect
12. #23456C2semidirect
14. #21174C2semidirect
15. #21067C2semidirect
16. #21022C2semidirect
18. #20882C2semidirect
19. #20765C2semidirect
34. #2568C22semidirect
35. #2566C22semidirect
36. #2565C22semidirect
37. #2563C22semidirect
59. #2074C22semidirect
61. #2073C22semidirect
62. #2072C22semidirect
64. #2071C22semidirect
65. #2059C22semidirect
66. #2057C22semidirect
75. #1949C22semidirect
76. #1948C22semidirect
77. #1928C22semidirect
78. #1925C22semidirect
79. #1917C22semidirect
80. #1916C22semidirect
83. #1865C22semidirect
84. #1864C22semidirect
85. #1836C22semidirect
86. #1835C22semidirect

Maximal subgroups

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

subgroupindex
#356202
#355582
#324532
#324522
#324372
#324362
C2×#20922
#236322
#236272
#235552
#235442
#234562
#212362
#211742
#210672
#210222
#209322
#208822
#207652