Finite Group Databasebeta

← #7511146

Subgroups — #7511146

Order 512.

1790 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35515C211yesmaximal, characteristic, complemented
2#35334C211yesmaximal, characteristic, complemented
3#32663C214yesmaximal, complemented
4#32659C214yesmaximal, complemented
5#32654C214yesmaximal, complemented
6#32652C214yesmaximal, complemented
7#23737C211yesmaximal, characteristic, complemented
8C2×#2092C211yesmaximal, characteristic, complemented
9#23661C211yesmaximal, characteristic, complemented
10#23659C211yesmaximal, characteristic, complemented
11#23614C211yesmaximal, characteristic, complemented
12#23610C211yesmaximal, characteristic, complemented
13#21304C211yesmaximal, characteristic, complemented
14#21235C211yesmaximal, characteristic
15#21158C211yesmaximal, characteristic, complemented
16#20929C211yesmaximal, characteristic
17#20928C211yesmaximal, characteristic
18#20879C211yesmaximal, characteristic, complemented
19#20872C211yesmaximal, characteristic, complemented
20C22×AGL(1,ℤ/8)C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×#518C2211yescharacteristic
23C2×C4×#41C2211yescharacteristic
24C2×#428C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C2×#424C2211yescharacteristic
30C2×#417C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C2×#416C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C2×#405C2211yescharacteristic
37#2851—22no—
38#2628—22no—
39#2581C2214yescomplemented
40#2580C2214yes—
41#2580C2214yescomplemented
42#2564—24no—
43#2538C2214yes—
44#2454—22no—
45#2452—22no—
46#2437—24no—
47#2317C2211yescharacteristic
48#2316C2211yescharacteristic
49#2316C2211yescharacteristic
50#2316C2211yescharacteristic
51#2311C2211yescharacteristic
52#2311C2211yescharacteristic
53#2309C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#390C2211yescharacteristic
56C2×#382C2211yescharacteristic
57#2097C2212yescomplemented
58#2095C2212yescomplemented
59#2093C2212yescomplemented
60#2092C2214yes—
61#2092C2214yes—
62#2092C2212yescomplemented
63#2085C2214yescomplemented
64#2085C2214yescomplemented
65#2085C2214yescomplemented
66#2084C2214yescomplemented
67#2067C2214yes—
68#2066C2214yescomplemented
69#2065C2214yes—
70#2065C2214yescomplemented
71#1985C2214yescomplemented
72#1984C2214yescomplemented
73#1973C2212yes—
74#1972C2212yes—
75#1970C2212yes—
76#1968C2212yes—
77#1943C2214yescomplemented
78#1942C2214yescomplemented
79#1880C2214yes—
80#1879C2214yes—
81#1879C2214yes—
82#1877C2214yes—
83#1865C2212yescomplemented
84#1864C2214yes—
85#1864C2212yescomplemented
86#1863C2212yescomplemented
87#1862C2214yes—
88#1862C2212yescomplemented
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C2×AGL(1,ℤ/8)—42no—
100C2×AGL(1,ℤ/8)—42no—
101C2×AGL(1,ℤ/8)Q818yescomplemented
102C22×SD16C2312yes—
103C22×D8C2312yes—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#533—42no—
109#533—42no—
110C2×C4⋊Q8D412yes—
111C2×#125D412yes—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8D414yes—
114#518—42no—
115#518—42no—
116C4×#41—42no—
117C4×#41—42no—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2314yes—
120C2×C4×D4—22no—
121C2×C4×D4—22no—
122C2×C4×D4C2314yes—
123C2×C4×D4C2312yes—
124C2×#117—21no—
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C22⋊C4—22no—
134C22×C22⋊C4—22no—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137#428—42no—
138#428—42no—
139#427—42no—
140C2×#107—24no—
141C2×#107C2312yes—
142C2×#107C2312yes—
143C2×#107C2312yes—
144C2×#107C2312yes—
145C2×#107C2312yes—
146C2×#106—42no—
147C2×#106C2312yes—
148C2×#106C2314yes—
149C2×#106C2312yes—
150C2×#106C2312yes—
151#424—42no—
152C2×C4⋊C8C2314yes—
153#418—42no—
154#417—42no—
155#416—24no—
156#416—42no—
157C2×#103C2311yescharacteristic
158C2×#103C2311yescharacteristic
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#102—24no—
162C2×#102—42no—
163C2×#102C2312yes—
164C2×#102C2312yes—
165C2×#102C2311yescharacteristic
166C2×#102C2311yescharacteristic
167C2×#102C2314yes—
168C2×#102C2314yes—
169C2×#102C2312yes—
170C2×#102C2312yes—
171#405—42no—
172C2×#97C2312yes—
173C2×C4×C8C2312yes—
174#400—24no—
175#399—24no—
176#399—24no—
177#396—24no—
178#396—24no—
179#395—24no—
180#393—24no—
181#393—24no—
182#391C2312yes—
183#391C2312yes—
184#390—22no—
185#390—21no—
186#390—21no—
187#390C2312yes—
188#390C2312yes—
189#390C2312yes—
190#390C2312yes—
191#390C2314yes—
192#389—22no—
193#389—22no—
194#389—22no—
195#389—22no—
196#389C2314yes—
197#389C2314yes—
198#389C2314yes—
199#389C2312yes—
200#389C2312yes—
201#387—24no—
202#387—24no—
203#386—22no—
204#386—22no—
205#385—22no—
206#385—24no—
207#385—24no—
208#385—24no—
209#385—24no—
210#385—22no—
211#384C2312yes—
212#382—21no—
213#382—21no—
214#382C2312yes—
215#382C2314yes—
216#380—22no—
217#380—22no—
218#380—24no—
219#380—22no—
220#380—22no—
221#380—22no—
222#379—22no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C2×#95—22no—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237C25—21no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41C2×Q814yes—
244C22×Q8C2×D414yes—
245C22×Q8C2411yescharacteristic
246C22×D4—21no—
247C22×D4—21no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4C2×Q814yes—
254C22×D4C2411yescharacteristic
255C22×D4C2412yes—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—24no—
261C23×C4—22no—
262C23×C4C2×D412yes—
263C23×C4C2411yescharacteristic
264AGL(1,ℤ/8)—416no—
265C2×SD16—42no—
266C2×SD16—42no—
267C2×SD16C2×Q818yes—
268C2×D8—42no—
269C2×D8—42no—
270C2×D8C2×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#4114yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2411yescharacteristic
287C22×C8C2411yescharacteristic
288C4⋊Q8—44no—
289#125—44no—
290C22⋊Q8—48no—
291C22⋊Q8—88no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×Q8—42no—
295C4×Q8—42no—
296C4×D4—42no—
297C4×D4—42no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4—48no—
301C4×D4—48no—
302#117—22no—
303#117—22no—
304#117—22no—
305#117—22no—
306#117—22no—
307#117—22no—
308#117—22no—
309#117—22no—
310#117—22no—
311#117—22no—
312#117—44no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D414yes—
332C2×C4⋊C4C2×D414yes—
333C2×C4⋊C4C2×D414yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—48no—
353C2×C22⋊C4—84no—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—22no—
362C2×C42—44no—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2412yes—
366C2×C42C2412yes—
367C2×C42C2412yes—
368#107—24no—
369#107—24no—
370#107—42no—
371#107—42no—
372#107—42no—
373#106—44no—
374#106—42no—
375#106—42no—
376#106—42no—
377C4⋊C8—44no—
378#103—22no—
379#103—22no—
380#103—42no—
381#103—42no—
382#102—24no—
383#102—24no—
384#102—22no—
385#102—22no—
386#102—44no—
387#102—42no—
388#102—42no—
389#102—44no—
390#97—42no—
391C4×C8—42no—
392#95—24no—
393#95—24no—
394#95—22no—
395#95—22no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402C24—21no—
403C24—22no—
404C24—24no—
405C24—42no—
406C24—42no—
407C24—82no—
408C24—82no—
409C24C22×D411yescharacteristic
410#41—28no—
411#41—28no—
412#41—44no—
413#41—44no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8C22×Q812yes—
423C2×D4—24no—
424C2×D4—28no—
425C2×D4—24no—
426C2×D4—28no—
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—22no—
433C2×D4—21no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—21no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—44no—
441C2×D4—42no—
442C2×D4—44no—
443C2×D4C22×Q812yes—
444C2×D4C22×Q814yes—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—48no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D414yes—
474C22×C4C22×D414yes—
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
483SD16—48no—
484D8—48no—
485M16—28no—
486M16—44no—
487C2×C8—28no—
488C2×C8—21no—
489C2×C8—22no—
490C2×C8—21no—
491C2×C8—22no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—44no—
501C2×C8C22×Q814yes—
502C2×C8C2×#4112yes—
503C2×C8C2×#4112yes—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—82no—
529C4⋊C4—84no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C4⋊C4—84no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—44no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C22⋊C4—84no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—44no—
555C42—42no—
556C42—42no—
557C42—42no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—24no—
562C23—42no—
563C23—44no—
564C23—44no—
565C23—42no—
566C23—41no—
567C23—42no—
568C23—41no—
569C23—42no—
570C23—48no—
571C23—88no—
572C23—164no—
573C23—162no—
574C23—162no—
575C23#53412yes—
576C23D4×Q812yes—
577C23#55012yes—
578C23C23×D411yescharacteristic
579Q8—22no—
580Q8—22no—
581Q8—41no—
582Q8—41no—
583Q8—88no—
584D4—22no—
585D4—24no—
586D4—24no—
587D4—22no—
588D4—48no—
589D4—42no—
590D4—41no—
591D4—48no—
592D4—42no—
593D4—41no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—48no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—84no—
627C2×C4—88no—
628C2×C4—168no—
629C2×C4—162no—
630C2×C4—162no—
631C2×C4—162no—
632C2×C4#53412yes—
633C2×C4#53914yes—
634C2×C4#54314yes—
635C2×C4D4×Q812yes—
636C2×C4#55012yes—
637C2×C4#55414yes—
638C2×C4#56214yes—
639C2×C4C23×Q811yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4C2×H(5,𝔽2)11yescharacteristic
642C2×C4C2×H(5,𝔽2)11yescharacteristic
643C2×C4C2×H(5,𝔽2)11yescharacteristic
644C2×C4C2×2−1+411yescharacteristic
645C8—22no—
646C8—22no—
647C8—24no—
648C8—42no—
649C8—42no—
650C22—24no—
651C22—24no—
652C22—22no—
653C22—22no—
654C22—42no—
655C22—44no—
656C22—44no—
657C22—48no—
658C22—82no—
659C22—84no—
660C22—81no—
661C22—82no—
662C22—81no—
663C22—82no—
664C22—168no—
665C22#273214yes—
666C22C2×#53411yescharacteristic
667C22C2×D4×Q811yescharacteristic
668C22C2×#55011yescharacteristic
669C22#358412yes—
670C22#358512yes—
671C4—42no—
672C4—42no—
673C4—44no—
674C4—44no—
675C4—42no—
676C4—82no—
677C4—82no—
678C4—81no—
679C4—82no—
680C4—81no—
681C4—82no—
682C4—82no—
683C4#273214yes—
684C4C2×#53911yescharacteristic
685C4C2×#54311yescharacteristic
686C4C2×#55411yescharacteristic
687C4C2×#56211yescharacteristic
688C2—24no—
689C2—82no—
690C2—84no—
691C2C2×#273211yescharacteristic
692C2#6076712yes—
693C2#6138512yes—
694C2#6141311yescharacteristic
695C2#6141711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×AGL(1,ℤ/8)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35515C2semidirect
2. #35334C2semidirect
3. #32663C2semidirect
4. #32659C2semidirect
5. #32654C2semidirect
6. #32652C2semidirect
7. #23737C2semidirect
8. C2×#2092C2semidirect
9. #23661C2semidirect
10. #23659C2semidirect
11. #23614C2semidirect
12. #23610C2semidirect
13. #21304C2semidirect
15. #21158C2semidirect
18. #20879C2semidirect
19. #20872C2semidirect
39. #2581C22semidirect
41. #2580C22semidirect
57. #2097C22semidirect
58. #2095C22semidirect
59. #2093C22semidirect
62. #2092C22semidirect
63. #2085C22semidirect
64. #2085C22semidirect
65. #2085C22semidirect
66. #2084C22semidirect
68. #2066C22semidirect
70. #2065C22semidirect
71. #1985C22semidirect
72. #1984C22semidirect
77. #1943C22semidirect
78. #1942C22semidirect
83. #1865C22semidirect
85. #1864C22semidirect
86. #1863C22semidirect
88. #1862C22semidirect
101. C2×AGL(1,ℤ/8)Q8semidirect

Maximal subgroups

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

subgroupindex
#355152
#353342
#326632
#326592
#326542
#326522
#237372
C2×#20922
#236612
#236592
#236142
#236102
#213042
#212352
#211582
#209292
#209282
#208792
#208722