Finite Group Databasebeta

← #7502479

Subgroups — #7502479

Order 512.

1848 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
1C22×#501C211yesmaximal, characteristic, complemented
2C2×#3084C211yesmaximal, characteristic
3#35639C211yesmaximal, characteristic
4#35576C211yesmaximal, characteristic, complemented
5#30972C214yesmaximal, complemented
6#30970C214yesmaximal, complemented
7#30965C214yesmaximal, complemented
8#30963C214yesmaximal, complemented
9#23737C211yesmaximal, characteristic
10#23667C211yesmaximal, characteristic, complemented
11#23661C211yesmaximal, characteristic, complemented
12#23555C211yesmaximal, characteristic, complemented
13#23553C211yesmaximal, characteristic, complemented
14#23452C211yesmaximal, characteristic
15#21086C211yesmaximal, characteristic
16#21067C211yesmaximal, characteristic
17#21038C211yesmaximal, characteristic, complemented
18#21021C211yesmaximal, characteristic, complemented
19#20882C211yesmaximal, characteristic
20C2×#544C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×#125C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C2×#501—28no—
25C2×#501C2214yes—
26C2×#501C2214yes—
27#3084—28no—
28C2×#479C2212yes—
29C2×#475C2212yes—
30C2×#478C2212yes—
31C2×#476C2212yes—
32C2×#428C2211yescharacteristic
33C2×#428C2211yescharacteristic
34C22×#107C2211yescharacteristic
35C22×#107C2211yescharacteristic
36C22×#106C2211yescharacteristic
37C22×#106C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C2×#417C2211yescharacteristic
40C2×#417C2211yescharacteristic
41C2×#416C2211yescharacteristic
42C2×#416C2211yescharacteristic
43C22×#97C2211yescharacteristic
44#2583—22no—
45#2579—22no—
46#2551—22no—
47#2551C2214yes—
48#2550—22no—
49#2550C2214yes—
50#2487—22no—
51#2454C2214yes—
52#2453C2214yes—
53#2442—22no—
54#2441—22no—
55#2434—22no—
56#2317C2211yescharacteristic
57#2317C2211yescharacteristic
58#2317C2211yescharacteristic
59#2316C2211yescharacteristic
60#2311C2211yescharacteristic
61#2297C2211yescharacteristic
62#2296C2211yescharacteristic
63#2296C2211yescharacteristic
64#2274C2211yescharacteristic
65#2097C2212yes—
66#2095C2212yes—
67#2093C2212yes—
68#2092C2212yes—
69#2087C2214yes—
70#2086C2214yes—
71#2085C2214yes—
72#2084C2214yes—
73#2059C2214yes—
74#2059C2214yes—
75#2059C2214yes—
76#2057C2214yes—
77#2024C2214yes—
78#2023C2214yes—
79#1936C2212yes—
80#1936C2212yes—
81#1935C2212yes—
82#1935C2212yes—
83#1928C2214yes—
84#1925C2214yes—
85#1920C2214yes—
86#1919C2214yes—
87#1917C2214yes—
88#1917C2214yes—
89#1865C2214yes—
90#1864C2214yes—
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1735C2211yescharacteristic
96#1735C2211yescharacteristic
97#1729C2211yescharacteristic
98#1729C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×Q8—21no—
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C23×C8C2311yescharacteristic
106C23×C8C2311yescharacteristic
107#544—24no—
108#544—42no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—24no—
111C2×C4⋊Q8—24no—
112C2×C4⋊Q8—22no—
113C2×C4⋊Q8—42no—
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2314yes—
116C2×#125—21no—
117C2×#125—24no—
118C2×#125—22no—
119C2×#125—42no—
120C2×#125C2312yes—
121C2×#125C2314yes—
122C2×C22⋊Q8—22no—
123#520—24no—
124#520—42no—
125C2×C4×Q8C2312yes—
126C2×C4×D4—22no—
127C2×C4×D4C2312yes—
128C2×#117—21no—
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2311yescharacteristic
142C22×C42C2311yescharacteristic
143#501—216no—
144#501—216no—
145#479—28no—
146#478—28no—
147#476—28no—
148#475—28no—
149#428—24no—
150#428—42no—
151C2×#107—24no—
152C2×#107—24no—
153C2×#107C2312yes—
154C2×#107C2314yes—
155C2×#107C2312yes—
156C2×#107C2312yes—
157C2×#107C2312yes—
158C2×#106—24no—
159C2×#106D418yes—
160C2×#106C2312yes—
161C2×#106C2314yes—
162C2×#106C2312yes—
163C2×#106C2312yes—
164C2×#106C2312yes—
165#424—24no—
166C2×C4⋊C8C2312yes—
167C2×C4⋊C8C2312yes—
168#417—24no—
169#417—42no—
170#416—24no—
171#416Q818yescomplemented
172C2×#103C2311yescharacteristic
173C2×#103C2312yes—
174C2×#103C2311yescharacteristic
175C2×#103C2312yes—
176C2×#103C2312yes—
177C2×#102C2311yescharacteristic
178C2×#102C2312yes—
179C2×#102C2311yescharacteristic
180C2×#102C2312yes—
181C2×#102C2312yes—
182C2×#97—24no—
183C2×#97C2312yes—
184C2×#97C2312yes—
185#400—24no—
186#399—24no—
187#398—24no—
188#398—24no—
189#396—24no—
190#395—24no—
191#395—24no—
192#395—24no—
193#391—24no—
194#391C2312yes—
195#391C2312yes—
196#390C2312yes—
197#390C2312yes—
198#389—22no—
199#389—22no—
200#389—24no—
201#389—24no—
202#389—24no—
203#389D412yes—
204#389D412yes—
205#389C2314yes—
206#389C2312yes—
207#389C2312yes—
208#389C2314yes—
209#389C2314yes—
210#387—24no—
211#386—21no—
212#386—21no—
213#386—22no—
214#385—21no—
215#385—21no—
216#385—24no—
217#385—22no—
218#385—22no—
219#385—22no—
220#385D412yes—
221#385D412yes—
222#384—21no—
223#384—21no—
224#384C2312yes—
225#382—21no—
226#382—21no—
227#382C2312yes—
228#380—22no—
229#380—22no—
230#380—24no—
231C4×C4⋊C4—22no—
232C4×C4⋊C4—22no—
233C4×C4⋊C4C2314yes—
234C4×C4⋊C4C2314yes—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237#340C2312yes—
238#340C2314yes—
239#340C2312yes—
240#340C2314yes—
241#340C2314yes—
242#340C2314yes—
243#340C2312yes—
244#340C2312yes—
245#336C2314yes—
246#336C2314yes—
247#336C2314yes—
248#336C2312yes—
249#336C2312yes—
250C2×#41—22no—
251C2×#41—22no—
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41C2×Q814yes—
256C22×Q8—24no—
257C22×Q8—22no—
258C22×Q8—44no—
259C22×Q8—84no—
260C22×Q8C2411yescharacteristic
261C22×D4—21no—
262C22×D4C2411yescharacteristic
263C23×C4—21no—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—22no—
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2411yescharacteristic
272C2×M16—22no—
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—22no—
276C2×M16—22no—
277C2×M16—22no—
278C22×C8—22no—
279C22×C8—22no—
280C22×C8—22no—
281C22×C8—22no—
282C22×C8C2×D414yes—
283C22×C8C2×Q814yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2411yescharacteristic
287C22×C8C2411yescharacteristic
288C22×C8C2412yes—
289C4⋊Q8—28no—
290C4⋊Q8—24no—
291C4⋊Q8—44no—
292C4⋊Q8—42no—
293C4⋊Q8—48no—
294C4⋊Q8—44no—
295C4⋊Q8—88no—
296#125—28no—
297#125—24no—
298#125—44no—
299#125—42no—
300#125—48no—
301#125—44no—
302C22⋊Q8—48no—
303C4×Q8—24no—
304C4×Q8—42no—
305C4×D4—24no—
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—44no—
318#117C2×Q814yes—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—44no—
343C2×C4⋊C4—44no—
344C2×C4⋊C4—44no—
345C2×C4⋊C4—44no—
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D412yes—
349C2×C4⋊C4C2×D414yes—
350C2×C4⋊C4C2×D414yes—
351C2×C4⋊C4C2×D414yes—
352C2×C4⋊C4C2×D414yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4C2×D412yes—
369C2×C22⋊C4C2×D412yes—
370C2×C42—21no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—21no—
375C2×C42—21no—
376C2×C42—24no—
377C2×C42C2×D412yes—
378C2×C42C2412yes—
379C2×C42C2412yes—
380C2×C42C2412yes—
381#107—24no—
382#107—24no—
383#107—28no—
384#107—24no—
385#107—42no—
386#106—28no—
387#106—24no—
388#106—42no—
389#106C2×D418yes—
390#106C2×D418yes—
391C4⋊C8—24no—
392C4⋊C8—24no—
393#103—22no—
394#103—22no—
395#103—24no—
396#103—42no—
397#103C2×Q818yescomplemented
398#102—22no—
399#102—22no—
400#102—24no—
401#102—42no—
402#102C2×Q818yescomplemented
403#97—24no—
404#97—24no—
405#95—24no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—24no—
411#95—24no—
412#95—24no—
413#95—24no—
414#95—24no—
415#95C2×D414yes—
416#95C2×D414yes—
417C24—21no—
418C24C22×D411yescharacteristic
419#41—28no—
420#41—28no—
421#41—44no—
422#41—44no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—48no—
429C2×Q8—44no—
430C2×Q8—48no—
431C2×Q8—84no—
432C2×Q8—84no—
433C2×Q8—816no—
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—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—84no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4C22×D414yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×Q812yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C42−1+412yes—
480C22×C42−1+412yes—
481C22×C42−1+412yes—
482C22×C4C2511yescharacteristic
483M16—44no—
484M16—44no—
485C2×C8—28no—
486C2×C8—28no—
487C2×C8—22no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8C22×D412yes—
500C2×C8C22×D412yes—
501C2×C8C22×Q814yes—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
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—22no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—48no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—84no—
534C4⋊C4—84no—
535C4⋊C4—84no—
536C4⋊C4—84no—
537C4⋊C4C22×D414yes—
538C4⋊C4C22×D414yes—
539C4⋊C4C22×Q814yes—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—24no—
543C22⋊C4—44no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—84no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C42—24no—
550C42—24no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—44no—
558C42—42no—
559C42—42no—
560C42—84no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—42no—
565C23—42no—
566C23—41no—
567C23—41no—
568C23#53612yes—
569C23D4212yes—
570C23D4×Q812yes—
571C23C23×D411yescharacteristic
572Q8—22no—
573Q8—22no—
574Q8—41no—
575Q8—41no—
576Q8—816no—
577Q8—164no—
578D4—22no—
579D4—22no—
580D4—48no—
581D4—41no—
582D4—41no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
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—24no—
599C2×C4—22no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—48no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—164no—
625C2×C4#53612yes—
626C2×C4#53614yes—
627C2×C4#54414yes—
628C2×C4D4212yes—
629C2×C4D4×Q812yes—
630C2×C4D4×Q814yes—
631C2×C4#56314yes—
632C2×C4C23×D411yescharacteristic
633C2×C4C23×Q811yescharacteristic
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×2−1+411yescharacteristic
636C2×C4C2×2−1+411yescharacteristic
637C2×C4C2×2−1+411yescharacteristic
638C8—22no—
639C8—22no—
640C8—24no—
641C8—42no—
642C8—42no—
643C22—22no—
644C22—22no—
645C22—24no—
646C22—24no—
647C22—42no—
648C22—44no—
649C22—81no—
650C22—81no—
651C22—82no—
652C22#267714yes—
653C22C2×#53611yescharacteristic
654C22C2×D4211yescharacteristic
655C22C2×D4×Q811yescharacteristic
656C22#358512yes—
657C22#358612yes—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—44no—
662C4—42no—
663C4—42no—
664C4—44no—
665C4—44no—
666C4—44no—
667C4—82no—
668C4—82no—
669C4—81no—
670C4—81no—
671C4—82no—
672C4#267714yes—
673C4C2×#53611yescharacteristic
674C4C2×#54411yescharacteristic
675C4C2×D4×Q811yescharacteristic
676C4C2×#56311yescharacteristic
677C2—24no—
678C2—82no—
679C2C2×#267711yescharacteristic
680C2#6088712yes—
681C2#6105911yescharacteristic
682C2#6106011yescharacteristic
683C2#6138012yes—
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. C22×#501C2semidirect
4. #35576C2semidirect
5. #30972C2semidirect
6. #30970C2semidirect
7. #30965C2semidirect
8. #30963C2semidirect
10. #23667C2semidirect
11. #23661C2semidirect
12. #23555C2semidirect
13. #23553C2semidirect
17. #21038C2semidirect
18. #21021C2semidirect
171. #416Q8semidirect
397. #103C2×Q8semidirect
402. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C2×#30842
#356392
#355762
#309722
#309702
#309652
#309632
#237372
#236672
#236612
#235552
#235532
#234522
#210862
#210672
#210382
#210212
#208822