Finite Group Databasebeta

← #7511147

Subgroups — #7511147

Order 512.

1686 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35662C211yesmaximal, characteristic, complemented
2#35334C211yesmaximal, characteristic
3#32663C214yesmaximal, complemented
4#32660C214yesmaximal, complemented
5#32657C214yesmaximal, complemented
6#32653C214yesmaximal, complemented
7#23737C211yesmaximal, characteristic
8C2×#2097C211yesmaximal, characteristic, complemented
9#23661C211yesmaximal, characteristic, complemented
10#23659C211yesmaximal, characteristic, complemented
11#23615C211yesmaximal, characteristic, complemented
12#23610C211yesmaximal, characteristic
13#21304C211yesmaximal, characteristic, complemented
14#21234C211yesmaximal, characteristic
15#21158C211yesmaximal, characteristic, complemented
16#20930C211yesmaximal, characteristic
17#20927C211yesmaximal, characteristic
18#20880C211yesmaximal, characteristic, complemented
19#20872C211yesmaximal, characteristic
20C22×#137C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×#519C2211yescharacteristic
23C2×C4×#41C2211yescharacteristic
24C2×#428C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C2×#424C2211yescharacteristic
30C2×#417C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C2×#416C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C2×#405C2211yescharacteristic
37#2851—22no—
38#2628—22no—
39#2582C2214yes—
40#2581C2214yes—
41#2581C2214yes—
42#2580C2214yes—
43#2564—24no—
44#2453—22no—
45#2451—22no—
46#2438—24no—
47#2317C2211yescharacteristic
48#2316C2211yescharacteristic
49#2316C2211yescharacteristic
50#2316C2211yescharacteristic
51#2311C2211yescharacteristic
52#2311C2211yescharacteristic
53#2310C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#391C2211yescharacteristic
56C2×#382C2211yescharacteristic
57#2097C2212yes—
58#2097C2214yes—
59#2097C2214yes—
60#2095C2212yes—
61#2093C2212yes—
62#2092C2212yes—
63#2085C2214yes—
64#2085C2214yes—
65#2085C2214yes—
66#2084C2214yes—
67#2068C2214yes—
68#2066C2214yes—
69#2066C2214yes—
70#2065C2214yes—
71#1985C2214yes—
72#1984C2214yes—
73#1973C2212yes—
74#1971C2212yes—
75#1969C2212yes—
76#1968C2212yes—
77#1943C2214yes—
78#1942C2214yes—
79#1880C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1877C2214yes—
83#1865C2212yes—
84#1865C2214yes—
85#1864C2212yes—
86#1863C2212yes—
87#1863C2214yes—
88#1862C2212yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
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#533—42no—
109#533—42no—
110C2×C4⋊Q8D412yes—
111C2×#125D412yes—
112C2×C22⋊Q8D414yes—
113#519—42no—
114#519—42no—
115C4×#41—42no—
116C4×#41—42no—
117C2×C4×Q8—22no—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2314yes—
121C2×C4×D4—22no—
122C2×C4×D4C2312yes—
123C2×#117—22no—
124C2×#117—21no—
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135#428—42no—
136#428—42no—
137#427—42no—
138C2×#107—24no—
139C2×#107C2312yes—
140C2×#107C2312yes—
141C2×#107C2312yes—
142C2×#107C2312yes—
143C2×#107C2312yes—
144C2×#106—42no—
145C2×#106C2314yes—
146C2×#106C2312yes—
147C2×#106C2312yes—
148C2×#106C2312yes—
149#424—42no—
150C2×C4⋊C8C2314yes—
151#419—42no—
152#417—42no—
153#416—24no—
154#416—42no—
155C2×#103—24no—
156C2×#103—42no—
157C2×#103C2311yescharacteristic
158C2×#103C2312yes—
159C2×#103C2311yescharacteristic
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#102C2311yescharacteristic
166C2×#102C2311yescharacteristic
167C2×#102C2312yes—
168C2×#102C2312yes—
169#405—42no—
170C2×#97C2312yes—
171C2×C4×C8C2312yes—
172#400—24no—
173#400—24no—
174#399—24no—
175#398—24no—
176#398—24no—
177#395—24no—
178#395—24no—
179#393—24no—
180#391—24no—
181#391—22no—
182#391—21no—
183#391—21no—
184#391C2312yes—
185#391C2312yes—
186#391C2312yes—
187#391C2312yes—
188#391C2314yes—
189#390C2312yes—
190#390C2312yes—
191#389—22no—
192#389—22no—
193#389—22no—
194#389—22no—
195#389—24no—
196#389C2314yes—
197#389C2314yes—
198#389C2314yes—
199#389C2312yes—
200#389C2312yes—
201#387—24no—
202#387—24no—
203#387—24no—
204#387—24no—
205#386—22no—
206#386—22no—
207#385—22no—
208#385—24no—
209#385—24no—
210#385—22no—
211#384C2312yes—
212#382—21no—
213#382—21no—
214#382C2312yes—
215#382C2314yes—
216#381—22no—
217#380—22no—
218#380—22no—
219#380—22no—
220#380—22no—
221#380—22no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C2×#95—22no—
225C2×#95—22no—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
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×Q8C2×D414yes—
250C22×Q8C2×Q814yes—
251C22×Q8C2412yes—
252C22×Q8C2411yescharacteristic
253C22×D4—21no—
254C22×D4C2411yescharacteristic
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4C2×D412yes—
262C23×C4C2411yescharacteristic
263#137—416no—
264C2×Q16—42no—
265C2×Q16—42no—
266C2×Q16C2×Q818yes—
267C2×SD16—42no—
268C2×SD16—42no—
269C2×SD16C2×Q818yes—
270C2×M16—22no—
271C2×M16—22no—
272C2×M16—22no—
273C2×M16—22no—
274C2×M16—22no—
275C2×M16C2×Q814yes—
276C22×C8—22no—
277C22×C8—22no—
278C22×C8—22no—
279C22×C8—22no—
280C22×C8—22no—
281C22×C8#4114yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2411yescharacteristic
286C22×C8C2411yescharacteristic
287C4⋊Q8—44no—
288#125—44no—
289C22⋊Q8—48no—
290C4×Q8—44no—
291C4×Q8—44no—
292C4×Q8—42no—
293C4×Q8—42no—
294C4×Q8—44no—
295C4×Q8—44no—
296C4×Q8—48no—
297C4×D4—42no—
298C4×D4—42no—
299C4×D4—48no—
300#117—22no—
301#117—22no—
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—44no—
311#117—48no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
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—22no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—24no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—22no—
356C2×C42—44no—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2412yes—
360C2×C42C2412yes—
361C2×C42C2412yes—
362#107—24no—
363#107—24no—
364#107—42no—
365#107—42no—
366#107—42no—
367#106—44no—
368#106—42no—
369#106—42no—
370#106—42no—
371C4⋊C8—44no—
372#103—24no—
373#103—24no—
374#103—22no—
375#103—22no—
376#103—44no—
377#103—42no—
378#103—42no—
379#103—44no—
380#102—22no—
381#102—22no—
382#102—42no—
383#102—42no—
384#97—42no—
385C4×C8—42no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95—24no—
400C24—21no—
401C24C22×D411yescharacteristic
402#41—28no—
403#41—28no—
404#41—44no—
405#41—44no—
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—28no—
410C2×Q8—28no—
411C2×Q8—22no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—21no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8—42no—
423C2×Q8—44no—
424C2×Q8—42no—
425C2×Q8—44no—
426C2×Q8C22×Q814yes—
427C2×Q8C22×Q812yes—
428C2×D4—24no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4C22×Q812yes—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×Q812yes—
470C22×C4C2×#4112yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C42−1+412yes—
475C22×C4C2511yescharacteristic
476Q16—48no—
477SD16—48no—
478M16—28no—
479M16—44no—
480C2×C8—28no—
481C2×C8—21no—
482C2×C8—22no—
483C2×C8—21no—
484C2×C8—22no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—44no—
494C2×C8C22×Q814yes—
495C2×C8C2×#4112yes—
496C2×C8C2×#4112yes—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—48no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—84no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C4⋊C4—82no—
526C4⋊C4—84no—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—44no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—84no—
534C22⋊C4—84no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—44no—
544C42—48no—
545C42—42no—
546C42—42no—
547C42—42no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—42no—
552C23—42no—
553C23—41no—
554C23—41no—
555C23#53412yes—
556C23D4×Q812yes—
557C23#55012yes—
558C23C23×D411yescharacteristic
559Q8—24no—
560Q8—22no—
561Q8—24no—
562Q8—22no—
563Q8—42no—
564Q8—41no—
565Q8—48no—
566Q8—42no—
567Q8—41no—
568Q8—88no—
569D4—22no—
570D4—22no—
571D4—48no—
572D4—41no—
573D4—41no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—48no—
593C2×C4—48no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—84no—
611C2×C4—88no—
612C2×C4—162no—
613C2×C4—162no—
614C2×C4—162no—
615C2×C4#53412yes—
616C2×C4#53914yes—
617C2×C4#54314yes—
618C2×C4D4×Q812yes—
619C2×C4#55012yes—
620C2×C4#55414yes—
621C2×C4#56214yes—
622C2×C4C23×Q811yescharacteristic
623C2×C4C22×#4111yescharacteristic
624C2×C4C2×H(5,𝔽2)11yescharacteristic
625C2×C4C2×H(5,𝔽2)11yescharacteristic
626C2×C4C2×H(5,𝔽2)11yescharacteristic
627C2×C4C2×2−1+411yescharacteristic
628C8—22no—
629C8—22no—
630C8—24no—
631C8—42no—
632C8—42no—
633C22—24no—
634C22—24no—
635C22—22no—
636C22—22no—
637C22—42no—
638C22—44no—
639C22—82no—
640C22—81no—
641C22—81no—
642C22#273214yes—
643C22C2×#53411yescharacteristic
644C22C2×D4×Q811yescharacteristic
645C22C2×#55011yescharacteristic
646C22#358412yes—
647C22#358612yes—
648C4—42no—
649C4—42no—
650C4—44no—
651C4—44no—
652C4—42no—
653C4—44no—
654C4—82no—
655C4—82no—
656C4—81no—
657C4—82no—
658C4—82no—
659C4—81no—
660C4—82no—
661C4#273214yes—
662C4C2×#53911yescharacteristic
663C4C2×#54311yescharacteristic
664C4C2×#55411yescharacteristic
665C4C2×#56211yescharacteristic
666C2—24no—
667C2—82no—
668C2C2×#273211yescharacteristic
669C2#6076612yes—
670C2#6138612yes—
671C2#6141311yescharacteristic
672C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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. #35662C2semidirect
3. #32663C2semidirect
4. #32660C2semidirect
5. #32657C2semidirect
6. #32653C2semidirect
8. C2×#2097C2semidirect
9. #23661C2semidirect
10. #23659C2semidirect
11. #23615C2semidirect
13. #21304C2semidirect
15. #21158C2semidirect
18. #20880C2semidirect
101. C2×#137Q8semidirect

Maximal subgroups

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

subgroupindex
#356622
#353342
#326632
#326602
#326572
#326532
#237372
C2×#20972
#236612
#236592
#236152
#236102
#213042
#212342
#211582
#209302
#209272
#208802
#208722