Finite Group Databasebeta

← #6451247

Subgroups — #6451247

Order 512.

1778 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35558C211yesmaximal, characteristic, complemented
2#34188C211yesmaximal, characteristic, complemented
3#27504C214yesmaximal, complemented
4#27503C214yesmaximal, complemented
5#27489C214yesmaximal, complemented
6#27486C214yesmaximal, complemented
7#23737C211yesmaximal, characteristic, complemented
8C2×#2093C211yesmaximal, characteristic, complemented
9#23658C211yesmaximal, characteristic, complemented
10#23544C211yesmaximal, characteristic
11#21235C211yesmaximal, characteristic
12#21173C211yesmaximal, characteristic, complemented
13#21067C211yesmaximal, characteristic, complemented
14#21065C211yesmaximal, characteristic, complemented
15#21063C211yesmaximal, characteristic, complemented
16#21026C211yesmaximal, characteristic, complemented
17#20928C211yesmaximal, characteristic
18#20872C211yesmaximal, characteristic, complemented
19#20455C211yesmaximal, characteristic, complemented
20C22×AGL(1,ℤ/8)C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C2×#428C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#416C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C2×#405C2211yescharacteristic
35#2513—22no—
36#2513C2214yes—
37#2512—22no—
38#2512C2214yes—
39#2480—24no—
40#2355C2214yescomplemented
41#2354C2214yescomplemented
42#2352—24no—
43#2317C2211yescharacteristic
44#2314—22no—
45#2311C2211yescharacteristic
46#2311C2211yescharacteristic
47#2309C2211yescharacteristic
48#2309C2211yescharacteristic
49#2296C2211yescharacteristic
50C2×#390C2211yescharacteristic
51#2285—22no—
52#2285C2211yescharacteristic
53C2×#384C2211yescharacteristic
54#2274C2211yescharacteristic
55#2274C2211yescharacteristic
56#2274C2211yescharacteristic
57#2097C2212yescomplemented
58#2095C2212yescomplemented
59#2093C2214yes—
60#2093C2214yes—
61#2093C2212yescomplemented
62#2092C2212yescomplemented
63#2084C2214yescomplemented
64#2083C2214yescomplemented
65#2055C2214yes—
66#2054C2214yes—
67#1973C2212yes—
68#1972C2212yes—
69#1970C2212yes—
70#1968C2212yes—
71#1949C2214yescomplemented
72#1947C2214yescomplemented
73#1928C2214yescomplemented
74#1926C2214yes—
75#1926C2214yes—
76#1925C2214yes—
77#1925C2214yes—
78#1925C2214yescomplemented
79#1917C2214yescomplemented
80#1916C2214yescomplemented
81#1880C2214yes—
82#1879C2214yes—
83#1865C2212yescomplemented
84#1864C2212yescomplemented
85#1863C2212yescomplemented
86#1862C2212yescomplemented
87#1776C2214yescomplemented
88C4×#107C2214yescomplemented
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)—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
108C2×#126—24no—
109C4×#41—24no—
110C4×#41—24no—
111C2×C4×Q8C2312yes—
112C2×C4×D4—22no—
113C2×C4×D4C2312yes—
114C2×#117—24no—
115C2×#117—21no—
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C22×C4⋊C4—22no—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C22⋊C4—22no—
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128#428—42no—
129#428—42no—
130#427—24no—
131C2×#107—24no—
132C2×#107—42no—
133C2×#107C2312yes—
134C2×#107C2312yes—
135C2×#107C2312yes—
136C2×#107C2312yes—
137C2×#107C2312yes—
138C2×#107C2314yes—
139C2×#106C2312yes—
140C2×#106C2312yes—
141C2×#106C2312yes—
142#424—24no—
143C2×C4⋊C8C2314yes—
144#418—24no—
145#417—42no—
146#416—24no—
147#416—24no—
148C2×#103C2312yes—
149C2×#103C2311yescharacteristic
150C2×#103C2311yescharacteristic
151C2×#103C2312yes—
152C2×#102—24no—
153C2×#102—42no—
154C2×#102C2312yes—
155C2×#102C2311yescharacteristic
156C2×#102C2312yes—
157C2×#102C2312yes—
158C2×#102C2311yescharacteristic
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2312yes—
162#405—24no—
163C2×#97C2312yes—
164C2×C4×C8C2312yes—
165#397—24no—
166#397—24no—
167#397—24no—
168#396—24no—
169#391C2312yes—
170#391C2312yes—
171#390—22no—
172#390—22no—
173#390—21no—
174#390—21no—
175#390C2314yes—
176#390C2312yes—
177#390C2312yes—
178#390C2312yes—
179#390C2312yes—
180#390C2314yes—
181#389—22no—
182#389C2312yes—
183#389C2312yes—
184#388—24no—
185#388—24no—
186#388—24no—
187#388—24no—
188#388—24no—
189#387—24no—
190#387—24no—
191#386—22no—
192#386—22no—
193#385—22no—
194#385—22no—
195#384—21no—
196#384—21no—
197#384D412yes—
198#384C2312yes—
199#384C2314yes—
200#382—24no—
201#382D412yes—
202#382C2312yes—
203#380—22no—
204#380—22no—
205#380—24no—
206#380D414yes—
207#379—22no—
208#379—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—24no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2314yes—
217C4×C4⋊C4C2314yes—
218C4×C4⋊C4C2314yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#336C2314yes—
228#336C2312yes—
229#336C2312yes—
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×Q8C2411yescharacteristic
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4C2412yes—
250C22×D4C2411yescharacteristic
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4C2×D412yes—
257C23×C4C2411yescharacteristic
258AGL(1,ℤ/8)—416no—
259C2×SD16—24no—
260C2×SD16—24no—
261C2×SD16—42no—
262C2×D8—24no—
263C2×D8—24no—
264C2×D8—42no—
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—22no—
270C2×M16#4114yes—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8#4114yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2411yescharacteristic
280C22×C8C2411yescharacteristic
281C22×C8C2412yes—
282#126—88no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×D4—216no—
286C4×D4—24no—
287C4×D4—24no—
288#117—28no—
289#117—22no—
290#117—22no—
291#117—22no—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—88no—
299#117#4114yes—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—84no—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D414yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—84no—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—28no—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2412yes—
355C2×C42C2412yes—
356C2×C42C2412yes—
357#107—24no—
358#107—24no—
359#107—24no—
360#107—44no—
361#107—42no—
362#107—42no—
363#106—24no—
364#106—42no—
365#106—42no—
366C4⋊C8—28no—
367#103—24no—
368#103—22no—
369#103—22no—
370#103—42no—
371#102—24no—
372#102—24no—
373#102—24no—
374#102—22no—
375#102—22no—
376#102—28no—
377#102—44no—
378#102—42no—
379#97—24no—
380C4×C8—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95—24no—
389#95C2×D414yes—
390#95C2×D414yes—
391C24—21no—
392C24—22no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—82no—
397C24—82no—
398C24C22×D411yescharacteristic
399#41—28no—
400#41—28no—
401#41—44no—
402#41—44no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8C2×#4112yes—
409C2×D4—24no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4C2×#4112yes—
431C22×C4—28no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4C22×D414yes—
455C22×C4C22×D414yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C42−1+412yes—
465C22×C4C2511yescharacteristic
466SD16—48no—
467D8—48no—
468M16—28no—
469M16—44no—
470C2×C8—28no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
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—44no—
484C2×C8C2×#4114yes—
485C2×C8C2×#4112yes—
486C2×C8C2×#4112yes—
487C4⋊C4—28no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—28no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—48no—
509C4⋊C4—44no—
510C4⋊C4—82no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4C2×#4114yes—
514C22⋊C4—28no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—48no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C42—28no—
529C42—22no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—42no—
539C42—42no—
540C42—84no—
541C42—84no—
542C42—84no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—44no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—88no—
556C23—84no—
557C23—164no—
558C23—162no—
559C23—162no—
560C23#52512yes—
561C23#54712yes—
562C23#55012yes—
563C23C23×D411yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—41no—
567Q8—41no—
568D4—22no—
569D4—22no—
570D4—48no—
571D4—41no—
572D4—42no—
573D4—41no—
574D4—42no—
575D4—42no—
576D4—42no—
577D4—84no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—48no—
592C2×C4—416no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—48no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—162no—
612C2×C4—162no—
613C2×C4—162no—
614C2×C4#52512yes—
615C2×C4#52914yes—
616C2×C4#54214yes—
617C2×C4#54712yes—
618C2×C4#55012yes—
619C2×C4#55114yes—
620C2×C4#55214yes—
621C2×C4#57614yes—
622C2×C4C22×#4111yescharacteristic
623C2×C4C22×#4111yescharacteristic
624C2×C4C22×#4111yescharacteristic
625C2×C4C22×#4111yescharacteristic
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—84no—
643C22—82no—
644C22—84no—
645C22—82no—
646C22—82no—
647C22—168no—
648C22#247914yes—
649C22C2×#52511yescharacteristic
650C22C2×#54711yescharacteristic
651C22C2×#55011yescharacteristic
652C22C2×#57612yes—
653C22#358612yes—
654C4—24no—
655C4—44no—
656C4—44no—
657C4—42no—
658C4—42no—
659C4—44no—
660C4—82no—
661C4—81no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—81no—
666C4#247914yes—
667C4C2×#52911yescharacteristic
668C4C2×#54211yescharacteristic
669C4C2×#55111yescharacteristic
670C4C2×#55211yescharacteristic
671C2—24no—
672C2—82no—
673C2—162no—
674C2C2×#247911yescharacteristic
675C2#6058412yes—
676C2#6127812yes—
677C2#6140911yescharacteristic
678C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×AGL(1,ℤ/8)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35558C2semidirect
2. #34188C2semidirect
3. #27504C2semidirect
4. #27503C2semidirect
5. #27489C2semidirect
6. #27486C2semidirect
7. #23737C2semidirect
8. C2×#2093C2semidirect
9. #23658C2semidirect
12. #21173C2semidirect
13. #21067C2semidirect
14. #21065C2semidirect
15. #21063C2semidirect
16. #21026C2semidirect
18. #20872C2semidirect
19. #20455C2semidirect
40. #2355C22semidirect
41. #2354C22semidirect
57. #2097C22semidirect
58. #2095C22semidirect
61. #2093C22semidirect
62. #2092C22semidirect
63. #2084C22semidirect
64. #2083C22semidirect
71. #1949C22semidirect
72. #1947C22semidirect
73. #1928C22semidirect
78. #1925C22semidirect
79. #1917C22semidirect
80. #1916C22semidirect
83. #1865C22semidirect
84. #1864C22semidirect
85. #1863C22semidirect
86. #1862C22semidirect
87. #1776C22semidirect
88. C4×#107C22semidirect

Maximal subgroups

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

subgroupindex
#355582
#341882
#275042
#275032
#274892
#274862
#237372
C2×#20932
#236582
#235442
#212352
#211732
#210672
#210652
#210632
#210262
#209282
#208722
#204552