Finite Group Databasebeta

← #6433961

Subgroups — #6433961

Order 512.

1590 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35334C211yesmaximal, characteristic
2#35307C211yesmaximal, characteristic
3#30374C214yesmaximal, complemented
4#30370C214yesmaximal, complemented
5#30363C214yesmaximal, complemented
6#30361C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8C2×#2097C211yesmaximal, characteristic, complemented
9#23640C211yesmaximal, characteristic
10#23639C211yesmaximal, characteristic
11#23551C211yesmaximal, characteristic, complemented
12#23487C211yesmaximal, characteristic
13#23487C211yesmaximal, characteristic
14#23455C211yesmaximal, characteristic, complemented
15#21158C211yesmaximal, characteristic, complemented
16#21157C211yesmaximal, characteristic, complemented
17#21088C211yesmaximal, characteristic, complemented
18#21088C211yesmaximal, characteristic, complemented
19#20378C211yesmaximal, characteristic
20C2×#536C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23C2×#427C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C2×#408C2211yescharacteristic
32C2×#408C2211yescharacteristic
33C2×#405C2211yescharacteristic
34C2×#405C2211yescharacteristic
35#2855C2214yes—
36#2851C2214yes—
37#2628C2214yes—
38#2625C2214yes—
39#2581—22no—
40#2580—22no—
41#2564—24no—
42#2447—22no—
43#2446—22no—
44#2419—24no—
45#2316C2211yescharacteristic
46#2316C2211yescharacteristic
47#2316C2211yescharacteristic
48#2311C2211yescharacteristic
49#2311C2211yescharacteristic
50#2311C2211yescharacteristic
51#2310C2211yescharacteristic
52#2296C2211yescharacteristic
53C2×#391C2211yescharacteristic
54C2×#382C2211yescharacteristic
55#2101C2214yes—
56#2101C2214yes—
57#2097C2212yes—
58#2097C2212yes—
59#2097C2212yes—
60#2097C2212yes—
61#2077C2214yes—
62#2077C2214yes—
63#2076C2214yes—
64#2075C2214yes—
65#2059C2212yes—
66#2059C2212yes—
67#2058C2212yes—
68#2057C2212yes—
69#2036C2214yes—
70#2036C2214yes—
71#2035C2214yes—
72#2035C2214yes—
73#2026C2214yes—
74#2024C2214yes—
75#1944C2214yes—
76#1943C2214yes—
77#1942C2214yes—
78#1941C2214yes—
79#1935C2214yes—
80#1935C2214yes—
81#1933C2214yes—
82#1933C2214yes—
83#1750C2212yes—
84#1750C2212yes—
85#1749C2212yes—
86C4×C22⋊C8C2212yes—
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103#536—42no—
104#536—42no—
105#533—42no—
106#533—42no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8C2312yes—
109C2×#125C2312yes—
110C2×#124—21no—
111C2×#123—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C2314yes—
114C4×#41—42no—
115C4×#41—42no—
116C2×C4×Q8—22no—
117C2×C4×Q8D412yes—
118C2×C4×D4D414yes—
119C2×C4×D4D412yes—
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2312yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131#427—42no—
132#427—42no—
133C2×#107—24no—
134C2×#107C2312yes—
135C2×#107C2312yes—
136C2×#107C2312yes—
137C2×#107C2312yes—
138C2×#106—24no—
139C2×#106C2312yes—
140C2×#106C2312yes—
141C2×#106C2312yes—
142C2×#106C2312yes—
143#419—42no—
144#419—42no—
145C2×#103—24no—
146C2×#103—24no—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153#408—24no—
154#408—24no—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159#405—42no—
160#405—42no—
161C2×#97C2312yes—
162C2×#97C2312yes—
163C2×C4×C8C2312yes—
164C2×C4×C8C2312yes—
165#400—24no—
166#400—24no—
167#399—24no—
168#398—24no—
169#397—24no—
170#397—24no—
171#395—24no—
172#393—24no—
173#391—21no—
174#391—21no—
175#391—21no—
176#391—22no—
177#391—22no—
178#391C2314yes—
179#391C2312yes—
180#391C2312yes—
181#390—21no—
182#390—21no—
183#390—21no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389C2312yes—
188#389C2312yes—
189#389C2314yes—
190#389C2314yes—
191#389C2314yes—
192#387—24no—
193#387—24no—
194#387—24no—
195#387—24no—
196#386—22no—
197#386—22no—
198#386—22no—
199#385—24no—
200#385—24no—
201#385—24no—
202#385—22no—
203#382—22no—
204#382C2312yes—
205#382C2312yes—
206#381—24no—
207#381—22no—
208#380—22no—
209#380—22no—
210#380—22no—
211#380C2314yes—
212#380C2314yes—
213#380C2314yes—
214C4×C4⋊C4—22no—
215C2×#95—22no—
216C2×#95—22no—
217#340C2312yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#336C2311yescharacteristic
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2312yes—
233C2×#41—22no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—22no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8C2×D411yescharacteristic
248C22×Q8C2412yes—
249C22×D4C2×D412yes—
250C22×D4C2×D411yescharacteristic
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4C2×D412yes—
256C23×C4C2×D414yes—
257C23×C4C2412yes—
258C23×C4C2411yescharacteristic
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
264C2×M16—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8#4114yes—
270C22×C8#4114yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2411yescharacteristic
274C22×C8C2411yescharacteristic
275C22×C8C2411yescharacteristic
276C22×C8C2411yescharacteristic
277C4⋊Q8—42no—
278C4⋊Q8—42no—
279C4⋊Q8—44no—
280#125—42no—
281#125—42no—
282#124—44no—
283#123—48no—
284C22⋊Q8—44no—
285C22⋊Q8—44no—
286C22⋊Q8—48no—
287C4×Q8—44no—
288C4×Q8—48no—
289C4×D4—44no—
290C4×D4—48no—
291#117—22no—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117—42no—
302#117—42no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D414yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D414yes—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—42no—
347C2×C42—42no—
348C2×C42C2×D412yes—
349C2×C42C2×D414yes—
350C2×C42C2411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353#107—24no—
354#107—24no—
355#107—42no—
356#107—42no—
357#106—24no—
358#106—24no—
359#106—42no—
360#106—42no—
361#103—24no—
362#103—24no—
363#103—24no—
364#103—24no—
365#103—44no—
366#103—44no—
367C22⋊C8—24no—
368C22⋊C8—24no—
369C22⋊C8—24no—
370C22⋊C8—24no—
371#97—42no—
372#97—42no—
373C4×C8—42no—
374C4×C8—42no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389C24C22×D412yes—
390C24C22×D411yescharacteristic
391#41—44no—
392#41—44no—
393#41—44no—
394#41—44no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—82no—
413C2×Q8—82no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C42−1+412yes—
459C22×C42−1+412yes—
460C22×C4C2511yescharacteristic
461M16—44no—
462M16—44no—
463C2×C8—28no—
464C2×C8—28no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8C2×#4112yes—
480C2×C8C2×#4112yes—
481C2×C8C2×#4112yes—
482C2×C8C2×#4112yes—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C4⋊C4—82no—
510C4⋊C4—84no—
511C22⋊C4—24no—
512C22⋊C4—44no—
513C22⋊C4—48no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—44no—
529C42—42no—
530C42—42no—
531C42—42no—
532C42—42no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—24no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23#53212yes—
541C23#53412yes—
542C23#53612yes—
543C23C23×D411yescharacteristic
544Q8—44no—
545Q8—42no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—88no—
550Q8—84no—
551D4—48no—
552D4—44no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—84no—
591C2×C4—88no—
592C2×C4—84no—
593C2×C4—162no—
594C2×C4—162no—
595C2×C4—162no—
596C2×C4#53212yes—
597C2×C4#53412yes—
598C2×C4#53612yes—
599C2×C4#53914yes—
600C2×C4#53914yes—
601C2×C4#54214yes—
602C2×C4#54214yes—
603C2×C4C22×#4111yescharacteristic
604C2×C4C22×#4111yescharacteristic
605C2×C4C2×H(5,𝔽2)11yescharacteristic
606C2×C4C2×H(5,𝔽2)11yescharacteristic
607C2×C4C2×2−1+411yescharacteristic
608C2×C4C2×2−1+411yescharacteristic
609C8—22no—
610C8—22no—
611C8—22no—
612C8—22no—
613C8—42no—
614C8—42no—
615C22—22no—
616C22—22no—
617C22—24no—
618C22—24no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—44no—
624C22#266314yes—
625C22C2×#53211yescharacteristic
626C22C2×#53411yescharacteristic
627C22C2×#53611yescharacteristic
628C22#358412yes—
629C22#358612yes—
630C4—42no—
631C4—44no—
632C4—42no—
633C4—42no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4#266314yes—
643C4C2×#53911yescharacteristic
644C4C2×#53911yescharacteristic
645C4C2×#54211yescharacteristic
646C4C2×#54211yescharacteristic
647C2—24no—
648C2—82no—
649C2C2×#266311yescharacteristic
650C2#6071311yescharacteristic
651C2#6072011yescharacteristic
652C2#6076612yes—
653C2#6089112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30374C2semidirect
4. #30370C2semidirect
5. #30363C2semidirect
6. #30361C2semidirect
7. #23746C2semidirect
8. C2×#2097C2semidirect
11. #23551C2semidirect
14. #23455C2semidirect
15. #21158C2semidirect
16. #21157C2semidirect
17. #21088C2semidirect
18. #21088C2semidirect

Maximal subgroups

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

subgroupindex
#353342
#353072
#303742
#303702
#303632
#303612
#237462
C2×#20972
#236402
#236392
#235512
#234872
#234552
#211582
#211572
#210882
#203782