Finite Group Databasebeta

← #6457449

Subgroups — #6457449

Order 512.

1640 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35629C211yesmaximal, characteristic
2#35628C211yesmaximal, characteristic
3#28093C214yesmaximal, complemented
4#28084C214yesmaximal, complemented
5#28073C214yesmaximal, complemented
6#28064C214yesmaximal, complemented
7C2×#2097C211yesmaximal, characteristic, complemented
8#23639C211yesmaximal, characteristic
9#23551C211yesmaximal, characteristic, complemented
10#23487C211yesmaximal, characteristic
11#21323C211yesmaximal, characteristic, complemented
12#21094C211yesmaximal, characteristic, complemented
13#21088C211yesmaximal, characteristic, complemented
14#21066C211yesmaximal, characteristic, complemented
15#21038C211yesmaximal, characteristic, complemented
16#20974C211yesmaximal, characteristic
17#20662C211yesmaximal, characteristic
18#20455C211yesmaximal, characteristic, complemented
19#20387C211yesmaximal, characteristic
20C22×#131C2211yescharacteristic
21C2×#536C2211yescharacteristic
22C2×#533C2211yescharacteristic
23C2×#427C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C2×#408C2211yescharacteristic
32C2×#405C2211yescharacteristic
33C2×C4×M16C2211yescharacteristic
34#2592—22no—
35#2568—22no—
36#2514—22no—
37#2513—22no—
38#2510—24no—
39#2495C2214yes—
40#2494C2214yes—
41#2492C2214yes—
42#2489C2214yes—
43#2488—24no—
44#2316C2211yescharacteristic
45#2311C2211yescharacteristic
46#2311C2211yescharacteristic
47#2311C2211yescharacteristic
48#2310C2211yescharacteristic
49#2297C2211yescharacteristic
50#2296C2211yescharacteristic
51C2×#391C2211yescharacteristic
52C2×#382C2211yescharacteristic
53#2274C2211yescharacteristic
54#2274C2211yescharacteristic
55#2097C2212yes—
56#2097C2212yes—
57#2097C2212yes—
58#2097C2212yes—
59#2076C2214yes—
60#2075C2214yes—
61#2059C2212yes—
62#2059C2212yes—
63#2058C2212yes—
64#2057C2212yes—
65#2036C2214yes—
66#2035C2214yes—
67#1993C2214yes—
68#1991C2214yes—
69#1935C2214yes—
70#1935C2214yes—
71#1933C2214yes—
72#1933C2214yes—
73#1928C2214yes—
74#1927C2214yes—
75#1920C2214yes—
76#1919C2214yes—
77#1894C2214yes—
78#1892C2214yes—
79#1809C2214yes—
80#1807C2214yes—
81#1776C2214yes—
82C4×#107C2214yes—
83#1753C2212yes—
84#1753C2212yes—
85#1752C2212yes—
86C8×C22⋊C4C2212yes—
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C2×#131—24no—
100C2×#131—24no—
101C2×#131—42no—
102C22×M16C2311yescharacteristic
103C22×M16C2312yes—
104C22×M16C2311yescharacteristic
105C23×C8C2312yes—
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#536—24no—
109#536—42no—
110#533—24no—
111#533—42no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8C2312yes—
114C2×#125C2312yes—
115C2×#124—21no—
116C2×#123—22no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8C2314yes—
119C2×C4×Q8—22no—
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#427—24no—
130#427—42no—
131C2×#107—24no—
132C2×#107C2312yes—
133C2×#107C2312yes—
134C2×#107C2312yes—
135C2×#107C2312yes—
136C2×#106—24no—
137C2×#106C2312yes—
138C2×#106C2312yes—
139C2×#106C2312yes—
140C2×#106C2312yes—
141#419—24no—
142#419—42no—
143C2×#103—24no—
144C2×#103—24no—
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151#408—24no—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154#405—24no—
155C4×M16—42no—
156C2×#97C2312yes—
157C2×#97C2312yes—
158C2×C4×C8C2312yes—
159C2×C4×C8C2312yes—
160#401—24no—
161#400—24no—
162#400—24no—
163#400—24no—
164#397—24no—
165#397—24no—
166#397—24no—
167#393—24no—
168#391—21no—
169#391—21no—
170#391—21no—
171#391—22no—
172#391—22no—
173#391—24no—
174#391D412yes—
175#391C2314yes—
176#391C2312yes—
177#391C2312yes—
178#390—21no—
179#390—21no—
180#390—21no—
181#390D412yes—
182#389—22no—
183#389C2312yes—
184#389C2312yes—
185#389C2314yes—
186#388—24no—
187#388—24no—
188#388—24no—
189#388—24no—
190#387—24no—
191#386—22no—
192#386—22no—
193#386—22no—
194#386D414yes—
195#385—22no—
196#383—24no—
197#382—22no—
198#382—24no—
199#382C2312yes—
200#382C2312yes—
201#381—22no—
202#380—22no—
203#380C2314yes—
204#380C2314yes—
205#380C2314yes—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2314yes—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4C2312yes—
214C4×C22⋊C4C2312yes—
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×C4C2412yes—
257C23×C4C2411yescharacteristic
258#131—416no—
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
264C2×M16—24no—
265C2×M16—24no—
266C2×M16—42no—
267C2×M16#4114yes—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—24no—
273C22×C8—24no—
274C22×C8—42no—
275C22×C8#4114yes—
276C22×C8#4114yes—
277C22×C8C2412yes—
278C22×C8C2411yescharacteristic
279C22×C8C2411yescharacteristic
280C22×C8C2412yes—
281C22×C8C2411yescharacteristic
282C22×C8C2411yescharacteristic
283C4⋊Q8—24no—
284C4⋊Q8—42no—
285C4⋊Q8—44no—
286#125—24no—
287#125—42no—
288#124—44no—
289#123—48no—
290C22⋊Q8—28no—
291C22⋊Q8—48no—
292C22⋊Q8—44no—
293C4×Q8—48no—
294#117—22no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117—22no—
302#117—22no—
303#117#4114yes—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D414yes—
338C2×C22⋊C4C2×D414yes—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42C2×D412yes—
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—24no—
352#107—24no—
353#107—24no—
354#107—42no—
355#106—24no—
356#106—24no—
357#106—24no—
358#106—42no—
359#103—24no—
360#103—24no—
361#103—24no—
362#103—24no—
363#103—28no—
364#103—44no—
365C22⋊C8—24no—
366C22⋊C8—24no—
367#97—24no—
368#97—42no—
369C4×C8—24no—
370C4×C8—42no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95C2×D414yes—
386#95C2×D414yes—
387C24C22×D412yes—
388C24C22×D411yescharacteristic
389#41—44no—
390#41—44no—
391#41—44no—
392#41—44no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—82no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
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—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4C22×D412yes—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C42−1+412yes—
455C22×C4C2511yescharacteristic
456M16—28no—
457M16—48no—
458M16—44no—
459C2×C8—28no—
460C2×C8—28no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—48no—
475C2×C8C2×#4114yes—
476C2×C8C2×#4112yes—
477C2×C8C2×#4112yes—
478C2×C8C2×#4112yes—
479C2×C8C2×#4112yes—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
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—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—82no—
502C4⋊C4—84no—
503C4⋊C4—82no—
504C4⋊C4C2×#4114yes—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—48no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C42—24no—
517C42—24no—
518C42—22no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—22no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—84no—
528C42—84no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23#53212yes—
537C23#54612yes—
538C23#54712yes—
539C23C23×D411yescharacteristic
540Q8—44no—
541Q8—42no—
542Q8—42no—
543Q8—42no—
544Q8—42no—
545Q8—88no—
546Q8—84no—
547D4—48no—
548D4—44no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
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—84no—
587C2×C4—84no—
588C2×C4—162no—
589C2×C4—162no—
590C2×C4—162no—
591C2×C4#52914yes—
592C2×C4#53212yes—
593C2×C4#54214yes—
594C2×C4#54612yes—
595C2×C4#54712yes—
596C2×C4#55114yes—
597C2×C4#55614yes—
598C2×C4C22×#4111yescharacteristic
599C2×C4C22×#4111yescharacteristic
600C2×C4C22×#4111yescharacteristic
601C2×C4C22×#4111yescharacteristic
602C2×C4C2×H(5,𝔽2)11yescharacteristic
603C2×C4C2×2−1+411yescharacteristic
604C8—22no—
605C8—22no—
606C8—22no—
607C8—22no—
608C8—24no—
609C8—42no—
610C22—24no—
611C22—22no—
612C22—22no—
613C22—24no—
614C22—44no—
615C22—42no—
616C22—42no—
617C22—44no—
618C22—42no—
619C22#252114yes—
620C22C2×#53211yescharacteristic
621C22C2×#54611yescharacteristic
622C22C2×#54711yescharacteristic
623C22#358412yes—
624C22#358612yes—
625C4—42no—
626C4—42no—
627C4—44no—
628C4—44no—
629C4—42no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4#252114yes—
638C4C2×#52911yescharacteristic
639C4C2×#54211yescharacteristic
640C4C2×#55111yescharacteristic
641C4C2×#55611yescharacteristic
642C2—24no—
643C2—82no—
644C2C2×#252111yescharacteristic
645C2#6071311yescharacteristic
646C2#6072011yescharacteristic
647C2#6119812yes—
648C2#6128012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #28093C2semidirect
4. #28084C2semidirect
5. #28073C2semidirect
6. #28064C2semidirect
7. C2×#2097C2semidirect
9. #23551C2semidirect
11. #21323C2semidirect
12. #21094C2semidirect
13. #21088C2semidirect
14. #21066C2semidirect
15. #21038C2semidirect
18. #20455C2semidirect

Maximal subgroups

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

subgroupindex
#356292
#356282
#280932
#280842
#280732
#280642
C2×#20972
#236392
#235512
#234872
#213232
#210942
#210882
#210662
#210382
#209742
#206622
#204552
#203872