Finite Group Databasebeta

← #6526632

Subgroups — #6526632

Order 512.

1870 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic, complemented
2C2×#2990C211yesmaximal, characteristic
3#35297C211yesmaximal, characteristic
4#34971C211yesmaximal, characteristic
5#28102C214yesmaximal, complemented
6#27669C214yesmaximal, complemented
7#25476C214yesmaximal, complemented
8#25343C214yesmaximal, complemented
9#24838C212yesmaximal, complemented
10#24837C212yesmaximal, complemented
11#24711C212yesmaximal, complemented
12#24710C212yesmaximal, complemented
13#21963C211yesmaximal, characteristic, complemented
14#21837C211yesmaximal, characteristic, complemented
15#20734C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19#3078—28no—
20C2×#479C2214yes—
21C2×#477C2212yes—
22C2×#475C2212yes—
23#2990—28no—
24C2×#446C2212yes—
25C2×#435C2214yes—
26C2×C8×D4C2212yes—
27C2×#432C2212yes—
28C2×#431C2211yescharacteristic
29C2×#428C2212yes—
30C2×#423C2211yescharacteristic
31C22×C4⋊C8C2212yes—
32C2×#419C2212yes—
33#2857C2212yes—
34#2857C2212yes—
35#2625—21no—
36#2616—21no—
37#2599—22no—
38#2584C2212yes—
39#2581—21no—
40#2581C2212yes—
41#2562C2212yes—
42#2536—21no—
43#2536C2212yes—
44#2486C2212yes—
45#2485C2214yes—
46#2484—21no—
47#2415—22no—
48#2409C2212yes—
49#2406—21no—
50#2399—21no—
51#2398—21no—
52#2323—22no—
53C4×C22⋊Q8C2212yes—
54C4×C22⋊Q8C2212yes—
55C4×#123—22no—
56#2285C2211yescharacteristic
57#2283C2211yescharacteristic
58#2088C2214yes—
59#2087C2214yes—
60#2077C2214yes—
61#2075C2214yes—
62#2068C2214yes—
63#2066C2214yes—
64#2036C2214yes—
65#2035C2214yes—
66#1991C2214yes—
67#1988C2211yescharacteristic
68#1988C2211yescharacteristic
69#1987C2211yescharacteristic
70#1987C2211yescharacteristic
71#1949C2214yes—
72#1948C2212yes—
73#1947C2212yes—
74#1944C2214yes—
75#1943C2214yes—
76#1935C2214yes—
77#1935C2214yes—
78#1892C2214yes—
79#1892C2214yes—
80#1825C2214yes—
81#1824C2212yes—
82#1809C2214yes—
83#1808C2214yes—
84#1807C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1787C2212yes—
88C4×#103C2214yes—
89C22×#41D411yescharacteristic
90C23×Q8C2311yescharacteristic
91C22×M16C2312yes—
92C23×C8C2314yes—
93#533—24no—
94#533—24no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×#125C2312yes—
98C2×#123—22no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2312yes—
105C4×#41—24no—
106C4×#41—24no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8—24no—
112C2×C4×Q8—24no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D412yes—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C2×#117—21no—
122C2×#117D412yes—
123C2×#117D411yescharacteristic
124C2×#117C2312yes—
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2311yescharacteristic
127C22×C42—21no—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#479—216no—
131#477—28no—
132#475—28no—
133#446—28no—
134#435—216no—
135C8×D4—28no—
136#432—28no—
137#431—24no—
138#428—28no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143#423D418yes—
144C2×C4⋊C8—28no—
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2311yescharacteristic
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150#419—28no—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2312yes—
157C2×#97C2314yes—
158C2×C4×C8C2312yes—
159#400—22no—
160#400—22no—
161#398—22no—
162#398—22no—
163#398—22no—
164#397—22no—
165#397—24no—
166#395—22no—
167#395—22no—
168#395—22no—
169#394—22no—
170#394—24no—
171#394—22no—
172#393—22no—
173#391—22no—
174#391—22no—
175#391C2312yes—
176#391C2312yes—
177#389—24no—
178#389—22no—
179#389—22no—
180#389C2314yes—
181#389C2314yes—
182#389C2312yes—
183#389C2312yes—
184#388—24no—
185#388—24no—
186#388—22no—
187#388—22no—
188#387—24no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#386—22no—
194#385—22no—
195#385—22no—
196#385—22no—
197#384C2312yes—
198#383—21no—
199#382—21no—
200#382C2312yes—
201#381—21no—
202#381—22no—
203#380—22no—
204#380—22no—
205#380—21no—
206#380C2314yes—
207#380C2314yes—
208#380C2312yes—
209#380C2312yes—
210#380C2312yes—
211#380C2312yes—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221#376—21no—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230C2×#41—24no—
231C2×#41—44no—
232C22×Q8—21no—
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8—24no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—42no—
239C22×Q8C2×D411yescharacteristic
240C22×Q8C2412yes—
241C22×D4C2×D411yescharacteristic
242C22×D4C2×D412yes—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4C2×D412yes—
246C23×C4C2×D412yes—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C2×M16—44no—
252C2×M16C2×D418yes—
253C22×C8—28no—
254C22×C8—44no—
255C22×C8—44no—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263#125—24no—
264#125—24no—
265#123—48no—
266C22⋊Q8—24no—
267C22⋊Q8—24no—
268C22⋊Q8—42no—
269C22⋊Q8—42no—
270C22⋊Q8—48no—
271C22⋊Q8—44no—
272C22⋊Q8—44no—
273C22⋊Q8—84no—
274C4×Q8—24no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×D4—28no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—22no—
290C4×D4—22no—
291#117—24no—
292#117—28no—
293#117—24no—
294#117—24no—
295#117—82no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2414yes—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—44no—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D414yes—
350C2×C42#4114yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—28no—
355#107—44no—
356#106—28no—
357#106—44no—
358C4⋊C8—28no—
359C4⋊C8—22no—
360C4⋊C8—22no—
361C4⋊C8C2×D418yes—
362C4⋊C8C2×D418yes—
363#103—28no—
364#103—28no—
365#103—28no—
366#103—44no—
367C22⋊C8—24no—
368C22⋊C8—28no—
369#97—28no—
370C4×C8—24no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—21no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390C24C22×D411yescharacteristic
391C24C22×D412yes—
392#41—416no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—28no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—48no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q82−1+414yes—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—42no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
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—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4#12114yes—
445C22×C4#12114yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C2×#4112yes—
456C22×C4C2511yescharacteristic
457M16—48no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—88no—
465C2×C8C22×D418yes—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—84no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—82no—
508C22⋊C4—84no—
509C42—24no—
510C42—24no—
511C42—22no—
512C42—22no—
513C42—44no—
514C42—44no—
515C42—44no—
516C42—42no—
517C42—42no—
518C42—82no—
519C42#12114yes—
520C42#12114yes—
521C42C22×D412yes—
522C42C22×D412yes—
523C42C2×#4112yes—
524C42C2×#4112yes—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—21no—
532C23—22no—
533C23—21no—
534C23—22no—
535C23C2×#12112yes—
536C23#57514yes—
537C23C23×D411yescharacteristic
538Q8—24no—
539Q8—24no—
540Q8—48no—
541Q8—44no—
542Q8—42no—
543Q8—42no—
544Q8—84no—
545Q8—84no—
546D4—28no—
547D4—48no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—44no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—48no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—84no—
595C2×C4C2×#12112yes—
596C2×C4C2×#12112yes—
597C2×C4C2×#12112yes—
598C2×C4C2×#12112yes—
599C2×C4C2×#12112yes—
600C2×C4#57712yes—
601C2×C4#57812yes—
602C2×C4C23×D411yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C2×C4C2×2−1+412yes—
605C2×C4#58512yes—
606C8—28no—
607C8—42no—
608C8—42no—
609C22—22no—
610C22—24no—
611C22—22no—
612C22—22no—
613C22—24no—
614C22—24no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22C22×#12111yescharacteristic
622C22#348912yes—
623C22C2×#57512yes—
624C22C2×#57711yescharacteristic
625C22C2×#57811yescharacteristic
626C4—22no—
627C4—22no—
628C4—22no—
629C4—22no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4—44no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4—84no—
639C4—82no—
640C4#306314yes—
641C4#348511yescharacteristic
642C4#348611yescharacteristic
643C4#349512yes—
644C2—42no—
645C2—44no—
646C2#3563311yescharacteristic
647C2#6044411yescharacteristic
648C2C2×#306311yescharacteristic
649C2#6212912yes—
650C2#6213012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
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
1. C2×#3078C2semidirect
5. #28102C2semidirect
6. #27669C2semidirect
7. #25476C2semidirect
8. #25343C2semidirect
9. #24838C2semidirect
10. #24837C2semidirect
11. #24711C2semidirect
12. #24710C2semidirect
13. #21963C2semidirect
14. #21837C2semidirect
15. #20734C2semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
C2×#29902
#352972
#349712
#281022
#276692
#254762
#253432
#248382
#248372
#247112
#247102
#219632
#218372
#207342