Finite Group Databasebeta

← #116810

Subgroups — #116810

Order 512.

960 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21170C211yesmaximal, characteristic, complemented
2#20732C211yesmaximal, characteristic
3C2×#1486C211yesmaximal, characteristic
4#13905C211yesmaximal, characteristic
5#8026C211yesmaximal, characteristic, complemented
6C2×#1286C211yesmaximal, characteristic, complemented
7#7892C211yesmaximal, characteristic, complemented
8C22×C4×D4C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#428—21no—
12C2×#423C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#382C2211yescharacteristic
15C2×#380—21no—
16C2×#380—21no—
17C2×C4×C4⋊C4C2211yescharacteristic
18#1949—22no—
19#1948—21no—
20#1947—21no—
21#1823—21no—
22#1802—21no—
23#1802—21no—
24#1786—21no—
25C4×#102—22no—
26#1736—21no—
27#1486—22no—
28#1486C414yescomplemented
29C2×D4⋊C8—22no—
30#1445—21no—
31#1438—21no—
32#1437—22no—
33#1294—22no—
34#1286—22no—
35#1279—22no—
36C23×D4D411yescharacteristic
37C24×C4—21no—
38C22×M16—21no—
39C2×C4×D4—21no—
40C2×C4×D4—21no—
41C2×C4×D4—21no—
42C2×C4×D4—21no—
43C2×C4×D4—21no—
44C2×C4×D4—21no—
45C2×C4×D4—21no—
46C2×C4×D4—21no—
47C2×C4×D4—21no—
48C2×C4×D4—21no—
49C2×C4×D4C2×C412yes—
50C2×C4×D4Q812yes—
51C2×#117—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431C2×C412yes—
69#428—42no—
70#428—42no—
71C2×#107—22no—
72C2×#106—22no—
73#423—21no—
74#423—21no—
75#423C2×C412yes—
76#423D412yes—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8D412yes—
81C2×#102—22no—
82C2×#102—22no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8D412yes—
87#384D411yescharacteristic
88#384D411yescharacteristic
89#382—21no—
90#382—21no—
91#382—21no—
92#382—21no—
93#382—21no—
94#380—21no—
95#380—21no—
96#380—42no—
97#380—42no—
98#380—42no—
99#380—42no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95D412yes—
111#340—22no—
112#340—22no—
113#340—22no—
114#340—22no—
115D4⋊C8—28no—
116C25—21no—
117C22×D4—21no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4—21no—
121C22×D4—21no—
122C22×D4—42no—
123C22×D4—42no—
124C22×D4C4⋊C412yes—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—41no—
132C23×C4—41no—
133C23×C4—41no—
134C23×C4—41no—
135C23×C4—41no—
136C23×C4—41no—
137C23×C4—41no—
138C23×C4—41no—
139C23×C4—41no—
140C23×C4—41no—
141C23×C4—41no—
142C23×C4—41no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—41no—
149C2×M16—41no—
150C2×M16—41no—
151C2×M16—41no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—41no—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—41no—
164C4×D4—41no—
165C4×D4—41no—
166C4×D4—41no—
167C4×D4—41no—
168C4×D4—41no—
169C4×D4—41no—
170C4×D4—41no—
171#117—41no—
172#117—41no—
173#117—41no—
174#117—41no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4C4⋊C412yes—
203C2×C4⋊C4D812yes—
204C2×C4⋊C4SD1612yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—82no—
217C2×C22⋊C4—82no—
218C2×C42—41no—
219C2×C42—41no—
220C2×C42—41no—
221C2×C42—82no—
222C2×C42C22×C411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×Q811yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42#4111yescharacteristic
230C2×C42#4111yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42#4111yescharacteristic
233C2×C42#4111yescharacteristic
234#107—44no—
235#106—44no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—41no—
241C4⋊C8—41no—
242C4⋊C8—41no—
243C4⋊C8C22⋊C414yescomplemented
244#102—44no—
245#102—44no—
246C22⋊C8—22no—
247C22⋊C8—41no—
248C22⋊C8—41no—
249C22⋊C8—41no—
250C4×C8—22no—
251C4×C8—22no—
252C4×C8—22no—
253C4×C8—41no—
254#95—22no—
255#95—22no—
256#95—44no—
257#95—44no—
258#95—41no—
259#95—41no—
260#95—41no—
261#95—41no—
262#95—41no—
263#95—41no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24#12011yescharacteristic
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—41no—
295C2×D4—41no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—81no—
339C22×C4—81no—
340C22×C4—81no—
341C22×C4—81no—
342C22×C4—81no—
343C22×C4—81no—
344C22×C4—81no—
345C22×C4—81no—
346C22×C4—81no—
347C22×C4—81no—
348C22×C4—81no—
349C22×C4—81no—
350C22×C4—81no—
351C22×C4—81no—
352C22×C4—81no—
353C22×C4—81no—
354C22×C4C2×C4⋊C411yescharacteristic
355C22×C4C4×D411yescharacteristic
356C22×C4C4×D411yescharacteristic
357C22×C4#12011yescharacteristic
358C22×C4#12011yescharacteristic
359C22×C4#12111yescharacteristic
360C22×C4#12111yescharacteristic
361C22×C4#12111yescharacteristic
362C22×C4#12111yescharacteristic
363C22×C4#12111yescharacteristic
364C22×C4#12111yescharacteristic
365C22×C4C22⋊Q811yescharacteristic
366C22×C4C22⋊Q811yescharacteristic
367C22×C4#12311yescharacteristic
368C22×C4#12311yescharacteristic
369C22×C4#12311yescharacteristic
370C22×C4#12411yescharacteristic
371C22×C4#12711yescharacteristic
372C22×C4C2×D811yescharacteristic
373C22×C4C2×SD1611yescharacteristic
374C22×C4#13511yescharacteristic
375C22×C4#13511yescharacteristic
376C22×C4AGL(1,ℤ/8)11yescharacteristic
377C22×C4AGL(1,ℤ/8)11yescharacteristic
378C22×C4AGL(1,ℤ/8)11yescharacteristic
379C22×C4#13711yescharacteristic
380M16—42no—
381M16—42no—
382M16—42no—
383M16—42no—
384C2×C8—24no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—81no—
392C2×C8—81no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—81no—
412C4⋊C4—81no—
413C4⋊C4—81no—
414C4⋊C4—81no—
415C4⋊C4—81no—
416C4⋊C4—81no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—88no—
422C22⋊C4—88no—
423C22⋊C4—81no—
424C22⋊C4—81no—
425C42—82no—
426C42—82no—
427C42—81no—
428C42—81no—
429C42C2×C22⋊C411yescharacteristic
430C42#11711yescharacteristic
431C42C4×D411yescharacteristic
432C42C4×D411yescharacteristic
433C42#12111yescharacteristic
434C42#12111yescharacteristic
435C42C22⋊Q811yescharacteristic
436C42C22⋊Q811yescharacteristic
437C42C22⋊Q811yescharacteristic
438C42C22⋊Q811yescharacteristic
439C42#12311yescharacteristic
440C42#12311yescharacteristic
441C42#12411yescharacteristic
442C42#12411yescharacteristic
443C42#12611yescharacteristic
444C42#12611yescharacteristic
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—24no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—81no—
469C23—81no—
470C23—81no—
471C23—81no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—81no—
476C23—81no—
477C23—81no—
478C23—81no—
479C23—81no—
480C23—81no—
481C23—81no—
482C23—81no—
483C23—81no—
484C23#38511yescharacteristic
485C23#39211yescharacteristic
486C23#39411yescharacteristic
487C23#44711yescharacteristic
488C23#44911yescharacteristic
489C23#45011yescharacteristic
490C23#45211yescharacteristic
491D4—24no—
492D4—24no—
493D4—42no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—42no—
498D4—42no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—82no—
553C2×C4#38011yescharacteristic
554C2×C4#38511yescharacteristic
555C2×C4#38611yescharacteristic
556C2×C4#38811yescharacteristic
557C2×C4#38811yescharacteristic
558C2×C4#39011yescharacteristic
559C2×C4#39311yescharacteristic
560C2×C4#39411yescharacteristic
561C2×C4#39611yescharacteristic
562C2×C4#39611yescharacteristic
563C2×C4#39711yescharacteristic
564C2×C4#39911yescharacteristic
565C2×C4C4×D811yescharacteristic
566C2×C4C4×SD1611yescharacteristic
567C2×C4#44011yescharacteristic
568C2×C4#44211yescharacteristic
569C2×C4#44711yescharacteristic
570C2×C4#44811yescharacteristic
571C2×C4#44911yescharacteristic
572C2×C4#44911yescharacteristic
573C2×C4#45911yescharacteristic
574C2×C4#46011yescharacteristic
575C2×C4#46311yescharacteristic
576C2×C4#46411yescharacteristic
577C2×C4#46511yescharacteristic
578C2×C4#46611yescharacteristic
579C2×C4#46811yescharacteristic
580C2×C4#46911yescharacteristic
581C2×C4#47411yescharacteristic
582C2×C4#47611yescharacteristic
583C2×C4#47811yescharacteristic
584C2×C4#47811yescharacteristic
585C2×C4#48611yescharacteristic
586C2×C4#48811yescharacteristic
587C2×C4#48911yescharacteristic
588C2×C4#49011yescharacteristic
589C8—44no—
590C8—82no—
591C8—82no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22#143911yescharacteristic
620C22#186211yescharacteristic
621C22#186411yescharacteristic
622C22#199911yescharacteristic
623C22#200011yescharacteristic
624C22#200811yescharacteristic
625C22#200911yescharacteristic
626C22#201011yescharacteristic
627C22#201011yescharacteristic
628C22#204111yescharacteristic
629C22#204211yescharacteristic
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4—162no—
642C4—161no—
643C4—161no—
644C4#187911yescharacteristic
645C4#188011yescharacteristic
646C4#193411yescharacteristic
647C4#193611yescharacteristic
648C4#202311yescharacteristic
649C4#202511yescharacteristic
650C4#206511yescharacteristic
651C4#206711yescharacteristic
652C2—21no—
653C2—21no—
654C2—21no—
655C2—21no—
656C2—84no—
657C2#999611yescharacteristic
658C2#999711yescharacteristic
659C2#1000011yescharacteristic
660C2#1000111yescharacteristic
661C2#1106611yescharacteristic
662C2#1110011yescharacteristic
663C2#1111711yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21170C2semidirect
5. #8026C2semidirect
6. C2×#1286C2semidirect
7. #7892C2semidirect
28. #1486C4semidirect
243. C4⋊C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
#211702
#207322
C2×#14862
#139052
#80262
C2×#12862
#78922