Finite Group Databasebeta

← #467080

Subgroups — #467080

Order 512.

1324 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2492C211yesmaximal, characteristic
2C2×#2072C211yesmaximal, characteristic, complemented
3C2×#1947C211yesmaximal, characteristic, complemented
4#15195C211yesmaximal, characteristic
5#11384C212yesmaximal, complemented
6#11124C212yesmaximal, complemented
7#11096C212yesmaximal
8#11077C211yesmaximal, characteristic, complemented
9#11042C211yesmaximal, characteristic
10#10970C212yesmaximal
11#10807C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#107C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2212yes—
19#2492—24no—
20#2492—24no—
21C2×#400—21no—
22C2×#393—21no—
23C2×#391—21no—
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#388—21no—
27C2×#384C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×#380C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32#2072—24no—
33#1947—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1442—21no—
42#1442—21no—
43#1439—21no—
44#1439—21no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×Q8C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—42no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—42no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4D411yescharacteristic
73C22×C22⋊C4D411yescharacteristic
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#107—24no—
78C2×#107—42no—
79C2×#107—42no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—22no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103—42no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—42no—
90C2×C22⋊C8—42no—
91#400—28no—
92#393—28no—
93#391—28no—
94#391—22no—
95#391—22no—
96#391—22no—
97#391D414yes—
98#388—28no—
99#384—22no—
100#384—22no—
101#380—22no—
102#380—22no—
103#380—42no—
104#380—42no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4D414yes—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95D411yescharacteristic
121C2×#95D411yescharacteristic
122#340—24no—
123#340—24no—
124#340—24no—
125#340—42no—
126#340—42no—
127#336—22no—
128#336—42no—
129C25C2×D411yescharacteristic
130C22×Q8—24no—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—44no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—82no—
144C22×Q8C2×D412yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8#4114yes—
181C4⋊Q8—48no—
182C4⋊Q8—44no—
183C4⋊Q8—44no—
184C22⋊Q8—48no—
185C22⋊Q8—44no—
186C22⋊Q8—44no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4#4114yes—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—82no—
231C2×C22⋊C4—82no—
232C2×C22⋊C4—82no—
233C2×C22⋊C4Q1614yes—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—24no—
241C2×C42—82no—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244#107—48no—
245C4⋊C8—28no—
246#103—48no—
247#103—48no—
248C22⋊C8—48no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—22no—
256#95—22no—
257#95—42no—
258#95—44no—
259#95—42no—
260#95—82no—
261#95—82no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—22no—
266C24—22no—
267C24—21no—
268C24—21no—
269C24—21no—
270C24—22no—
271C24—22no—
272C24C2×Q1612yes—
273C24C22×D411yescharacteristic
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—48no—
279C2×Q8—48no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—88no—
295C2×Q8—84no—
296C2×Q8—88no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4#12012yes—
369C22×C4#12012yes—
370C22×C4C2×Q1612yes—
371C22×C4C2×Q1612yes—
372C22×C4#13712yes—
373C22×C4#13712yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C2×#4112yes—
377C22×C4C2×#4112yes—
378C22×C4H(5,𝔽2)11yescharacteristic
379C22×C4H(5,𝔽2)11yescharacteristic
380C22×C4H(5,𝔽2)11yescharacteristic
381C22×C4H(5,𝔽2)12yes—
382C22×C42−1+412yes—
383C22×C42−1+411yescharacteristic
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—48no—
404C4⋊C4—48no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—84no—
418C4⋊C4—84no—
419C4⋊C4#12414yes—
420C22⋊C4—28no—
421C22⋊C4—44no—
422C22⋊C4—48no—
423C22⋊C4—84no—
424C22⋊C4—88no—
425C42—22no—
426C42—22no—
427C42—24no—
428C42—41no—
429C42—41no—
430C42—42no—
431C42—84no—
432C23—24no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—41no—
449C23—42no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23C2×#12011yescharacteristic
456C23#53511yescharacteristic
457C23#53711yescharacteristic
458C23#54612yes—
459C23#54712yes—
460C23C22×Q1611yescharacteristic
461C23C2×#13711yescharacteristic
462C23C2×#13711yescharacteristic
463C23#57511yescharacteristic
464C23#57612yes—
465C23#57712yes—
466Q8—44no—
467Q8—44no—
468Q8—816no—
469Q8—81no—
470Q8—81no—
471Q8—81no—
472Q8—81no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4#45114yes—
538C2×C4#52511yescharacteristic
539C2×C4C2×#12411yescharacteristic
540C2×C4#53511yescharacteristic
541C2×C4#53511yescharacteristic
542C2×C4#53511yescharacteristic
543C2×C4#53611yescharacteristic
544C2×C4#53711yescharacteristic
545C2×C4#53711yescharacteristic
546C2×C4#53812yes—
547C2×C4#53812yes—
548C2×C4#54212yes—
549C2×C4#54212yes—
550C2×C4#54612yes—
551C2×C4#54712yes—
552C2×C4#55612yes—
553C2×C4#55912yes—
554C2×C4#56212yes—
555C2×C4#56312yes—
556C8—44no—
557C8—82no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—44no—
562C22—42no—
563C22—81no—
564C22—81no—
565C22—82no—
566C22—81no—
567C22—82no—
568C22—81no—
569C22#240611yescharacteristic
570C22#243511yescharacteristic
571C22#261511yescharacteristic
572C22#271012yes—
573C22#271512yes—
574C22C2×#45111yescharacteristic
575C22#300711yescharacteristic
576C22#300811yescharacteristic
577C22#301011yescharacteristic
578C22#320911yescharacteristic
579C22#321611yescharacteristic
580C22#322412yes—
581C22#324911yescharacteristic
582C22#325511yescharacteristic
583C22#325712yes—
584C22#329912yes—
585C22#331412yes—
586C22#331912yes—
587C22#332912yes—
588C22#333812yes—
589C22#334612yes—
590C22#335012yes—
591C22#335612yes—
592C4—24no—
593C4—42no—
594C4—42no—
595C4—44no—
596C4—42no—
597C4—82no—
598C4—81no—
599C4—82no—
600C4—82no—
601C4—162no—
602C4—162no—
603C4#259211yescharacteristic
604C4#263111yescharacteristic
605C4#264011yescharacteristic
606C4#271012yes—
607C4#271512yes—
608C4#282011yescharacteristic
609C2—82no—
610C2#1655811yescharacteristic
611C2#2409811yescharacteristic
612C2#2410811yescharacteristic
613C2#2588811yescharacteristic
614C2#2589711yescharacteristic
615C2#2966411yescharacteristic
616C2#2966911yescharacteristic
617C2#3205212yes—
618C2#3206312yes—
619C2#3220012yes—
620C2#3221812yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2072C2semidirect
3. C2×#1947C2semidirect
5. #11384C2semidirect
6. #11124C2semidirect
8. #11077C2semidirect
11. #10807C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24922
C2×#20722
C2×#19472
#151952
#113842
#111242
#110962
#110772
#110422
#109702
#108072