Finite Group Databasebeta

← #360666

Subgroups — #360666

Order 512.

1498 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2283C211yesmaximal, characteristic
2C2×#1846C211yesmaximal, characteristic, complemented
3C2×#1835C211yesmaximal, characteristic, complemented
4C2×C4×C22⋊C8C211yesmaximal, characteristic
5#14682C211yesmaximal, characteristic
6#11096C212yesmaximal
7#11095C212yesmaximal
8#10908C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10#10842C211yesmaximal, characteristic, complemented
11#10842C211yesmaximal, characteristic, complemented
12C23×C42C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2283—24no—
20#2283—24no—
21C2×#383—21no—
22C2×#382C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#381—21no—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C22⋊C4—21no—
31C2×#376—21no—
32#1846—24no—
33#1835—24no—
34C4×C22⋊C8C418yescomplemented
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43C4⋊#95C2212yes—
44#1434—21no—
45#1434—21no—
46C22⋊#95—21no—
47C22⋊#95—21no—
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C42—22no—
62C22×C42—22no—
63C22×C42—22no—
64C22×C42—22no—
65C22×C42—22no—
66C22×C42C2×C414yes—
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42Q814yes—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C2×#107—24no—
73C2×#107—24no—
74C2×#107—42no—
75C2×#106—24no—
76C2×#106—24no—
77C2×#106—42no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8C2×C414yes—
83C2×C22⋊C8C2×C414yes—
84C2×C4×C8—22no—
85C2×C4×C8—22no—
86C2×C4×C8—22no—
87C2×C4×C8—22no—
88C2×C4×C8C2×C414yes—
89C2×C4×C8C2×C414yes—
90#383—28no—
91#382—22no—
92#382—22no—
93#382—22no—
94#382—22no—
95#382—22no—
96#382—22no—
97#382D414yes—
98#382D414yes—
99#381—28no—
100#380—24no—
101#380—24no—
102#380—42no—
103#380—42no—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—24no—
106C4×C22⋊C4—28no—
107#376—28no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—22no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—42no—
126#340—42no—
127#336C2×C414yes—
128#336C2×C414yes—
129C25C2×D411yescharacteristic
130C23×C4—24no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—24no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4C4⋊C414yes—
155C23×C4C4⋊C414yes—
156C23×C4C22×C412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×Q812yes—
163C23×C4C2411yescharacteristic
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8C22×C412yes—
185C22×C8C22×C412yes—
186C22×C8C22×C412yes—
187C22×C8C22×C412yes—
188C22×C8#4112yes—
189C22×C8#4112yes—
190C22×C8#4112yes—
191C22×C8#4112yes—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
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—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4C2×D414yes—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—28no—
231C2×C42—28no—
232C2×C42—28no—
233C2×C42—28no—
234C2×C42—28no—
235C2×C42—44no—
236C2×C42—44no—
237C2×C42C4⋊C414yes—
238C2×C42C4⋊C414yes—
239C2×C42C22×C412yes—
240C2×C42C22×C412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×Q812yes—
244C2×C42C2×Q812yes—
245C2×C42#4112yes—
246C2×C42#4112yes—
247#107—48no—
248#106—48no—
249C22⋊C8—28no—
250C22⋊C8—28no—
251C4×C8—28no—
252C4×C8—28no—
253#95—28no—
254#95—28no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—44no—
260#95—44no—
261#95—42no—
262#95—42no—
263#95—42no—
264#95—42no—
265#95—44no—
266C24—21no—
267C24—21no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24C2×C4⋊C412yes—
276C24#13514yes—
277C24C22×D411yescharacteristic
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—28no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—28no—
309C22×C4—28no—
310C22×C4—28no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—48no—
316C22×C4—48no—
317C22×C4—48no—
318C22×C4—48no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
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×C4C2×C4⋊C412yes—
362C22×C4C2×C4⋊C412yes—
363C22×C4C2×C4⋊C412yes—
364C22×C4C2×C4⋊C412yes—
365C22×C4C2×C4⋊C412yes—
366C22×C4#12012yes—
367C22×C4#12012yes—
368C22×C4#12312yes—
369C22×C4#12312yes—
370C22×C4#13514yes—
371C22×C4C23×C411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×Q811yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4H(5,𝔽2)11yescharacteristic
382C22×C4H(5,𝔽2)11yescharacteristic
383C22×C42−1+411yescharacteristic
384C22×C42−1+411yescharacteristic
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8#11714yes—
412C2×C8#11714yes—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C22⋊C4—416no—
430C22⋊C4—88no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—28no—
436C42—28no—
437C42—416no—
438C42C4×D414yes—
439C42C4×D414yes—
440C42C22⋊Q814yes—
441C42C22⋊Q814yes—
442C23—22no—
443C23—24no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—22no—
453C23—22no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—41no—
458C23—41no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—41no—
463C23—41no—
464C23#42714yes—
465C23C22×C4⋊C411yescharacteristic
466C23C2×#12011yescharacteristic
467C23C2×#12311yescharacteristic
468C23#53211yescharacteristic
469C23#53211yescharacteristic
470C23#53511yescharacteristic
471C23#53611yescharacteristic
472C23C2×#13512yes—
473C23C2×#13512yes—
474C23#57512yes—
475C23#57512yes—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—416no—
495C2×C4—416no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
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—82no—
538C2×C4#38514yes—
539C2×C4#38514yes—
540C2×C4#42714yes—
541C2×C4C2×#11711yescharacteristic
542C2×C4C2×#11711yescharacteristic
543C2×C4C2×C4×D411yescharacteristic
544C2×C4C2×C4×D411yescharacteristic
545C2×C4#52011yescharacteristic
546C2×C4#52011yescharacteristic
547C2×C4C2×C22⋊Q811yescharacteristic
548C2×C4C2×C22⋊Q811yescharacteristic
549C2×C4#53311yescharacteristic
550C2×C4#53311yescharacteristic
551C2×C4#54411yescharacteristic
552C2×C4#54411yescharacteristic
553C2×C4#54611yescharacteristic
554C2×C4#54611yescharacteristic
555C2×C4#54711yescharacteristic
556C2×C4#54711yescharacteristic
557C2×C4#54811yescharacteristic
558C2×C4#54811yescharacteristic
559C2×C4#55011yescharacteristic
560C2×C4#55011yescharacteristic
561C2×C4#55211yescharacteristic
562C2×C4#55211yescharacteristic
563C2×C4#55311yescharacteristic
564C2×C4#55311yescharacteristic
565C2×C4#55511yescharacteristic
566C2×C4#55511yescharacteristic
567C2×C4#55611yescharacteristic
568C2×C4#55611yescharacteristic
569C8—24no—
570C8—24no—
571C8—42no—
572C8—42no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—44no—
577C22—42no—
578C22—44no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—44no—
583C22—44no—
584C22#228511yescharacteristic
585C22C2×#38511yescharacteristic
586C22#231711yescharacteristic
587C22#243111yescharacteristic
588C22#243211yescharacteristic
589C22#249911yescharacteristic
590C22#250011yescharacteristic
591C22C2×#42711yescharacteristic
592C22C2×#42711yescharacteristic
593C22#291111yescharacteristic
594C22#291111yescharacteristic
595C22#300212yes—
596C22#300212yes—
597C22#309112yes—
598C22#309112yes—
599C22#316812yes—
600C22#316812yes—
601C22#318312yes—
602C22#318312yes—
603C22#321112yes—
604C22#321212yes—
605C22#323212yes—
606C22#323312yes—
607C4—24no—
608C4—24no—
609C4—24no—
610C4—42no—
611C4—42no—
612C4—84no—
613C4—84no—
614C4—162no—
615C4—162no—
616C4#235211yescharacteristic
617C4#235211yescharacteristic
618C4#235411yescharacteristic
619C4#235411yescharacteristic
620C4#243911yescharacteristic
621C4#243911yescharacteristic
622C4#244111yescharacteristic
623C4#244111yescharacteristic
624C2—44no—
625C2#1387911yescharacteristic
626C2#2081811yescharacteristic
627C2#2082511yescharacteristic
628C2#2084311yescharacteristic
629C2#2084311yescharacteristic
630C2#2198311yescharacteristic
631C2#2198511yescharacteristic
632C2#2560112yes—
633C2#2567512yes—
634C2#2787912yes—
635C2#2789312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1846C2semidirect
3. C2×#1835C2semidirect
8. #10908C2semidirect
9. #10907C2semidirect
10. #10842C2semidirect
11. #10842C2semidirect
34. C4×C22⋊C8C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22832
C2×#18462
C2×#18352
C2×C4×C22⋊C82
#146822
#110962
#110952
#109082
#109072
#108422