Finite Group Databasebeta

← #368436

Subgroups — #368436

Order 512.

1784 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#107C211yesmaximal, characteristic
2C2×#1928C212yesmaximal, complemented
3C2×#1892C211yesmaximal, characteristic
4C2×#1872C211yesmaximal, characteristic
5#14139C211yesmaximal, characteristic
6C22⋊C4×C4⋊C4C211yesmaximal, characteristic
7#11256C212yesmaximal, complemented
8#11124C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#107—28no—
17C22×#107—24no—
18C22×#107C2212yes—
19C22×#107C2212yes—
20C22×#103C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C2×#391C2212yes—
25C2×#380C2211yescharacteristic
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C22⋊C4—21no—
28C2×C4×C22⋊C4C2211yescharacteristic
29#1928—28no—
30#1892C418yescomplemented
31#1872—24no—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39#1441—22no—
40#1436—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C22⋊#95—21no—
44C22⋊#95—21no—
45C22⋊#95—21no—
46C23×Q8C2311yescharacteristic
47C24×C4—21no—
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C23×C8—22no—
51C23×C8—22no—
52C23×C8—24no—
53C23×C8—24no—
54C23×C8C2×C414yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2312yes—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8C2×C414yes—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—42no—
73C22×C4⋊C4—42no—
74C22×C4⋊C4D414yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4D411yescharacteristic
83C22×C22⋊C4D411yescharacteristic
84C22×C42—22no—
85C22×C42C2312yes—
86C2×#107—232no—
87C2×#107—24no—
88C2×#107—24no—
89C2×#107—24no—
90C2×#107—24no—
91C2×#107—24no—
92C2×#107D418yes—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—24no—
98C2×#103—42no—
99C2×#103C2×C418yescomplemented
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103#391—24no—
104#391—42no—
105#380—22no—
106#380C2×C414yes—
107C4×C4⋊C4—24no—
108C4×C4⋊C4—42no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—44no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95D412yes—
121#340—22no—
122#340—22no—
123#340—24no—
124#340—42no—
125#340C2×C418yescomplemented
126#336—24no—
127#336—24no—
128C25C2×D411yescharacteristic
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8C22×C412yes—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—82no—
158C23×C4—82no—
159C23×C4C22×C412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4#4112yes—
165C23×C4#4112yes—
166C23×C4#4112yes—
167C23×C4#4112yes—
168C23×C4C2411yescharacteristic
169C22×C8—216no—
170C22×C8—28no—
171C22×C8—28no—
172C22×C8—28no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—216no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8C22×C414yes—
189C22×C8C2×D414yes—
190C22×C8#4114yes—
191C22⋊Q8—28no—
192C22⋊Q8—28no—
193C22⋊Q8—44no—
194C22⋊Q8—44no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—48no—
221C2×C4⋊C4—48no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C414yes—
230C2×C4⋊C4C2×D414yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×D412yes—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4Q1614yes—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—44no—
253C2×C42—44no—
254C2×C42—82no—
255#107—216no—
256#107—216no—
257#103—216no—
258#103—44no—
259#103—44no—
260C22⋊C8—44no—
261#95—24no—
262#95—24no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—82no—
268#95—82no—
269#95—82no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24C4×D412yes—
275C24#12112yes—
276C24#12312yes—
277C24C2×Q1612yes—
278C24#13512yes—
279C24AGL(1,ℤ/8)12yes—
280C24#13712yes—
281C24C22×D411yescharacteristic
282C2×Q8—22no—
283C2×Q8—24no—
284C2×Q8—22no—
285C2×Q8—24no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—42no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—48no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—48no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
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—48no—
366C22×C4—84no—
367C22×C4—88no—
368C22×C4—88no—
369C22×C4—88no—
370C22×C4—88no—
371C22×C4—84no—
372C22×C4—84no—
373C22×C4#11714yes—
374C22×C4C4×D414yes—
375C22×C4C4×D412yes—
376C22×C4#12112yes—
377C22×C4#12312yes—
378C22×C4#12414yes—
379C22×C4#12614yes—
380C22×C4C2×Q1614yes—
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D412yes—
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4112yes—
388C22×C4C2×#4111yescharacteristic
389C22×C4H(5,𝔽2)12yes—
390C22×C42−1+412yes—
391C2×C8—232no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8C4×D418yes—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—48no—
429C4⋊C4#12014yes—
430C22⋊C4—28no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—44no—
434C42—42no—
435C42—42no—
436C42—42no—
437C42—42no—
438C42—88no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—22no—
446C23—22no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—22no—
455C23#38814yes—
456C23C4×Q1614yes—
457C23#44014yes—
458C23#46214yes—
459C23#46314yes—
460C23#48414yes—
461C23#48514yes—
462C23C2×C4×D411yescharacteristic
463C23C2×#12111yescharacteristic
464C23C2×#12311yescharacteristic
465C23D4212yes—
466C23#54712yes—
467C23C22×Q1611yescharacteristic
468C23C2×#13511yescharacteristic
469C23C2×AGL(1,ℤ/8)11yescharacteristic
470C23C2×#13711yescharacteristic
471C23#57612yes—
472C23#57812yes—
473Q8—42no—
474Q8—44no—
475Q8—42no—
476Q8—81no—
477Q8—81no—
478Q8—82no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—24no—
487C2×C4—28no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—816no—
522C2×C4—816no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—81no—
527C2×C4—81no—
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—81no—
537C2×C4—82no—
538C2×C4—81no—
539C2×C4#38814yes—
540C2×C4C2×#11711yescharacteristic
541C2×C4C2×C4×D411yescharacteristic
542C2×C4C2×C4×D412yes—
543C2×C4#52012yes—
544C2×C4C2×#12011yescharacteristic
545C2×C4#52511yescharacteristic
546C2×C4#52512yes—
547C2×C4C2×#12411yescharacteristic
548C2×C4C2×#12611yescharacteristic
549C2×C4#53411yescharacteristic
550C2×C4#53611yescharacteristic
551C2×C4#53912yes—
552C2×C4#54612yes—
553C2×C4#54812yes—
554C2×C4#54812yes—
555C2×C4#55212yes—
556C2×C4#55312yes—
557C2×C4#55312yes—
558C8—28no—
559C8—81no—
560C8—81no—
561C22—24no—
562C22—24no—
563C22—28no—
564C22—28no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—24no—
569C22—28no—
570C22—28no—
571C22#192818yescomplemented
572C22C2×#38811yescharacteristic
573C22#236112yes—
574C22#241011yescharacteristic
575C22#241311yescharacteristic
576C22#244312yes—
577C22C2×C4×Q1611yescharacteristic
578C22C2×#44011yescharacteristic
579C22#295112yes—
580C22C2×#46311yescharacteristic
581C22C2×#46211yescharacteristic
582C22#304111yescharacteristic
583C22#304411yescharacteristic
584C22C2×#48511yescharacteristic
585C22C2×#48411yescharacteristic
586C22#309812yes—
587C22#326412yes—
588C22#327012yes—
589C22#328012yes—
590C22D4×Q1612yes—
591C22#333612yes—
592C22#333812yes—
593C22#334312yes—
594C22#334612yes—
595C4—24no—
596C4—44no—
597C4—44no—
598C4—82no—
599C4—84no—
600C4—82no—
601C4—82no—
602C4—82no—
603C4—84no—
604C4C4×#12011yescharacteristic
605C4#232211yescharacteristic
606C4#235212yes—
607C4#243611yescharacteristic
608C4#248112yes—
609C4#259111yescharacteristic
610C2—28no—
611C2#1401911yescharacteristic
612C2C2×#192811yescharacteristic
613C2#2106711yescharacteristic
614C2#2302412yes—
615C2#2303212yes—
616C2#2435511yescharacteristic
617C2#2436011yescharacteristic
618C2#2457711yescharacteristic
619C2#2458211yescharacteristic
620C2#2613212yes—
621C2#2613412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×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×#1928C2semidirect
7. #11256C2semidirect
8. #11124C2semidirect
9. #10907C2semidirect
30. #1892C4semidirect
99. C2×#103C2×C4semidirect
125. #340C2×C4semidirect
571. C22#1928semidirect

Maximal subgroups

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

subgroupindex
C23×#1072
C2×#19282
C2×#18922
C2×#18722
#141392
C22⋊C4×C4⋊C42
#112562
#111242
#109072
#107302
#107142