Finite Group Databasebeta

← #373739

Subgroups — #373739

Order 512.

1672 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×#2345C211yesmaximal, characteristic
2C2×#1852C211yesmaximal, characteristic
3C2×#1824C211yesmaximal, characteristic
4C22×#340C211yesmaximal, characteristic
5#14130C211yesmaximal, characteristic
6#11328C212yesmaximal, complemented
7#11293C212yesmaximal, complemented
8#10907C212yesmaximal, complemented
9#10885C212yesmaximal, complemented
10#10721C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17#2345—24no—
18#2345—24no—
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#389C2212yes—
22C2×#387—21no—
23C2×#387—21no—
24C2×#385—21no—
25C2×#380—21no—
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4—21no—
31C2×C4×C22⋊C4C2211yescharacteristic
32#1852—24no—
33#1824C418yescomplemented
34C2×#340—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1444—21no—
45#1442—21no—
46C4⋊#95C2212yes—
47C22⋊#95—21no—
48C24×C4—21no—
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C23×C8—22no—
52C23×C8—24no—
53C23×C8—22no—
54C23×C8—22no—
55C23×C8—42no—
56C23×C8C2312yes—
57C23×C8C2312yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—42no—
70C22×C4⋊C4C2×C414yes—
71C22×C4⋊C4D414yes—
72C22×C4⋊C4Q814yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4D411yescharacteristic
82C22×C22⋊C4D411yescharacteristic
83C22×C42—21no—
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C2×#107—24no—
88C2×#107—24no—
89C2×#107—42no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8C2×C414yes—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—24no—
98#389—42no—
99#387—28no—
100#387—28no—
101#385—44no—
102#380—28no—
103#380—22no—
104#380—22no—
105#380C2×C414yes—
106#380D414yes—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C22⋊C4—28no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95D411yescharacteristic
124C2×#95D411yescharacteristic
125#340—24no—
126#340—24no—
127#340—24no—
128#340—24no—
129#340—42no—
130#340C2×C418yescomplemented
131#340C2×C418yescomplemented
132#336—24no—
133#336—42no—
134C25C2×D411yescharacteristic
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—44no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—82no—
153C23×C4—82no—
154C23×C4—82no—
155C23×C4C22×C412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×Q812yes—
164C23×C4#4112yes—
165C23×C4#4112yes—
166C23×C4C2411yescharacteristic
167C22×C8—28no—
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—48no—
181C22×C8—48no—
182C22×C8—44no—
183C22×C8—44no—
184C22×C8C22×C414yes—
185C22×C8#4114yes—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—28no—
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—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—48no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4C4⋊C4116yescomplemented
227C2×C4⋊C4C22×C414yes—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×Q812yes—
235C2×C4⋊C4C2×Q812yes—
236C2×C4⋊C4#4114yes—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4Q1614yes—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—82no—
260C2×C42—82no—
261#107—48no—
262C22⋊C8—28no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—22no—
270#95—22no—
271#95—44no—
272#95—44no—
273#95—82no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24C4×D412yes—
279C24#12012yes—
280C24C22⋊Q812yes—
281C24C2×Q1612yes—
282C24#13512yes—
283C24AGL(1,ℤ/8)12yes—
284C24#13712yes—
285C24C22×D411yescharacteristic
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—216no—
317C22×C4—24no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—48no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—88no—
357C22×C4—88no—
358C22×C4—84no—
359C22×C4—84no—
360C22×C4—88no—
361C22×C4—88no—
362C22×C4—84no—
363C22×C4—84no—
364C22×C4C2×C4⋊C414yes—
365C22×C4C4×D414yes—
366C22×C4C4×D412yes—
367C22×C4#12012yes—
368C22×C4C22⋊Q814yes—
369C22×C4C22⋊Q812yes—
370C22×C4#12314yes—
371C22×C4C2×Q1612yes—
372C22×C4C2×Q1612yes—
373C22×C4C23×C411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×Q811yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4112yes—
379C22×C4C2×#4112yes—
380C22×C4C2×#4111yescharacteristic
381C22×C4H(5,𝔽2)12yes—
382C22×C42−1+412yes—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—416no—
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—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—216no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—48no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—42no—
416C4⋊C4—41no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—84no—
421C4⋊C4—84no—
422C4⋊C4—88no—
423C4⋊C4C2×C4⋊C414yes—
424C4⋊C4C2×C4⋊C414yes—
425C22⋊C4—28no—
426C22⋊C4—48no—
427C22⋊C4—44no—
428C22⋊C4—48no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—42no—
434C42—42no—
435C42—84no—
436C42—84no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—22no—
444C23—22no—
445C23—24no—
446C23—24no—
447C23—22no—
448C23—22no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23#38514yes—
454C23C4×Q1614yes—
455C23#44014yes—
456C23#44914yes—
457C23#45114yes—
458C23#47714yes—
459C23#47914yes—
460C23C2×C4×D411yescharacteristic
461C23C2×#12011yescharacteristic
462C23C2×C22⋊Q811yescharacteristic
463C23#54612yes—
464C23#54712yes—
465C23C22×Q1611yescharacteristic
466C23C2×#13511yescharacteristic
467C23C2×AGL(1,ℤ/8)11yescharacteristic
468C23C2×#13711yescharacteristic
469C23#57612yes—
470C23#57712yes—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—28no—
477C2×C4—28no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—41no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—816no—
524C2×C4—816no—
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#38514yes—
538C2×C4C22×C4⋊C411yescharacteristic
539C2×C4C2×C4×D411yescharacteristic
540C2×C4C4×#4112yes—
541C2×C4#52012yes—
542C2×C4C2×C22⋊Q811yescharacteristic
543C2×C4C2×#12311yescharacteristic
544C2×C4#52511yescharacteristic
545C2×C4#52511yescharacteristic
546C2×C4#53312yes—
547C2×C4#53511yescharacteristic
548C2×C4#53611yescharacteristic
549C2×C4#54412yes—
550C2×C4#54612yes—
551C2×C4#54712yes—
552C2×C4#55112yes—
553C2×C4#55112yes—
554C2×C4#55212yes—
555C2×C4#55212yes—
556C8—44no—
557C8—44no—
558C22—24no—
559C22—24no—
560C22—24no—
561C22—28no—
562C22—28no—
563C22—24no—
564C22—28no—
565C22—28no—
566C22—24no—
567C22—24no—
568C22#186318yes—
569C22C2×#38511yescharacteristic
570C22#235912yes—
571C22#240411yescharacteristic
572C22#240611yescharacteristic
573C22#243812yes—
574C22C2×C4×Q1611yescharacteristic
575C22C2×#44011yescharacteristic
576C22#294612yes—
577C22C2×#45111yescharacteristic
578C22C2×#44911yescharacteristic
579C22#300811yescharacteristic
580C22#300911yescharacteristic
581C22C2×#47711yescharacteristic
582C22C2×#47911yescharacteristic
583C22#308112yes—
584C22#329712yes—
585C22#329912yes—
586C22#331612yes—
587C22#331812yes—
588C22#333712yes—
589C22#333812yes—
590C22#334512yes—
591C22#334612yes—
592C4—24no—
593C4—42no—
594C4—44no—
595C4—44no—
596C4—42no—
597C4—81no—
598C4—82no—
599C4—84no—
600C4—81no—
601C4—82no—
602C4—84no—
603C4#229611yescharacteristic
604C4#231711yescharacteristic
605C4#235912yes—
606C4#243812yes—
607C4#248211yescharacteristic
608C4#248411yescharacteristic
609C2—28no—
610C2#1412811yescharacteristic
611C2C2×#186311yescharacteristic
612C2#2087611yescharacteristic
613C2#2292212yes—
614C2#2293612yes—
615C2#2396311yescharacteristic
616C2#2397011yescharacteristic
617C2#2409111yescharacteristic
618C2#2409811yescharacteristic
619C2#2599412yes—
620C2#2600012yes—
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
6. #11328C2semidirect
7. #11293C2semidirect
8. #10907C2semidirect
9. #10885C2semidirect
33. #1824C4semidirect
130. #340C2×C4semidirect
131. #340C2×C4semidirect
226. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23452
C2×#18522
C2×#18242
C22×#3402
#141302
#113282
#112932
#109072
#108852
#107212
#107142