Finite Group Databasebeta

← #360623

Subgroups — #360623

Order 512.

1672 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2345C211yesmaximal, characteristic
2C2×#1824C211yesmaximal, characteristic
3C2×#1807C211yesmaximal, characteristic
4C22×#340C211yesmaximal, characteristic
5#13907C211yesmaximal, characteristic
6#11327C212yesmaximal, complemented
7#11292C212yesmaximal, complemented
8#10908C212yesmaximal, complemented
9#10884C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#106C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17#2345—24no—
18#2345—24no—
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#387—22no—
22C2×#380—21no—
23C2×#380—21no—
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C22⋊C4—21no—
28C2×C4×C22⋊C4C2211yescharacteristic
29#1824—24no—
30#1807—24no—
31C2×#340C418yescomplemented
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1441—22no—
44C4⋊#95C2212yes—
45C22⋊#95—21no—
46C24×C4—21no—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C23×C8—22no—
50C23×C8—22no—
51C23×C8—24no—
52C23×C8—42no—
53C23×C8C2×C414yes—
54C23×C8C2312yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—42no—
67C22×C4⋊C4C2×C414yes—
68C22×C4⋊C4C2×C414yes—
69C22×C4⋊C4D414yes—
70C22×C4⋊C4Q814yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4D411yescharacteristic
80C22×C22⋊C4D411yescharacteristic
81C22×C42—22no—
82C22×C42C2312yes—
83C2×#106—24no—
84C2×#106—24no—
85C2×#106—42no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89#389—24no—
90#389—24no—
91#389—24no—
92#389—42no—
93#387—216no—
94#380—28no—
95#380—22no—
96#380—22no—
97#380—22no—
98#380—44no—
99#380D414yes—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—24no—
102C4×C22⋊C4—28no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95D412yes—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—42no—
118#340C2×C414yes—
119#340C2×C414yes—
120#340C2×C414yes—
121#340C2×C414yes—
122#336—24no—
123#336—42no—
124C25C2×D411yescharacteristic
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—44no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—82no—
143C23×C4—82no—
144C23×C4—82no—
145C23×C4C22×C412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4C2×Q812yes—
152C23×C4#4112yes—
153C23×C4#4112yes—
154C23×C4C2411yescharacteristic
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—28no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—28no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—44no—
172C22×C8—48no—
173C22×C8—48no—
174C22×C8C22×C412yes—
175C22×C8C22×C412yes—
176C22×C8#4114yes—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—28no—
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—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
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—28no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4C4⋊C4116yescomplemented
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C2×D414yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×Q812yes—
225C2×C4⋊C4C2×Q812yes—
226C2×C4⋊C4#4114yes—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4SD1614yes—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—28no—
246C2×C42—82no—
247C2×C42—82no—
248#106—48no—
249C22⋊C8—44no—
250#95—28no—
251#95—28no—
252#95—24no—
253#95—24no—
254#95—44no—
255#95—42no—
256#95—44no—
257#95—82no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24C4×D412yes—
263C24#12012yes—
264C24C22⋊Q812yes—
265C24C2×SD1612yes—
266C24#13512yes—
267C24#13712yes—
268C24#13712yes—
269C24C22×D411yescharacteristic
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—216no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—48no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—48no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—88no—
337C22×C4—84no—
338C22×C4—84no—
339C22×C4—84no—
340C22×C4—88no—
341C22×C4—88no—
342C22×C4—88no—
343C22×C4C2×C4⋊C414yes—
344C22×C4C4×D414yes—
345C22×C4C4×D412yes—
346C22×C4#12012yes—
347C22×C4C22⋊Q814yes—
348C22×C4C22⋊Q812yes—
349C22×C4#12314yes—
350C22×C4C2×SD1614yes—
351C22×C4C23×C411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×Q811yescharacteristic
355C22×C4C2×#4112yes—
356C22×C4C2×#4112yes—
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4H(5,𝔽2)12yes—
360C22×C42−1+412yes—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—416no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8#11714yes—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—216no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—48no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—88no—
401C4⋊C4—88no—
402C4⋊C4C2×C4⋊C414yes—
403C4⋊C4C2×C4⋊C414yes—
404C4⋊C4#11714yes—
405C22⋊C4—28no—
406C22⋊C4—44no—
407C22⋊C4—48no—
408C22⋊C4—84no—
409C42—42no—
410C42—42no—
411C42—42no—
412C42—42no—
413C42—88no—
414C23—24no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23—24no—
419C23—22no—
420C23—22no—
421C23—24no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23#38514yes—
431C23C4×SD1614yes—
432C23#44114yes—
433C23#44814yes—
434C23#45214yes—
435C23#47514yes—
436C23#47914yes—
437C23C2×C4×D411yescharacteristic
438C23C2×#12011yescharacteristic
439C23C2×C22⋊Q811yescharacteristic
440C23#54612yes—
441C23#54712yes—
442C23C22×SD1611yescharacteristic
443C23C2×#13511yescharacteristic
444C23C2×#13711yescharacteristic
445C23C2×#13711yescharacteristic
446C23#57612yes—
447C23#57712yes—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—816no—
494C2×C4—816no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4#38514yes—
509C2×C4C22×C4⋊C411yescharacteristic
510C2×C4C2×#11711yescharacteristic
511C2×C4C2×#11711yescharacteristic
512C2×C4C2×C4×D411yescharacteristic
513C2×C4#51811yescharacteristic
514C2×C4#51911yescharacteristic
515C2×C4C2×C22⋊Q811yescharacteristic
516C2×C4C2×#12311yescharacteristic
517C2×C4#53212yes—
518C2×C4#53312yes—
519C2×C4#53712yes—
520C2×C4#54412yes—
521C2×C4#54612yes—
522C2×C4#54712yes—
523C2×C4#55112yes—
524C2×C4#55112yes—
525C2×C4#55212yes—
526C2×C4#55212yes—
527C8—24no—
528C8—42no—
529C8—82no—
530C22—24no—
531C22—24no—
532C22—24no—
533C22—24no—
534C22—24no—
535C22—28no—
536C22—24no—
537C22—28no—
538C22—28no—
539C22—28no—
540C22#186518yes—
541C22C2×#38511yescharacteristic
542C22#234211yescharacteristic
543C22#234411yescharacteristic
544C22#243012yes—
545C22#243812yes—
546C22C2×C4×SD1611yescharacteristic
547C22C2×#44111yescharacteristic
548C22#294811yescharacteristic
549C22#294911yescharacteristic
550C22C2×#44811yescharacteristic
551C22C2×#45211yescharacteristic
552C22#301212yes—
553C22C2×#47511yescharacteristic
554C22C2×#47911yescharacteristic
555C22#308112yes—
556C22#329512yes—
557C22#329712yes—
558C22#331612yes—
559C22#332012yes—
560C22#333512yes—
561C22#333712yes—
562C22#334612yes—
563C22#334612yes—
564C4—24no—
565C4—44no—
566C4—44no—
567C4—44no—
568C4—44no—
569C4—82no—
570C4—82no—
571C4—84no—
572C4—162no—
573C4#228511yescharacteristic
574C4#231611yescharacteristic
575C4#234211yescharacteristic
576C4#234411yescharacteristic
577C4#243812yes—
578C4#249812yes—
579C2—28no—
580C2#1387811yescharacteristic
581C2C2×#186511yescharacteristic
582C2#2087911yescharacteristic
583C2#2256711yescharacteristic
584C2#2257411yescharacteristic
585C2#2260911yescharacteristic
586C2#2261811yescharacteristic
587C2#2553512yes—
588C2#2554312yes—
589C2#2599212yes—
590C2#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. #11327C2semidirect
7. #11292C2semidirect
8. #10908C2semidirect
9. #10884C2semidirect
31. C2×#340C4semidirect
216. 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×#18242
C2×#18072
C22×#3402
#139072
#113272
#112922
#109082
#108842
#107302
#107142