Finite Group Databasebeta

← #360621

Subgroups — #360621

Order 512.

1780 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2344C211yesmaximal, characteristic
2C2×#1823C211yesmaximal, characteristic
3C2×#1808C211yesmaximal, characteristic
4C22×#340C211yesmaximal, characteristic
5#13907C211yesmaximal, characteristic
6#11328C212yesmaximal, complemented
7#11292C212yesmaximal, complemented
8#10907C212yesmaximal, complemented
9#10884C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×#117—21no—
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#107C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18#2344—24no—
19#2344—24no—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#387—22no—
23C2×#381—21no—
24C2×#380—21no—
25C2×#380C2211yescharacteristic
26C2×#380C2211yescharacteristic
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29#1823—24no—
30#1808—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
57C2×#117—28no—
58C2×#117—44no—
59C2×#117—44no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—42no—
70C22×C4⋊C4C2×C414yes—
71C22×C4⋊C4C2×C414yes—
72C22×C4⋊C4D414yes—
73C22×C4⋊C4Q814yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4D411yescharacteristic
83C22×C22⋊C4D411yescharacteristic
84C22×C42—22no—
85C22×C42C2312yes—
86C2×#107—24no—
87C2×#107—24no—
88C2×#107—42no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92#389—24no—
93#389—24no—
94#389—24no—
95#389—42no—
96#387—216no—
97#381—28no—
98#380—22no—
99#380—22no—
100#380—22no—
101#380—44no—
102#380D414yes—
103C4×C4⋊C4—24no—
104C4×C4⋊C4—24no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95D412yes—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—42no—
120#340C2×C414yes—
121#340C2×C414yes—
122#340C2×C414yes—
123#340C2×C414yes—
124#336—24no—
125#336—42no—
126C25C2×D411yescharacteristic
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—44no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—82no—
149C23×C4C22×C412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×Q812yes—
156C23×C4#4112yes—
157C23×C4#4112yes—
158C23×C4C2411yescharacteristic
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—28no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—22no—
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—44no—
176C22×C8—48no—
177C22×C8—48no—
178C22×C8C22×C412yes—
179C22×C8C22×C412yes—
180C22×C8#4114yes—
181#117—432no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
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—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
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—22no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4C4⋊C4116yescomplemented
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C2×D414yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×Q812yes—
230C2×C4⋊C4C2×Q812yes—
231C2×C4⋊C4#4114yes—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4D814yes—
244C2×C42—28no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—44no—
252C2×C42—44no—
253#107—48no—
254C22⋊C8—44no—
255#95—28no—
256#95—28no—
257#95—24no—
258#95—24no—
259#95—44no—
260#95—42no—
261#95—44no—
262#95—82no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24C4×D412yes—
268C24#12012yes—
269C24C22⋊Q812yes—
270C24C2×D812yes—
271C24#13512yes—
272C24AGL(1,ℤ/8)12yes—
273C24#13712yes—
274C24C22×D411yescharacteristic
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—216no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—48no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—48no—
328C22×C4—48no—
329C22×C4—48no—
330C22×C4—48no—
331C22×C4—48no—
332C22×C4—48no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—48no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—84no—
352C22×C4—88no—
353C22×C4—88no—
354C22×C4C2×C4⋊C414yes—
355C22×C4C4×D414yes—
356C22×C4C4×D412yes—
357C22×C4#12012yes—
358C22×C4C22⋊Q814yes—
359C22×C4C22⋊Q812yes—
360C22×C4#12314yes—
361C22×C4C2×D814yes—
362C22×C4C23×C411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×Q811yescharacteristic
366C22×C4C2×#4112yes—
367C22×C4C2×#4112yes—
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4H(5,𝔽2)12yes—
371C22×C42−1+412yes—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—416no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8#11714yescomplemented
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—216no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—48no—
404C4⋊C4—416no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—88no—
413C4⋊C4C2×C4⋊C414yes—
414C4⋊C4C2×C4⋊C414yes—
415C4⋊C4#11714yescomplemented
416C22⋊C4—28no—
417C22⋊C4—48no—
418C22⋊C4—44no—
419C22⋊C4—84no—
420C42—416no—
421C42—42no—
422C42—42no—
423C42—42no—
424C42—42no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—22no—
431C23—22no—
432C23—24no—
433C23—22no—
434C23—24no—
435C23—22no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23#38514yes—
442C23C4×D814yes—
443C23#44014yes—
444C23#44714yes—
445C23#45214yes—
446C23#47414yes—
447C23#47814yes—
448C23C2×C4×D411yescharacteristic
449C23C2×#12011yescharacteristic
450C23C2×C22⋊Q811yescharacteristic
451C23#54612yes—
452C23#54712yes—
453C23C22×D811yescharacteristic
454C23C2×#13511yescharacteristic
455C23C2×AGL(1,ℤ/8)11yescharacteristic
456C23C2×#13711yescharacteristic
457C23#57712yes—
458C23#57812yes—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—28no—
467C2×C4—28no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—432no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—816no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4#38514yes—
524C2×C4C22×C4⋊C411yescharacteristic
525C2×C4C2×#11711yescharacteristic
526C2×C4C2×#11711yescharacteristic
527C2×C4C2×C4×D411yescharacteristic
528C2×C4#51811yescharacteristic
529C2×C4#51911yescharacteristic
530C2×C4C2×C22⋊Q811yescharacteristic
531C2×C4C2×#12311yescharacteristic
532C2×C4#53212yes—
533C2×C4#53312yes—
534C2×C4#53712yes—
535C2×C4#54412yes—
536C2×C4#54612yes—
537C2×C4#54712yes—
538C2×C4#55112yes—
539C2×C4#55112yes—
540C2×C4#55212yes—
541C2×C4#55212yes—
542C8—24no—
543C8—42no—
544C8—82no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—24no—
549C22—24no—
550C22—28no—
551C22—28no—
552C22—24no—
553C22—28no—
554C22—28no—
555C22#186218yes—
556C22C2×#38511yescharacteristic
557C22#234211yescharacteristic
558C22#234411yescharacteristic
559C22#243012yes—
560C22#243812yes—
561C22C2×C4×D811yescharacteristic
562C22C2×#44011yescharacteristic
563C22#294911yescharacteristic
564C22#295011yescharacteristic
565C22C2×#44711yescharacteristic
566C22C2×#45211yescharacteristic
567C22#301312yes—
568C22C2×#47411yescharacteristic
569C22C2×#47811yescharacteristic
570C22#308212yes—
571C22#329412yes—
572C22#329612yes—
573C22#330812yes—
574C22#331212yes—
575C22#333412yes—
576C22#333612yes—
577C22#334412yes—
578C22#334512yes—
579C4—24no—
580C4—44no—
581C4—44no—
582C4—44no—
583C4—44no—
584C4—82no—
585C4—84no—
586C4—82no—
587C4—162no—
588C4#228511yescharacteristic
589C4#231611yescharacteristic
590C4#234211yescharacteristic
591C4#234411yescharacteristic
592C4#243812yes—
593C4#249812yes—
594C2—28no—
595C2#1387811yescharacteristic
596C2C2×#186211yescharacteristic
597C2#2088011yescharacteristic
598C2#2256311yescharacteristic
599C2#2257011yescharacteristic
600C2#2261111yescharacteristic
601C2#2261511yescharacteristic
602C2#2553312yes—
603C2#2554112yes—
604C2#2598812yes—
605C2#2599612yes—
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. #11292C2semidirect
8. #10907C2semidirect
9. #10884C2semidirect
31. C2×#340C4semidirect
221. C2×C4⋊C4C4⋊C4semidirect
389. C2×C8#117semidirect
415. C4⋊C4#117semidirect

Maximal subgroups

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

subgroupindex
C2×#23442
C2×#18232
C2×#18082
C22×#3402
#139072
#113282
#112922
#109072
#108842
#107302
#107142