Finite Group Databasebeta

← #356140

Subgroups — #356140

Order 512.

1914 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×C8×C22⋊C4C211yesmaximal, characteristic
4C22×#340C211yesmaximal, characteristic
5C22⋊C4×C4⋊C4C211yesmaximal, characteristic
6#10884C212yesmaximal, complemented
7#10880C212yesmaximal, complemented
8#10806C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10#10721C211yesmaximal, characteristic
11C4×#340C212yesmaximal, complemented
12C24×C8C2211yescharacteristic
13C22×#117—21no—
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×C4×C8C2212yes—
18#2344—24no—
19#2344C418yescomplemented
20C2×#389C2212yes—
21C2×#387—22no—
22C2×#381—21no—
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C22⋊C4—21no—
28C2×C4×C22⋊C4C2211yescharacteristic
29#1823C418yescomplemented
30C8×C22⋊C4—24no—
31C2×#340C418yescomplemented
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43#1436—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—42no—
69C22×C4⋊C4C2×C414yes—
70C22×C4⋊C4C2×C414yes—
71C22×C4⋊C4C2×C414yes—
72C22×C4⋊C4C2×C414yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4D411yescharacteristic
82C22×C22⋊C4D411yescharacteristic
83C22×C42—22no—
84C22×C42—22no—
85C22×C42C2312yes—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8C2×C414yes—
89C2×C4×C8—24no—
90C2×C4×C8—24no—
91C2×C4×C8—42no—
92#389—24no—
93#389C2×C418yescomplemented
94#387—216no—
95#381—28no—
96#380—22no—
97#380—22no—
98#380C2×C414yes—
99#380C2×C414yes—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—42no—
103C4×C4⋊C4C2×C418yescomplemented
104C4×C22⋊C4—44no—
105C4×C22⋊C4C2×C414yes—
106C4×C22⋊C4D414yes—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95D412yes—
115#340—24no—
116#340—24no—
117#340—42no—
118#340—42no—
119#340C2×C414yes—
120#340C2×C414yes—
121#340C2×C414yes—
122#340C2×C414yes—
123#340C2×C418yescomplemented
124#340C2×C418yescomplemented
125#336—24no—
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—22no—
139C23×C4—21no—
140C23×C4—24no—
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×C4C22×C412yes—
151C23×C4C22×C412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
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—22no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—28no—
174C22×C8—44no—
175C22×C8—48no—
176C22×C8—48no—
177C22×C8C22×C412yes—
178C22×C8C22×C414yes—
179C22×C8C22×C412yes—
180C22×C8#4114yes—
181#117—432no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
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—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4C42116yescomplemented
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C414yes—
220C2×C4⋊C4C22×C414yes—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4#4114yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4C22⋊C414yes—
238C2×C22⋊C4C22⋊C414yes—
239C2×C22⋊C4D814yes—
240C2×C22⋊C4SD1614yes—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—28no—
246C2×C42—44no—
247C2×C42—44no—
248C2×C42—44no—
249C2×C42—44no—
250C2×C42—82no—
251C2×C42C22×C414yes—
252C2×C42C2×D414yes—
253C22⋊C8—28no—
254C4×C8—48no—
255#95—28no—
256#95—24no—
257#95—28no—
258#95—24no—
259#95—44no—
260#95—44no—
261#95—82no—
262#95C22⋊C418yes—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24C2×C22⋊C412yes—
268C24C4×D412yes—
269C24C4×D412yes—
270C24C2×D812yes—
271C24C2×SD1612yes—
272C24#13512yes—
273C24#13512yes—
274C24C22×D411yescharacteristic
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
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—216no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—48no—
314C22×C4—41no—
315C22×C4—41no—
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—42no—
325C22×C4—48no—
326C22×C4—48no—
327C22×C4—48no—
328C22×C4—48no—
329C22×C4—48no—
330C22×C4—48no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—48no—
348C22×C4—84no—
349C22×C4—88no—
350C22×C4—88no—
351C22×C4C2×C4214yes—
352C22×C4C2×C22⋊C414yes—
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C414yes—
355C22×C4#11714yes—
356C22×C4C4×D414yes—
357C22×C4C4×D414yes—
358C22×C4C4×D412yes—
359C22×C4C4×D412yes—
360C22×C4C2×D814yes—
361C22×C4C2×SD1614yes—
362C22×C4C23×C411yescharacteristic
363C22×C4C23×C411yescharacteristic
364C22×C4C23×C411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4112yes—
369C22×C4C2×#4112yes—
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—24no—
378C2×C8—22no—
379C2×C8—24no—
380C2×C8—22no—
381C2×C8—24no—
382C2×C8—416no—
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—24no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—48no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—416no—
412C4⋊C4C2×C4214yes—
413C4⋊C4C2×C4214yes—
414C4⋊C4#11714yescomplemented
415C22⋊C4—28no—
416C22⋊C4—48no—
417C22⋊C4—48no—
418C22⋊C4#102116yes—
419C42—24no—
420C42—24no—
421C42—24no—
422C42—24no—
423C42—416no—
424C42—88no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—22no—
431C23—22no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—22no—
436C23—22no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23C4×C22⋊C414yes—
442C23C2×#10214yes—
443C23#41614yes—
444C23C4×D814yes—
445C23C4×D814yes—
446C23C4×SD1614yes—
447C23C4×SD1614yes—
448C23C22×C22⋊C411yescharacteristic
449C23C2×C4×D411yescharacteristic
450C23C2×C4×D411yescharacteristic
451C23#54612yes—
452C23#54712yes—
453C23C22×D811yescharacteristic
454C23C22×SD1611yescharacteristic
455C23C2×#13511yescharacteristic
456C23C2×#13511yescharacteristic
457C23#57712yes—
458C23#57812yes—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—28no—
471C2×C4—24no—
472C2×C4—28no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
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—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—432no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—816no—
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×C4C4×C22⋊C414yes—
523C2×C4C2×#10218yes—
524C2×C4C22×C4211yescharacteristic
525C2×C4C2×#11711yescharacteristic
526C2×C4C2×#11711yescharacteristic
527C2×C4C2×#11711yescharacteristic
528C2×C4C2×C4×D411yescharacteristic
529C2×C4C2×C4×D411yescharacteristic
530C2×C4C4×#4112yes—
531C2×C4C4×#4112yes—
532C2×C4#51811yescharacteristic
533C2×C4#51911yescharacteristic
534C2×C4#52012yes—
535C2×C4#52012yes—
536C2×C4#54612yes—
537C2×C4#54712yes—
538C2×C4#55112yes—
539C2×C4#55112yes—
540C2×C4#55212yes—
541C2×C4#55212yes—
542C8—24no—
543C8—42no—
544C8—44no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—24no—
549C22—28no—
550C22—28no—
551C22—24no—
552C22—28no—
553C22—28no—
554C22—24no—
555C22C4×#10218yes—
556C22C2×C4×C22⋊C411yescharacteristic
557C22#233912yes—
558C22#234211yescharacteristic
559C22#234411yescharacteristic
560C22#235912yes—
561C22C22×#10211yescharacteristic
562C22C2×#41611yescharacteristic
563C22#289812yes—
564C22C2×C4×D811yescharacteristic
565C22C2×C4×D811yescharacteristic
566C22C2×C4×SD1611yescharacteristic
567C22C2×C4×SD1611yescharacteristic
568C22#294712yes—
569C22#294911yescharacteristic
570C22#295011yescharacteristic
571C22#329412yes—
572C22#329612yes—
573C22#329812yes—
574C22#330012yes—
575C22#333412yes—
576C22#333512yes—
577C22#333612yes—
578C22#333712yes—
579C4—28no—
580C4—24no—
581C4—44no—
582C4—44no—
583C4—44no—
584C4—44no—
585C4—82no—
586C4—84no—
587C4—84no—
588C4C4×#11711yescharacteristic
589C4#227411yescharacteristic
590C4#234211yescharacteristic
591C4#234312yes—
592C4#234411yescharacteristic
593C4#235912yes—
594C2—28no—
595C2#1379911yescharacteristic
596C2C2×C4×#10211yescharacteristic
597C2#2041711yescharacteristic
598C2#2248512yes—
599C2#2248612yes—
600C2#2257011yescharacteristic
601C2#2257111yescharacteristic
602C2#2260911yescharacteristic
603C2#2261111yescharacteristic
604C2#2291612yes—
605C2#2292012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C2×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. #10884C2semidirect
7. #10880C2semidirect
8. #10806C2semidirect
11. C4×#340C2semidirect
19. #2344C4semidirect
29. #1823C4semidirect
31. C2×#340C4semidirect
93. #389C2×C4semidirect
103. C4×C4⋊C4C2×C4semidirect
123. #340C2×C4semidirect
124. #340C2×C4semidirect
217. C2×C4⋊C4C42semidirect
389. C2×C8#117semidirect
414. 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×C8×C22⋊C42
C22×#3402
C22⋊C4×C4⋊C42
#108842
#108802
#108062
#107302
#107212
C4×#3402