Finite Group Databasebeta

← #411957

Subgroups — #411957

Order 512.

1320 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2C2×#2026C212yesmaximal
3C2×#1803C211yesmaximal, characteristic
4C2×#1763C211yesmaximal, characteristic
5#13942C211yesmaximal, characteristic
6#11385C212yesmaximal
7#11293C212yesmaximal
8#11067C212yesmaximal
9#10850C211yesmaximal, characteristic
10#10843C211yesmaximal, characteristic
11#10835C211yesmaximal, characteristic
12C22×#125C2212yes—
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×#97C2211yescharacteristic
19#2552—24no—
20#2552—24no—
21C2×#398—22no—
22C2×#391C2212yes—
23C2×#389—21no—
24C2×#389C2212yes—
25C2×#384—21no—
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2026—28no—
31#1803—24no—
32#1763—24no—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1441—21no—
43#1441—21no—
44C4⋊#95C2212yes—
45#1434—22no—
46C23×Q8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×#125—24no—
51C2×#125—24no—
52C2×#125—24no—
53C2×#125—24no—
54C2×#125—42no—
55C2×#125—42no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8Q814yes—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#103—22no—
85C2×#103—22no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—42no—
89C2×#103D414yes—
90C2×#103Q814yes—
91C2×#97—22no—
92C2×#97—22no—
93C2×#97—22no—
94#398—216no—
95#391—24no—
96#391—42no—
97#389—28no—
98#389—24no—
99#389—24no—
100#384—28no—
101#384—22no—
102#384D414yes—
103#382—22no—
104#382—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4D414yes—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95D412yes—
118#340—24no—
119#340—24no—
120#340—22no—
121#340—22no—
122#340—24no—
123#340—24no—
124#340—42no—
125#336—24no—
126#336—42no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8C2×D412yes—
139C22×Q8C2×Q812yes—
140C22×Q8C2×Q812yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2411yescharacteristic
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8C2×D412yes—
172C22×C8C2×Q812yes—
173#125—48no—
174#125—48no—
175#125—48no—
176#125—48no—
177C4×Q8—28no—
178C4×Q8—28no—
179C4×Q8—44no—
180C4×Q8—44no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
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—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
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—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4C2×D414yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×Q812yes—
228C2×C4⋊C4C2×Q812yes—
229C2×C4⋊C4#4112yes—
230C2×C4⋊C4#4112yes—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—44no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×Q812yes—
245C2×C42#4112yes—
246C2×C42#4112yes—
247C2×C42#4112yes—
248C4⋊C8—44no—
249#103—28no—
250#103—28no—
251#103—48no—
252#97—44no—
253#95—28no—
254#95—24no—
255#95—24no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—42no—
261#95—42no—
262C24C22×D411yescharacteristic
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8C4⋊Q814yes—
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—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4#12012yes—
346C22×C4#12012yes—
347C22×C4C22⋊Q812yes—
348C22×C4C22⋊Q812yes—
349C22×C4#12412yes—
350C22×C4#12412yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×Q811yescharacteristic
355C22×C4C22×Q811yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4H(5,𝔽2)11yescharacteristic
362C22×C4H(5,𝔽2)11yescharacteristic
363C22×C42−1+411yescharacteristic
364C22×C42−1+411yescharacteristic
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—48no—
399C4⋊C4—416no—
400C4⋊C4—416no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C4⋊C4—48no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—84no—
411C4⋊C4—88no—
412C4⋊C4#12114yes—
413C4⋊C4C22⋊Q814yes—
414C42—48no—
415C42—42no—
416C42—42no—
417C42—42no—
418C42—42no—
419C42—48no—
420C42C22⋊Q814yes—
421C42C22⋊Q814yes—
422C42#12414yes—
423C42#12414yes—
424C23C2×#12011yescharacteristic
425C23C2×C22⋊Q811yescharacteristic
426C23C2×#12411yescharacteristic
427C23D4211yescharacteristic
428C23#54611yescharacteristic
429C23#54811yescharacteristic
430C23D4×Q811yescharacteristic
431C23#57512yes—
432C23#57512yes—
433C23#57712yes—
434C23#57812yes—
435Q8—24no—
436Q8—24no—
437Q8—42no—
438Q8—42no—
439Q8—42no—
440Q8—42no—
441Q8—42no—
442Q8—42no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
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—82no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4#39314yes—
502C2×C4#39314yes—
503C2×C4C2×#12111yescharacteristic
504C2×C4C2×C22⋊Q811yescharacteristic
505C2×C4C2×C22⋊Q811yescharacteristic
506C2×C4C2×C22⋊Q811yescharacteristic
507C2×C4C2×#12411yescharacteristic
508C2×C4C2×#12411yescharacteristic
509C2×C4C2×C4⋊Q811yescharacteristic
510C2×C4#53211yescharacteristic
511C2×C4#53311yescharacteristic
512C2×C4#53311yescharacteristic
513C2×C4#53511yescharacteristic
514C2×C4#53611yescharacteristic
515C2×C4#54211yescharacteristic
516C2×C4#54211yescharacteristic
517C2×C4#54411yescharacteristic
518C2×C4#54411yescharacteristic
519C2×C4D4211yescharacteristic
520C2×C4#54611yescharacteristic
521C2×C4#54711yescharacteristic
522C2×C4#54711yescharacteristic
523C2×C4#54811yescharacteristic
524C2×C4D4×Q811yescharacteristic
525C2×C4#55411yescharacteristic
526C2×C4#55411yescharacteristic
527C2×C4#55511yescharacteristic
528C2×C4#55511yescharacteristic
529C2×C4#55611yescharacteristic
530C2×C4#55611yescharacteristic
531C8—42no—
532C8—42no—
533C8—82no—
534C22C2×#39311yescharacteristic
535C22#240811yescharacteristic
536C22#240911yescharacteristic
537C22#243911yescharacteristic
538C22#244011yescharacteristic
539C22#249111yescharacteristic
540C22#249211yescharacteristic
541C22#300511yescharacteristic
542C22#300511yescharacteristic
543C22#300911yescharacteristic
544C22#301111yescharacteristic
545C22#308012yes—
546C22#308212yes—
547C22#313712yes—
548C22#314212yes—
549C22#326912yes—
550C22#327112yes—
551C22#331512yes—
552C22#331712yes—
553C22#337612yes—
554C22#337712yes—
555C22#338812yes—
556C22#339012yes—
557C4—24no—
558C4—24no—
559C4—44no—
560C4—44no—
561C4—44no—
562C4—82no—
563C4—82no—
564C4—82no—
565C4—84no—
566C4—162no—
567C4#243911yescharacteristic
568C4#244011yescharacteristic
569C4#249111yescharacteristic
570C4#249211yescharacteristic
571C4#254711yescharacteristic
572C4#256311yescharacteristic
573C4#262811yescharacteristic
574C4#285711yescharacteristic
575C2#1512511yescharacteristic
576C2#2345311yescharacteristic
577C2#2345711yescharacteristic
578C2#2425311yescharacteristic
579C2#2425411yescharacteristic
580C2#2434311yescharacteristic
581C2#2434411yescharacteristic
582C2#2603912yes—
583C2#2605712yes—
584C2#2778112yes—
585C2#2783112yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#20262
C2×#18032
C2×#17632
#139422
#113852
#112932
#110672
#108502
#108432
#108352