Finite Group Databasebeta

← #403659

Subgroups — #403659

Order 512.

1292 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#125C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic
3C2×#2083C211yesmaximal, characteristic
4C2×#2059C212yesmaximal
5#14712C211yesmaximal, characteristic
6#11208C212yesmaximal
7#11207C211yesmaximal, characteristic
8#11207C211yesmaximal, characteristic
9#10908C212yesmaximal
10#10907C212yesmaximal
11#10795C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C4×#125—24no—
16C4×#125—24no—
17C2×#389—21no—
18C2×#389—21no—
19C2×#389C2212yes—
20C2×#389C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#387—21no—
23C2×#387—21no—
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C43C2211yescharacteristic
32#2084—24no—
33#2083—24no—
34#2059—28no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44C2×#336C2212yes—
45C4⋊#95—21no—
46C4⋊#95—21no—
47C4⋊#95C2212yes—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125D414yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C42—21no—
72C22×C42—21no—
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#107—24no—
81C2×#107—24no—
82C2×#107—24no—
83C2×#106—24no—
84C2×#106—24no—
85C2×#106—24no—
86#389—22no—
87#389—22no—
88#389—24no—
89#389—24no—
90#389—44no—
91#389—44no—
92#389D414yes—
93#389D414yes—
94#387—28no—
95#387—28no—
96#384—24no—
97#384—22no—
98#384—22no—
99#382—22no—
100#382—22no—
101C4×C4⋊C4—28no—
102C4×C4⋊C4—28no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C43—22no—
117C43—22no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124#336—24no—
125#336—22no—
126#336—22no—
127#336Q818yes—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8C2×Q814yes—
154C22×C8C2×Q814yes—
155#125—28no—
156#125—28no—
157#125—28no—
158#125—28no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4#4114yes—
216C2×C4⋊C4#4112yes—
217C2×C4⋊C4#4112yes—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—41no—
240C2×C42—41no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2×Q814yes—
249C2×C42#4112yes—
250#107—48no—
251#106—48no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—44no—
259#95—44no—
260#95—44no—
261C24C22×D411yescharacteristic
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4#12012yes—
335C22×C4#12012yes—
336C22×C4#12112yes—
337C22×C4#12112yes—
338C22×C4C4⋊Q812yes—
339C22×C4C4⋊Q812yes—
340C22×C4#13512yes—
341C22×C4#13512yes—
342C22×C4AGL(1,ℤ/8)12yes—
343C22×C4AGL(1,ℤ/8)12yes—
344C22×C4#13712yes—
345C22×C4#13712yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×Q812yes—
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4112yes—
354C22×C42−1+412yes—
355C22×C42−1+412yes—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C4⋊C4—216no—
371C4⋊C4—216no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—48no—
383C4⋊C4—48no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—84no—
395C4⋊C4—88no—
396C4⋊C4—84no—
397C4⋊C4#12114yes—
398C4⋊C4#12114yes—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—24no—
404C42—24no—
405C42—24no—
406C42—24no—
407C42—44no—
408C42—44no—
409C42—48no—
410C42—48no—
411C42—44no—
412C42—44no—
413C23C2×#12011yescharacteristic
414C23C2×#12111yescharacteristic
415C23#52511yescharacteristic
416C23#52511yescharacteristic
417C23C2×C4⋊Q811yescharacteristic
418C23#53611yescharacteristic
419C23#53611yescharacteristic
420C23C2×#13511yescharacteristic
421C23C2×#13511yescharacteristic
422C23C2×AGL(1,ℤ/8)11yescharacteristic
423C23C2×AGL(1,ℤ/8)11yescharacteristic
424C23C2×#13711yescharacteristic
425C23C2×#13711yescharacteristic
426C23#57511yescharacteristic
427C23#57511yescharacteristic
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—82no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—81no—
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—82no—
509C2×C4#44914yes—
510C2×C4#45214yes—
511C2×C4#50114yes—
512C2×C4C2×#12111yescharacteristic
513C2×C4C2×#12111yescharacteristic
514C2×C4#52511yescharacteristic
515C2×C4#52511yescharacteristic
516C2×C4#53312yes—
517C2×C4#53312yes—
518C2×C4#54112yes—
519C2×C4#54112yes—
520C2×C4#54712yes—
521C2×C4#54712yes—
522C2×C4#54812yes—
523C2×C4D4×Q812yes—
524C2×C4D4×Q812yes—
525C2×C4D4×Q812yes—
526C2×C4#55212yes—
527C2×C4#55712yes—
528C8—44no—
529C8—44no—
530C22#239911yescharacteristic
531C22#240611yescharacteristic
532C22#243411yescharacteristic
533C22#253611yescharacteristic
534C22#254611yescharacteristic
535C22#263411yescharacteristic
536C22#284711yescharacteristic
537C22C2×#44911yescharacteristic
538C22C2×#45211yescharacteristic
539C22#300611yescharacteristic
540C22#300711yescharacteristic
541C22#303511yescharacteristic
542C22#303511yescharacteristic
543C22#303811yescharacteristic
544C22#303911yescharacteristic
545C22#310511yescharacteristic
546C22#310611yescharacteristic
547C22#310711yescharacteristic
548C22#310811yescharacteristic
549C22#311411yescharacteristic
550C22#311511yescharacteristic
551C22#311611yescharacteristic
552C22#311711yescharacteristic
553C22#316011yescharacteristic
554C22C2×#50111yescharacteristic
555C22#316511yescharacteristic
556C22#316611yescharacteristic
557C22#322811yescharacteristic
558C22#323111yescharacteristic
559C22#323211yescharacteristic
560C22#323211yescharacteristic
561C22#323411yescharacteristic
562C22#323611yescharacteristic
563C22#323811yescharacteristic
564C22#323911yescharacteristic
565C4—24no—
566C4—24no—
567C4—24no—
568C4—42no—
569C4—44no—
570C4—42no—
571C4—44no—
572C4—42no—
573C4—82no—
574C4—82no—
575C4—84no—
576C4—162no—
577C4#242912yes—
578C4#242912yes—
579C4#246212yes—
580C4#246212yes—
581C2#1488011yescharacteristic
582C2#2383811yescharacteristic
583C2#2384111yescharacteristic
584C2#2406611yescharacteristic
585C2#2406711yescharacteristic
586C2#2581211yescharacteristic
587C2#2581911yescharacteristic
588C2#2820711yescharacteristic
589C2#2820911yescharacteristic
590C2#2869011yescharacteristic
591C2#2869411yescharacteristic
592C2#3009811yescharacteristic
593C2#3009911yescharacteristic
594C2#3290411yescharacteristic
595C2#3290711yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ 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×C4×#1252
C2×#20842
C2×#20832
C2×#20592
#147122
#112082
#112072
#109082
#109072
#107952