Finite Group Databasebeta

← #401675

Subgroups — #401675

Order 512.

1430 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic
3C2×#1863C212yesmaximal
4C2×#1803C211yesmaximal, characteristic
5#15196C211yesmaximal, characteristic
6#11228C211yesmaximal, characteristic
7#11219C212yesmaximal
8#11207C212yesmaximal
9#11202C211yesmaximal, characteristic
10#10789C212yesmaximal
11#10789C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#107C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19Q8×C4⋊C4—24no—
20Q8×C4⋊C4—24no—
21C2×#389—21no—
22C2×#389—21no—
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2211yescharacteristic
26C2×#384—21no—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31#2083—24no—
32#1863—28no—
33#1803—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1444—21no—
43#1444—21no—
44#1441—21no—
45#1441—21no—
46C4⋊#95C2212yes—
47C4⋊#95C2211yescharacteristic
48C23×Q8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—24no—
60C2×C4×Q8—24no—
61C2×C4×Q8—42no—
62C2×C4×Q8—42no—
63C2×C4×Q8D414yes—
64C2×C4×Q8D414yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C42—21no—
83C22×C42—21no—
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C2×#107—24no—
89C2×#107—24no—
90C2×#107—24no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—42no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103Q818yes—
99#389—28no—
100#389—28no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—24no—
105#389—24no—
106#389D414yes—
107#384—28no—
108#384—22no—
109#384—22no—
110#384—22no—
111#384—22no—
112C4×C4⋊C4—28no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95D411yescharacteristic
123C2×#95D411yescharacteristic
124#340—24no—
125#340—24no—
126#340—24no—
127#340—42no—
128#340—42no—
129#336—22no—
130#336—24no—
131#336—42no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8C2×D412yes—
138C22×Q8C2×D412yes—
139C22×Q8C2×Q814yes—
140C22×Q8#4112yes—
141C22×Q8#4114yes—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8C2×Q814yes—
169C4×Q8—28no—
170C4×Q8—28no—
171C4×Q8—28no—
172C4×Q8—28no—
173C4×Q8—28no—
174C4×Q8—44no—
175C4×Q8—48no—
176C4×Q8—48no—
177C4×Q8—44no—
178C4×Q8—44no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
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—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4Q1614yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×Q814yes—
233C2×C4⋊C4#4112yes—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—44no—
250C2×C42—44no—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42#4112yes—
255#107—48no—
256C4⋊C8—44no—
257#103—216no—
258#103—48no—
259#95—24no—
260#95—24no—
261#95—22no—
262#95—22no—
263#95—42no—
264#95—44no—
265#95—42no—
266#95—82no—
267#95—82no—
268C24C22×D411yescharacteristic
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8#12114yes—
283C2×Q8C22⋊Q818yes—
284C2×Q8#12314yes—
285C2×Q8#12518yes—
286C2×Q8C4⋊Q814yes—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
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#12012yes—
345C22×C4#12012yes—
346C22×C4#12112yes—
347C22×C4#12112yes—
348C22×C4C2×Q1612yes—
349C22×C4C2×Q1612yes—
350C22×C4C2×Q1612yes—
351C22×C4#13712yes—
352C22×C4#13712yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×Q812yes—
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4112yes—
360C22×C4C2×#4111yescharacteristic
361C22×C4H(5,𝔽2)12yes—
362C22×C42−1+412yes—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C4⋊C4—216no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—28no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—44no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C4⋊C4—88no—
404C4⋊C4—84no—
405C4⋊C4—84no—
406C4⋊C4#12114yes—
407C42—216no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—24no—
415C42—48no—
416C42—48no—
417C42—41no—
418C42—41no—
419C42—42no—
420C23C2×#12011yescharacteristic
421C23C2×#12111yescharacteristic
422C23#52511yescharacteristic
423C23#54612yes—
424C23D4×Q812yes—
425C23C22×Q1611yescharacteristic
426C23C2×#13711yescharacteristic
427C23C2×#13711yescharacteristic
428C23#57511yescharacteristic
429C23#57612yes—
430C23#57712yes—
431Q8—24no—
432Q8—24no—
433Q8—28no—
434Q8—28no—
435Q8—24no—
436Q8#39818yes—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—28no—
444C2×C4—24no—
445C2×C4—28no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—44no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
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#45114yes—
502C2×C4#46214yes—
503C2×C4C2×#12111yescharacteristic
504C2×C4C2×#12111yescharacteristic
505C2×C4C2×C22⋊Q812yes—
506C2×C4C2×#12311yescharacteristic
507C2×C4#52511yescharacteristic
508C2×C4#52511yescharacteristic
509C2×C4C2×#12512yes—
510C2×C4C2×C4⋊Q811yescharacteristic
511C2×C4#53411yescharacteristic
512C2×C4#53611yescharacteristic
513C2×C4#53712yes—
514C2×C4#53912yes—
515C2×C4#54612yes—
516C2×C4D4×Q812yes—
517C2×C4#55212yes—
518C2×C4#55212yes—
519C2×C4#55412yes—
520C2×C4#55412yes—
521C8—44no—
522C8—82no—
523C22#240511yescharacteristic
524C22#241712yes—
525C22#243411yescharacteristic
526C22#247312yes—
527C22#284711yescharacteristic
528C22C2×#45111yescharacteristic
529C22#300711yescharacteristic
530C22#300811yescharacteristic
531C22#301011yescharacteristic
532C22C2×#46211yescharacteristic
533C22#303711yescharacteristic
534C22#304512yes—
535C22#310411yescharacteristic
536C22#311311yescharacteristic
537C22#311912yes—
538C22#329912yes—
539C22#330912yes—
540C22#332012yes—
541C22#333312yes—
542C22Q8×Q1612yes—
543C22#338512yes—
544C22#338712yes—
545C22#338812yes—
546C4—24no—
547C4—28no—
548C4—24no—
549C4—42no—
550C4—44no—
551C4—42no—
552C4—42no—
553C4—81no—
554C4—82no—
555C4—82no—
556C4—84no—
557C4—162no—
558C4C2×#39811yescharacteristic
559C4#241712yes—
560C4#247312yes—
561C4#248411yescharacteristic
562C4#258311yescharacteristic
563C4#261811yescharacteristic
564C2#1482111yescharacteristic
565C2#2404211yescharacteristic
566C2#2404611yescharacteristic
567C2#2481712yes—
568C2#2482412yes—
569C2#2584111yescharacteristic
570C2#2584811yescharacteristic
571C2#2730012yes—
572C2#2731312yes—
573C2#3290611yescharacteristic
574C2#3291311yescharacteristic
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×Q8×C4⋊C42
C2×#20832
C2×#18632
C2×#18032
#151962
#112282
#112192
#112072
#112022
#107892