Finite Group Databasebeta

← #415006

Subgroups — #415006

Order 512.

1340 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×#2565C211yesmaximal, characteristic
2C2×#1992C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5C2×#1763C211yesmaximal, characteristic
6#13910C211yesmaximal, characteristic
7#11208C212yesmaximal
8#11207C212yesmaximal
9#11067C212yesmaximal
10#11067C212yesmaximal
11#10835C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#97C2211yescharacteristic
18#2565—24no—
19#2565—24no—
20C2×#398—21no—
21C2×#398—21no—
22C2×#391C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#387—21no—
27C2×#387—21no—
28C2×#384—21no—
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1992—24no—
35#1989—24no—
36#1803—24no—
37#1763—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1444—21no—
47#1444—21no—
48C4⋊#95—21no—
49C23×Q8C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125D414yes—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—42no—
92C2×#103—42no—
93C2×#103Q818yes—
94C2×#97—22no—
95C2×#97—22no—
96C2×#97—22no—
97#398—28no—
98#398—28no—
99#391—22no—
100#391—22no—
101#391—22no—
102#391—22no—
103#389—24no—
104#389—24no—
105#389—42no—
106#389—42no—
107#387—28no—
108#387—28no—
109#384—22no—
110#384—22no—
111#384—22no—
112#384—22no—
113#384—44no—
114#382—22no—
115#382—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4D414yes—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95D411yescharacteristic
130C2×#95D411yescharacteristic
131#340—24no—
132#340—24no—
133#340—24no—
134#340—24no—
135#340—24no—
136#340—24no—
137#336—42no—
138#336—42no—
139C22×Q8—21no—
140C22×Q8—22no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8C2×D412yes—
148C22×Q8C2×Q814yes—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8C2×Q814yes—
176#125—28no—
177#125—28no—
178#125—44no—
179#125—44no—
180C4×Q8—28no—
181C4×Q8—28no—
182C4×Q8—44no—
183C4×Q8—44no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
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—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—82no—
238C2×C4⋊C4—82no—
239C2×C4⋊C4—82no—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D414yes—
244C2×C4⋊C4C2×Q814yes—
245C2×C4⋊C4#4114yes—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42#4112yes—
264C2×C42#4112yes—
265C2×C42#4112yes—
266C4⋊C8—44no—
267#103—216no—
268#103—48no—
269#97—44no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—22no—
277#95—22no—
278#95—24no—
279#95—44no—
280#95—44no—
281C24C22×D411yescharacteristic
282C2×Q8—22no—
283C2×Q8—24no—
284C2×Q8—22no—
285C2×Q8—24no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—24no—
289C2×Q8—22no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8C4⋊Q814yes—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4#12012yes—
363C22×C4#12012yes—
364C22×C4#12112yes—
365C22×C4#12112yes—
366C22×C4#12412yes—
367C22×C4#12412yes—
368C22×C4AGL(1,ℤ/8)12yes—
369C22×C4#13712yes—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×Q812yes—
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4112yes—
377C22×C4C2×#4111yescharacteristic
378C22×C4H(5,𝔽2)12yes—
379C22×C42−1+412yes—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—48no—
406C4⋊C4—48no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—88no—
421C4⋊C4—84no—
422C4⋊C4—84no—
423C4⋊C4—88no—
424C4⋊C4C22⋊Q818yes—
425C42—48no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—42no—
431C42—42no—
432C42—48no—
433C42#12114yes—
434C42#12114yes—
435C42#12314yes—
436C42#12314yes—
437C23C2×#12011yescharacteristic
438C23C2×#12111yescharacteristic
439C23C2×#12411yescharacteristic
440C23#54612yes—
441C23D4×Q812yes—
442C23C2×AGL(1,ℤ/8)11yescharacteristic
443C23C2×#13711yescharacteristic
444C23#57511yescharacteristic
445C23#57511yescharacteristic
446C23#57712yes—
447C23#57812yes—
448Q8—24no—
449Q8—24no—
450Q8—42no—
451Q8—44no—
452Q8—42no—
453Q8—44no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
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#39414yes—
523C2×C4#39414yes—
524C2×C4C2×#12111yescharacteristic
525C2×C4C2×#12111yescharacteristic
526C2×C4C2×C22⋊Q812yes—
527C2×C4C2×#12311yescharacteristic
528C2×C4C2×#12311yescharacteristic
529C2×C4C2×C4⋊Q811yescharacteristic
530C2×C4#53211yescharacteristic
531C2×C4#53312yes—
532C2×C4#53511yescharacteristic
533C2×C4#53611yescharacteristic
534C2×C4#53712yes—
535C2×C4#54212yes—
536C2×C4D4×Q812yes—
537C2×C4#55012yes—
538C2×C4#55212yes—
539C2×C4#55412yes—
540C2×C4#55412yes—
541C2×C4#55612yes—
542C8—44no—
543C8—82no—
544C22C2×#39411yescharacteristic
545C22#241712yes—
546C22#243111yescharacteristic
547C22#243411yescharacteristic
548C22#249212yes—
549C22#300611yescharacteristic
550C22#300711yescharacteristic
551C22#301011yescharacteristic
552C22#301111yescharacteristic
553C22#303611yescharacteristic
554C22#303711yescharacteristic
555C22#304612yes—
556C22#313712yes—
557C22#314111yescharacteristic
558C22#314211yescharacteristic
559C22#331012yes—
560C22#331212yes—
561C22#331512yes—
562C22#331712yes—
563C22#338612yes—
564C22#338712yes—
565C22#338812yes—
566C22#339012yes—
567C4—24no—
568C4—24no—
569C4—44no—
570C4—42no—
571C4—42no—
572C4—48no—
573C4—82no—
574C4—82no—
575C4—82no—
576C4—81no—
577C4—81no—
578C4—162no—
579C4#242312yes—
580C4#249911yescharacteristic
581C4#250211yescharacteristic
582C4#253812yes—
583C4#262011yescharacteristic
584C4#285711yescharacteristic
585C2#1518211yescharacteristic
586C2#2350811yescharacteristic
587C2#2351011yescharacteristic
588C2#2482512yes—
589C2#2482612yes—
590C2#2558311yescharacteristic
591C2#2558411yescharacteristic
592C2#2584811yescharacteristic
593C2#2584911yescharacteristic
594C2#2782712yes—
595C2#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×#25652
C2×#19922
C2×#19892
C2×#18032
C2×#17632
#139102
#112082
#112072
#110672
#108352