Finite Group Databasebeta

← #404541

Subgroups — #404541

Order 512.

1326 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2295C211yesmaximal, characteristic
2C2×#2100C211yesmaximal, characteristic
3C2×#2095C211yesmaximal, characteristic
4#14712C211yesmaximal, characteristic
5#11328C212yesmaximal
6#11207C211yesmaximal, characteristic
7#11067C212yesmaximal
8#11050C211yesmaximal, characteristic
9#10907C212yesmaximal
10#10795C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4×Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16#2295—24no—
17#2295—24no—
18C2×#391C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#389—21no—
21C2×#389—21no—
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#387—21no—
26C2×#387—21no—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×#376—21no—
32C2×C43C2211yescharacteristic
33#2100—24no—
34#2095—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43C2×#336C2211yescharacteristic
44C4⋊#95—21no—
45C4⋊#95—21no—
46C4⋊#95C2212yes—
47C23×Q8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8Q814yes—
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—22no—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C42—21no—
70C22×C42—21no—
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#107—24no—
79C2×#107—24no—
80C2×#107—42no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—42no—
86C2×#103—42no—
87#391—22no—
88#391—22no—
89#391—22no—
90#391D414yes—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—42no—
95#389—42no—
96#389—44no—
97#389—44no—
98#389D414yes—
99#387—28no—
100#387—28no—
101#384—24no—
102#384—22no—
103#384—22no—
104C4×C4⋊C4—28no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107#376—28no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C43—22no—
119C43—22no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—42no—
125#336—24no—
126#336—22no—
127#336—22no—
128#336—24no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8C2×D412yes—
137C22×Q8C2×Q812yes—
138C22×Q8#4112yes—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2411yescharacteristic
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
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#4114yes—
166C4×Q8—28no—
167C4×Q8—28no—
168C4×Q8—28no—
169C4×Q8—28no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×Q812yes—
218C2×C4⋊C4#4114yes—
219C2×C4⋊C4#4112yes—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—22no—
223C2×C42—24no—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—24no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—41no—
244C2×C42—41no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42Q1614yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×Q812yes—
251C2×C42#4112yes—
252#107—48no—
253#103—48no—
254#103—48no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—44no—
263#95—44no—
264#95—44no—
265C24C22×D411yescharacteristic
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8C22⋊Q814yes—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
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—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4#12012yes—
355C22×C4#12012yes—
356C22×C4C22⋊Q812yes—
357C22×C4C22⋊Q812yes—
358C22×C4C2×Q1612yes—
359C22×C4C2×Q1612yes—
360C22×C4C2×Q1612yes—
361C22×C4#13512yes—
362C22×C4AGL(1,ℤ/8)12yes—
363C22×C4AGL(1,ℤ/8)12yes—
364C22×C4#13712yes—
365C22×C4#13712yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×Q811yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4112yes—
373C22×C4C2×#4112yes—
374C22×C4H(5,𝔽2)12yes—
375C22×C42−1+412yes—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C4⋊C4—216no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—48no—
399C4⋊C4—48no—
400C4⋊C4—48no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—48no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—84no—
411C4⋊C4—84no—
412C4⋊C4—88no—
413C4⋊C4C22⋊Q814yes—
414C42—216no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—24no—
420C42—24no—
421C42—28no—
422C42—44no—
423C42—44no—
424C42—48no—
425C42—42no—
426C42—42no—
427C42—44no—
428C23C2×#12011yescharacteristic
429C23C2×C22⋊Q811yescharacteristic
430C23#52511yescharacteristic
431C23#52511yescharacteristic
432C23#53211yescharacteristic
433C23#53511yescharacteristic
434C23#53611yescharacteristic
435C23C22×Q1611yescharacteristic
436C23C2×#13511yescharacteristic
437C23C2×AGL(1,ℤ/8)11yescharacteristic
438C23C2×AGL(1,ℤ/8)11yescharacteristic
439C23C2×#13711yescharacteristic
440C23C2×#13711yescharacteristic
441C23#57511yescharacteristic
442C23#57511yescharacteristic
443Q8—28no—
444Q8—44no—
445Q8—44no—
446Q8—44no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
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—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4#45114yes—
529C2×C4#45214yes—
530C2×C4#47714yes—
531C2×C4C2×C22⋊Q811yescharacteristic
532C2×C4C2×C22⋊Q811yescharacteristic
533C2×C4#52511yescharacteristic
534C2×C4#52511yescharacteristic
535C2×C4#52912yes—
536C2×C4#53312yes—
537C2×C4#54212yes—
538C2×C4#54412yes—
539C2×C4#54612yes—
540C2×C4#54712yes—
541C2×C4#54712yes—
542C2×C4#55012yes—
543C2×C4#55112yes—
544C2×C4#55212yes—
545C2×C4#55212yes—
546C2×C4#55312yes—
547C8—44no—
548C8—82no—
549C22#239611yescharacteristic
550C22#240611yescharacteristic
551C22#243111yescharacteristic
552C22#259811yescharacteristic
553C22#261111yescharacteristic
554C22#262811yescharacteristic
555C22#284811yescharacteristic
556C22C2×#45111yescharacteristic
557C22C2×#45211yescharacteristic
558C22#300611yescharacteristic
559C22#300611yescharacteristic
560C22C2×#47711yescharacteristic
561C22#307611yescharacteristic
562C22#307811yescharacteristic
563C22#307911yescharacteristic
564C22#310411yescharacteristic
565C22#310411yescharacteristic
566C22#310511yescharacteristic
567C22#310611yescharacteristic
568C22#311011yescharacteristic
569C22#311211yescharacteristic
570C22#311611yescharacteristic
571C22#311711yescharacteristic
572C22#317011yescharacteristic
573C22#317111yescharacteristic
574C22#318011yescharacteristic
575C22#318511yescharacteristic
576C22#320911yescharacteristic
577C22#321011yescharacteristic
578C22#321711yescharacteristic
579C22#321911yescharacteristic
580C22#323011yescharacteristic
581C22#323211yescharacteristic
582C22#323611yescharacteristic
583C22#323911yescharacteristic
584C4—24no—
585C4—24no—
586C4—24no—
587C4—42no—
588C4—42no—
589C4—44no—
590C4—42no—
591C4—44no—
592C4—82no—
593C4—82no—
594C4—84no—
595C4—162no—
596C4#243812yes—
597C4#244112yes—
598C4#247012yes—
599C4#247912yes—
600C2#1489311yescharacteristic
601C2#2377711yescharacteristic
602C2#2378211yescharacteristic
603C2#2406411yescharacteristic
604C2#2406711yescharacteristic
605C2#2560011yescharacteristic
606C2#2560511yescharacteristic
607C2#2936711yescharacteristic
608C2#2937011yescharacteristic
609C2#2957311yescharacteristic
610C2#2957911yescharacteristic
611C2#2998411yescharacteristic
612C2#2998911yescharacteristic
613C2#3295711yescharacteristic
614C2#3295811yescharacteristic
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×#22952
C2×#21002
C2×#20952
#147122
#113282
#112072
#110672
#110502
#109072
#107952
#107892