Finite Group Databasebeta

← #412747

Subgroups — #412747

Order 512.

1360 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2068C212yesmaximal, complemented
2C2×#2067C212yesmaximal
3C2×#1983C211yesmaximal, characteristic
4C2×#1983C211yesmaximal, characteristic
5C2×#1947C211yesmaximal, characteristic, complemented
6C2×#1843C211yesmaximal, characteristic, complemented
7#14273C211yesmaximal, characteristic
8#13783C211yesmaximal, characteristic, complemented
9#11077C212yesmaximal, complemented
10#11076C212yesmaximal
11#10849C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×#107C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C4⋊C8C2212yes—
17C22×C4⋊C8C2211yescharacteristic
18C22×#103C2211yescharacteristic
19C22×#103C2212yes—
20C22×#103C2211yescharacteristic
21C22×#102C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C2×#391C2212yes—
25C2×#390C2212yes—
26C2×#389C2212yes—
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×#376C2211yescharacteristic
31#2068—28no—
32#2067—28no—
33#1983—24no—
34#1983—24no—
35#1947—24no—
36#1843—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40#1439—21no—
41#1439—21no—
42#1439—22no—
43#1436—22no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46#1434—21no—
47C23×Q8C2311yescharacteristic
48C23×D4C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
52C2×#124—22no—
53C2×#124—22no—
54C2×#124—22no—
55C2×#124—22no—
56C2×#124—22no—
57C2×#124Q814yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C42—21no—
70C22×C42—21no—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73C2×#107—24no—
74C2×#107—24no—
75C2×#107—24no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—42no—
86C2×C4⋊C8—42no—
87C2×C4⋊C8D414yes—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—22no—
92C2×#103—22no—
93C2×#103—42no—
94C2×#103—42no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—42no—
101C2×#102—42no—
102#391—24no—
103#391—24no—
104#390—24no—
105#390—24no—
106#389—24no—
107#389D418yes—
108#384—22no—
109#384—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4D414yes—
114#376—22no—
115#376—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95D411yescharacteristic
124C2×#95D411yescharacteristic
125C2×#95D412yes—
126#340—24no—
127#340—24no—
128#340—24no—
129C25—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8C2×Q812yes—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4C2×Q812yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2411yescharacteristic
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×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8C2×D414yes—
177#124—28no—
178#124—28no—
179#124—44no—
180#124—44no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
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—28no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—81no—
204C2×C4⋊C4—81no—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D414yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D414yes—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—82no—
218C2×C22⋊C4—82no—
219C2×C22⋊C4—82no—
220C2×C22⋊C4—82no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—24no—
226C2×C42—42no—
227C2×C42—82no—
228C2×C42—81no—
229C2×C42—81no—
230C2×C42C2×D414yes—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×Q812yes—
234C2×C42#4114yes—
235C2×C42#4112yes—
236C2×C42#4112yes—
237C2×C42#4112yes—
238#107—48no—
239C4⋊C8—28no—
240C4⋊C8—48no—
241C4⋊C8—48no—
242C4⋊C8—44no—
243#103—48no—
244#103—44no—
245#103—44no—
246#102—48no—
247#102—44no—
248#102—44no—
249#95—24no—
250#95—28no—
251#95—22no—
252#95—22no—
253#95—24no—
254#95—42no—
255#95—44no—
256#95—44no—
257#95—82no—
258#95—82no—
259C24—22no—
260C24—41no—
261C24—41no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24—42no—
269C24—41no—
270C24C22×D411yescharacteristic
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—42no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4#12114yes—
355C22×C4#12114yes—
356C22×C4#12112yes—
357C22×C4#12112yes—
358C22×C4#12112yes—
359C22×C4#12112yes—
360C22×C4C22⋊Q812yes—
361C22×C4C22⋊Q812yes—
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D412yes—
365C22×C4C22×Q811yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4112yes—
370C22×C4H(5,𝔽2)12yes—
371C22×C42−1+412yes—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—88no—
403C4⋊C4—84no—
404C4⋊C4—84no—
405C4⋊C4#12714yes—
406C22⋊C4—48no—
407C22⋊C4—48no—
408C22⋊C4—88no—
409C22⋊C4—88no—
410C42—24no—
411C42—24no—
412C42—24no—
413C42—24no—
414C42—84no—
415C42—84no—
416C42#12114yes—
417C42C22⋊Q814yes—
418C42C22⋊Q814yes—
419C42#12614yes—
420C23—44no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—44no—
425C23—81no—
426C23—81no—
427C23—82no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23—81no—
434C23—81no—
435C23—82no—
436C23—82no—
437C23—81no—
438C23—81no—
439C23—82no—
440C23—82no—
441C23—81no—
442C23—81no—
443C23C2×#12111yescharacteristic
444C23C2×#12112yes—
445C23C2×#12111yescharacteristic
446C23C2×C22⋊Q811yescharacteristic
447C23#53712yes—
448C23#57612yes—
449C23#57611yescharacteristic
450C23#57711yescharacteristic
451C23#57711yescharacteristic
452C23#57812yes—
453C23#57811yescharacteristic
454Q8—42no—
455Q8—44no—
456Q8—42no—
457Q8—82no—
458Q8—81no—
459Q8—81no—
460D4—42no—
461D4—44no—
462D4—42no—
463D4—82no—
464D4—81no—
465D4—81no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4#39614yes—
520C2×C4#39614yes—
521C2×C4C2×#12111yescharacteristic
522C2×C4C2×C22⋊Q811yescharacteristic
523C2×C4C2×C22⋊Q811yescharacteristic
524C2×C4C2×#12611yescharacteristic
525C2×C4C2×#12711yescharacteristic
526C2×C4#53211yescharacteristic
527C2×C4#53212yes—
528C2×C4#53511yescharacteristic
529C2×C4#53611yescharacteristic
530C2×C4#54212yes—
531C2×C4D4212yes—
532C2×C4#54712yes—
533C2×C4#54812yes—
534C2×C4D4×Q812yes—
535C2×C4#55012yes—
536C2×C4#55312yes—
537C2×C4#55412yes—
538C2×C4#55612yes—
539C8—44no—
540C8—81no—
541C8—81no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—84no—
546C22—84no—
547C22—161no—
548C22—161no—
549C22—161no—
550C22—161no—
551C22—162no—
552C22—162no—
553C22C2×#39611yescharacteristic
554C22#241612yes—
555C22#253812yes—
556C22#262011yescharacteristic
557C22#285611yescharacteristic
558C22#304111yescharacteristic
559C22#304112yes—
560C22#304411yescharacteristic
561C22#304412yes—
562C22#304511yescharacteristic
563C22#304512yes—
564C22#304611yescharacteristic
565C22#304612yes—
566C22#306112yes—
567C22#306211yescharacteristic
568C22#306311yescharacteristic
569C22#308312yes—
570C22#308311yescharacteristic
571C22#308411yescharacteristic
572C22#325712yes—
573C22#325812yes—
574C22#326212yes—
575C22#326212yes—
576C4—24no—
577C4—24no—
578C4—44no—
579C4—44no—
580C4—44no—
581C4—82no—
582C4—82no—
583C4—84no—
584C4—162no—
585C4#245212yes—
586C4#251212yes—
587C4#254111yescharacteristic
588C4#254211yescharacteristic
589C4#254511yescharacteristic
590C4#256611yescharacteristic
591C2—162no—
592C2#1513411yescharacteristic
593C2#2361111yescharacteristic
594C2#2361611yescharacteristic
595C2#2476312yes—
596C2#2476512yes—
597C2#2828812yes—
598C2#2829312yes—
599C2#2983911yescharacteristic
600C2#2984011yescharacteristic
601C2#3316611yescharacteristic
602C2#3316911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#2068C2semidirect
5. C2×#1947C2semidirect
6. C2×#1843C2semidirect
8. #13783C2semidirect
9. #11077C2semidirect
11. #10849C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20682
C2×#20672
C2×#19832
C2×#19472
C2×#18432
#142732
#137832
#110772
#110762
#108492