Finite Group Databasebeta

← #412745

Subgroups — #412745

Order 512.

1522 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2067C212yesmaximal, complemented
2C2×#2065C212yesmaximal, complemented
3C2×#1982C211yesmaximal, characteristic
4C2×#1982C211yesmaximal, characteristic
5C2×#1949C211yesmaximal, characteristic, complemented
6C2×#1843C211yesmaximal, characteristic, complemented
7#14272C211yesmaximal, characteristic
8#13783C211yesmaximal, characteristic, complemented
9#11076C212yesmaximal, complemented
10#11076C212yesmaximal, complemented
11#10850C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×C4⋊C8C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×#102C2212yes—
22C22×#102C2212yes—
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#389C2212yes—
26C2×#382C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#2067—28no—
31#2065—28no—
32#1982—24no—
33#1982—24no—
34#1949—24no—
35#1843—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40#1438—21no—
41#1438—21no—
42#1438—22no—
43#1436—22no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46#1434—21no—
47C23×D4—21no—
48C23×D4—21no—
49C23×D4C2312yes—
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×#127—22no—
54C2×#127—22no—
55C2×#127—22no—
56C2×#127—22no—
57C2×#127—22no—
58C2×#127Q814yes—
59C22×C4⋊C4—22no—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C42—21no—
70C22×C42—21no—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73C2×#107—24no—
74C2×#107—22no—
75C2×#106—24no—
76C2×#106—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—24no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—42no—
87C2×C4⋊C8—42no—
88C2×C4⋊C8D414yes—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—42no—
98C2×#102—42no—
99C2×#102—42no—
100C2×#102—42no—
101#390—24no—
102#390—24no—
103#390—24no—
104#390—24no—
105#389—24no—
106#389D418yescomplemented
107#382—22no—
108#382—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4D414yes—
113#376—22no—
114#376—22no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95D411yescharacteristic
123C2×#95D411yescharacteristic
124C2×#95D412yes—
125#340—24no—
126#340—24no—
127#340—22no—
128#340—22no—
129C25—22no—
130C22×D4—24no—
131C22×D4—24no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—82no—
150C22×D4—81no—
151C22×D4—81no—
152C22×D4—82no—
153C22×D4—81no—
154C22×D4—81no—
155C22×D4C2×Q814yes—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2411yescharacteristic
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8C2×D412yes—
182C22×C8C2×D412yes—
183#127—28no—
184#127—28no—
185#127—44no—
186#127—44no—
187C2×C4⋊C4—21no—
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—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D414yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D414yes—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—82no—
217C2×C22⋊C4—82no—
218C2×C22⋊C4—82no—
219C2×C22⋊C4—82no—
220C2×C22⋊C4—82no—
221C2×C22⋊C4—82no—
222C2×C22⋊C4—82no—
223C2×C22⋊C4—82no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—42no—
230C2×C42—82no—
231C2×C42—81no—
232C2×C42—81no—
233C2×C42C2×D414yes—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×Q812yes—
237C2×C42#4114yes—
238C2×C42#4112yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241#107—44no—
242#106—44no—
243C4⋊C8—28no—
244C4⋊C8—48no—
245C4⋊C8—48no—
246C4⋊C8—44no—
247#102—48no—
248#102—48no—
249#102—48no—
250#102—48no—
251#95—24no—
252#95—28no—
253#95—22no—
254#95—22no—
255#95—24no—
256#95—42no—
257#95—42no—
258#95—42no—
259#95—82no—
260#95—82no—
261C24—24no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24C22×D411yescharacteristic
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—48no—
306C2×D4—48no—
307C2×D4—84no—
308C2×D4—84no—
309C2×D4—88no—
310C2×D4—84no—
311C2×D4—84no—
312C2×D4—84no—
313C2×D4—84no—
314C2×D4—84no—
315C2×D4—84no—
316C2×D4—88no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4#12114yes—
363C22×C4#12114yes—
364C22×C4#12112yes—
365C22×C4#12112yes—
366C22×C4#12112yes—
367C22×C4#12112yes—
368C22×C4C22⋊Q812yes—
369C22×C4C22⋊Q812yes—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D412yes—
373C22×C4C22×Q811yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4112yes—
378C22×C4H(5,𝔽2)12yes—
379C22×C42−1+412yes—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—88no—
413C4⋊C4#12714yes—
414C22⋊C4—88no—
415C22⋊C4—88no—
416C22⋊C4—88no—
417C22⋊C4—88no—
418C42—24no—
419C42—24no—
420C42—24no—
421C42—24no—
422C42—84no—
423C42—84no—
424C42#12114yes—
425C42C22⋊Q814yes—
426C42C22⋊Q814yes—
427C42#12614yes—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23—82no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23C2×#12111yescharacteristic
464C23C2×#12112yes—
465C23C2×#12111yescharacteristic
466C23C2×C22⋊Q811yescharacteristic
467C23#53712yes—
468C23#57611yescharacteristic
469C23#57612yes—
470C23#57611yescharacteristic
471C23#57711yescharacteristic
472C23#57712yes—
473C23#57711yescharacteristic
474D4—44no—
475D4—44no—
476D4—44no—
477D4—44no—
478D4—816no—
479D4—816no—
480D4—82no—
481D4—82no—
482D4—82no—
483D4—82no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—82no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4#39614yes—
536C2×C4#39614yes—
537C2×C4C2×#12111yescharacteristic
538C2×C4C2×C22⋊Q811yescharacteristic
539C2×C4C2×C22⋊Q811yescharacteristic
540C2×C4C2×#12611yescharacteristic
541C2×C4C2×#12711yescharacteristic
542C2×C4#53211yescharacteristic
543C2×C4#53212yes—
544C2×C4#53511yescharacteristic
545C2×C4#53611yescharacteristic
546C2×C4#54212yes—
547C2×C4D4212yes—
548C2×C4#54712yes—
549C2×C4#54812yes—
550C2×C4D4×Q812yes—
551C2×C4#55012yes—
552C2×C4#55312yes—
553C2×C4#55412yes—
554C2×C4#55612yes—
555C8—42no—
556C8—42no—
557C8—82no—
558C22—84no—
559C22—84no—
560C22—84no—
561C22—84no—
562C22—84no—
563C22—84no—
564C22—84no—
565C22—162no—
566C22—162no—
567C22—162no—
568C22—162no—
569C22—162no—
570C22—162no—
571C22—162no—
572C22—162no—
573C22C2×#39611yescharacteristic
574C22#241612yes—
575C22#253812yes—
576C22#262011yescharacteristic
577C22#285611yescharacteristic
578C22#304111yescharacteristic
579C22#304112yes—
580C22#304311yescharacteristic
581C22#304312yes—
582C22#304511yescharacteristic
583C22#304511yescharacteristic
584C22#304512yes—
585C22#304512yes—
586C22#305711yescharacteristic
587C22#305811yescharacteristic
588C22#306212yes—
589C22#308111yescharacteristic
590C22#308111yescharacteristic
591C22#308112yes—
592C22#325712yes—
593C22#325712yes—
594C22#325912yes—
595C22#326012yes—
596C4—24no—
597C4—24no—
598C4—44no—
599C4—44no—
600C4—44no—
601C4—82no—
602C4—84no—
603C4—161no—
604C4—161no—
605C4#245212yes—
606C4#251212yes—
607C4#254111yescharacteristic
608C4#254211yescharacteristic
609C4#254511yescharacteristic
610C4#256611yescharacteristic
611C2—164no—
612C2#1513411yescharacteristic
613C2#2361211yescharacteristic
614C2#2361411yescharacteristic
615C2#2476212yes—
616C2#2476812yes—
617C2#2829012yes—
618C2#2829212yes—
619C2#2983411yescharacteristic
620C2#2983511yescharacteristic
621C2#3316611yescharacteristic
622C2#3316711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
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×#2067C2semidirect
2. C2×#2065C2semidirect
5. C2×#1949C2semidirect
6. C2×#1843C2semidirect
8. #13783C2semidirect
9. #11076C2semidirect
10. #11076C2semidirect
11. #10850C2semidirect
106. #389D4semidirect

Maximal subgroups

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

subgroupindex
C2×#20672
C2×#20652
C2×#19822
C2×#19492
C2×#18432
#142722
#137832
#110762
#108502