Finite Group Databasebeta

← #416504

Subgroups — #416504

Order 512.

1514 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic, complemented
3C2×#1988C211yesmaximal, characteristic, complemented
4C2×#1955C211yesmaximal, characteristic
5C2×#1949C211yesmaximal, characteristic, complemented
6#13905C211yesmaximal, characteristic
7#11200C212yesmaximal, complemented
8#11199C212yesmaximal, complemented
9#11096C212yesmaximal, complemented
10#11095C212yesmaximal, complemented
11#10703C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×#97C2211yescharacteristic
20#2563—24no—
21#2563—24no—
22C2×#393—22no—
23C2×#390—21no—
24C2×#382—21no—
25C2×#382C2211yescharacteristic
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1993—24no—
33#1988—24no—
34#1955—24no—
35#1949—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2212yes—
44#1445—21no—
45#1438—21no—
46#1437—22no—
47C23×Q8—21no—
48C23×D4D411yescharacteristic
49C24×C4C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2312yes—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8D414yes—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4D412yes—
78C22×C42—21no—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C2×#107—22no—
82C2×#107—42no—
83C2×#106—22no—
84C2×#106—42no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8D414yes—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—42no—
93C2×C22⋊C8—42no—
94C2×#97—22no—
95C2×#97—22no—
96C2×#97—22no—
97#393—216no—
98#390—28no—
99#382—28no—
100#382—22no—
101#382—22no—
102#380—24no—
103#380—24no—
104#380—24no—
105#380—24no—
106#380—24no—
107#380—24no—
108#380—42no—
109#380—42no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4D414yes—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95—21no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—22no—
126#340—22no—
127#340—24no—
128#340—24no—
129#340—24no—
130C25C2×D412yes—
131C22×Q8—24no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—81no—
135C22×Q8—81no—
136C22×Q8—81no—
137C22×Q8—81no—
138C22×D4—24no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—42no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8C2×D412yes—
182C22×C8C2×D412yes—
183C22×C8#4114yes—
184C4⋊Q8—28no—
185C4⋊Q8—28no—
186C4⋊Q8—44no—
187C4⋊Q8—44no—
188C4×D4—44no—
189C4×D4—44no—
190C4×D4—44no—
191C4×D4—44no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4C2×D414yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4#4114yes—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—82no—
236C2×C22⋊C4—82no—
237C2×C22⋊C4—82no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4—82no—
240C2×C22⋊C4—82no—
241C2×C42—24no—
242C2×C42—44no—
243C2×C42—42no—
244C2×C42—81no—
245C2×C42—81no—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42#4112yes—
250C2×C42#4112yes—
251C2×C42#4112yes—
252#107—44no—
253#106—44no—
254C4⋊C8—28no—
255C22⋊C8—48no—
256C22⋊C8—48no—
257#97—44no—
258#95—28no—
259#95—28no—
260#95—24no—
261#95—44no—
262#95—44no—
263#95—42no—
264#95—42no—
265#95—42no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24#13714yes—
281C24C22×D411yescharacteristic
282C2×Q8—48no—
283C2×Q8—84no—
284C2×Q8—84no—
285C2×Q8—84no—
286C2×Q8—84no—
287C2×Q8—84no—
288C2×Q8—84no—
289C2×D4—28no—
290C2×D4—48no—
291C2×D4—48no—
292C2×D4—44no—
293C2×D4—44no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
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—42no—
345C22×C4—42no—
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—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4#12012yes—
370C22×C4#12012yes—
371C22×C4#12112yes—
372C22×C4#12112yes—
373C22×C4#12312yes—
374C22×C4#12312yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4112yes—
382C22×C4C2×#4112yes—
383C22×C4H(5,𝔽2)12yes—
384C22×C42−1+412yes—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—42no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—44no—
417C4⋊C4—48no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C22⋊C4—48no—
425C22⋊C4—88no—
426C22⋊C4—88no—
427C22⋊C4—88no—
428C22⋊C4—88no—
429C42—48no—
430C42—84no—
431C42#12114yes—
432C42#12114yes—
433C42#12314yes—
434C42#12414yes—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23C2×#12011yescharacteristic
471C23C2×#12111yescharacteristic
472C23C2×#12311yescharacteristic
473C23#54612yes—
474C23#54712yes—
475C23C2×#13712yes—
476C23#57512yes—
477C23#57611yescharacteristic
478C23#57611yescharacteristic
479C23#57711yescharacteristic
480C23#57711yescharacteristic
481Q8—816no—
482D4—416no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—82no—
513C2×C4—82no—
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—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4#39414yes—
543C2×C4#39414yes—
544C2×C4C2×#12111yescharacteristic
545C2×C4C2×#12111yescharacteristic
546C2×C4C2×#12311yescharacteristic
547C2×C4C2×#12411yescharacteristic
548C2×C4#52912yes—
549C2×C4#52912yes—
550C2×C4#53211yescharacteristic
551C2×C4#53211yescharacteristic
552C2×C4#53511yescharacteristic
553C2×C4#53611yescharacteristic
554C2×C4#54212yes—
555C2×C4#54212yes—
556C2×C4#54612yes—
557C2×C4#54812yes—
558C2×C4#54812yes—
559C2×C4#55012yes—
560C2×C4#55612yes—
561C2×C4#55912yes—
562C8—44no—
563C8—42no—
564C8—42no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22C2×#39411yescharacteristic
581C22#242712yes—
582C22#243111yescharacteristic
583C22#243211yescharacteristic
584C22#244712yes—
585C22#300411yescharacteristic
586C22#300511yescharacteristic
587C22#301211yescharacteristic
588C22#301211yescharacteristic
589C22#305612yes—
590C22#305711yescharacteristic
591C22#305811yescharacteristic
592C22#309512yes—
593C22#309611yescharacteristic
594C22#309611yescharacteristic
595C22#331412yes—
596C22#331612yes—
597C22#331912yes—
598C22#332112yes—
599C22#334412yes—
600C22#334612yes—
601C22#334712yes—
602C22#334812yes—
603C4—24no—
604C4—24no—
605C4—84no—
606C4—82no—
607C4—82no—
608C4—84no—
609C4—82no—
610C4—161no—
611C4—161no—
612C4#242612yes—
613C4#244812yes—
614C4#254411yescharacteristic
615C4#254511yescharacteristic
616C4#264011yescharacteristic
617C4#286111yescharacteristic
618C2—84no—
619C2#1522611yescharacteristic
620C2#2348111yescharacteristic
621C2#2348611yescharacteristic
622C2#2535012yes—
623C2#2535612yes—
624C2#2557811yescharacteristic
625C2#2557911yescharacteristic
626C2#2571011yescharacteristic
627C2#2571211yescharacteristic
628C2#2631312yes—
629C2#2631512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1993C2semidirect
3. C2×#1988C2semidirect
5. C2×#1949C2semidirect
7. #11200C2semidirect
8. #11199C2semidirect
9. #11096C2semidirect
10. #11095C2semidirect
11. #10703C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#19932
C2×#19882
C2×#19552
C2×#19492
#139052
#112002
#111992
#110962
#110952
#107032