Finite Group Databasebeta

← #405541

Subgroups — #405541

Order 512.

1586 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2862C211yesmaximal, characteristic
2C2×#2024C211yesmaximal, characteristic
3C2×#2016C211yesmaximal, characteristic
4C2×#1878C212yesmaximal
5C4×#391C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic
7#11190C212yesmaximal
8#11024C212yesmaximal
9#11010C211yesmaximal, characteristic
10#10932C212yesmaximal
11#10795C211yesmaximal, characteristic
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20#2862—28no—
21C2×#391—21no—
22C2×#391—21no—
23C2×#391C2211yescharacteristic
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#391C2211yescharacteristic
27C2×#382—21no—
28C2×#382—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C43C2211yescharacteristic
31#2024—24no—
32#2016—24no—
33#1878—28no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1436—21no—
42#1436—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C4×#95—21no—
46C4×#95—21no—
47C23×Q8—21no—
48C23×Q8C2312yes—
49C23×Q8C2311yescharacteristic
50C22×Q16—24no—
51C22×Q16—24no—
52C22×Q16—24no—
53C22×Q16—24no—
54C22×Q16—24no—
55C22×Q16—42no—
56C22×Q16—42no—
57C23×C8C2312yes—
58C23×C8C2312yes—
59C2×C4⋊Q8—24no—
60C2×C4⋊Q8—24no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—42no—
66C2×C4⋊Q8—42no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—24no—
70C2×C4×Q8—42no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C42—21no—
80C22×C42—21no—
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—42no—
96C2×#103—42no—
97C2×#103—42no—
98C2×#103—42no—
99C2×#103—42no—
100#391—28no—
101#391—28no—
102#391—24no—
103#391—22no—
104#391—24no—
105#391—22no—
106#391—24no—
107#391—22no—
108#391D418yes—
109#391D414yes—
110#382—28no—
111#382—28no—
112C4×C4⋊C4—42no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C43—24no—
127#340—24no—
128#340—24no—
129#340—42no—
130#336—24no—
131#336—24no—
132#336—22no—
133#336—22no—
134C22×Q8—24no—
135C22×Q8—22no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—42no—
156C22×Q8—42no—
157C22×Q8—42no—
158C22×Q8—42no—
159C22×Q8—42no—
160C22×Q8—42no—
161C22×Q8C2×D414yes—
162C22×Q8C2×D412yes—
163C22×Q8#4112yes—
164C22×Q8#4112yes—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2411yescharacteristic
178C2×Q16—216no—
179C2×Q16—48no—
180C2×Q16—48no—
181C2×Q16—48no—
182C2×Q16—48no—
183C2×Q16—48no—
184C2×Q16—48no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8#4114yes—
199C4⋊Q8—48no—
200C4⋊Q8—48no—
201C4⋊Q8—48no—
202C4⋊Q8—48no—
203C4×Q8—48no—
204C4×Q8—48no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
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—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4C2×D414yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4#4112yes—
234C2×C4⋊C4#4112yes—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—41no—
255C2×C42—41no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42Q1614yescomplemented
262C2×C42C2×D414yes—
263C2×C42C2×D412yes—
264C2×C42#4112yes—
265#103—48no—
266#103—48no—
267#103—48no—
268#103—48no—
269#95—28no—
270#95—28no—
271#95—24no—
272#95—24no—
273#95—28no—
274#95—28no—
275#95—24no—
276#95—24no—
277#95—42no—
278#95—42no—
279#95—42no—
280#95—42no—
281#95—42no—
282C24C22×D411yescharacteristic
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—48no—
299C2×Q8—48no—
300C2×Q8—48no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—84no—
330C2×Q8—84no—
331C2×Q8—84no—
332C2×Q8—84no—
333C2×Q8#12414yes—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4#12012yes—
407C22×C4#12012yes—
408C22×C4#12112yes—
409C22×C4#12112yes—
410C22×C4C2×Q1612yes—
411C22×C4C2×Q1612yes—
412C22×C4C2×Q1612yes—
413C22×C4#13512yes—
414C22×C4#13712yes—
415C22×C4#13712yes—
416C22×C4#13712yes—
417C22×C4#13712yes—
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4112yes—
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4111yescharacteristic
426C22×C4H(5,𝔽2)12yes—
427C22×C4H(5,𝔽2)12yes—
428Q16—432no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C42—28no—
458C42—24no—
459C42—24no—
460C42—24no—
461C42—24no—
462C42—28no—
463C42—44no—
464C42—44no—
465C42—48no—
466C42—42no—
467C42—42no—
468C42—84no—
469C23C2×#12011yescharacteristic
470C23C2×#12111yescharacteristic
471C23#52511yescharacteristic
472C23#52511yescharacteristic
473C23#53211yescharacteristic
474C23#53511yescharacteristic
475C23#53511yescharacteristic
476C23C22×Q1611yescharacteristic
477C23C2×#13511yescharacteristic
478C23C2×#13711yescharacteristic
479C23C2×#13711yescharacteristic
480C23C2×#13711yescharacteristic
481C23C2×#13711yescharacteristic
482C23#57511yescharacteristic
483C23#57511yescharacteristic
484Q8—28no—
485Q8—44no—
486Q8—44no—
487Q8—44no—
488Q8—44no—
489Q8—44no—
490Q8—44no—
491Q8—44no—
492Q8—816no—
493Q8—82no—
494Q8—82no—
495Q8—82no—
496Q8—82no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—82no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—82no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4#45114yes—
579C2×C4#46214yes—
580C2×C4#46414yes—
581C2×C4#52512yes—
582C2×C4C2×#12411yescharacteristic
583C2×C4#52911yescharacteristic
584C2×C4#52911yescharacteristic
585C2×C4#53211yescharacteristic
586C2×C4#53212yes—
587C2×C4#53812yes—
588C2×C4#53812yes—
589C2×C4D4212yes—
590C2×C4#54612yes—
591C2×C4#54612yes—
592C2×C4#54612yes—
593C2×C4#54612yes—
594C2×C4#55112yes—
595C2×C4#55512yes—
596C2×C4#55912yes—
597C8—44no—
598C8—82no—
599C22#238811yescharacteristic
600C22#240411yescharacteristic
601C22#240811yescharacteristic
602C22#254411yescharacteristic
603C22#261011yescharacteristic
604C22#263111yescharacteristic
605C22#285011yescharacteristic
606C22C2×#45111yescharacteristic
607C22#300211yescharacteristic
608C22#300411yescharacteristic
609C22#300711yescharacteristic
610C22C2×#46211yescharacteristic
611C22C2×#46411yescharacteristic
612C22#303911yescharacteristic
613C22#304011yescharacteristic
614C22#310411yescharacteristic
615C22#310411yescharacteristic
616C22#310711yescharacteristic
617C22#310711yescharacteristic
618C22#311311yescharacteristic
619C22#311311yescharacteristic
620C22#311611yescharacteristic
621C22#311611yescharacteristic
622C22#317011yescharacteristic
623C22#317211yescharacteristic
624C22#318111yescharacteristic
625C22#318511yescharacteristic
626C22#320911yescharacteristic
627C22#320911yescharacteristic
628C22#321011yescharacteristic
629C22#321111yescharacteristic
630C22#321511yescharacteristic
631C22#321511yescharacteristic
632C22#321611yescharacteristic
633C22#321811yescharacteristic
634C4—24no—
635C4—24no—
636C4—24no—
637C4—42no—
638C4—42no—
639C4—44no—
640C4—42no—
641C4—84no—
642C4—82no—
643C4—82no—
644C4—84no—
645C4—82no—
646C4#245912yes—
647C4#247212yes—
648C4#255512yes—
649C4#255712yes—
650C2#1491511yescharacteristic
651C2#2357111yescharacteristic
652C2#2357711yescharacteristic
653C2#2393411yescharacteristic
654C2#2393511yescharacteristic
655C2#2421211yescharacteristic
656C2#2421711yescharacteristic
657C2#2861611yescharacteristic
658C2#2862011yescharacteristic
659C2#2955311yescharacteristic
660C2#2955911yescharacteristic
661C2#3004211yescharacteristic
662C2#3004611yescharacteristic
663C2#3302711yescharacteristic
664C2#3303311yescharacteristic
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

Complements

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

NH ≅ G/Ntype
261. C2×C42Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#28622
C2×#20242
C2×#20162
C2×#18782
C4×#3912
#113842
#111902
#110242
#110102
#109322
#107952