Finite Group Databasebeta

← #441161

Subgroups — #441161

Order 512.

1466 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2429C211yesmaximal, characteristic
2C2×#2100C211yesmaximal, characteristic, complemented
3C2×#1956C211yesmaximal, characteristic, complemented
4#15195C211yesmaximal, characteristic
5#11190C212yesmaximal, complemented
6#11010C211yesmaximal, characteristic, complemented
7#10970C212yesmaximal
8#10970C212yesmaximal
9#10921C211yesmaximal, characteristic
10#10880C211yesmaximal, characteristic, complemented
11#10789C212yesmaximal, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×C22⋊C8C2212yes—
21C22×C4×C8C2211yescharacteristic
22#2429—24no—
23#2429—24no—
24C2×#398—21no—
25C2×#396—21no—
26C2×#391C2211yescharacteristic
27C2×#389—21no—
28C2×#389C2211yescharacteristic
29C2×#388—21no—
30C2×#384C2211yescharacteristic
31C2×#380C2212yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33#2100—24no—
34#1956—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1442—21no—
42#1442—21no—
43#1439—21no—
44#1439—21no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2212yes—
47C23×Q8C2311yescharacteristic
48C23×Q8C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C2×C4⋊Q8D414yes—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—42no—
63C2×C22⋊Q8—42no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—24no—
67C2×C4×Q8—42no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4D411yescharacteristic
84C22×C22⋊C4D411yescharacteristic
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—42no—
94C2×#103—42no—
95C2×#103—42no—
96C2×#103—42no—
97C2×#103—42no—
98C2×#103—42no—
99C2×#103—42no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—42no—
102C2×C22⋊C8—42no—
103C2×C4×C8—22no—
104C2×C4×C8—42no—
105#398—28no—
106#396—28no—
107#391—22no—
108#391—22no—
109#389—28no—
110#389—22no—
111#389D414yes—
112#388—28no—
113#384—22no—
114#384—22no—
115#380—42no—
116#380—42no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95D411yescharacteristic
128C2×#95D411yescharacteristic
129#340—24no—
130#340—24no—
131#340—24no—
132#340—42no—
133#336—22no—
134#336—42no—
135C25C2×D411yescharacteristic
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—22no—
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×Q8C2×D412yes—
150C22×Q8C2×D412yes—
151C22×Q8#4114yes—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2411yescharacteristic
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×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8#4114yes—
186C4⋊Q8—28no—
187C4⋊Q8—44no—
188C4⋊Q8—48no—
189C22⋊Q8—48no—
190C22⋊Q8—48no—
191C22⋊Q8—44no—
192C22⋊Q8—44no—
193C22⋊Q8—44no—
194C22⋊Q8—44no—
195C4×Q8—48no—
196C4×Q8—44no—
197C4×Q8—44no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—21no—
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—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4#4114yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—82no—
246C2×C22⋊C4—82no—
247C2×C22⋊C4Q1614yes—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—24no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—44no—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258#103—48no—
259#103—48no—
260#103—48no—
261#103—48no—
262C22⋊C8—48no—
263C4×C8—44no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—22no—
268#95—22no—
269#95—44no—
270#95—42no—
271#95—42no—
272#95—82no—
273#95—82no—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—22no—
278C24—22no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—22no—
283C24—22no—
284C24C2×Q1612yes—
285C24C22×D411yescharacteristic
286C2×Q8—22no—
287C2×Q8—24no—
288C2×Q8—22no—
289C2×Q8—24no—
290C2×Q8—22no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
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#12314yes—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4#12012yes—
387C22×C4#12012yes—
388C22×C4C2×Q1612yes—
389C22×C4C2×Q1612yes—
390C22×C4#13512yes—
391C22×C4#13712yes—
392C22×C4#13712yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C2×#4112yes—
396C22×C4C2×#4112yes—
397C22×C4C2×#4112yes—
398C22×C4H(5,𝔽2)12yes—
399C22×C4H(5,𝔽2)12yes—
400C22×C4H(5,𝔽2)12yes—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—48no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—84no—
435C4⋊C4—84no—
436C4⋊C4#12414yes—
437C22⋊C4—28no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—44no—
441C22⋊C4—88no—
442C42—22no—
443C42—22no—
444C42—24no—
445C42—48no—
446C42—41no—
447C42—41no—
448C42—42no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—24no—
453C23—24no—
454C23—42no—
455C23—42no—
456C23—41no—
457C23—42no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23C2×#12011yescharacteristic
473C23#53211yescharacteristic
474C23#53511yescharacteristic
475C23#54612yes—
476C23#54612yes—
477C23C22×Q1611yescharacteristic
478C23C2×#13511yescharacteristic
479C23C2×#13711yescharacteristic
480C23C2×#13711yescharacteristic
481C23#57712yes—
482C23#57712yes—
483Q8—28no—
484Q8—44no—
485Q8—44no—
486Q8—44no—
487Q8—44no—
488Q8—44no—
489Q8—82no—
490Q8—82no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4#45114yes—
553C2×C4#45214yes—
554C2×C4C2×#12311yescharacteristic
555C2×C4C2×#12411yescharacteristic
556C2×C4#52912yes—
557C2×C4#53411yescharacteristic
558C2×C4#53511yescharacteristic
559C2×C4#53812yes—
560C2×C4#53812yes—
561C2×C4#54012yes—
562C2×C4#54612yes—
563C2×C4#54612yes—
564C2×C4#54612yes—
565C2×C4#55012yes—
566C2×C4#55312yes—
567C2×C4#55312yes—
568C2×C4#55312yes—
569C2×C4#56012yes—
570C8—44no—
571C8—82no—
572C22—42no—
573C22—42no—
574C22—44no—
575C22—44no—
576C22—42no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—82no—
581C22—82no—
582C22—81no—
583C22#240511yescharacteristic
584C22#243111yescharacteristic
585C22#243511yescharacteristic
586C22#255912yes—
587C22#270312yes—
588C22C2×#45111yescharacteristic
589C22C2×#45211yescharacteristic
590C22#301011yescharacteristic
591C22#301011yescharacteristic
592C22#317011yescharacteristic
593C22#317412yes—
594C22#318511yescharacteristic
595C22#320911yescharacteristic
596C22#321011yescharacteristic
597C22#322212yes—
598C22#329912yes—
599C22#329912yes—
600C22#330112yes—
601C22#330312yes—
602C22#331612yes—
603C22#332012yes—
604C22#332912yes—
605C22#333112yes—
606C4—24no—
607C4—24no—
608C4—44no—
609C4—42no—
610C4—44no—
611C4—81no—
612C4—81no—
613C4—82no—
614C4—82no—
615C4—162no—
616C4—162no—
617C4#244412yes—
618C4#249012yes—
619C4#268812yes—
620C4#271412yes—
621C2—82no—
622C2#1567411yescharacteristic
623C2#2403611yescharacteristic
624C2#2404211yescharacteristic
625C2#2560011yescharacteristic
626C2#2560711yescharacteristic
627C2#2588611yescharacteristic
628C2#2588811yescharacteristic
629C2#2904112yes—
630C2#2907012yes—
631C2#3183712yes—
632C2#3184012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2100C2semidirect
3. C2×#1956C2semidirect
5. #11190C2semidirect
6. #11010C2semidirect
10. #10880C2semidirect
11. #10789C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24292
C2×#21002
C2×#19562
#151952
#111902
#110102
#109702
#109212
#108802
#107892