Finite Group Databasebeta

← #369318

Subgroups — #369318

Order 512.

1758 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2592C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#1985C211yesmaximal, characteristic, complemented
4C2×#1983C211yesmaximal, characteristic
5C2×#1959C211yesmaximal, characteristic
6#11050C212yesmaximal, complemented
7#11049C212yesmaximal
8#10932C212yesmaximal, complemented
9#10931C212yesmaximal
10#10849C211yesmaximal, characteristic, complemented
11#10703C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×C4×Q8—21no—
14C22×C4×Q8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×#102C2211yescharacteristic
20C22×#102C2212yes—
21C22×#102C2211yescharacteristic
22C22×#97C2211yescharacteristic
23#2592—24no—
24#2592C418yescomplemented
25C2×#397—21no—
26C2×#393—21no—
27Q8×C4⋊C4—24no—
28Q8×C4⋊C4—24no—
29C2×#391C2212yes—
30C2×#390C2212yes—
31C2×#389—21no—
32C2×#389—22no—
33C2×#388—22no—
34C2×#384C2212yes—
35C2×#384C2211yescharacteristic
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39#1985—24no—
40#1983—24no—
41#1959—24no—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#336C2212yes—
47C2×#336C2212yes—
48C23×Q8D411yescharacteristic
49C23×Q8C2311yescharacteristic
50C23×D4C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2312yes—
54C2×#124—22no—
55C2×#124—22no—
56C2×#124—22no—
57C2×#124—22no—
58C2×#124—22no—
59C2×#124C2×C414yes—
60C2×C4×Q8—28no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—48no—
66C2×C4×Q8C2×C414yes—
67C2×C4×Q8D414yes—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D412yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C42—22no—
83C22×C42—21no—
84C22×C42—21no—
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—22no—
93C2×#103—22no—
94C2×#103—24no—
95C2×#103—42no—
96C2×#103—42no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—24no—
102C2×#102—42no—
103C2×#102—42no—
104C2×#97—22no—
105C2×#97—22no—
106C2×#97—22no—
107#397—28no—
108#393—28no—
109#391—24no—
110#391C2×C418yes—
111#390—24no—
112#390C2×C418yes—
113#389—28no—
114#389—216no—
115#388—216no—
116#384—24no—
117#384—24no—
118#384—22no—
119#384D414yes—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4C2×C414yes—
125C4×C4⋊C4D414yes—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—21no—
131#340—24no—
132#340—24no—
133#340—42no—
134#340—42no—
135#336—24no—
136#336—24no—
137C25—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—24no—
141C22×Q8—24no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8C22⋊C414yes—
152C22×Q8C22×C412yes—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4C22×C412yes—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—21no—
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2411yescharacteristic
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8#4114yes—
190#124—28no—
191#124—28no—
192#124—44no—
193#124—44no—
194C4×Q8—28no—
195C4×Q8—28no—
196C4×Q8—28no—
197C4×Q8—28no—
198C4×Q8—432no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
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—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—81no—
226C2×C4⋊C4—81no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4C22⋊C418yes—
230C2×C4⋊C4C22⋊C414yes—
231C2×C4⋊C4C22×C414yes—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4#4114yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—82no—
247C2×C22⋊C4—82no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—28no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—44no—
256C2×C42—44no—
257C2×C42—44no—
258C2×C42—81no—
259C2×C42—81no—
260C2×C42C22×C414yes—
261C2×C42C22×C412yes—
262C2×C42C2×D414yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42#4112yes—
267C2×C42#4112yes—
268C4⋊C8—44no—
269#103—48no—
270#103—44no—
271#103—44no—
272#102—48no—
273#102—44no—
274#102—44no—
275#97—44no—
276#95—28no—
277#95—24no—
278#95—28no—
279#95—28no—
280#95—24no—
281C24—22no—
282C24—41no—
283C24—41no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24—41no—
289C24—41no—
290C24—41no—
291C24—41no—
292C24C22×D411yescharacteristic
293C2×Q8—28no—
294C2×Q8—216no—
295C2×Q8—28no—
296C2×Q8—28no—
297C2×Q8—216no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—41no—
307C2×Q8—42no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—42no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
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—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4C2×C22⋊C414yes—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C414yes—
383C22×C4#12012yes—
384C22×C4#12012yes—
385C22×C4#12312yes—
386C22×C4#12312yes—
387C22×C4AGL(1,ℤ/8)14yes—
388C22×C4#13714yes—
389C22×C4C23×C411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4112yes—
396C22×C4C2×#4112yes—
397C22×C4H(5,𝔽2)12yes—
398C22×C42−1+412yes—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—216no—
419C4⋊C4—28no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—48no—
427C4⋊C4—416no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—84no—
434C4⋊C4—88no—
435C4⋊C4#12414yes—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—416no—
439C22⋊C4—88no—
440C42—24no—
441C42—24no—
442C42—24no—
443C42—24no—
444C42—48no—
445C42—416no—
446C42—84no—
447C42C4×D414yes—
448C42C4×D414yes—
449C42#12114yes—
450C42#12114yes—
451C23—44no—
452C23—44no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—82no—
457C23—81no—
458C23—81no—
459C23—82no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23—81no—
468C23—81no—
469C23—82no—
470C23—81no—
471C23—81no—
472C23—81no—
473C23—81no—
474C23C22×C22⋊C411yescharacteristic
475C23C2×#12011yescharacteristic
476C23C2×#12311yescharacteristic
477C23#54812yes—
478C23#55012yes—
479C23C2×AGL(1,ℤ/8)12yes—
480C23C2×#13712yes—
481C23#57611yescharacteristic
482C23#57711yescharacteristic
483C23#57711yescharacteristic
484C23#57811yescharacteristic
485Q8—232no—
486Q8—42no—
487Q8—44no—
488Q8—42no—
489Q8—82no—
490Q8—81no—
491Q8—81no—
492D4—42no—
493D4—44no—
494D4—42no—
495D4—82no—
496D4—81no—
497D4—81no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
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—81no—
548C2×C4—81no—
549C2×C4—82no—
550C2×C4#38614yes—
551C2×C4#38614yes—
552C2×C4#41718yes—
553C2×C4C2×C4×D411yescharacteristic
554C2×C4C2×C4×D411yescharacteristic
555C2×C4C4×#4112yes—
556C2×C4#52012yes—
557C2×C4C2×#12111yescharacteristic
558C2×C4C2×#12111yescharacteristic
559C2×C4C2×#12411yescharacteristic
560C2×C4#52912yes—
561C2×C4#53211yescharacteristic
562C2×C4#53511yescharacteristic
563C2×C4#53611yescharacteristic
564C2×C4#54212yes—
565C2×C4#54612yes—
566C2×C4#54812yes—
567C2×C4#54812yes—
568C2×C4#55012yes—
569C2×C4#55612yes—
570C2×C4#55912yes—
571C8—44no—
572C8—81no—
573C8—81no—
574C22—84no—
575C22—84no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—161no—
580C22—161no—
581C22—161no—
582C22—161no—
583C22—162no—
584C22—162no—
585C22C2×#38611yescharacteristic
586C22#234112yes—
587C22#243111yescharacteristic
588C22#243611yescharacteristic
589C22#244812yes—
590C22C2×#41712yes—
591C22#289711yescharacteristic
592C22#289811yescharacteristic
593C22#300311yescharacteristic
594C22#300411yescharacteristic
595C22#301211yescharacteristic
596C22#301311yescharacteristic
597C22#309212yes—
598C22#309811yescharacteristic
599C22#309911yescharacteristic
600C22#336712yes—
601C22#336912yes—
602C22#337312yes—
603C22#337512yes—
604C22#339712yes—
605C22#339812yes—
606C22#339912yes—
607C22#340012yes—
608C4—24no—
609C4—28no—
610C4—24no—
611C4—44no—
612C4—44no—
613C4—84no—
614C4—82no—
615C4—84no—
616C4—82no—
617C4C4×#12411yescharacteristic
618C4#232611yescharacteristic
619C4#236712yes—
620C4#242612yes—
621C4#254511yescharacteristic
622C4#254811yescharacteristic
623C2—162no—
624C2#1402611yescharacteristic
625C2#2090911yescharacteristic
626C2#2091511yescharacteristic
627C2#2254412yes—
628C2#2254612yes—
629C2#2563711yescharacteristic
630C2#2564011yescharacteristic
631C2#2593911yescharacteristic
632C2#2594211yescharacteristic
633C2#2632812yes—
634C2#2633012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×Q8×C4⋊C4C2semidirect
3. C2×#1985C2semidirect
6. #11050C2semidirect
8. #10932C2semidirect
10. #10849C2semidirect
11. #10703C2semidirect
24. #2592C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25922
C2×Q8×C4⋊C42
C2×#19852
C2×#19832
C2×#19592
#110502
#110492
#109322
#109312
#108492
#107032