Finite Group Databasebeta

← #416054

Subgroups — #416054

Order 512.

1422 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2592C211yesmaximal, characteristic
2C2×#1984C211yesmaximal, characteristic, complemented
3C2×#1983C211yesmaximal, characteristic
4C2×#1958C211yesmaximal, characteristic
5C2×C4×#107C211yesmaximal, characteristic, complemented
6#13925C211yesmaximal, characteristic, complemented
7#11190C212yesmaximal, complemented
8#11189C212yesmaximal
9#11124C212yesmaximal, complemented
10#11123C212yesmaximal
11#10850C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#107C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×#102C2211yescharacteristic
20C22×#102C2212yes—
21C22×#102C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#2592—24no—
24#2592—24no—
25C2×#397—21no—
26C2×#393—21no—
27C2×#391C2212yes—
28C2×#390C2212yes—
29C2×#388—22no—
30C2×#382C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1984—24no—
34#1983—24no—
35#1958—24no—
36C4×#107—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1446—21no—
43#1445—21no—
44#1436—22no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8D411yescharacteristic
48C23×Q8C2311yescharacteristic
49C23×D4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×#124—22no—
54C2×#124—22no—
55C2×#124—22no—
56C2×#124—22no—
57C2×#124—22no—
58C2×#124D414yes—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D412yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79C2×#107—24no—
80C2×#107—24no—
81C2×#107—24no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×#103—24no—
86C2×#103—22no—
87C2×#103—22no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—42no—
91C2×#103—42no—
92C2×#102—24no—
93C2×#102—22no—
94C2×#102—22no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—42no—
98C2×#102—42no—
99C2×C4×C8—22no—
100C2×C4×C8—22no—
101C2×C4×C8D414yes—
102#397—28no—
103#393—28no—
104#391—24no—
105#391—24no—
106#390—24no—
107#390—24no—
108#388—216no—
109#382—22no—
110#382—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4D414yes—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125#340—24no—
126#340—42no—
127#340—42no—
128#336—24no—
129#336—24no—
130C25—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—24no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—42no—
145C22×Q8C2×D412yes—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4C2×D412yes—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—22no—
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—21no—
170C22×C8—21no—
171C22×C8—22no—
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×C8C2×D412yes—
181C22×C8C2×D412yes—
182C22×C8#4114yes—
183#124—28no—
184#124—28no—
185#124—44no—
186#124—44no—
187C4×Q8—44no—
188C4×Q8—44no—
189C4×Q8—44no—
190C4×Q8—44no—
191C2×C4⋊C4—22no—
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—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4#4114yes—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—82no—
228C2×C22⋊C4—82no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—42no—
237C2×C42—81no—
238C2×C42—81no—
239C2×C42—81no—
240C2×C42—81no—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42#4112yes—
245C2×C42#4112yes—
246C2×C42#4112yes—
247#107—216no—
248C4⋊C8—44no—
249#103—48no—
250#103—44no—
251#103—44no—
252#102—48no—
253#102—44no—
254#102—44no—
255C4×C8—28no—
256#95—28no—
257#95—28no—
258#95—24no—
259#95—44no—
260#95—42no—
261#95—42no—
262#95—42no—
263#95—44no—
264#95—82no—
265#95—82no—
266C24—22no—
267C24—41no—
268C24—41no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24C22×D411yescharacteristic
278C2×Q8—28no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—48no—
284C2×Q8—48no—
285C2×Q8—44no—
286C2×Q8—41no—
287C2×Q8—42no—
288C2×Q8—41no—
289C2×Q8—42no—
290C2×Q8—44no—
291C2×Q8—41no—
292C2×Q8—42no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—24no—
304C2×D4—24no—
305C2×D4—41no—
306C2×D4—42no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—42no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
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—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4#12012yes—
368C22×C4#12012yes—
369C22×C4#12112yes—
370C22×C4#12112yes—
371C22×C4#12312yes—
372C22×C4#12312yes—
373C22×C4#13514yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4112yes—
381C22×C4C2×#4112yes—
382C22×C4H(5,𝔽2)12yes—
383C22×C42−1+412yes—
384C2×C8—22no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8#12414yes—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—48no—
410C4⋊C4—44no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C22⋊C4—416no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—88no—
421C42—42no—
422C42—42no—
423C42—42no—
424C42—42no—
425C42—84no—
426C42—84no—
427C42#12114yes—
428C42#12114yes—
429C42#12314yes—
430C42#12414yes—
431C23—42no—
432C23—44no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—82no—
437C23—81no—
438C23—81no—
439C23—82no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—81no—
449C23—81no—
450C23—81no—
451C23—81no—
452C23—81no—
453C23—81no—
454C23C2×#12011yescharacteristic
455C23C2×#12111yescharacteristic
456C23C2×#12311yescharacteristic
457C23#54812yes—
458C23#55012yes—
459C23C2×#13512yes—
460C23#57512yes—
461C23#57611yescharacteristic
462C23#57611yescharacteristic
463C23#57811yescharacteristic
464C23#57811yescharacteristic
465Q8—416no—
466Q8—44no—
467Q8—42no—
468Q8—42no—
469Q8—82no—
470Q8—81no—
471Q8—81no—
472D4—44no—
473D4—42no—
474D4—42no—
475D4—82no—
476D4—81no—
477D4—81no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4#39414yes—
532C2×C4#39414yes—
533C2×C4C2×#12111yescharacteristic
534C2×C4C2×#12111yescharacteristic
535C2×C4C2×#12311yescharacteristic
536C2×C4#52512yes—
537C2×C4C2×#12411yescharacteristic
538C2×C4C2×#12411yescharacteristic
539C2×C4#52912yes—
540C2×C4#53211yescharacteristic
541C2×C4#53511yescharacteristic
542C2×C4#53611yescharacteristic
543C2×C4#53712yes—
544C2×C4#54212yes—
545C2×C4#54612yes—
546C2×C4#54712yes—
547C2×C4#55612yes—
548C2×C4#55612yes—
549C2×C4#55912yes—
550C2×C4#55912yes—
551C8—24no—
552C8—42no—
553C8—82no—
554C22—82no—
555C22—84no—
556C22—84no—
557C22—82no—
558C22—82no—
559C22—161no—
560C22—161no—
561C22—162no—
562C22—162no—
563C22—161no—
564C22—161no—
565C22C2×#39411yescharacteristic
566C22#240712yes—
567C22#244812yes—
568C22#254511yescharacteristic
569C22#254811yescharacteristic
570C22#300212yes—
571C22#300811yescharacteristic
572C22#301111yescharacteristic
573C22#305011yescharacteristic
574C22#305411yescharacteristic
575C22#306111yescharacteristic
576C22#306111yescharacteristic
577C22#309112yes—
578C22#309811yescharacteristic
579C22#309811yescharacteristic
580C22#336112yes—
581C22#336412yes—
582C22#337012yes—
583C22#337712yes—
584C22#339312yes—
585C22#339612yes—
586C22#340212yes—
587C22#340312yes—
588C4—24no—
589C4—24no—
590C4—44no—
591C4—82no—
592C4—84no—
593C4—84no—
594C4—82no—
595C4—161no—
596C4—161no—
597C4#244712yes—
598C4#254411yescharacteristic
599C4#254711yescharacteristic
600C4#259212yes—
601C4#262011yescharacteristic
602C4#285011yescharacteristic
603C2—162no—
604C2#1521711yescharacteristic
605C2#2347911yescharacteristic
606C2#2348811yescharacteristic
607C2#2419212yes—
608C2#2419512yes—
609C2#2632512yes—
610C2#2632612yes—
611C2#2864111yescharacteristic
612C2#2864311yescharacteristic
613C2#2874011yescharacteristic
614C2#2874111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×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×#1984C2semidirect
5. C2×C4×#107C2semidirect
6. #13925C2semidirect
7. #11190C2semidirect
9. #11124C2semidirect
11. #10850C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25922
C2×#19842
C2×#19832
C2×#19582
C2×C4×#1072
#139252
#111902
#111892
#111242
#111232
#108502