Finite Group Databasebeta

← #396577

Subgroups — #396577

Order 512.

1486 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2075C211yesmaximal, characteristic, complemented
2C2×#1869C211yesmaximal, characteristic
3C2×#1808C211yesmaximal, characteristic, complemented
4C2×#1735C211yesmaximal, characteristic, complemented
5#13871C212yesmaximal
6#11379C211yesmaximal, characteristic, complemented
7#11316C212yesmaximal, complemented
8#11256C211yesmaximal, characteristic, complemented
9#11200C212yesmaximal, complemented
10#11096C211yesmaximal, characteristic, complemented
11#10731C212yesmaximal, complemented
12C24×C8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#117C2212yes—
15C22×#107C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C2×#380C2212yes—
21C2×#380C2212yes—
22C2×#380C2212yes—
23#2075—24no—
24#1869—24no—
25#1808—24no—
26#1735—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1440—22no—
38#1438—22no—
39#1437—22no—
40#1436—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C23×D4D411yescharacteristic
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C23×C8—24no—
47C23×C8—22no—
48C23×C8—22no—
49C23×C8—42no—
50C23×C8D414yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—24no—
58C2×C4×D4—24no—
59C2×#117—24no—
60C2×#117—24no—
61C2×#117—24no—
62C2×#117—24no—
63C2×#117—42no—
64C2×#117—42no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D411yescharacteristic
75C22×C22⋊C4D411yescharacteristic
76C22×C42—22no—
77C22×C42C2311yescharacteristic
78C2×#107—22no—
79C2×#107—22no—
80C2×#107—22no—
81C2×#107—22no—
82C2×#107—42no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—42no—
90#380—24no—
91#380—24no—
92#380—24no—
93#380—24no—
94#380—42no—
95#380—42no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103#340—24no—
104#340—22no—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—42no—
109#340—42no—
110#340D414yescomplemented
111#336—22no—
112#336—22no—
113C25C2×D411yescharacteristic
114C25C2×D411yescharacteristic
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—24no—
118C22×D4—22no—
119C22×D4—42no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4#4112yes—
146C23×C4#4114yes—
147C23×C4#4114yes—
148C23×C4C2411yescharacteristic
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—28no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—28no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—44no—
168C22×C8—48no—
169C22×C8—48no—
170C22×C8C2×D412yes—
171C22×C8C2×D412yes—
172C22×C8C2×D412yes—
173C22×C8#4112yes—
174C4×D4—44no—
175C4×D4—44no—
176C4×D4—48no—
177#117—216no—
178#117—48no—
179#117—48no—
180#117—48no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
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—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4C2×D414yes—
206C2×C4⋊C4#4114yes—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—82no—
220C2×C22⋊C4—82no—
221C2×C22⋊C4—82no—
222C2×C22⋊C4—82no—
223C2×C42—21no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42—44no—
228C2×C42—44no—
229C2×C42—44no—
230#107—44no—
231#107—44no—
232C22⋊C8—44no—
233C22⋊C8—44no—
234C22⋊C8—44no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—82no—
241#95—82no—
242C24—21no—
243C24—22no—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—22no—
248C24—21no—
249C24—21no—
250C24—22no—
251C24—21no—
252C24—22no—
253C24—21no—
254C24—22no—
255C24—21no—
256C24—21no—
257C24—22no—
258C24—21no—
259C24—21no—
260C24—22no—
261C24#12112yes—
262C24#12314yes—
263C24#13512yes—
264C24C22×D411yescharacteristic
265C2×D4—28no—
266C2×D4—48no—
267C2×D4—48no—
268C2×D4—44no—
269C2×D4—44no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
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#12112yes—
349C22×C4#12314yes—
350C22×C4#12314yes—
351C22×C4#12414yes—
352C22×C4#12618yes—
353C22×C4#13512yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4112yes—
358C22×C4C2×#4112yes—
359C22×C4C2×#4112yes—
360C22×C4C2×#4111yescharacteristic
361C22×C4H(5,𝔽2)12yes—
362C22×C4H(5,𝔽2)11yescharacteristic
363C22×C42−1+411yescharacteristic
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—416no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—41no—
375C2×C8—42no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—42no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8#12114yes—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C22⋊C4—416no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—88no—
406C22⋊C4—88no—
407C22⋊C4—88no—
408C42—416no—
409C42—41no—
410C42—41no—
411C42—42no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23—24no—
416C23—22no—
417C23—24no—
418C23—24no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—24no—
426C23—22no—
427C23—24no—
428C23—22no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23—42no—
433C23—41no—
434C23—41no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—41no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—41no—
445C23—41no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—41no—
450C23—42no—
451C23—42no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23#39714yes—
458C23C2×#12111yescharacteristic
459C23C2×#12312yes—
460C23D4211yescharacteristic
461C23#54612yes—
462C23#54711yescharacteristic
463C23C2×#13511yescharacteristic
464C23C2×#13511yescharacteristic
465C23#57512yes—
466C23#57611yescharacteristic
467C23#57611yescharacteristic
468C23#57812yes—
469D4—416no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—28no—
478C2×C4—24no—
479C2×C4—28no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—44no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—82no—
513C2×C4—81no—
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#39714yes—
530C2×C4C2×#12111yescharacteristic
531C2×C4C2×#12311yescharacteristic
532C2×C4#52512yes—
533C2×C4C2×#12411yescharacteristic
534C2×C4C2×#12612yes—
535C2×C4#52912yes—
536C2×C4#53211yescharacteristic
537C2×C4#53411yescharacteristic
538C2×C4#53711yescharacteristic
539C2×C4#53812yes—
540C2×C4#53912yes—
541C2×C4#54612yes—
542C2×C4#54811yescharacteristic
543C2×C4#55112yes—
544C2×C4#55112yes—
545C2×C4#55212yes—
546C2×C4#55312yes—
547C2×C4#55311yescharacteristic
548C8—24no—
549C8—42no—
550C8—81no—
551C8—81no—
552C22—24no—
553C22—24no—
554C22—24no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—44no—
564C22—42no—
565C22—44no—
566C22—42no—
567C22—42no—
568C22—82no—
569C22—81no—
570C22—82no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22C2×#39711yescharacteristic
575C22#241611yescharacteristic
576C22#241812yes—
577C22#241912yes—
578C22#254311yescharacteristic
579C22#305011yescharacteristic
580C22#305411yescharacteristic
581C22#305811yescharacteristic
582C22#305911yescharacteristic
583C22#309112yes—
584C22#309112yes—
585C22#309812yes—
586C22#309812yes—
587C22#326912yes—
588C22#328011yescharacteristic
589C22#328711yescharacteristic
590C22#330012yes—
591C22#330212yes—
592C22#330912yes—
593C22#331912yes—
594C22#333911yescharacteristic
595C22#334111yescharacteristic
596C22#334912yes—
597C4—24no—
598C4—42no—
599C4—84no—
600C4—84no—
601C4—81no—
602C4—81no—
603C4—162no—
604C4—162no—
605C4#241211yescharacteristic
606C4#248112yes—
607C4#249811yescharacteristic
608C4#251012yes—
609C4#263911yescharacteristic
610C4#285011yescharacteristic
611C2—44no—
612C2—82no—
613C2#1473511yescharacteristic
614C2#2363811yescharacteristic
615C2#2363911yescharacteristic
616C2#2477811yescharacteristic
617C2#2478511yescharacteristic
618C2#2486612yes—
619C2#2487212yes—
620C2#2493412yes—
621C2#2494812yes—
622C2#2855411yescharacteristic
623C2#2856211yescharacteristic
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
1. C2×#2075C2semidirect
3. C2×#1808C2semidirect
4. C2×#1735C2semidirect
6. #11379C2semidirect
7. #11316C2semidirect
8. #11256C2semidirect
9. #11200C2semidirect
10. #11096C2semidirect
11. #10731C2semidirect
110. #340D4semidirect

Maximal subgroups

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

subgroupindex
C2×#20752
C2×#18692
C2×#18082
C2×#17352
#138712
#113792
#113162
#112562
#112002
#110962
#107312