Finite Group Databasebeta

← #401240

Subgroups — #401240

Order 512.

1504 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2068C212yesmaximal, complemented
2C2×#2042C212yesmaximal
3C2×#1983C212yesmaximal
4C2×#1981C211yesmaximal, characteristic, complemented
5C2×#1925C211yesmaximal, characteristic
6#15243C211yesmaximal, characteristic
7#13821C211yesmaximal, characteristic, complemented
8#11322C211yesmaximal, characteristic, complemented
9#11076C211yesmaximal, characteristic
10#11010C211yesmaximal, characteristic, complemented
11#10932C212yesmaximal, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2212yes—
14C22×#107C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#102C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C2×#391C2212yes—
25C2×#390C2211yescharacteristic
26C2×#389C2212yes—
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30#2068—28no—
31#2042—28no—
32#1983—28no—
33#1981—24no—
34#1925—24no—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39#1439—21no—
40#1439—21no—
41#1439—22no—
42#1436—22no—
43C4⋊#95C2211yescharacteristic
44#1434—21no—
45C4×#95C2211yescharacteristic
46C23×Q8C2312yes—
47C23×D4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C2×#124—24no—
56C2×#124—24no—
57C2×#124—24no—
58C2×#124—24no—
59C2×#124—42no—
60C2×#124—42no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C42—21no—
73C22×C42—22no—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76C2×#107—24no—
77C2×#107—22no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—42no—
82C2×C4⋊C8—42no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—42no—
94C2×#103—42no—
95C2×#102—22no—
96C2×#102—22no—
97C2×#102—24no—
98C2×#102—22no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—42no—
102C2×#102D414yes—
103#391—24no—
104#391—24no—
105#390—22no—
106#390—22no—
107#389—24no—
108#389D418yes—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—44no—
112C4×C4⋊C4—42no—
113C4×C4⋊C4D414yes—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95D411yescharacteristic
122C2×#95D411yescharacteristic
123C2×#95D412yes—
124#340—24no—
125#340—22no—
126#336—22no—
127#336—24no—
128C25—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8#4114yes—
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—22no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4C2×D412yes—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2411yescharacteristic
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—21no—
179C22×C8C2×D412yes—
180C4⋊Q8—28no—
181C4⋊Q8—48no—
182C4⋊Q8—44no—
183#124—216no—
184#124—48no—
185#124—48no—
186#124—48no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4C2×D414yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D414yes—
213C2×C4⋊C4#4112yes—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—24no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—44no—
235C2×C42—44no—
236C2×C42—82no—
237C2×C42C2×D414yes—
238C2×C42C2×D412yes—
239C2×C42#4114yes—
240C2×C42#4112yes—
241#107—44no—
242C4⋊C8—48no—
243C4⋊C8—44no—
244#103—48no—
245#103—48no—
246#103—48no—
247#103—48no—
248#102—28no—
249#102—48no—
250#102—44no—
251#95—24no—
252#95—28no—
253#95—22no—
254#95—22no—
255#95—42no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—42no—
260#95—81no—
261C24—22no—
262C24—41no—
263C24—42no—
264C24—42no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24—42no—
269C24—41no—
270C24—42no—
271C24—41no—
272C24C22×D411yescharacteristic
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×D4—22no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—22no—
305C2×D4—41no—
306C2×D4—42no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—42no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—41no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4#12114yes—
369C22×C4#12112yes—
370C22×C4#12114yes—
371C22×C4#12112yes—
372C22×C4#13512yes—
373C22×C4AGL(1,ℤ/8)12yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D412yes—
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4112yes—
379C22×C4C2×#4112yes—
380C22×C4C2×#4111yescharacteristic
381C22×C4H(5,𝔽2)12yes—
382C22×C4H(5,𝔽2)11yescharacteristic
383C22×C42−1+411yescharacteristic
384C2×C8—22no—
385C2×C8—24no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—41no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—416no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—42no—
416C4⋊C4#12114yes—
417C4⋊C4#12714yes—
418C22⋊C4—416no—
419C22⋊C4—416no—
420C22⋊C4—84no—
421C22⋊C4—84no—
422C42—22no—
423C42—24no—
424C42—22no—
425C42—24no—
426C42—24no—
427C42—24no—
428C42—24no—
429C42—48no—
430C42—88no—
431C23—42no—
432C23—44no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—81no—
437C23—81no—
438C23—81no—
439C23—82no—
440C23—81no—
441C23—82no—
442C23—81no—
443C23—82no—
444C23—82no—
445C23—81no—
446C23—82no—
447C23—81no—
448C23—82no—
449C23—82no—
450C23—81no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23C2×#12111yescharacteristic
455C23C2×#12112yes—
456C23D4211yescharacteristic
457C23#54612yes—
458C23#54811yescharacteristic
459C23C2×#13511yescharacteristic
460C23C2×AGL(1,ℤ/8)11yescharacteristic
461C23#57512yes—
462C23#57711yescharacteristic
463C23#57811yescharacteristic
464C23#57812yes—
465Q8—44no—
466Q8—44no—
467Q8—44no—
468Q8—44no—
469Q8—82no—
470Q8—82no—
471Q8—82no—
472Q8—82no—
473D4—42no—
474D4—44no—
475D4—42no—
476D4—81no—
477D4—81no—
478D4—82no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
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—81no—
529C2×C4—81no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—81no—
533C2×C4#46314yes—
534C2×C4C2×#12111yescharacteristic
535C2×C4#52512yes—
536C2×C4#52511yescharacteristic
537C2×C4#52912yes—
538C2×C4C2×#12711yescharacteristic
539C2×C4#53211yescharacteristic
540C2×C4#53212yes—
541C2×C4#53511yescharacteristic
542C2×C4#53711yescharacteristic
543C2×C4#53812yes—
544C2×C4#54212yes—
545C2×C4D4212yes—
546C2×C4#54612yes—
547C2×C4#54711yescharacteristic
548C2×C4#54812yes—
549C2×C4#55012yes—
550C2×C4#55112yes—
551C2×C4#55311yescharacteristic
552C8—42no—
553C8—81no—
554C8—82no—
555C22—84no—
556C22—84no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—161no—
561C22—161no—
562C22—161no—
563C22—161no—
564C22—162no—
565C22—162no—
566C22#241111yescharacteristic
567C22#241611yescharacteristic
568C22#242412yes—
569C22#242612yes—
570C22#285611yescharacteristic
571C22C2×#46311yescharacteristic
572C22#303512yes—
573C22#303612yes—
574C22#304011yescharacteristic
575C22#304412yes—
576C22#304411yescharacteristic
577C22#304612yes—
578C22#304611yescharacteristic
579C22#326912yes—
580C22#327211yescharacteristic
581C22#328311yescharacteristic
582C22#330012yes—
583C22#330812yes—
584C22#331312yes—
585C22#332612yes—
586C22#336011yescharacteristic
587C22#336711yescharacteristic
588C22#337712yes—
589C4—24no—
590C4—44no—
591C4—44no—
592C4—44no—
593C4—44no—
594C4—42no—
595C4—84no—
596C4—84no—
597C4—161no—
598C4#239211yescharacteristic
599C4#246112yes—
600C4#254011yescharacteristic
601C4#254111yescharacteristic
602C4#255412yes—
603C4#266211yescharacteristic
604C2—162no—
605C2#1481511yescharacteristic
606C2#2448111yescharacteristic
607C2#2449711yescharacteristic
608C2#2472711yescharacteristic
609C2#2474311yescharacteristic
610C2#2508912yes—
611C2#2510412yes—
612C2#2528312yes—
613C2#2528612yes—
614C2#3316411yescharacteristic
615C2#3316911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2068C2semidirect
4. C2×#1981C2semidirect
7. #13821C2semidirect
8. #11322C2semidirect
10. #11010C2semidirect
11. #10932C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20682
C2×#20422
C2×#19832
C2×#19812
C2×#19252
#152432
#138212
#113222
#110762
#110102
#109322