Finite Group Databasebeta

← #415936

Subgroups — #415936

Order 512.

1842 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#489C211yesmaximal, characteristic
2C22×#477C211yesmaximal, characteristic, complemented
3C2×#2545C211yesmaximal, characteristic
4C2×#1983C211yesmaximal, characteristic
5C2×#1763C211yesmaximal, characteristic, complemented
6#14155C211yesmaximal, characteristic, complemented
7#11124C212yesmaximal, complemented
8#11123C212yesmaximal
9#11010C212yesmaximal, complemented
10#11009C212yesmaximal
11#10835C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×#124C2211yescharacteristic
15C22×#124C2211yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17C2×#489—24no—
18C2×#489—24no—
19C2×#489—24no—
20C2×#477—24no—
21C2×#477—24no—
22C2×#477—24no—
23C22×#107C2212yes—
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2212yes—
26C22×#103C2212yes—
27C22×#103C2212yes—
28C22×#103C2212yes—
29C22×#102C2212yes—
30C22×#102C2212yes—
31C22×#97C2211yescharacteristic
32#2545—24no—
33#2545—24no—
34C2×#396—22no—
35C2×#391—22no—
36C2×#391C2211yescharacteristic
37C2×#390C2211yescharacteristic
38C2×#388—22no—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41#1983—24no—
42#1763—24no—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1445—22no—
47C4⋊#95C2212yes—
48C4⋊#95C2212yes—
49C23×Q8—21no—
50C23×Q8C2311yescharacteristic
51C23×Q8C2311yescharacteristic
52C23×D4C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—24no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—42no—
64C2×C4⋊Q8D414yes—
65C2×#124—22no—
66C2×#124—22no—
67C2×#124—24no—
68C2×#124—22no—
69C2×#124—22no—
70C2×#124—22no—
71C2×#124—22no—
72C2×#124—22no—
73C2×#124—42no—
74C2×#124D414yes—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8—22no—
78C2×C4×Q8—22no—
79C2×C4×Q8—22no—
80C2×C4×Q8D414yes—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4D412yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—22no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—21no—
92C22×C42—21no—
93C22×C42—21no—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97#489—232no—
98#477—232no—
99C2×#107—24no—
100C2×#107—24no—
101C2×#107—24no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8D414yes—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103—24no—
110C2×#103—24no—
111C2×#103—24no—
112C2×#103—24no—
113C2×#103—24no—
114C2×#103—24no—
115C2×#103—24no—
116C2×#103—42no—
117C2×#102—24no—
118C2×#102—24no—
119C2×#102—24no—
120C2×#102—24no—
121C2×#102—24no—
122C2×#102—42no—
123C2×#97—22no—
124C2×#97—22no—
125C2×#97D414yes—
126#396—216no—
127#391—22no—
128#391—22no—
129#391—48no—
130#390—22no—
131#390—22no—
132#388—216no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C2×#95—22no—
138C2×#95—24no—
139C2×#95—24no—
140C2×#95—21no—
141C2×#95D412yes—
142#340—24no—
143#336—42no—
144#336—42no—
145C25—21no—
146C22×Q8—24no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—22no—
158C22×Q8—21no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—44no—
165C22×Q8—44no—
166C22×Q8—42no—
167C22×Q8—42no—
168C22×Q8C2×D412yes—
169C22×Q8C2×D412yes—
170C22×Q8#4114yes—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4C2×D412yes—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2411yescharacteristic
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8C2×D414yes—
204C22×C8C2×D414yes—
205C4⋊Q8—28no—
206C4⋊Q8—28no—
207C4⋊Q8—48no—
208C4⋊Q8—44no—
209C4⋊Q8—44no—
210C4⋊Q8—44no—
211C4⋊Q8—44no—
212#124—28no—
213#124—28no—
214#124—48no—
215#124—44no—
216#124—44no—
217#124—44no—
218#124—44no—
219C4×Q8—28no—
220C4×Q8—28no—
221C4×Q8—44no—
222C4×Q8—44no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4#4114yes—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—44no—
263C2×C42—24no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42—44no—
275C2×C42—42no—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42#4112yes—
282#107—216no—
283C4⋊C8—28no—
284#103—216no—
285#103—216no—
286#103—216no—
287#103—48no—
288#102—216no—
289#102—48no—
290#97—28no—
291#95—28no—
292#95—24no—
293#95—48no—
294#95—42no—
295#95—48no—
296C24—22no—
297C24—41no—
298C24—41no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24—42no—
303C24—44no—
304C24C22×D411yescharacteristic
305C2×Q8—22no—
306C2×Q8—24no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—416no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—48no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—84no—
343C2×Q8—84no—
344C2×Q8—88no—
345C2×Q8#12414yes—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—42no—
353C2×D4—41no—
354C2×D4—42no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4#12012yes—
402C22×C4#12012yes—
403C22×C4#12112yes—
404C22×C4#12112yes—
405C22×C4#12712yes—
406C22×C4#12712yes—
407C22×C4AGL(1,ℤ/8)14yes—
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4112yes—
415C22×C4C2×#4112yes—
416C22×C4H(5,𝔽2)12yes—
417C22×C42−1+412yes—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—48no—
439C4⋊C4—416no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C22⋊C4—416no—
449C22⋊C4—416no—
450C22⋊C4—48no—
451C22⋊C4—48no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—42no—
457C42—42no—
458C42—48no—
459C42—84no—
460C42#12014yes—
461C42#12014yes—
462C42#12114yes—
463C42#12114yes—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—44no—
468C23—44no—
469C23—84no—
470C23—84no—
471C23—81no—
472C23—81no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—84no—
479C23—84no—
480C23C2×#12011yescharacteristic
481C23C2×#12111yescharacteristic
482C23C2×#12711yescharacteristic
483C23#54612yes—
484C23#54812yes—
485C23C2×AGL(1,ℤ/8)12yes—
486C23#57512yes—
487C23#57814yes—
488Q8—24no—
489Q8—24no—
490Q8—42no—
491Q8—44no—
492Q8—44no—
493Q8—42no—
494Q8—44no—
495Q8—44no—
496Q8—816no—
497Q8—81no—
498Q8—81no—
499Q8—81no—
500Q8—81no—
501D4—44no—
502D4—44no—
503D4—81no—
504D4—81no—
505D4—81no—
506D4—81no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—48no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—48no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—82no—
539C2×C4—81no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—84no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—84no—
549C2×C4#39214yes—
550C2×C4#39214yes—
551C2×C4C2×#12011yescharacteristic
552C2×C4C2×#12011yescharacteristic
553C2×C4C2×#12111yescharacteristic
554C2×C4C2×#12111yescharacteristic
555C2×C4#52512yes—
556C2×C4C2×#12411yescharacteristic
557C2×C4#52912yes—
558C2×C4#53211yescharacteristic
559C2×C4#53511yescharacteristic
560C2×C4#53611yescharacteristic
561C2×C4#53712yes—
562C2×C4#54212yes—
563C2×C4#54612yes—
564C2×C4#54612yes—
565C2×C4#54712yes—
566C2×C4#54812yes—
567C2×C4#54812yes—
568C2×C4#55012yes—
569C8—44no—
570C8—44no—
571C22—84no—
572C22—84no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—162no—
577C22—162no—
578C22—164no—
579C22C2×#39211yescharacteristic
580C22#242612yes—
581C22#243211yescharacteristic
582C22#243511yescharacteristic
583C22#254012yes—
584C22#300612yes—
585C22#301112yes—
586C22#303612yes—
587C22#304412yes—
588C22#315111yescharacteristic
589C22#315311yescharacteristic
590C22#315612yes—
591C22#330814yes—
592C22#331314yes—
593C22#336714yes—
594C22#337714yes—
595C4—24no—
596C4—24no—
597C4—44no—
598C4—44no—
599C4—84no—
600C4—82no—
601C4—82no—
602C4—82no—
603C4—84no—
604C4#240712yes—
605C4#242712yes—
606C4#243111yescharacteristic
607C4#243611yescharacteristic
608C4#262011yescharacteristic
609C4#285011yescharacteristic
610C2—162no—
611C2#1521511yescharacteristic
612C2#2342511yescharacteristic
613C2#2342811yescharacteristic
614C2#2528614yes—
615C2#2566112yes—
616C2#2589612yes—
617C2#2842412yes—
618C2#2842612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#477C2semidirect
5. C2×#1763C2semidirect
6. #14155C2semidirect
7. #11124C2semidirect
9. #11010C2semidirect
11. #10835C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4892
C22×#4772
C2×#25452
C2×#19832
C2×#17632
#141552
#111242
#111232
#110102
#110092
#108352