Finite Group Databasebeta

← #472399

Subgroups — #472399

Order 512.

1426 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2035C212yesmaximal, complemented
2C2×#1954C211yesmaximal, characteristic
3C2×#1933C211yesmaximal, characteristic, complemented
4#15177C211yesmaximal, characteristic
5#15129C211yesmaximal, characteristic
6#11227C211yesmaximal, characteristic, complemented
7#11200C212yesmaximal, complemented
8#11095C212yesmaximal, complemented
9#11077C211yesmaximal, characteristic, complemented
10#10970C212yesmaximal, complemented
11#10861C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8C2212yes—
13C22×C4×D4C2211yescharacteristic
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2212yes—
21C22×C4×C8C2211yescharacteristic
22C2×#394—21no—
23C2×#391C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28#2035—28no—
29#1954—24no—
30#1933—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37#1444—21no—
38#1441—21no—
39#1440—22no—
40#1439—22no—
41#1438—21no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C23×Q8C2311yescharacteristic
45C23×D4D411yescharacteristic
46C24×C4C2312yes—
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8—42no—
53C2×C22⋊Q8—42no—
54C2×C22⋊Q8—42no—
55C2×C22⋊Q8—42no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—42no—
59C2×C4×D4—42no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4D412yes—
73C22×C42C2311yescharacteristic
74C2×#106—24no—
75C2×#106—24no—
76C2×#106—42no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—42no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—24no—
82C2×#103—22no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103D414yescomplemented
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—42no—
92C2×C22⋊C8—42no—
93C2×C4×C8—42no—
94C2×C4×C8D414yes—
95#394—44no—
96#391—22no—
97#391—22no—
98#389—22no—
99#389—22no—
100#380—24no—
101#380—24no—
102#380—42no—
103#380—42no—
104#380—42no—
105#380—42no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—21no—
116#340—22no—
117#340—22no—
118#340—24no—
119#340—42no—
120#340—42no—
121#340—42no—
122C25C2×D412yes—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—21no—
132C22×Q8C2×D412yes—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4SD1614yes—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—22no—
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×C8C2×D412yes—
176C22×C8C2×D412yes—
177C22×C8C2×D412yes—
178C22×C8#4112yes—
179C22⋊Q8—48no—
180C22⋊Q8—48no—
181C22⋊Q8—48no—
182C22⋊Q8—48no—
183C4×D4—44no—
184C4×D4—44no—
185C4×D4—48no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
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—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4#4114yes—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4—82no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42C2×D412yes—
238C2×C42#4112yes—
239#106—48no—
240C4⋊C8—44no—
241#103—28no—
242#103—44no—
243#103—48no—
244C22⋊C8—48no—
245C22⋊C8—48no—
246C4×C8—44no—
247#95—44no—
248#95—44no—
249#95—44no—
250#95—42no—
251#95—42no—
252#95—42no—
253#95—42no—
254#95—42no—
255#95—81no—
256#95—81no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24C2×SD1614yes—
272C24C22×D411yescharacteristic
273C2×Q8—22no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—42no—
279C2×Q8—41no—
280C2×Q8—42no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×D4—28no—
292C2×D4—48no—
293C2×D4—44no—
294C2×D4—48no—
295C2×D4—44no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
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—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
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—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4#12112yes—
374C22×C4#12112yes—
375C22×C4C2×SD1612yes—
376C22×C4#13512yes—
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4112yes—
381C22×C4C2×#4111yescharacteristic
382C22×C4H(5,𝔽2)12yes—
383C22×C4H(5,𝔽2)12yes—
384C22×C4H(5,𝔽2)12yes—
385C22×C4H(5,𝔽2)11yescharacteristic
386C22×C42−1+411yescharacteristic
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8#12114yes—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—44no—
409C4⋊C4—416no—
410C4⋊C4—42no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—84no—
425C22⋊C4—48no—
426C22⋊C4—416no—
427C22⋊C4—88no—
428C22⋊C4—88no—
429C22⋊C4—88no—
430C42—22no—
431C42—22no—
432C42—24no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23C2×#12111yescharacteristic
469C23#53412yes—
470C23D4211yescharacteristic
471C23#54612yes—
472C23#54711yescharacteristic
473C23C22×SD1611yescharacteristic
474C23C2×#13511yescharacteristic
475C23#57512yes—
476C23#57711yescharacteristic
477C23#57711yescharacteristic
478C23#57812yes—
479Q8—42no—
480Q8—44no—
481Q8—42no—
482Q8—81no—
483Q8—82no—
484Q8—81no—
485D4—416no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
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—81no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4#46514yes—
547C2×C4C2×#12111yescharacteristic
548C2×C4#53211yescharacteristic
549C2×C4#53411yescharacteristic
550C2×C4#53511yescharacteristic
551C2×C4#53512yes—
552C2×C4#53511yescharacteristic
553C2×C4#53711yescharacteristic
554C2×C4#53812yes—
555C2×C4#53812yes—
556C2×C4#54012yes—
557C2×C4#54212yes—
558C2×C4#54612yes—
559C2×C4#54711yescharacteristic
560C2×C4#55312yes—
561C2×C4#55511yescharacteristic
562C2×C4#55912yes—
563C2×C4#56012yes—
564C2×C4#56212yes—
565C8—24no—
566C8—42no—
567C8—82no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22#253711yescharacteristic
584C22#254411yescharacteristic
585C22#261711yescharacteristic
586C22#267812yes—
587C22#270112yes—
588C22C2×#46511yescharacteristic
589C22#305411yescharacteristic
590C22#305911yescharacteristic
591C22#306011yescharacteristic
592C22#318712yes—
593C22#319112yes—
594C22#320112yes—
595C22#320112yes—
596C22#327712yes—
597C22D4×SD1611yescharacteristic
598C22#328411yescharacteristic
599C22#329812yes—
600C22#330212yes—
601C22#332312yes—
602C22#332612yes—
603C22#333511yescharacteristic
604C22#334011yescharacteristic
605C22#335212yes—
606C4—24no—
607C4—42no—
608C4—84no—
609C4—82no—
610C4—82no—
611C4—82no—
612C4—161no—
613C4—162no—
614C4—161no—
615C4#254311yescharacteristic
616C4#261911yescharacteristic
617C4#266211yescharacteristic
618C4#271012yes—
619C4#280312yes—
620C4#281311yescharacteristic
621C2—84no—
622C2#1681911yescharacteristic
623C2#2822311yescharacteristic
624C2#2823411yescharacteristic
625C2#2860011yescharacteristic
626C2#2860411yescharacteristic
627C2#2973411yescharacteristic
628C2#2973711yescharacteristic
629C2#3102712yes—
630C2#3102912yes—
631C2#3177212yes—
632C2#3177712yes—
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×#2035C2semidirect
3. C2×#1933C2semidirect
6. #11227C2semidirect
7. #11200C2semidirect
8. #11095C2semidirect
9. #11077C2semidirect
10. #10970C2semidirect
11. #10861C2semidirect
86. C2×#103D4semidirect

Maximal subgroups

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

subgroupindex
C2×#20352
C2×#19542
C2×#19332
#151772
#151292
#112272
#112002
#110952
#110772
#109702
#108612