Finite Group Databasebeta

← #367003

Subgroups — #367003

Order 512.

1726 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic
2C2×#1981C211yesmaximal, characteristic
3C2×#1956C212yesmaximal
4C2×#1956C211yesmaximal, characteristic
5C2×#1803C211yesmaximal, characteristic
6C2×C8×C4⋊C4C211yesmaximal, characteristic
7C2×C4×#103C211yesmaximal, characteristic
8#13910C211yesmaximal, characteristic
9#11050C212yesmaximal
10#11010C212yesmaximal
11#10926C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2212yes—
14C22×C4×Q8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2212yes—
23C22×C4×C8C2212yes—
24C22×C4×C8C2211yescharacteristic
25#2546—24no—
26#2546C418yescomplemented
27C2×#398—22no—
28C2×#391C2212yes—
29C2×#389—22no—
30C2×#389—22no—
31C2×#384C2211yescharacteristic
32C2×#384C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1981—24no—
36#1956—28no—
37#1956—24no—
38#1803—24no—
39C8×C4⋊C4—24no—
40C4×#103—24no—
41C2×#340C2211yescharacteristic
42C2×#340C2212yes—
43C2×#340C2211yescharacteristic
44C2×#336C2212yes—
45#1444—22no—
46C4⋊#95C2212yes—
47C23×Q8C2312yes—
48C23×Q8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—42no—
61C2×C4⋊Q8—42no—
62C2×C4⋊Q8C2×C414yes—
63C2×C4⋊Q8C2×C418yes—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8C2×C414yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—24no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D412yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C42—21no—
83C22×C42—21no—
84C22×C42C2312yes—
85C22×C42C2311yescharacteristic
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8D414yes—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—22no—
92C2×#103—22no—
93C2×#103—22no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—42no—
105C2×#103—42no—
106C2×#103—42no—
107C2×#103—42no—
108C2×#103—42no—
109C2×#103D414yes—
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112C2×C4×C8—24no—
113C2×C4×C8—24no—
114C2×C4×C8—42no—
115C2×C4×C8D414yes—
116#398—216no—
117#391—24no—
118#391C2×C418yes—
119#389—216no—
120#389—48no—
121#384—24no—
122#384—22no—
123#384—22no—
124#384—42no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4C2×C414yes—
128C4×C4⋊C4D414yes—
129C2×#95—24no—
130C2×#95—24no—
131C2×#95—21no—
132C2×#95D412yes—
133#340—24no—
134#340—22no—
135#340—22no—
136#336—24no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—21no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—21no—
160C22×Q8—22no—
161C22×Q8C22×C414yes—
162C22×Q8C22×C412yes—
163C22×Q8C2×D412yes—
164C22×Q8#4112yes—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—22no—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D414yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2411yescharacteristic
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8C2×D412yes—
192C22×C8C2×D414yes—
193C22×C8C2×D412yes—
194C22×C8C2×D412yes—
195C22×C8#4112yes—
196C4⋊Q8—216no—
197C4⋊Q8—28no—
198C4⋊Q8—28no—
199C4⋊Q8—48no—
200C4⋊Q8—48no—
201C4⋊Q8—48no—
202C4⋊Q8—44no—
203C4⋊Q8—44no—
204C4×Q8—28no—
205C4×Q8—28no—
206C4×Q8—44no—
207C4×Q8—44no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—216no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4—48no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—48no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—84no—
247C2×C4⋊C4—82no—
248C2×C4⋊C4C22⋊C418yes—
249C2×C4⋊C4SD1618yes—
250C2×C4⋊C4Q1618yes—
251C2×C4⋊C4C22×C412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4#4112yes—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—44no—
263C2×C42C22×C414yes—
264C2×C42C22×C412yes—
265C2×C42C2×D414yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42#4112yes—
270C2×C42#4112yes—
271C4⋊C8—28no—
272#103—28no—
273#103—28no—
274#103—48no—
275#103—48no—
276#103—48no—
277#103—48no—
278#103—48no—
279C4×C8—28no—
280C4×C8—48no—
281#95—216no—
282#95—42no—
283#95—48no—
284#95C22⋊C418yes—
285C24C22×D411yescharacteristic
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—22no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—22no—
298C2×Q8—24no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—42no—
316C2×Q8—41no—
317C2×Q8—42no—
318C2×Q8—41no—
319C2×Q8—42no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—42no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8C4×D414yes—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C22⋊C418yes—
384C22×C4C2×C22⋊C412yes—
385C22×C4#12412yes—
386C22×C4#12412yes—
387C22×C4#12712yes—
388C22×C4#12712yes—
389C22×C4C2×SD1618yes—
390C22×C4C2×SD1614yes—
391C22×C4C2×Q1618yes—
392C22×C4C2×Q1614yes—
393C22×C4C23×C411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4H(5,𝔽2)11yescharacteristic
404C22×C4H(5,𝔽2)11yescharacteristic
405C22×C4H(5,𝔽2)11yescharacteristic
406C22×C42−1+411yescharacteristic
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—24no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—42no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8#12114yes—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—432no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—816no—
449C4⋊C4—84no—
450C4⋊C4—84no—
451C4⋊C4#103132yescomplemented
452C42—24no—
453C42—24no—
454C42—24no—
455C42—24no—
456C42—48no—
457C42—84no—
458C42C4×D414yes—
459C42C4×D414yes—
460C42#12114yes—
461C42#12114yes—
462C23C22×C22⋊C411yescharacteristic
463C23C2×#12411yescharacteristic
464C23C2×#12711yescharacteristic
465C23D4211yescharacteristic
466C23#54611yescharacteristic
467C23#54811yescharacteristic
468C23#55011yescharacteristic
469C23C22×SD1612yes—
470C23C22×Q1612yes—
471C23#57712yes—
472C23#57812yes—
473Q8—24no—
474Q8—24no—
475Q8—42no—
476Q8—42no—
477Q8—42no—
478Q8—42no—
479Q8—42no—
480Q8—42no—
481Q8—44no—
482Q8—44no—
483Q8—44no—
484Q8—44no—
485Q8—82no—
486Q8—82no—
487Q8—82no—
488Q8—82no—
489C2×C4—28no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—84no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—84no—
532C2×C4—82no—
533C2×C4#39014yes—
534C2×C4#39014yes—
535C2×C4C2×#103116yes—
536C2×C4C2×#10318yes—
537C2×C4#48718yes—
538C2×C4#49214yes—
539C2×C4#49414yes—
540C2×C4C2×C4×D411yescharacteristic
541C2×C4C2×C4×D411yescharacteristic
542C2×C4C2×C4×D411yescharacteristic
543C2×C4C4×#4111yescharacteristic
544C2×C4#51811yescharacteristic
545C2×C4#52011yescharacteristic
546C2×C4C2×#12111yescharacteristic
547C2×C4C2×#12111yescharacteristic
548C2×C4C2×#12111yescharacteristic
549C2×C4#52511yescharacteristic
550C2×C4#52911yescharacteristic
551C2×C4#53211yescharacteristic
552C2×C4#53511yescharacteristic
553C2×C4#53711yescharacteristic
554C2×C4#54011yescharacteristic
555C2×C4#54211yescharacteristic
556C2×C4D4211yescharacteristic
557C2×C4D4211yescharacteristic
558C2×C4#54611yescharacteristic
559C2×C4#54611yescharacteristic
560C2×C4#54711yescharacteristic
561C2×C4#54811yescharacteristic
562C2×C4#54811yescharacteristic
563C2×C4#55011yescharacteristic
564C2×C4#55011yescharacteristic
565C2×C4#55311yescharacteristic
566C2×C4#55311yescharacteristic
567C2×C4#55511yescharacteristic
568C8—24no—
569C8—42no—
570C8—44no—
571C22C2×#39011yescharacteristic
572C22D4×C22⋊C411yescharacteristic
573C22#234111yescharacteristic
574C22#249011yescharacteristic
575C22#249111yescharacteristic
576C22#254011yescharacteristic
577C22#254111yescharacteristic
578C22C22×#10312yes—
579C22#289812yes—
580C22C2×#48712yes—
581C22#313712yes—
582C22C2×#49211yescharacteristic
583C22C2×#49411yescharacteristic
584C22#315411yescharacteristic
585C22#315611yescharacteristic
586C22D4×SD1612yes—
587C22D4×Q1612yes—
588C22#329812yes—
589C22#329912yes—
590C22#335912yes—
591C22#336312yes—
592C22#339212yes—
593C22#339512yes—
594C4—216no—
595C4—24no—
596C4—44no—
597C4—82no—
598C4—84no—
599C4—82no—
600C4—82no—
601C4—162no—
602C4#195618yescomplemented
603C4C4×#12111yescharacteristic
604C4#232411yescharacteristic
605C4#236111yescharacteristic
606C4#236311yescharacteristic
607C4#241411yescharacteristic
608C4#241611yescharacteristic
609C4#242611yescharacteristic
610C4#242811yescharacteristic
611C2#1399811yescharacteristic
612C2C2×#195611yescharacteristic
613C2#2120511yescharacteristic
614C2D4×#10312yes—
615C2#2254212yes—
616C2#2774912yes—
617C2#2776912yes—
618C2#2840011yescharacteristic
619C2#2840111yescharacteristic
620C2#2846111yescharacteristic
621C2#2846211yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×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
26. #2546C4semidirect
451. C4⋊C4#103semidirect
602. C4#1956semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#19812
C2×#19562
C2×#18032
C2×C8×C4⋊C42
C2×C4×#1032
#139102
#110502
#110102
#109262