Finite Group Databasebeta

← #360434

Subgroups — #360434

Order 512.

1280 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×#1927C212yesmaximal
2C2×#1844C211yesmaximal, characteristic
3C2×#1803C211yesmaximal, characteristic
4C2×C4×#103C211yesmaximal, characteristic
5#13926C211yesmaximal, characteristic
6C4⋊C42C211yesmaximal, characteristic
7#11257C212yesmaximal
8#11122C212yesmaximal
9#10908C212yesmaximal
10#10849C211yesmaximal, characteristic
11#10834C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#106C2212yes—
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C2×#391C2212yes—
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×C4×C4⋊C4—21no—
24C2×C4×C4⋊C4C2212yes—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1927—28no—
28#1844—24no—
29#1803—24no—
30C4×#103C418yescomplemented
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#336C2212yes—
37C2×#336C2212yes—
38#1441—22no—
39#1436—22no—
40C4⋊#95—21no—
41C4⋊#95—21no—
42C4⋊#95—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C23×Q8C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8C2×C414yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C42—21no—
68C22×C42—21no—
69C22×C42—22no—
70C22×C42C2312yes—
71C22×C42C2311yescharacteristic
72C2×#106—24no—
73C2×#106—24no—
74C2×#106—24no—
75C2×#106—24no—
76C2×#106—42no—
77C2×#106—42no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—42no—
86C2×#103C2×C414yes—
87C2×#103C2×C414yes—
88C2×#103D414yes—
89C2×#103Q814yes—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92C2×C4×C8C2×C414yes—
93#391—24no—
94#391—42no—
95#384—22no—
96#384—22no—
97#384—22no—
98#384D414yes—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—42no—
104C4×C4⋊C4—44no—
105C4×C4⋊C4C2×C414yes—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95D412yes—
115#340—24no—
116#340—24no—
117#340—42no—
118#340C2×C414yes—
119#340C2×C414yes—
120#336—24no—
121#336—24no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8C22×C412yes—
134C22×Q8C2×D412yes—
135C22×Q8C2×Q812yes—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8C22×C412yes—
163C22×C8C22×C412yes—
164C22×C8C2×D412yes—
165C22×C8C2×Q812yes—
166C22×C8#4112yes—
167C22×C8#4112yes—
168C4×Q8—28no—
169C4×Q8—28no—
170C4×Q8—44no—
171C4×Q8—44no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
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—24no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—81no—
213C2×C4⋊C4—81no—
214C2×C4⋊C4—81no—
215C2×C4⋊C4—81no—
216C2×C4⋊C4SD1614yes—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C2×D414yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×Q812yes—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42—44no—
236C2×C42—44no—
237C2×C42—82no—
238C2×C42C22×C412yes—
239C2×C42C2×D412yes—
240C2×C42#4112yes—
241C2×C42#4112yes—
242C2×C42#4112yes—
243C2×C42#4112yes—
244#106—48no—
245#106—48no—
246C4⋊C8—44no—
247#103—28no—
248#103—48no—
249#103C4⋊C4116yescomplemented
250C4×C8—28no—
251#95—24no—
252#95—24no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—82no—
258#95—82no—
259#95—82no—
260C24C22×D411yescharacteristic
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8C2×C4⋊C414yes—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4C4×D412yes—
349C22×C4C4×D412yes—
350C22×C4#12112yes—
351C22×C4#12112yes—
352C22×C4#12312yes—
353C22×C4#12312yes—
354C22×C4C2×SD1614yes—
355C22×C4C2×SD1612yes—
356C22×C4#13512yes—
357C22×C4#13712yes—
358C22×C4#13712yes—
359C22×C4C23×C411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×Q811yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4H(5,𝔽2)11yescharacteristic
370C22×C4H(5,𝔽2)11yescharacteristic
371C22×C42−1+411yescharacteristic
372C22×C42−1+411yescharacteristic
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8C2×C4⋊C414yes—
393C2×C8#11714yes—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—28no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—44no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—84no—
418C4⋊C4—84no—
419C4⋊C4—84no—
420C4⋊C4C2×C4⋊C414yes—
421C42—42no—
422C42—42no—
423C42—42no—
424C42—42no—
425C42—48no—
426C42—48no—
427C42—88no—
428C42#11714yes—
429C42C4×D414yes—
430C42#12414yes—
431C42#12614yes—
432C23C2×C4×D411yescharacteristic
433C23C2×#12111yescharacteristic
434C23C2×#12311yescharacteristic
435C23D4211yescharacteristic
436C23#54711yescharacteristic
437C23D4×Q811yescharacteristic
438C23#55011yescharacteristic
439C23C22×SD1611yescharacteristic
440C23C2×#13511yescharacteristic
441C23C2×#13711yescharacteristic
442C23C2×#13711yescharacteristic
443C23#57711yescharacteristic
444C23#57711yescharacteristic
445C23#57711yescharacteristic
446C23#57711yescharacteristic
447Q8—24no—
448Q8—24no—
449Q8—42no—
450Q8—42no—
451Q8—42no—
452Q8—42no—
453Q8—42no—
454Q8—42no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4—81no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4#38814yes—
518C2×C4#38814yes—
519C2×C4C4×SD1614yes—
520C2×C4#44114yes—
521C2×C4#46114yes—
522C2×C4#46414yes—
523C2×C4#48314yes—
524C2×C4#48514yes—
525C2×C4C22×C4⋊C411yescharacteristic
526C2×C4C2×#11711yescharacteristic
527C2×C4C2×#11711yescharacteristic
528C2×C4C2×C4×D411yescharacteristic
529C2×C4#52011yescharacteristic
530C2×C4#52011yescharacteristic
531C2×C4C2×#12411yescharacteristic
532C2×C4C2×#12611yescharacteristic
533C2×C4#53211yescharacteristic
534C2×C4#53211yescharacteristic
535C2×C4#53311yescharacteristic
536C2×C4#53311yescharacteristic
537C2×C4#54011yescharacteristic
538C2×C4#54111yescharacteristic
539C2×C4#54211yescharacteristic
540C2×C4#54211yescharacteristic
541C2×C4D4211yescharacteristic
542C2×C4#54611yescharacteristic
543C2×C4#54711yescharacteristic
544C2×C4#54711yescharacteristic
545C2×C4#54811yescharacteristic
546C2×C4D4×Q811yescharacteristic
547C2×C4#55211yescharacteristic
548C2×C4#55311yescharacteristic
549C2×C4#55411yescharacteristic
550C2×C4#55411yescharacteristic
551C2×C4#55511yescharacteristic
552C2×C4#55611yescharacteristic
553C8—24no—
554C8—24no—
555C8—82no—
556C22C2×#38811yescharacteristic
557C22D4×C4⋊C411yescharacteristic
558C22#235411yescharacteristic
559C22#242211yescharacteristic
560C22#242311yescharacteristic
561C22#242811yescharacteristic
562C22#242911yescharacteristic
563C22C2×C4×SD1611yescharacteristic
564C22C2×#44111yescharacteristic
565C22#295211yescharacteristic
566C22#295211yescharacteristic
567C22C2×#46111yescharacteristic
568C22C2×#46411yescharacteristic
569C22#304211yescharacteristic
570C22#304311yescharacteristic
571C22C2×#48311yescharacteristic
572C22C2×#48511yescharacteristic
573C22#309911yescharacteristic
574C22#309911yescharacteristic
575C22#327811yescharacteristic
576C22#327811yescharacteristic
577C22D4×SD1611yescharacteristic
578C22#328411yescharacteristic
579C22#333511yescharacteristic
580C22#334011yescharacteristic
581C22#335611yescharacteristic
582C22#335611yescharacteristic
583C22Q8×SD1611yescharacteristic
584C22#338111yescharacteristic
585C22#338711yescharacteristic
586C22#338711yescharacteristic
587C22#339211yescharacteristic
588C22#339411yescharacteristic
589C22#340011yescharacteristic
590C22#340011yescharacteristic
591C4—24no—
592C4—44no—
593C4—44no—
594C4—44no—
595C4—82no—
596C4—84no—
597C4—82no—
598C4—82no—
599C4—162no—
600C4#192718yescomplemented
601C4#228511yescharacteristic
602C4#231411yescharacteristic
603C4D4×C4⋊C411yescharacteristic
604C4#235211yescharacteristic
605C4#249111yescharacteristic
606C4#249211yescharacteristic
607C4#249611yescharacteristic
608C4#249711yescharacteristic
609C2#1387611yescharacteristic
610C2C2×#192711yescharacteristic
611C2#2106811yescharacteristic
612C2#2267411yescharacteristic
613C2C4⋊C4×SD1611yescharacteristic
614C2#2280511yescharacteristic
615C2#2281411yescharacteristic
616C2#2501411yescharacteristic
617C2#2501511yescharacteristic
618C2#2506011yescharacteristic
619C2#2506111yescharacteristic
620C2#2536811yescharacteristic
621C2#2536911yescharacteristic
622C2#2545311yescharacteristic
623C2#2545411yescharacteristic
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
30. C4×#103C4semidirect
249. #103C4⋊C4semidirect
600. C4#1927semidirect

Maximal subgroups

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

subgroupindex
C2×#19272
C2×#18442
C2×#18032
C2×C4×#1032
#139262
C4⋊C422
#112572
#111222
#109082
#108492
#108342