Finite Group Databasebeta

← #360451

Subgroups — #360451

Order 512.

1300 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1933C212yesmaximal
2C2×#1803C211yesmaximal, characteristic
3C2×C8×C4⋊C4C211yesmaximal, characteristic
4C2×C4×#103C211yesmaximal, characteristic
5#13926C211yesmaximal, characteristic
6#13910C211yesmaximal, characteristic
7#11293C212yesmaximal
8#11067C212yesmaximal
9#10880C212yesmaximal
10#10849C211yesmaximal, characteristic
11#10841C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×C4×C8C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#391C2212yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#384—21no—
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1933—28no—
28#1803—24no—
29C8×C4⋊C4—24no—
30C4×#103C418yescomplemented
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#336C2212yes—
39#1444—22no—
40#1441—22no—
41C4⋊#95—21no—
42C4⋊#95—21no—
43C4⋊#95—21no—
44C4⋊#95C2212yes—
45C23×Q8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2312yes—
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—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C42—21no—
70C22×C42—21no—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8D414yes—
76C2×#103—22no—
77C2×#103—22no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—42no—
81C2×#103C2×C414yes—
82C2×#103C2×C414yes—
83C2×#103D414yes—
84C2×#103Q814yes—
85C2×C4×C8—22no—
86C2×C4×C8—22no—
87C2×C4×C8—24no—
88C2×C4×C8—24no—
89C2×C4×C8—42no—
90C2×C4×C8C2×C414yes—
91#391—24no—
92#391—42no—
93#389—24no—
94#389—24no—
95#389—42no—
96#389—42no—
97#384—22no—
98#384—22no—
99#384—22no—
100#384—22no—
101#384—44no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
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×#95D412yes—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—42no—
119#340C2×C414yes—
120#340C2×C414yes—
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—21no—
139C23×C4—21no—
140C23×C4—22no—
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—22no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8C22×C412yes—
164C22×C8C22×C412yes—
165C22×C8C2×D414yes—
166C22×C8C2×D412yes—
167C22×C8C2×Q812yes—
168C4×Q8—28no—
169C4×Q8—28no—
170C4×Q8—44no—
171C4×Q8—44no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4—81no—
217C2×C4⋊C4—81no—
218C2×C4⋊C4—81no—
219C2×C4⋊C4—81no—
220C2×C4⋊C4SD1614yes—
221C2×C4⋊C4Q1614yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×Q812yes—
227C2×C4⋊C4#4112yes—
228C2×C4⋊C4#4112yes—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42C22×C412yes—
242C2×C42C2×D412yes—
243C2×C42#4112yes—
244C2×C42#4112yes—
245C2×C42#4112yes—
246C2×C42#4112yes—
247C4⋊C8—28no—
248#103—28no—
249#103—48no—
250#103C4⋊C4116yescomplemented
251C4×C8—28no—
252C4×C8—48no—
253#95—24no—
254#95—24no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—82no—
260#95—82no—
261C24C22×D411yescharacteristic
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—22no—
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×Q8—41no—
289C2×Q8C2×C4⋊C414yes—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
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—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
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—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
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×C4C4×D412yes—
349C22×C4C4×D412yes—
350C22×C4#12112yes—
351C22×C4#12112yes—
352C22×C4#12412yes—
353C22×C4#12412yes—
354C22×C4C2×SD1614yes—
355C22×C4C2×SD1612yes—
356C22×C4C2×Q1614yes—
357C22×C4C2×Q1612yes—
358C22×C4#13512yes—
359C22×C4#13512yes—
360C22×C4C23×C411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×Q811yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4H(5,𝔽2)11yescharacteristic
371C22×C4H(5,𝔽2)11yescharacteristic
372C22×C42−1+411yescharacteristic
373C22×C42−1+411yescharacteristic
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—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—41no—
389C2×C8—42no—
390C2×C8—41no—
391C2×C8—42no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8C2×C4⋊C414yes—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—28no—
402C4⋊C4—28no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—48no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—88no—
417C4⋊C4—88no—
418C4⋊C4—84no—
419C4⋊C4—84no—
420C4⋊C4—84no—
421C4⋊C4—84no—
422C4⋊C4C2×C4⋊C414yes—
423C4⋊C4#11714yes—
424C42—42no—
425C42—42no—
426C42—42no—
427C42—42no—
428C42—48no—
429C42—48no—
430C42#11714yes—
431C42C4×D414yes—
432C42#12314yes—
433C42#12614yes—
434C23C2×C4×D411yescharacteristic
435C23C2×#12111yescharacteristic
436C23C2×#12411yescharacteristic
437C23D4211yescharacteristic
438C23#54611yescharacteristic
439C23#54811yescharacteristic
440C23D4×Q811yescharacteristic
441C23C22×SD1611yescharacteristic
442C23C22×Q1611yescharacteristic
443C23C2×#13511yescharacteristic
444C23C2×#13511yescharacteristic
445C23#57611yescharacteristic
446C23#57711yescharacteristic
447C23#57711yescharacteristic
448C23#57811yescharacteristic
449Q8—24no—
450Q8—24no—
451Q8—42no—
452Q8—42no—
453Q8—42no—
454Q8—42no—
455Q8—42no—
456Q8—42no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
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—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4#38814yes—
520C2×C4#38814yes—
521C2×C4C4×SD1614yes—
522C2×C4C4×Q1614yes—
523C2×C4#46514yes—
524C2×C4#46714yes—
525C2×C4#48714yes—
526C2×C4#48814yes—
527C2×C4C22×C4⋊C411yescharacteristic
528C2×C4C2×#11711yescharacteristic
529C2×C4C2×#11711yescharacteristic
530C2×C4C2×C4×D411yescharacteristic
531C2×C4#52011yescharacteristic
532C2×C4#52011yescharacteristic
533C2×C4C2×#12311yescharacteristic
534C2×C4C2×#12611yescharacteristic
535C2×C4#53211yescharacteristic
536C2×C4#53211yescharacteristic
537C2×C4#53311yescharacteristic
538C2×C4#53311yescharacteristic
539C2×C4#54011yescharacteristic
540C2×C4#54111yescharacteristic
541C2×C4#54211yescharacteristic
542C2×C4#54211yescharacteristic
543C2×C4D4211yescharacteristic
544C2×C4#54711yescharacteristic
545C2×C4#54711yescharacteristic
546C2×C4#54711yescharacteristic
547C2×C4D4×Q811yescharacteristic
548C2×C4#55011yescharacteristic
549C2×C4#55211yescharacteristic
550C2×C4#55311yescharacteristic
551C2×C4#55411yescharacteristic
552C2×C4#55411yescharacteristic
553C2×C4#55511yescharacteristic
554C2×C4#55611yescharacteristic
555C8—24no—
556C8—42no—
557C8—44no—
558C22C2×#38811yescharacteristic
559C22D4×C4⋊C411yescharacteristic
560C22#235211yescharacteristic
561C22#242811yescharacteristic
562C22#242911yescharacteristic
563C22#249111yescharacteristic
564C22#249211yescharacteristic
565C22C2×C4×SD1611yescharacteristic
566C22C2×C4×Q1611yescharacteristic
567C22#295111yescharacteristic
568C22#295211yescharacteristic
569C22C2×#46511yescharacteristic
570C22C2×#46711yescharacteristic
571C22#306211yescharacteristic
572C22#306311yescharacteristic
573C22C2×#48711yescharacteristic
574C22C2×#48811yescharacteristic
575C22#313611yescharacteristic
576C22#313711yescharacteristic
577C22D4×SD1611yescharacteristic
578C22#328411yescharacteristic
579C22D4×Q1611yescharacteristic
580C22#328711yescharacteristic
581C22#329511yescharacteristic
582C22#329911yescharacteristic
583C22#330211yescharacteristic
584C22#330411yescharacteristic
585C22#335911yescharacteristic
586C22#336111yescharacteristic
587C22#336311yescharacteristic
588C22#336511yescharacteristic
589C22Q8×SD1611yescharacteristic
590C22#338111yescharacteristic
591C22Q8×Q1611yescharacteristic
592C22#338311yescharacteristic
593C4—24no—
594C4—44no—
595C4—44no—
596C4—44no—
597C4—44no—
598C4—44no—
599C4—82no—
600C4—162no—
601C4—162no—
602C4#193318yescomplemented
603C4#228511yescharacteristic
604C4#231411yescharacteristic
605C4D4×C4⋊C411yescharacteristic
606C4#235411yescharacteristic
607C4#242211yescharacteristic
608C4#242311yescharacteristic
609C4#249611yescharacteristic
610C4#249711yescharacteristic
611C2#1387611yescharacteristic
612C2C2×#193311yescharacteristic
613C2#2108711yescharacteristic
614C2C4⋊C4×SD1611yescharacteristic
615C2C4⋊C4×Q1611yescharacteristic
616C2#2277011yescharacteristic
617C2#2277111yescharacteristic
618C2#2538611yescharacteristic
619C2#2538711yescharacteristic
620C2#2547111yescharacteristic
621C2#2547211yescharacteristic
622C2#2776911yescharacteristic
623C2#2777011yescharacteristic
624C2#2781511yescharacteristic
625C2#2781611yescharacteristic
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
250. #103C4⋊C4semidirect
602. C4#1933semidirect

Maximal subgroups

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

subgroupindex
C2×#19332
C2×#18032
C2×C8×C4⋊C42
C2×C4×#1032
#139262
#139102
#112932
#110672
#108802
#108492
#108412