Finite Group Databasebeta

← #414478

Subgroups — #414478

Order 512.

1152 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2085C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic
3C2×#1993C211yesmaximal, characteristic
4C2×#1943C211yesmaximal, characteristic
5#15231C211yesmaximal, characteristic
6#15143C211yesmaximal, characteristic
7#11325C211yesmaximal, characteristic
8#11324C211yesmaximal, characteristic
9#11323C211yesmaximal, characteristic
10#11323C211yesmaximal, characteristic
11#10863C211yesmaximal, characteristic
12#10861C211yesmaximal, characteristic
13#10843C211yesmaximal, characteristic
14#10841C211yesmaximal, characteristic
15#10696C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×#97C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C2×#395—22no—
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#382C2211yescharacteristic
34#2085—24no—
35#2058—24no—
36#1993—24no—
37#1943—24no—
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#336C2211yescharacteristic
49C2×#336C2211yescharacteristic
50C2×#336C2211yescharacteristic
51#1444—22no—
52#1442—22no—
53#1441—22no—
54C4⋊#95C2211yescharacteristic
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4×#95C2211yescharacteristic
59C23×Q8—21no—
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—41no—
69C2×C4⋊Q8—41no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C2×#107—22no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#107—22no—
92C2×#107—41no—
93C2×#106—22no—
94C2×#106—22no—
95C2×#106—22no—
96C2×#106—22no—
97C2×#106—22no—
98C2×#106—22no—
99C2×#106—22no—
100C2×#106—41no—
101C2×#106—41no—
102C2×#106—41no—
103C2×#106—41no—
104C2×#106—41no—
105C2×#97—41no—
106C2×#97—41no—
107C2×#97—41no—
108C2×#97—41no—
109C2×#97D414yes—
110C2×#97Q814yescomplemented
111C2×C4×C8—41no—
112C2×C4×C8—41no—
113C2×C4×C8Q814yescomplemented
114#395—48no—
115#389—22no—
116#389—22no—
117#389—22no—
118#389—22no—
119#384—22no—
120#384—22no—
121#384—22no—
122#384—41no—
123#384—41no—
124#384—41no—
125#382—22no—
126#382—22no—
127#382—22no—
128#382—41no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—22no—
138C2×#95D412yes—
139C2×#95D412yes—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#340—22no—
146#340—22no—
147#340—22no—
148#340—22no—
149#340—22no—
150#336D414yes—
151#336Q814yescomplemented
152#336Q814yescomplemented
153C22×Q8—42no—
154C22×Q8—42no—
155C22×Q8—44no—
156C22×Q8—42no—
157C22×Q8—82no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2411yescharacteristic
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8C2×D412yes—
188C22×C8C2×D412yes—
189C22×C8C2×D412yes—
190C22×C8C2×D412yes—
191C22×C8C2×Q812yes—
192C22×C8C2×Q812yes—
193C22×C8C2×Q812yes—
194C22×C8C2×Q812yes—
195C22×C8C2×Q812yes—
196C22×C8C2×Q812yes—
197C22×C8C2×Q812yes—
198C22×C8C2×Q812yes—
199C4⋊Q8—28no—
200C4⋊Q8—44no—
201C4⋊Q8—44no—
202C4⋊Q8—44no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
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—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—82no—
260C2×C4⋊C4—82no—
261C2×C4⋊C4—82no—
262C2×C4⋊C4—82no—
263C2×C4⋊C4#4112yes—
264C2×C4⋊C4#4112yes—
265C2×C4⋊C4#4112yes—
266C2×C4⋊C4#4112yes—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×Q812yes—
282C2×C42C2×Q812yes—
283C2×C42C2×Q812yes—
284C2×C42C2×Q812yes—
285#107—28no—
286#107—44no—
287#106—44no—
288#106—44no—
289#106—44no—
290#106—44no—
291#97—44no—
292#97—44no—
293C4×C8—44no—
294#95—24no—
295#95—24no—
296#95—44no—
297#95—44no—
298#95—42no—
299#95—42no—
300#95—42no—
301#95—42no—
302#95—44no—
303#95—44no—
304#95—82no—
305C24C22×D411yescharacteristic
306C2×Q8—48no—
307C2×Q8—84no—
308C2×Q8—84no—
309C2×Q8—88no—
310C2×Q8—88no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
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×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4#12712yes—
370C22×C4#12712yes—
371C22×C4C4⋊Q814yes—
372C22×C4C4⋊Q814yes—
373C22×C4AGL(1,ℤ/8)14yes—
374C22×C4#13714yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×Q811yescharacteristic
378C22×C4C22×Q811yescharacteristic
379C22×C4C22×Q811yescharacteristic
380C22×C4C22×Q811yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4H(5,𝔽2)11yescharacteristic
386C22×C4H(5,𝔽2)11yescharacteristic
387C22×C4H(5,𝔽2)11yescharacteristic
388C22×C42−1+411yescharacteristic
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8C4⋊Q814yes—
414C2×C8C4⋊Q814yes—
415C2×C8C4⋊Q814yes—
416C2×C8C4⋊Q814yes—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—48no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—88no—
451C4⋊C4—88no—
452C4⋊C4—84no—
453C4⋊C4—84no—
454C4⋊C4—88no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C23C2×#12711yescharacteristic
468C23C2×C4⋊Q811yescharacteristic
469C23C2×C4⋊Q811yescharacteristic
470C23#54611yescharacteristic
471C23#54611yescharacteristic
472C23#54611yescharacteristic
473C23#54811yescharacteristic
474C23C2×AGL(1,ℤ/8)12yes—
475C23C2×#13712yes—
476C23#57712yes—
477C23#57712yes—
478Q8—816no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—82no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—82no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4#50118yes—
543C2×C4#52511yescharacteristic
544C2×C4#52511yescharacteristic
545C2×C4C2×C4⋊Q811yescharacteristic
546C2×C4C2×C4⋊Q811yescharacteristic
547C2×C4C2×C4⋊Q811yescharacteristic
548C2×C4C2×C4⋊Q811yescharacteristic
549C2×C4#53311yescharacteristic
550C2×C4#53311yescharacteristic
551C2×C4#53311yescharacteristic
552C2×C4#53311yescharacteristic
553C2×C4#53411yescharacteristic
554C2×C4#53711yescharacteristic
555C2×C4#54311yescharacteristic
556C2×C4#54311yescharacteristic
557C2×C4#54411yescharacteristic
558C2×C4#54411yescharacteristic
559C2×C4#54611yescharacteristic
560C2×C4#54611yescharacteristic
561C2×C4#54611yescharacteristic
562C2×C4#54811yescharacteristic
563C2×C4#55411yescharacteristic
564C2×C4#55411yescharacteristic
565C2×C4#55411yescharacteristic
566C2×C4#55411yescharacteristic
567C2×C4#55411yescharacteristic
568C2×C4#55411yescharacteristic
569C2×C4#55711yescharacteristic
570C2×C4#55711yescharacteristic
571C8—24no—
572C8—24no—
573C8—24no—
574C8—24no—
575C22#253911yescharacteristic
576C22#254011yescharacteristic
577C22#254911yescharacteristic
578C22#254911yescharacteristic
579C22#255011yescharacteristic
580C22#255011yescharacteristic
581C22#286811yescharacteristic
582C22#315111yescharacteristic
583C22#315211yescharacteristic
584C22#315411yescharacteristic
585C22#315411yescharacteristic
586C22C2×#50112yes—
587C22#316312yes—
588C22#316312yes—
589C22#316512yes—
590C22#330612yes—
591C22#331012yes—
592C22#331612yes—
593C22#332012yes—
594C22#332412yes—
595C22#332912yes—
596C22#336912yes—
597C22#337312yes—
598C4—28no—
599C4—44no—
600C4—42no—
601C4—42no—
602C4—82no—
603C4—82no—
604C4—82no—
605C4—82no—
606C4—161no—
607C4—161no—
608C4—161no—
609C4—161no—
610C4#239911yescharacteristic
611C4#254911yescharacteristic
612C4#254911yescharacteristic
613C4#255011yescharacteristic
614C4#255011yescharacteristic
615C4#258311yescharacteristic
616C4#258411yescharacteristic
617C4#265911yescharacteristic
618C2#1517311yescharacteristic
619C2#2836511yescharacteristic
620C2#2836711yescharacteristic
621C2#2843011yescharacteristic
622C2#2843211yescharacteristic
623C2#2877612yes—
624C2#2878212yes—
625C2#2879912yes—
626C2#2880312yes—
627C2#3331611yescharacteristic
628C2#3331811yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
110. C2×#97Q8semidirect
113. C2×C4×C8Q8semidirect
151. #336Q8semidirect
152. #336Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20852
C2×#20582
C2×#19932
C2×#19432
#152312
#151432
#113252
#113242
#113232
#108632
#108612
#108432
#108412
#106962