Finite Group Databasebeta

← #626974

Subgroups — #626974

Order 512.

2512 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×#2868C211yesmaximal, characteristic, complemented
2C2×#2629C211yesmaximal, characteristic
3#15790C214yesmaximal, complemented
4#15783C214yesmaximal, complemented
5#15771C214yesmaximal
6#15322C214yesmaximal
7#15146C212yesmaximal, complemented
8#15125C212yesmaximal
9#14709C211yesmaximal, characteristic, complemented
10#14340C211yesmaximal, characteristic
11#14283C211yesmaximal, characteristic, complemented
12#14053C212yesmaximal, complemented
13#13876C212yesmaximal
14#13779C211yesmaximal, characteristic, complemented
15#13761C211yesmaximal, characteristic
16C22×C4⋊Q8C2212yes—
17C22×C4⋊Q8C2214yes—
18C22×#125C2211yescharacteristic
19C22×#117C2211yescharacteristic
20#2868—28no—
21#2629—28no—
22C2×#400C2214yes—
23C2×#398C2214yes—
24C2×#396C2214yes—
25C2×#393C2214yes—
26C2×#391C2214yes—
27C2×#389C2212yes—
28C2×#388C2214yes—
29C2×#384C2211yescharacteristic
30C2×#384C2212yes—
31C2×#384C2214yes—
32C2×#384C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C22⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×#376C2211yescharacteristic
40C2×C43C2211yescharacteristic
41#1446C2214yes—
42#1446C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1437C2212yes—
56#1437C2212yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2212yes—
63#1436C2212yes—
64#1436C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4×#95C2211yescharacteristic
72C4×#95C2211yescharacteristic
73C23×Q8C2314yes—
74C24×C4C2311yescharacteristic
75C2×C4⋊Q8—28no—
76C2×C4⋊Q8—216no—
77C2×C4⋊Q8—28no—
78C2×C4⋊Q8—44no—
79C2×C4⋊Q8—48no—
80C2×C4⋊Q8—48no—
81C2×#125—24no—
82C2×#125—24no—
83C2×#125—24no—
84C2×#117—24no—
85C2×#117—24no—
86C2×#117—24no—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2314yes—
113C22×C42C2311yescharacteristic
114#400—48no—
115#398—216no—
116#396—216no—
117#393—48no—
118#391—48no—
119#389—28no—
120#388—48no—
121#384—28no—
122#384—216no—
123#384—24no—
124#384D418yes—
125#382—24no—
126#380—28no—
127#380—44no—
128C4×C4⋊C4—44no—
129C4×C22⋊C4—24no—
130C4×C22⋊C4Q818yescomplemented
131#376—24no—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2311yescharacteristic
146C2×#95C2311yescharacteristic
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C43—24no—
159C25C2411yescharacteristic
160C22×Q8—28no—
161C22×Q8—28no—
162C22×Q8—28no—
163C22×Q8—28no—
164C22×Q8—44no—
165C22×Q8—44no—
166C22×Q8—44no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4C2×Q814yes—
174C23×C4C2414yes—
175C23×C4C2414yes—
176C23×C4C2412yes—
177C23×C4C2414yes—
178C23×C4C2411yescharacteristic
179C23×C4C2411yescharacteristic
180C23×C4C2414yes—
181C23×C4C2412yes—
182C23×C4C2411yescharacteristic
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C4⋊Q8—432no—
189C4⋊Q8—464no—
190#125—416no—
191#117—416no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4C2×D418yes—
227C2×C4⋊C4C2×D4116yes—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4C2×Q818yes—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42—44no—
265C2×C42C2×D414yes—
266C2×C42C2×Q814yes—
267C2×C42C2×Q814yes—
268C2×C42#4114yes—
269C2×C42#4114yes—
270C2×C42#4114yes—
271#95—28no—
272#95—28no—
273#95—24no—
274#95—24no—
275#95—22no—
276#95—22no—
277#95—28no—
278#95—28no—
279#95—24no—
280#95—28no—
281#95—24no—
282#95—24no—
283#95—28no—
284#95—24no—
285#95—28no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95C2×Q818yes—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—22no—
302C24—22no—
303C24—21no—
304C24—22no—
305C24—22no—
306C24—21no—
307C24C22×Q812yes—
308C24C2511yescharacteristic
309C2×Q8—416no—
310C2×Q8—416no—
311C2×Q8—416no—
312C2×Q8—416no—
313C2×Q8—88no—
314C2×Q8—88no—
315C2×Q8—88no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4C22×D418yes—
394C22×C4C22×D414yes—
395C22×C4C22×D412yes—
396C22×C4C22×Q812yes—
397C22×C4C22×Q814yes—
398C22×C4C22×Q814yes—
399C22×C4C22×Q812yes—
400C22×C4C2×#4114yes—
401C22×C4C2×#4114yes—
402C22×C4C2×#4112yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4112yes—
405C22×C4C2×#4112yes—
406C22×C4C2×#4112yes—
407C22×C4H(5,𝔽2)14yes—
408C22×C4H(5,𝔽2)14yes—
409C22×C4H(5,𝔽2)12yes—
410C22×C42−1+414yes—
411C22×C42−1+414yes—
412C22×C42−1+412yes—
413C4⋊C4—216no—
414C4⋊C4—416no—
415C4⋊C4—416no—
416C4⋊C4—416no—
417C4⋊C4—416no—
418C4⋊C4—416no—
419C4⋊C4—48no—
420C4⋊C4—416no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—88no—
425C22⋊C4—216no—
426C22⋊C4—48no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—88no—
430C22⋊C4—88no—
431C42—216no—
432C42—216no—
433C42—28no—
434C42—28no—
435C42—28no—
436C42—44no—
437C42—44no—
438C42—88no—
439C42#124116yes—
440C23—22no—
441C23—24no—
442C23—22no—
443C23—22no—
444C23—24no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—42no—
452C23—42no—
453C23—41no—
454C23—41no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23C23×D411yescharacteristic
464C23C23×Q811yescharacteristic
465C23C22×#4111yescharacteristic
466C23C22×#4111yescharacteristic
467C23C22×#4111yescharacteristic
468C23C2×H(5,𝔽2)12yes—
469C23C2×H(5,𝔽2)11yescharacteristic
470C23C2×H(5,𝔽2)12yes—
471C23C2×2−1+412yes—
472C23C2×2−1+411yescharacteristic
473C23C2×2−1+412yes—
474Q8—832no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
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—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—41no—
505C2×C4—44no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—416no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—41no—
520C2×C4—44no—
521C2×C4—41no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—41no—
533C2×C4—44no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4C2×#12418yes—
556C2×C4C2×#12414yes—
557C2×C4#53314yes—
558C2×C4#54014yes—
559C2×C4#54114yes—
560C2×C4#54214yes—
561C2×C4#54418yes—
562C2×C4#54614yes—
563C2×C4D4×Q814yes—
564C2×C4#55414yes—
565C2×C4#55614yes—
566C22—44no—
567C22—44no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—82no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—82no—
577C22C22×#12411yescharacteristic
578C22C2×#53311yescharacteristic
579C22C2×#54011yescharacteristic
580C22C2×#54111yescharacteristic
581C22C2×#54211yescharacteristic
582C22C2×#54412yes—
583C22C2×#54611yescharacteristic
584C22C2×D4×Q811yescharacteristic
585C22C2×#55411yescharacteristic
586C22C2×#55611yescharacteristic
587C22#348412yes—
588C22#348612yes—
589C22#350312yes—
590C22#350811yescharacteristic
591C22#350911yescharacteristic
592C22#351112yes—
593C22#351112yes—
594C22#351112yes—
595C22#351112yes—
596C22#354312yes—
597C22#354412yes—
598C22#355212yes—
599C22#355212yes—
600C4—28no—
601C4—28no—
602C4—24no—
603C4—24no—
604C4—48no—
605C4—44no—
606C4—42no—
607C4—82no—
608C4—84no—
609C4—84no—
610C4—81no—
611C4—81no—
612C4—84no—
613C4#249218yes—
614C2—82no—
615C2C2×#249211yescharacteristic
616C2#3528411yescharacteristic
617C2#3529411yescharacteristic
618C2#3559212yes—
619C2#3732712yes—
620C2#3768612yes—
621C2#3769411yescharacteristic
622C2#3769511yescharacteristic
623C2#3949712yes—
624C2#3951311yescharacteristic
625C2#3951411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2868C2semidirect
3. #15790C2semidirect
4. #15783C2semidirect
7. #15146C2semidirect
9. #14709C2semidirect
11. #14283C2semidirect
12. #14053C2semidirect
14. #13779C2semidirect
130. C4×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28682
C2×#26292
#157902
#157832
#157712
#153222
#151462
#151252
#147092
#143402
#142832
#140532
#138762
#137792
#137612