Finite Group Databasebeta

← #722921

Subgroups — #722921

Order 512.

2542 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic
2C2×#2581C211yesmaximal, characteristic
3C2×#2399C211yesmaximal, characteristic
4C2×Q8×C4⋊C4C211yesmaximal, characteristic
5#16594C214yesmaximal
6#15860C214yesmaximal
7#15347C214yesmaximal
8#15184C212yesmaximal
9#15133C212yesmaximal
10#14880C214yesmaximal
11#14276C211yesmaximal, characteristic
12#14046C212yesmaximal
13#14021C212yesmaximal
14#13925C211yesmaximal, characteristic
15#13786C211yesmaximal, characteristic
16C22×C4×Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20#2582—28no—
21#2581—28no—
22#2399—28no—
23C2×#400C2212yes—
24C2×#398C2214yes—
25C2×#398C2212yes—
26C2×#398C2212yes—
27C2×#398C2214yes—
28C2×#395C2212yes—
29C2×#395C2212yes—
30C2×#395C2212yes—
31Q8×C4⋊C4—28no—
32C2×#391C2211yescharacteristic
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#391C2211yescharacteristic
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2214yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#389C2214yes—
44C2×#387C2214yes—
45C2×#384C2212yes—
46C2×#384C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×#376C2211yescharacteristic
53#1446C2212yes—
54#1445C2212yes—
55#1443C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2211yescharacteristic
77#1434C2214yes—
78#1434C2214yes—
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C2×C4×Q8—28no—
83C2×C4×Q8—24no—
84C2×C4×Q8—24no—
85C2×C4×Q8—24no—
86C2×C4×Q8—24no—
87C2×C4×Q8—24no—
88C2×C4×Q8—24no—
89C2×C4×Q8—24no—
90C2×C4×Q8—24no—
91C2×C4×Q8—48no—
92C2×C4×Q8D418yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2314yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122#400—28no—
123#398—216no—
124#398—28no—
125#398—28no—
126#398—48no—
127#395—28no—
128#395—28no—
129#395—28no—
130#391—28no—
131#391—28no—
132#391—28no—
133#391—24no—
134#391Q818yescomplemented
135#389—28no—
136#389—28no—
137#389—216no—
138#389—28no—
139#389—28no—
140#389—28no—
141#389—48no—
142#387—216no—
143#384—28no—
144#384D418yes—
145#382—24no—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—28no—
150#376—24no—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—42no—
193C22×Q8C2×D414yes—
194C22×Q8C2×Q814yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C4×Q8—232no—
211C4×Q8—432no—
212C4×Q8—416no—
213C4×Q8—416no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C2×D418yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×D418yes—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×D418yes—
274C2×C4⋊C4C2×Q814yes—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—28no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42C2×D418yes—
293C2×C42C2×D414yes—
294C2×C42C2×D414yes—
295C2×C42C2×Q814yes—
296C2×C42#4114yes—
297C2×C42#4114yes—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95C2×Q8116yes—
325C24C2511yescharacteristic
326C2×Q8—28no—
327C2×Q8—28no—
328C2×Q8—216no—
329C2×Q8—28no—
330C2×Q8—28no—
331C2×Q8—28no—
332C2×Q8—216no—
333C2×Q8—28no—
334C2×Q8—28no—
335C2×Q8—48no—
336C2×Q8—48no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8—44no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4C22×D414yes—
416C22×C4C22×D418yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D414yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×Q814yes—
425C22×C4C22×Q814yes—
426C22×C4C22×Q814yes—
427C22×C4C22×Q812yes—
428C22×C4C2×#4114yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4H(5,𝔽2)14yes—
434C22×C42−1+414yes—
435C22×C42−1+414yes—
436C22×C42−1+412yes—
437C22×C42−1+412yes—
438C4⋊C4—216no—
439C4⋊C4—28no—
440C4⋊C4—216no—
441C4⋊C4—28no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—416no—
447C4⋊C4—416no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—416no—
455C4⋊C4—44no—
456C4⋊C4—48no—
457C4⋊C4—88no—
458C4⋊C4—88no—
459C42—216no—
460C42—28no—
461C42—28no—
462C42—416no—
463C42—48no—
464C42—48no—
465C42—48no—
466C42#121116yes—
467C23C23×D411yescharacteristic
468C23C23×D411yescharacteristic
469C23C23×Q811yescharacteristic
470C23C22×#4111yescharacteristic
471C23C22×#4111yescharacteristic
472C23C2×H(5,𝔽2)12yes—
473C23C2×2−1+412yes—
474C23C2×2−1+411yescharacteristic
475C23C2×2−1+411yescharacteristic
476C23C2×2−1+412yes—
477C23#58512yes—
478Q8—232no—
479Q8—416no—
480Q8—416no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4C2×#12114yes—
567C2×C4C2×#12114yes—
568C2×C4C2×#12114yes—
569C2×C4#53314yes—
570C2×C4#53718yes—
571C2×C4#54114yes—
572C2×C4#54714yes—
573C2×C4#54814yes—
574C2×C4D4×Q814yes—
575C2×C4D4×Q814yes—
576C22C22×#12111yescharacteristic
577C22C2×#53311yescharacteristic
578C22C2×#53712yes—
579C22C2×#54111yescharacteristic
580C22C2×#54711yescharacteristic
581C22C2×#54811yescharacteristic
582C22C2×D4×Q811yescharacteristic
583C22C2×D4×Q811yescharacteristic
584C22#348411yescharacteristic
585C22#348511yescharacteristic
586C22#348611yescharacteristic
587C22#348611yescharacteristic
588C22#348912yes—
589C22#349212yes—
590C22#349512yes—
591C22#350212yes—
592C22#350312yes—
593C22#350612yes—
594C22#350912yes—
595C22#355012yes—
596C22#355412yes—
597C22#355512yes—
598C22#355612yes—
599C4—24no—
600C4—24no—
601C4—28no—
602C4—48no—
603C4—44no—
604C4—44no—
605C4—44no—
606C4—44no—
607C4—44no—
608C4—44no—
609C4—82no—
610C4—84no—
611C4—82no—
612C4—82no—
613C4#242918yes—
614C2C2×#242911yescharacteristic
615C2#3522311yescharacteristic
616C2#3563311yescharacteristic
617C2#3593612yes—
618C2#3959912yes—
619C2#3972712yes—
620C2#4251112yes—
621C2#4493411yescharacteristic
622C2#4533411yescharacteristic
623C2#4535411yescharacteristic
624C2#4538911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
134. #391Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
C2×#25812
C2×#23992
C2×Q8×C4⋊C42
#165942
#158602
#153472
#151842
#151332
#148802
#142762
#140462
#140212
#139252
#137862