Finite Group Databasebeta

← #697411

Subgroups — #697411

Order 512.

2350 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2441C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3#16992C212yesmaximal, complemented
4#16991C212yesmaximal, complemented
5#15730C214yesmaximal
6#15593C214yesmaximal
7#15228C212yesmaximal, complemented
8#15207C212yesmaximal, complemented
9#15133C212yesmaximal, complemented
10#15068C212yesmaximal, complemented
11#14264C211yesmaximal, characteristic
12#14130C211yesmaximal, characteristic
13#14006C212yesmaximal, complemented
14#13909C212yesmaximal, complemented
15#13881C211yesmaximal, characteristic
16#13844C211yesmaximal, characteristic
17#13801C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2441—28no—
23C2×#399C2212yes—
24C2×#398C2214yes—
25C2×#396C2212yes—
26#2317—28no—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#387C2212yes—
35C2×#385C2214yes—
36C2×#385C2212yes—
37C2×#384C2212yes—
38C2×#384C2211yescharacteristic
39C2×#382C2212yes—
40C2×#382C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47#1444C2211yescharacteristic
48#1444C2211yescharacteristic
49#1444C2212yes—
50#1444C2214yes—
51#1444C2212yes—
52#1444C2214yes—
53#1442C2211yescharacteristic
54#1442C2211yescharacteristic
55#1442C2212yes—
56#1442C2212yes—
57#1442C2212yes—
58#1442C2212yes—
59#1442C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1439C2212yes—
69#1439C2212yes—
70#1439C2214yes—
71#1437C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2214yes—
80#1434C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2214yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C2×#125—24no—
88C2×#125—44no—
89C2×#117—24no—
90C2×#117—28no—
91C2×#117—44no—
92C2×#117D418yes—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2314yes—
128C22×C42C2312yes—
129#399—28no—
130#398—48no—
131#396—28no—
132#389—24no—
133#389—24no—
134#389—28no—
135#389—28no—
136#389—28no—
137#389—28no—
138#389—44no—
139#387—28no—
140#385—216no—
141#385—44no—
142#384—28no—
143#384—24no—
144#382—28no—
145#382—44no—
146#380—24no—
147#380Q818yes—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—44no—
150C4×C22⋊C4—24no—
151C4×C22⋊C4Q818yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2311yescharacteristic
179C2×#95C2312yes—
180C2×#95C2311yescharacteristic
181C25C2411yescharacteristic
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—44no—
186C23×C4—42no—
187C23×C4—44no—
188C23×C4—42no—
189C23×C4C2×D414yes—
190C23×C4C2×Q814yes—
191C23×C4C2×Q814yes—
192C23×C4C2411yescharacteristic
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2411yescharacteristic
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208#125—416no—
209#117—232no—
210#117—416no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
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—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—416no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×Q818yes—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4C2×D418yes—
281C2×C22⋊C4C2×Q818yes—
282C2×C22⋊C4C2×Q818yes—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42C2×D418yes—
296C2×C42C2×Q818yes—
297C2×C42#4118yes—
298#95—22no—
299#95—22no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—42no—
313#95—42no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—42no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95C2×Q814yes—
325#95C2×Q814yes—
326#95C2×Q818yes—
327C24—22no—
328C24—21no—
329C24—22no—
330C24—21no—
331C24—22no—
332C24—21no—
333C24—22no—
334C24—21no—
335C24C22×D412yes—
336C24C22×Q812yes—
337C24C22×Q812yes—
338C24C2511yescharacteristic
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—216no—
344C22×C4—216no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—48no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4—88no—
422C22×C4—88no—
423C22×C4C4⋊Q8116yes—
424C22×C4C22×D414yes—
425C22×C4C22×D414yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×Q814yes—
429C22×C4C22×Q814yes—
430C22×C4C22×Q818yes—
431C22×C4C22×Q812yes—
432C22×C4C22×Q812yes—
433C22×C4C22×Q812yes—
434C22×C4C22×Q812yes—
435C22×C4C2×#4114yes—
436C22×C4C2×#4114yes—
437C22×C4C2×#4114yes—
438C22×C4H(5,𝔽2)12yes—
439C22×C4H(5,𝔽2)14yes—
440C22×C42−1+412yes—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—28no—
444C4⋊C4—28no—
445C4⋊C4—216no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—88no—
457C4⋊C4—88no—
458C4⋊C4—84no—
459C4⋊C4—88no—
460C4⋊C4—84no—
461C4⋊C4—88no—
462C22⋊C4—216no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—84no—
469C42—216no—
470C42—216no—
471C42—48no—
472C42—44no—
473C42—44no—
474C42—88no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—22no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23C2×C4⋊Q814yes—
498C23C23×D411yescharacteristic
499C23C23×Q811yescharacteristic
500C23C23×Q811yescharacteristic
501C23C22×#4112yes—
502C23C2×H(5,𝔽2)12yes—
503C23C2×H(5,𝔽2)11yescharacteristic
504C23C2×2−1+411yescharacteristic
505C23#58512yes—
506C23#58512yes—
507C23#58511yescharacteristic
508C23#58511yescharacteristic
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—232no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—816no—
597C2×C4#52514yes—
598C2×C4C2×C4⋊Q814yes—
599C2×C4C2×C4⋊Q814yes—
600C2×C4#53414yes—
601C2×C4#55418yes—
602C2×C4#55418yes—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—44no—
607C22—44no—
608C22—42no—
609C22—42no—
610C22—44no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22C2×#52511yescharacteristic
615C22C22×C4⋊Q811yescharacteristic
616C22C2×#53411yescharacteristic
617C22#345812yes—
618C22C2×#55412yes—
619C22C2×#55412yes—
620C22#348811yescharacteristic
621C22#348911yescharacteristic
622C22#349412yes—
623C22#349512yes—
624C22#350011yescharacteristic
625C22#350011yescharacteristic
626C22#350111yescharacteristic
627C22#350211yescharacteristic
628C22#350311yescharacteristic
629C22#350412yes—
630C22#350411yescharacteristic
631C22#351212yes—
632C22#351212yes—
633C22#354612yes—
634C22#355412yes—
635C22#356212yes—
636C22#356212yes—
637C4—24no—
638C4—44no—
639C4—48no—
640C4—44no—
641C4—44no—
642C4—44no—
643C4—42no—
644C4—44no—
645C4—42no—
646C4—44no—
647C4—81no—
648C4—82no—
649C4—81no—
650C4—84no—
651C4—82no—
652C4#258318yes—
653C2—44no—
654C2C2×#258311yescharacteristic
655C2#3533011yescharacteristic
656C2#3556611yescharacteristic
657C2#3608011yescharacteristic
658C2#3608311yescharacteristic
659C2#3787812yes—
660C2#3949412yes—
661C2#4291311yescharacteristic
662C2#4296711yescharacteristic
663C2#4494212yes—
664C2#4527312yes—
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
3. #16992C2semidirect
4. #16991C2semidirect
7. #15228C2semidirect
8. #15207C2semidirect
9. #15133C2semidirect
10. #15068C2semidirect
13. #14006C2semidirect
14. #13909C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24412
C2×#23172
#169922
#169912
#157302
#155932
#152282
#152072
#151332
#150682
#142642
#141302
#140062
#139092
#138812
#138442
#138012