Finite Group Databasebeta

← #723279

Subgroups — #723279

Order 512.

2546 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2351C211yesmaximal, characteristic, complemented
2C2×#2326C211yesmaximal, characteristic
3#16704C214yesmaximal, complemented
4#15296C214yesmaximal, complemented
5#15292C214yesmaximal, complemented
6#15148C212yesmaximal
7#15146C212yesmaximal, complemented
8#15103C214yesmaximal, complemented
9#14276C211yesmaximal, characteristic, complemented
10#14026C212yesmaximal
11#14017C212yesmaximal, complemented
12#13941C211yesmaximal, characteristic
13#13926C211yesmaximal, characteristic, complemented
14#13906C211yesmaximal, characteristic
15#13848C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20C2×#400C2214yes—
21C2×#399C2214yes—
22#2351—28no—
23#2326—28no—
24C2×#391C2211yescharacteristic
25C2×#391C2212yes—
26C2×#390C2211yescharacteristic
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#389C2212yes—
30C2×#389C2214yes—
31C2×#389C2212yes—
32C2×#388C2214yes—
33C2×#385C2214yes—
34C2×#384C2211yescharacteristic
35C2×#384C2211yescharacteristic
36C2×#382C2212yes—
37C2×#382C2211yescharacteristic
38C2×#380C2214yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×#376C2211yescharacteristic
44#1444C2212yes—
45#1444C2214yes—
46#1442C2212yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2212yes—
51#1441C2212yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2212yes—
55#1441C2212yes—
56#1441C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2212yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2212yes—
65#1437C2214yes—
66#1437C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C4×#95C2212yes—
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4C2312yes—
80C2×#124—24no—
81C2×#124—24no—
82C2×#124Q818yes—
83C2×C4×Q8—28no—
84C2×C4×Q8—24no—
85C2×C4×Q8—24no—
86C2×C4×Q8—44no—
87C2×C4×Q8—44no—
88C2×C4×Q8D418yes—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2314yes—
120C22×C42C2311yescharacteristic
121#400—48no—
122#399—48no—
123#391—24no—
124#391—28no—
125#390—24no—
126#390—44no—
127#390—44no—
128#389—28no—
129#389—216no—
130#389—44no—
131#388—216no—
132#385—216no—
133#384—24no—
134#384—24no—
135#382—28no—
136#382—24no—
137#380—48no—
138C4×C4⋊C4—28no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—24no—
141C4×C4⋊C4—44no—
142#376—24no—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2311yescharacteristic
147C2×#95C2311yescharacteristic
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2311yescharacteristic
153C2×#95C2311yescharacteristic
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C25C2412yes—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8C2×D414yes—
176C22×Q8C2×Q814yes—
177C22×Q8#4114yes—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4C2×Q814yes—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4C2414yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204#124—232no—
205C4×Q8—232no—
206C4×Q8—432no—
207C4×D4—416no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4C2×D418yes—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4C2×Q8116yes—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—28no—
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—44no—
287C2×C42—44no—
288C2×C42C2×D418yes—
289C2×C42C2×D414yes—
290C2×C42C2×Q814yes—
291C2×C42#4114yes—
292C2×C42#4114yes—
293#95—24no—
294#95—24no—
295#95—22no—
296#95—22no—
297#95—24no—
298#95—28no—
299#95—24no—
300#95—28no—
301#95—22no—
302#95—22no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—24no—
307#95—24no—
308#95—42no—
309#95—44no—
310#95—44no—
311#95—42no—
312#95—42no—
313#95—42no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—24no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24C22×Q814yes—
331C24C2511yescharacteristic
332C2×Q8—28no—
333C2×Q8—28no—
334C2×Q8—28no—
335C2×Q8—28no—
336C2×Q8—28no—
337C2×Q8—28no—
338C2×Q8C22⋊Q8116yes—
339C2×D4—28no—
340C2×D4—28no—
341C2×D4—28no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—44no—
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—22no—
361C22×C4—22no—
362C22×C4—22no—
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—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—416no—
417C22×C4—48no—
418C22×C4—88no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4C22×D414yes—
422C22×C4C22×D414yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×Q814yes—
427C22×C4C22×Q814yes—
428C22×C4C22×Q812yes—
429C22×C4C2×#4114yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4H(5,𝔽2)14yes—
438C22×C4H(5,𝔽2)12yes—
439C22×C4H(5,𝔽2)12yes—
440C22×C42−1+414yes—
441C4⋊C4—216no—
442C4⋊C4—28no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—416no—
446C4⋊C4—48no—
447C4⋊C4—416no—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—84no—
453C4⋊C4—84no—
454C4⋊C4—88no—
455C4⋊C4—88no—
456C22⋊C4—232no—
457C22⋊C4—48no—
458C22⋊C4—88no—
459C22⋊C4—88no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C22⋊C4—88no—
463C42—28no—
464C42—216no—
465C42—416no—
466C42—44no—
467C42—48no—
468C42—48no—
469C42#123116yes—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—28no—
474C23—28no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23C23×D411yescharacteristic
494C23C23×Q811yescharacteristic
495C23C22×#4111yescharacteristic
496C23C22×#4111yescharacteristic
497C23C22×#4111yescharacteristic
498C23C2×H(5,𝔽2)11yescharacteristic
499C23C2×H(5,𝔽2)11yescharacteristic
500C23C2×H(5,𝔽2)12yes—
501C23C2×2−1+412yes—
502C23#58512yes—
503C23#58512yes—
504Q8—232no—
505D4—416no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—816no—
587C2×C4C2×C22⋊Q818yes—
588C2×C4C2×C22⋊Q814yes—
589C2×C4C2×#12314yes—
590C2×C4C2×#12314yes—
591C2×C4C2×#12314yes—
592C2×C4#53514yes—
593C2×C4#55314yes—
594C2×C4#55314yes—
595C2×C4#55414yes—
596C2×C4#55414yes—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—48no—
601C22—48no—
602C22—82no—
603C22—82no—
604C22—84no—
605C22—82no—
606C22—84no—
607C22—82no—
608C22C22×C22⋊Q811yescharacteristic
609C22C22×#12311yescharacteristic
610C22C2×#53511yescharacteristic
611C22#346012yes—
612C22C2×#55311yescharacteristic
613C22C2×#55311yescharacteristic
614C22C2×#55411yescharacteristic
615C22C2×#55411yescharacteristic
616C22#348812yes—
617C22#349512yes—
618C22#349712yes—
619C22#350312yes—
620C22#350412yes—
621C22#350811yescharacteristic
622C22#350911yescharacteristic
623C22#351011yescharacteristic
624C22#351011yescharacteristic
625C22#351612yes—
626C22#351712yes—
627C22#353612yes—
628C22#354012yes—
629C22#355412yes—
630C22#355812yes—
631C4—24no—
632C4—24no—
633C4—28no—
634C4—44no—
635C4—44no—
636C4—44no—
637C4—44no—
638C4—84no—
639C4—84no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4#248718yes—
645C2—84no—
646C2C2×#248711yescharacteristic
647C2#3525411yescharacteristic
648C2#3567311yescharacteristic
649C2#3581612yes—
650C2#3925912yes—
651C2#3967412yes—
652C2#4280512yes—
653C2#4531211yescharacteristic
654C2#4534911yescharacteristic
655C2#4542511yescharacteristic
656C2#4543411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2351C2semidirect
3. #16704C2semidirect
4. #15296C2semidirect
5. #15292C2semidirect
7. #15146C2semidirect
8. #15103C2semidirect
9. #14276C2semidirect
11. #14017C2semidirect
13. #13926C2semidirect
15. #13848C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23512
C2×#23262
#167042
#152962
#152922
#151482
#151462
#151032
#142762
#140262
#140172
#139412
#139262
#139062
#138482