Finite Group Databasebeta

← #895979

Subgroups — #895979

Order 512.

2946 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2733C211yesmaximal, characteristic
2C2×#2731C212yesmaximal, complemented
3C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
4#17077C212yesmaximal, complemented
5#16514C212yesmaximal, complemented
6#15617C214yesmaximal, complemented
7#15361C214yesmaximal, complemented
8#14883C212yesmaximal, complemented
9#14815C214yesmaximal, complemented
10#14700C212yesmaximal, complemented
11#14467C211yesmaximal, characteristic, complemented
12#14119C214yesmaximal, complemented
13#14026C212yesmaximal, complemented
14C22×C4⋊Q8C2211yescharacteristic
15C22×#124C2212yes—
16C22×C4×Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18#2733—28no—
19#2731—216no—
20C2×#396C2214yes—
21C2×#396C2214yes—
22C2×#394C2214yes—
23C2×#394C2212yes—
24C2×#394C2212yes—
25C2×#394C2214yes—
26C2×#393C2214yes—
27C2×#392C2214yes—
28C4×C4⋊Q8—28no—
29C2×#391C2212yes—
30C2×#391C2211yescharacteristic
31C2×#391C2211yescharacteristic
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#390C2212yes—
38C2×#389C2214yes—
39C2×#389C2212yes—
40C2×#388C2214yes—
41C2×#388C2214yes—
42C2×#388C2214yes—
43C2×#388C2214yes—
44C2×#387C2214yes—
45C2×#384C2211yescharacteristic
46C2×C4×C4⋊C4C2214yes—
47C2×C43C2211yescharacteristic
48#1444C2214yes—
49#1442C2214yes—
50#1441C2212yes—
51#1441C2214yes—
52#1441C2212yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1438C2214yes—
60#1437C2214yes—
61#1437C2214yes—
62#1437C2214yes—
63#1437C2214yes—
64#1436C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68#1434C2212yes—
69C4×#95C2212yes—
70C23×Q8C2311yescharacteristic
71C23×Q8C2311yescharacteristic
72C23×D4C2312yes—
73C23×D4C2312yes—
74C23×D4C2312yes—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8—24no—
77C2×C4⋊Q8D418yescomplemented
78C2×#124—28no—
79C2×#124—28no—
80C2×#124—28no—
81C2×C4×Q8—24no—
82C2×C4×Q8—24no—
83C2×C4×Q8—42no—
84C2×C4×Q8—42no—
85C2×C4×Q8—42no—
86C2×C4×Q8—42no—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120#396—216no—
121#396—48no—
122#394—216no—
123#394—28no—
124#394—48no—
125#394—44no—
126#393—216no—
127#392—216no—
128#391—28no—
129#391—24no—
130#391D418yes—
131#390—28no—
132#390—28no—
133#390—44no—
134#390—44no—
135#390—44no—
136#390—44no—
137#389—216no—
138#389—44no—
139#388—216no—
140#388—216no—
141#388—216no—
142#388—216no—
143#387—216no—
144#384—24no—
145C4×C4⋊C4—216no—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C43—24no—
162C25C2414yes—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—41no—
166C22×Q8—41no—
167C22×Q8—41no—
168C22×Q8—41no—
169C22×Q8—41no—
170C22×Q8—41no—
171C22×Q8—41no—
172C22×Q8—41no—
173C22×Q8C2×D414yes—
174C22×Q8C2×D414yes—
175C22×Q8C2×D414yes—
176C22×Q8#4114yes—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—42no—
185C22×D4—42no—
186C22×D4—42no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4—44no—
190C22×D4—42no—
191C22×D4—42no—
192C22×D4—42no—
193C22×D4—42no—
194C22×D4—42no—
195C22×D4—44no—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C4⋊Q8—232no—
209#124—432no—
210C4×Q8—416no—
211C4×Q8—416no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4C2×D4116yes—
246C2×C4⋊C4C2×D414yes—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42—41no—
303C2×C42—41no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42C2×D414yes—
308C2×C42C2×D414yes—
309C2×C42#4118yes—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—44no—
320#95—44no—
321#95—42no—
322#95—44no—
323#95—44no—
324#95C2×D4116yes—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24C2511yescharacteristic
341C2×Q8—28no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×Q8—44no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×Q8—44no—
354C2×Q8#121116yes—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—84no—
363C2×D4—84no—
364C2×D4—84no—
365C2×D4—84no—
366C2×D4—84no—
367C2×D4—84no—
368C2×D4—84no—
369C2×D4—84no—
370C2×D4—84no—
371C2×D4—88no—
372C2×D4—84no—
373C2×D4—88no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×D412yes—
436C22×C4C22×D414yes—
437C22×C4C22×D414yes—
438C22×C4C22×D412yes—
439C22×C4C22×D414yes—
440C22×C4C22×D414yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4114yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4114yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C4H(5,𝔽2)14yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C42−1+414yes—
450C4⋊C4—232no—
451C4⋊C4—44no—
452C4⋊C4—416no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—416no—
458C4⋊C4—84no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C4⋊C4—84no—
462C22⋊C4—416no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C22⋊C4—416no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C42—28no—
475C42—28no—
476C42—216no—
477C42—44no—
478C42—48no—
479C42—48no—
480C42—48no—
481C42—48no—
482C42—84no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23C23×D411yescharacteristic
519C23C23×D411yescharacteristic
520C23C22×#4111yescharacteristic
521C23C22×#4112yes—
522C23C2×H(5,𝔽2)11yescharacteristic
523C23C2×H(5,𝔽2)11yescharacteristic
524C23C2×H(5,𝔽2)12yes—
525C23C2×2−1+412yes—
526C23#58511yescharacteristic
527C23#58512yes—
528C23#58511yescharacteristic
529Q8—416no—
530Q8—416no—
531D4—816no—
532D4—816no—
533D4—816no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4C2×#12114yes—
606C2×C4C2×#12118yes—
607C2×C4#54818yes—
608C2×C4#55018yes—
609C2×C4#55514yes—
610C22—84no—
611C22—84no—
612C22—84no—
613C22—84no—
614C22—84no—
615C22—84no—
616C22—84no—
617C22—84no—
618C22—84no—
619C22—84no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22C22×#12111yescharacteristic
626C22#345512yes—
627C22#345911yescharacteristic
628C22#345911yescharacteristic
629C22C2×#54812yes—
630C22C2×#55012yes—
631C22C2×#55511yescharacteristic
632C22#348012yes—
633C22#348411yescharacteristic
634C22#348511yescharacteristic
635C22#348812yes—
636C22#349611yescharacteristic
637C22#349611yescharacteristic
638C22#349712yes—
639C22#350612yes—
640C22#351112yes—
641C22#352212yes—
642C22#352911yescharacteristic
643C22#354411yescharacteristic
644C22#354911yescharacteristic
645C22#355812yes—
646C22#356112yes—
647C22#356412yes—
648C4—28no—
649C4—28no—
650C4—24no—
651C4—28no—
652C4—24no—
653C4—42no—
654C4—44no—
655C4—44no—
656C4—82no—
657C4—84no—
658C4—84no—
659C4—84no—
660C2—164no—
661C2#3498411yescharacteristic
662C2#3657612yes—
663C2#3726711yescharacteristic
664C2#3770712yes—
665C2#3938112yes—
666C2#3983012yes—
667C2#4622511yescharacteristic
668C2#4631211yescharacteristic
669C2#5227011yescharacteristic
670C2#5913511yescharacteristic
671C2#5982211yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2733C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2731C2semidirect
3. C2×C4×C4⋊Q8C2semidirect
4. #17077C2semidirect
5. #16514C2semidirect
6. #15617C2semidirect
7. #15361C2semidirect
8. #14883C2semidirect
9. #14815C2semidirect
10. #14700C2semidirect
11. #14467C2semidirect
12. #14119C2semidirect
13. #14026C2semidirect
77. C2×C4⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#27332
C2×#27312
C2×C4×C4⋊Q82
#170772
#165142
#156172
#153612
#148832
#148152
#147002
#144672
#141192
#140262