Finite Group Databasebeta

← #719287

Subgroups — #719287

Order 512.

3278 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic, complemented
2C2×#2452C212yesmaximal
3C2×D4×C4⋊C4C211yesmaximal, characteristic
4#16520C214yesmaximal, complemented
5#15421C214yesmaximal, complemented
6#15265C214yesmaximal, complemented
7#15194C212yesmaximal, complemented
8#15128C212yesmaximal, complemented
9#14720C214yesmaximal, complemented
10#14276C211yesmaximal, characteristic, complemented
11#14150C211yesmaximal, characteristic, complemented
12#14018C212yesmaximal
13#13941C211yesmaximal, characteristic
14#13856C211yesmaximal, characteristic, complemented
15#13785C211yesmaximal, characteristic
16C22×C4⋊Q8C2212yes—
17C22×C4⋊Q8C2212yes—
18C22×C22⋊Q8C2214yes—
19C22×C4×D4C2212yes—
20C22×C4×D4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22#2551—28no—
23#2452—216no—
24C2×#396C2214yes—
25C2×#395C2214yes—
26C2×#395C2214yes—
27C2×#393C2214yes—
28D4×C4⋊C4—28no—
29C2×#390C2212yes—
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#385C2214yes—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#384C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×#376C2211yescharacteristic
49#1446C2214yes—
50#1445C2214yes—
51#1444C2214yes—
52#1444C2212yes—
53#1444C2212yes—
54#1444C2212yes—
55#1444C2212yes—
56#1444C2212yes—
57#1444C2214yes—
58#1442C2212yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1439C2212yes—
65#1439C2214yes—
66#1439C2212yes—
67#1439C2214yes—
68#1437C2214yes—
69#1436C2214yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2211yescharacteristic
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C22⋊#95C2214yes—
78C23×Q8C2314yes—
79C23×D4C2311yescharacteristic
80C24×C4C2314yes—
81C24×C4C2312yes—
82C2×C4⋊Q8—28no—
83C2×C4⋊Q8—28no—
84C2×C4⋊Q8—28no—
85C2×C4⋊Q8—44no—
86C2×C4⋊Q8—48no—
87C2×C22⋊Q8—216no—
88C2×C22⋊Q8—48no—
89C2×C22⋊Q8—48no—
90C2×C4×D4—28no—
91C2×C4×D4—28no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4—44no—
95C2×C4×D4Q818yescomplemented
96C22×C4⋊C4—28no—
97C22×C4⋊C4—48no—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#396—216no—
131#395—216no—
132#395—48no—
133#393—216no—
134#390—28no—
135#390D418yes—
136#390Q818yescomplemented
137#389—28no—
138#389—28no—
139#389—28no—
140#389—28no—
141#389—28no—
142#385—216no—
143#385—216no—
144#385—48no—
145#384D418yes—
146#382—24no—
147#380—28no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—44no—
153#376—24no—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C25C2412yes—
178C22×Q8—28no—
179C22×Q8—28no—
180C22×Q8—28no—
181C22×Q8—44no—
182C22×Q8—44no—
183C22×Q8—44no—
184C22×Q8—44no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4C2×D414yes—
190C22×D4C2×Q814yes—
191C22×D4C2×Q814yes—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—28no—
195C23×C4—28no—
196C23×C4—28no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4C2×Q818yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C4⋊Q8—432no—
219C4⋊Q8—432no—
220C22⋊Q8—464no—
221C4×D4—232no—
222C4×D4—432no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
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—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—432no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D418yes—
275C2×C4⋊C4C2×D418yes—
276C2×C4⋊C4C2×Q814yes—
277C2×C4⋊C4C2×Q814yes—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4C2×D4116yes—
294C2×C22⋊C4C2×Q818yes—
295C2×C22⋊C4C2×Q8116yes—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—44no—
308C2×C42C2×D414yes—
309C2×C42C2×D414yes—
310C2×C42C2×Q814yes—
311C2×C42C2×Q814yes—
312C2×C42#4114yes—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—42no—
330#95—42no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336C24—22no—
337C24—22no—
338C24—24no—
339C24—24no—
340C24—22no—
341C24—24no—
342C24—22no—
343C24—24no—
344C24C22×D414yes—
345C24C22×Q814yes—
346C24C22×Q814yes—
347C24C2511yescharacteristic
348C2×Q8—416no—
349C2×Q8—416no—
350C2×Q8—416no—
351C2×Q8—88no—
352C2×Q8—88no—
353C2×Q8—88no—
354C2×Q8—88no—
355C2×D4—28no—
356C2×D4—28no—
357C2×D4—28no—
358C2×D4—28no—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4C4⋊Q8116yescomplemented
362C22×C4—22no—
363C22×C4—22no—
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—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—216no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—416no—
430C22×C4—416no—
431C22×C4—48no—
432C22×C4—416no—
433C22×C4—48no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—88no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4C22×D414yes—
441C22×C4C22×D418yes—
442C22×C4C22×D414yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×Q814yes—
448C22×C4C22×Q814yes—
449C22×C4C22×Q814yes—
450C22×C4C22×Q812yes—
451C22×C4C22×Q812yes—
452C22×C4C22×Q812yes—
453C22×C4C22×Q812yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4H(5,𝔽2)14yes—
458C22×C4H(5,𝔽2)14yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C42−1+414yes—
461C22×C42−1+412yes—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—216no—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—216no—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—416no—
475C4⋊C4—416no—
476C4⋊C4—416no—
477C4⋊C4—416no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—416no—
481C4⋊C4—48no—
482C22⋊C4—216no—
483C22⋊C4—416no—
484C22⋊C4—416no—
485C22⋊C4—416no—
486C22⋊C4—416no—
487C22⋊C4—416no—
488C42—28no—
489C42—28no—
490C42—48no—
491C42—48no—
492C42—48no—
493C42C22⋊Q8116yescomplemented
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—44no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23C2×C4⋊Q818yes—
517C23C23×D411yescharacteristic
518C23C23×D411yescharacteristic
519C23C23×Q811yescharacteristic
520C23C23×Q811yescharacteristic
521C23C22×#4111yescharacteristic
522C23C2×H(5,𝔽2)11yescharacteristic
523C23C2×H(5,𝔽2)12yes—
524C23C2×H(5,𝔽2)12yes—
525C23C2×2−1+411yescharacteristic
526C23C2×2−1+412yes—
527C23#58512yes—
528Q8—832no—
529D4—232no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—432no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—832no—
611C2×C4C2×C22⋊Q814yes—
612C2×C4C2×C22⋊Q814yes—
613C2×C4C2×C22⋊Q814yes—
614C2×C4C2×C4⋊Q814yes—
615C2×C4#54414yes—
616C2×C4D4214yes—
617C2×C4#54714yes—
618C2×C4D4×Q814yes—
619C2×C4#55414yes—
620C22—216no—
621C22—28no—
622C22—48no—
623C22—48no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—48no—
629C22—44no—
630C22—44no—
631C22C22×C22⋊Q811yescharacteristic
632C22C22×C4⋊Q811yescharacteristic
633C22C2×#54411yescharacteristic
634C22#346012yes—
635C22C2×D4211yescharacteristic
636C22C2×#54711yescharacteristic
637C22C2×D4×Q811yescharacteristic
638C22C2×#55411yescharacteristic
639C22#348411yescharacteristic
640C22#348412yes—
641C22#348411yescharacteristic
642C22#348611yescharacteristic
643C22#348711yescharacteristic
644C22#348912yes—
645C22#350112yes—
646C22#350112yes—
647C22#350312yes—
648C22#350312yes—
649C22#351112yes—
650C22#353712yes—
651C22#354112yes—
652C22#355012yes—
653C22#355412yes—
654C4—24no—
655C4—24no—
656C4—44no—
657C4—44no—
658C4—48no—
659C4—44no—
660C4—44no—
661C4—44no—
662C4—44no—
663C4—44no—
664C4—44no—
665C4—84no—
666C4—82no—
667C4#244018yes—
668C2—48no—
669C2C2×#244011yescharacteristic
670C2#3517411yescharacteristic
671C2#3557411yescharacteristic
672C2#3592812yes—
673C2#3779412yes—
674C2#3968812yes—
675C2#4249612yes—
676C2#4493411yescharacteristic
677C2#4526711yescharacteristic
678C2#4535511yescharacteristic
679C2#4535711yescharacteristic
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×#2551C2semidirect
4. #16520C2semidirect
5. #15421C2semidirect
6. #15265C2semidirect
7. #15194C2semidirect
8. #15128C2semidirect
9. #14720C2semidirect
10. #14276C2semidirect
11. #14150C2semidirect
14. #13856C2semidirect
95. C2×C4×D4Q8semidirect
136. #390Q8semidirect
361. C2×D4C4⋊Q8semidirect
493. C42C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#24522
C2×D4×C4⋊C42
#165202
#154212
#152652
#151942
#151282
#147202
#142762
#141502
#140182
#139412
#138562
#137852