Finite Group Databasebeta

← #688354

Subgroups — #688354

Order 512.

3814 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic
2C2×#2551C212yesmaximal, complemented
3C2×#2439C211yesmaximal, characteristic
4#16608C212yesmaximal, complemented
5#16603C212yesmaximal, complemented
6#15276C214yesmaximal
7#15265C214yesmaximal
8#15228C212yesmaximal, complemented
9#15218C212yesmaximal, complemented
10#15190C212yesmaximal, complemented
11#14929C212yesmaximal, complemented
12#14264C211yesmaximal, characteristic
13#14130C211yesmaximal, characteristic
14#14053C212yesmaximal, complemented
15#13880C211yesmaximal, characteristic
16#13844C211yesmaximal, characteristic
17#13801C211yesmaximal, characteristic
18C2×#544—216no—
19C2×#544—28no—
20C22×C4⋊Q8C2212yes—
21C22×C4⋊Q8C2211yescharacteristic
22C22×C4⋊Q8C2212yes—
23C22×C4⋊Q8C2214yes—
24C22×#125C2212yes—
25C22×#125C2212yes—
26C22×C22⋊Q8C2214yes—
27C22×#117C2211yescharacteristic
28C22×#117C2211yescharacteristic
29C23×C4⋊C4C2211yescharacteristic
30#2551—216no—
31#2439—28no—
32C2×#396C2212yes—
33C2×#395C2214yes—
34C2×#395C2214yes—
35C2×#393C2212yes—
36C2×#391C2212yes—
37C2×#391C2214yes—
38C2×#389C2212yes—
39C2×#389C2214yes—
40C2×#389C2212yes—
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#387C2212yes—
44C2×#385C2212yes—
45C2×#385C2214yes—
46C2×#384C2212yes—
47C2×#382C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54#1446C2214yes—
55#1444C2211yescharacteristic
56#1444C2211yescharacteristic
57#1444C2212yes—
58#1444C2212yes—
59#1444C2212yes—
60#1444C2214yes—
61#1444C2214yes—
62#1444C2214yes—
63#1444C2212yes—
64#1444C2214yes—
65#1442C2211yescharacteristic
66#1442C2211yescharacteristic
67#1442C2212yes—
68#1442C2212yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1441C2214yes—
74#1439C2212yes—
75#1439C2212yes—
76#1439C2214yes—
77#1437C2214yes—
78#1436C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83#1434C2212yes—
84C22⋊#95C2212yes—
85C4×#95C2212yes—
86C4×#95C2212yes—
87C23×Q8C2312yes—
88C23×Q8C2314yes—
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91#544—2128no—
92C2×C4⋊Q8—216no—
93C2×C4⋊Q8—216no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—28no—
99C2×C4⋊Q8—48no—
100C2×C4⋊Q8—48no—
101C2×C4⋊Q8D4116yes—
102C2×#125—216no—
103C2×#125—28no—
104C2×#125—28no—
105C2×#125—28no—
106C2×#125—28no—
107C2×C22⋊Q8—216no—
108C2×C22⋊Q8—216no—
109C2×C22⋊Q8—216no—
110C2×#117—24no—
111C2×#117—28no—
112C2×#117—28no—
113C2×#117D418yes—
114C22×C4⋊C4—24no—
115C22×C4⋊C4—28no—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2314yes—
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2314yes—
134C22×C4⋊C4C2314yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2311yescharacteristic
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2311yescharacteristic
139C22×C22⋊C4C2311yescharacteristic
140C22×C22⋊C4C2312yes—
141C22×C22⋊C4C2312yes—
142C22×C22⋊C4C2312yes—
143C22×C22⋊C4C2312yes—
144C22×C42C2311yescharacteristic
145C22×C42C2311yescharacteristic
146C22×C42C2314yes—
147C22×C42C2312yes—
148C22×C42C2312yes—
149C22×C42C2312yes—
150#396—28no—
151#395—216no—
152#395—216no—
153#393—28no—
154#391—28no—
155#391—48no—
156#389—24no—
157#389—24no—
158#389—216no—
159#389—28no—
160#389—28no—
161#387—28no—
162#385—48no—
163#385—44no—
164#384—28no—
165#382—24no—
166#380—24no—
167#380Q818yescomplemented
168C4×C4⋊C4—28no—
169C4×C4⋊C4—28no—
170C4×C22⋊C4—24no—
171C4×C22⋊C4Q818yescomplemented
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2311yescharacteristic
186C2×#95C2311yescharacteristic
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2311yescharacteristic
197C2×#95C2312yes—
198C2×#95C2311yescharacteristic
199C25C2411yescharacteristic
200C22×Q8—28no—
201C22×Q8—28no—
202C22×Q8—28no—
203C22×Q8—24no—
204C22×Q8—24no—
205C22×Q8—44no—
206C22×Q8—44no—
207C22×Q8—44no—
208C22×Q8—44no—
209C22×Q8—42no—
210C22×Q8—44no—
211C22×Q8C2×D4116yes—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—28no—
216C23×C4—42no—
217C23×C4—44no—
218C23×C4—42no—
219C23×C4C2×D414yes—
220C23×C4C2×Q814yes—
221C23×C4C2×Q814yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C4⋊Q8—264no—
239C4⋊Q8—264no—
240C4⋊Q8—464no—
241C4⋊Q8—416no—
242#125—264no—
243#125—264no—
244C22⋊Q8—2128no—
245#117—232no—
246#117—232no—
247C2×C4⋊C4—232no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4C2×D418yes—
297C2×C4⋊C4C2×D414yes—
298C2×C4⋊C4C2×D4116yes—
299C2×C4⋊C4C2×D418yes—
300C2×C4⋊C4C2×D414yes—
301C2×C4⋊C4C2×Q818yes—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—41no—
315C2×C22⋊C4—41no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4C2×D418yes—
319C2×C22⋊C4C2×Q818yes—
320C2×C22⋊C4C2×Q818yes—
321C2×C42—28no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—28no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—28no—
331C2×C42—24no—
332C2×C42—44no—
333C2×C42C2×D418yes—
334C2×C42C2×D418yes—
335C2×C42C2×Q818yes—
336#95—22no—
337#95—22no—
338#95—24no—
339#95—28no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—28no—
347#95—24no—
348#95—28no—
349#95—24no—
350#95—28no—
351#95—24no—
352#95—42no—
353#95—44no—
354#95—44no—
355#95—42no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95C2×Q814yes—
361#95C2×Q814yes—
362#95C2×Q818yes—
363C24—22no—
364C24—22no—
365C24—21no—
366C24—22no—
367C24—21no—
368C24—22no—
369C24—21no—
370C24—21no—
371C24C22×D412yes—
372C24C22×Q812yes—
373C24C22×Q812yes—
374C24C2511yescharacteristic
375C2×Q8—232no—
376C2×Q8—416no—
377C2×Q8—416no—
378C2×Q8—416no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—88no—
382C2×Q8—88no—
383C2×Q8—88no—
384C2×Q8—88no—
385C2×Q8—88no—
386C2×Q8—84no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—216no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—88no—
464C22×C4—88no—
465C22×C4—84no—
466C22×C4—84no—
467C22×C4C4⋊Q8116yescomplemented
468C22×C4C22×D414yes—
469C22×C4C22×D414yes—
470C22×C4C22×D414yes—
471C22×C4C22×D418yes—
472C22×C4C22×D418yes—
473C22×C4C22×D414yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D414yes—
477C22×C4C22×Q814yes—
478C22×C4C22×Q818yes—
479C22×C4C22×Q812yes—
480C22×C4C22×Q812yes—
481C22×C4C22×Q814yes—
482C22×C4C22×Q812yes—
483C22×C4C22×Q812yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C4H(5,𝔽2)14yes—
486C22×C42−1+412yes—
487C22×C42−1+414yes—
488C22×C42−1+414yes—
489C4⋊C4—216no—
490C4⋊C4—216no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—232no—
494C4⋊C4—232no—
495C4⋊C4—216no—
496C4⋊C4—216no—
497C4⋊C4—28no—
498C4⋊C4—232no—
499C4⋊C4—216no—
500C4⋊C4—48no—
501C4⋊C4—416no—
502C4⋊C4—416no—
503C4⋊C4—416no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—84no—
508C4⋊C4—88no—
509C4⋊C4#120116yes—
510C22⋊C4—216no—
511C22⋊C4—216no—
512C22⋊C4—216no—
513C22⋊C4—48no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—84no—
517C42—216no—
518C42—216no—
519C42—216no—
520C42—416no—
521C42—44no—
522C42—44no—
523C23—22no—
524C23—22no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—22no—
536C23—22no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C2×C4⋊Q814yes—
546C23C23×D411yescharacteristic
547C23C23×D412yes—
548C23C23×Q811yescharacteristic
549C23C23×Q811yescharacteristic
550C23C2×H(5,𝔽2)12yes—
551C23C2×H(5,𝔽2)11yescharacteristic
552C23C2×2−1+412yes—
553C23C2×2−1+412yes—
554C23C2×2−1+411yescharacteristic
555C23#58511yescharacteristic
556C23#58511yescharacteristic
557Q8—832no—
558Q8—816no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—232no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—28no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—24no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—816no—
645C2×C4C2×#12018yes—
646C2×C4C2×#12014yes—
647C2×C4C2×C4⋊Q814yes—
648C2×C4C2×C4⋊Q814yes—
649C2×C4#53614yes—
650C2×C4#54418yescomplemented
651C2×C4D4×Q818yes—
652C2×C4D4×Q818yes—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—44no—
663C22—42no—
664C22C22×#12011yescharacteristic
665C22C22×C4⋊Q811yescharacteristic
666C22C2×#53611yescharacteristic
667C22C2×#54412yes—
668C22C2×D4×Q812yes—
669C22C2×D4×Q812yes—
670C22#348412yes—
671C22#348612yes—
672C22#348811yescharacteristic
673C22#348911yescharacteristic
674C22#349612yes—
675C22#349712yes—
676C22#349811yescharacteristic
677C22#349911yescharacteristic
678C22#350111yescharacteristic
679C22#350211yescharacteristic
680C22#350311yescharacteristic
681C22#350312yes—
682C22#350411yescharacteristic
683C22#354412yes—
684C22#354712yes—
685C22#355212yes—
686C22#355512yes—
687C4—28no—
688C4—28no—
689C4—24no—
690C4—48no—
691C4—48no—
692C4—44no—
693C4—42no—
694C4—44no—
695C4—44no—
696C4—42no—
697C4—44no—
698C4—82no—
699C4—81no—
700C4—81no—
701C4—82no—
702C4#243418yes—
703C2—44no—
704C2C2×#243411yescharacteristic
705C2#3527511yescharacteristic
706C2#3556611yescharacteristic
707C2#3568011yescharacteristic
708C2#3568711yescharacteristic
709C2#3768612yes—
710C2#3937012yes—
711C2#4291811yescharacteristic
712C2#4296811yescharacteristic
713C2#4487912yes—
714C2#4489412yes—
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
2. C2×#2551C2semidirect
4. #16608C2semidirect
5. #16603C2semidirect
8. #15228C2semidirect
9. #15218C2semidirect
10. #15190C2semidirect
11. #14929C2semidirect
14. #14053C2semidirect
167. #380Q8semidirect
171. C4×C22⋊C4Q8semidirect
467. C22×C4C4⋊Q8semidirect
650. C2×C4#544semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#25512
C2×#24392
#166082
#166032
#152762
#152652
#152282
#152182
#151902
#149292
#142642
#141302
#140532
#138802
#138442
#138012