Finite Group Databasebeta

← #674761

Subgroups — #674761

Order 512.

2618 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2446C212yesmaximal
2C2×Q8×C4⋊C4C211yesmaximal, characteristic
3C2×C4×C4⋊Q8C211yesmaximal, characteristic
4#15888C214yesmaximal
5#15347C214yesmaximal
6#15186C212yesmaximal
7#15152C212yesmaximal
8#15151C212yesmaximal
9#15133C212yesmaximal
10#14712C211yesmaximal, characteristic
11#14042C212yesmaximal
12#14042C212yesmaximal
13#13931C212yesmaximal
14#13926C211yesmaximal, characteristic
15#13926C211yesmaximal, characteristic
16#13926C211yesmaximal, characteristic
17C4×#389C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22#2446—216no—
23C2×#400C2214yes—
24C2×#398C2214yes—
25C2×#398C2214yes—
26Q8×C4⋊C4—28no—
27C4×C4⋊Q8—28no—
28C2×#391C2211yescharacteristic
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2214yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2211yescharacteristic
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2211yescharacteristic
42C2×#387C2214yes—
43C2×#387C2214yes—
44C2×#384C2211yescharacteristic
45C2×#384C2212yes—
46C2×#384C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C43C2211yescharacteristic
55#1445C2214yes—
56#1445C2214yes—
57#1445C2214yes—
58#1443C2214yes—
59#1443C2214yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1436C2212yes—
68#1436C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2211yescharacteristic
83#1434C2214yes—
84#1434C2214yes—
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C23×Q8C2311yescharacteristic
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8D418yes—
91C2×C4×Q8—28no—
92C2×C4×Q8—28no—
93C2×C4×Q8—28no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8D418yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#400—216no—
136#398—216no—
137#398—48no—
138#391—24no—
139#391—28no—
140#391—28no—
141#391—28no—
142#391—24no—
143#389—28no—
144#389—28no—
145#389—28no—
146#389—216no—
147#389—28no—
148#389—24no—
149#389—28no—
150#389—44no—
151#389Q818yescomplemented
152#387—216no—
153#387—216no—
154#384—24no—
155#384—28no—
156#384—24no—
157#384—24no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4—28no—
160C4×C4⋊C4—28no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4—28no—
163C4×C4⋊C4—24no—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2312yes—
192C43—24no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8C2×D414yes—
204C22×Q8C2×D414yes—
205C22×Q8C2×D414yes—
206C22×Q8#4114yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C4⋊Q8—232no—
225C4×Q8—232no—
226C4×Q8—432no—
227C4×Q8—416no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4C2×D414yes—
291C2×C4⋊C4C2×D418yes—
292C2×C4⋊C4C2×D418yes—
293C2×C4⋊C4C2×D418yes—
294C2×C4⋊C4C2×Q814yes—
295C2×C4⋊C4C2×Q818yes—
296C2×C4⋊C4C2×Q814yes—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—28no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—28no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—28no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—41no—
325C2×C42—41no—
326C2×C42C2×D418yes—
327C2×C42C2×D414yes—
328C2×C42C2×D414yes—
329C2×C42C2×Q814yes—
330C2×C42#4114yes—
331C2×C42#4114yes—
332C2×C42#4114yes—
333#95—24no—
334#95—28no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—24no—
349#95—24no—
350#95—28no—
351#95—42no—
352#95—44no—
353#95—42no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95C2×Q818yes—
361C24C2511yescharacteristic
362C2×Q8—28no—
363C2×Q8—28no—
364C2×Q8—28no—
365C2×Q8—28no—
366C2×Q8—28no—
367C2×Q8—28no—
368C2×Q8—28no—
369C2×Q8—28no—
370C2×Q8—28no—
371C2×Q8—44no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8#121116yes—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C22×D414yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×Q812yes—
474C22×C4C22×Q814yes—
475C22×C4C22×Q814yes—
476C22×C4C22×Q812yes—
477C22×C4C22×Q812yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C42−1+412yes—
491C22×C42−1+412yes—
492C22×C42−1+412yes—
493C22×C42−1+412yes—
494C22×C42−1+412yes—
495C4⋊C4—216no—
496C4⋊C4—216no—
497C4⋊C4—216no—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—216no—
501C4⋊C4—28no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—416no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—48no—
511C4⋊C4—44no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—416no—
515C4⋊C4—48no—
516C4⋊C4—84no—
517C4⋊C4—88no—
518C42—28no—
519C42—216no—
520C42—28no—
521C42—28no—
522C42—44no—
523C42—44no—
524C42—416no—
525C42—48no—
526C42—48no—
527C42—44no—
528C42—48no—
529C42#123116yes—
530C23C23×D411yescharacteristic
531C23C23×D411yescharacteristic
532C23C23×Q811yescharacteristic
533C23C22×#4111yescharacteristic
534C23C22×#4111yescharacteristic
535C23C22×#4111yescharacteristic
536C23C22×#4111yescharacteristic
537C23C2×H(5,𝔽2)11yescharacteristic
538C23C2×2−1+411yescharacteristic
539C23C2×2−1+411yescharacteristic
540C23C2×2−1+411yescharacteristic
541C23C2×2−1+411yescharacteristic
542C23C2×2−1+411yescharacteristic
543C23#58511yescharacteristic
544C23#58511yescharacteristic
545Q8—232no—
546Q8—416no—
547C2×C4—22no—
548C2×C4—22no—
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—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4C2×#12118yes—
649C2×C4C2×#12114yes—
650C2×C4C2×#12314yes—
651C2×C4C2×#12314yes—
652C2×C4C2×#12314yes—
653C2×C4#53314yes—
654C2×C4#53614yes—
655C2×C4#54114yes—
656C2×C4#54714yes—
657C2×C4#54714yes—
658C2×C4#54814yes—
659C2×C4D4×Q814yes—
660C2×C4D4×Q814yes—
661C2×C4#55014yes—
662C2×C4#55214yes—
663C2×C4#55214yes—
664C2×C4#55614yes—
665C2×C4#55714yes—
666C22C22×#12111yescharacteristic
667C22C22×#12311yescharacteristic
668C22C2×#53311yescharacteristic
669C22C2×#53611yescharacteristic
670C22C2×#54111yescharacteristic
671C22C2×#54711yescharacteristic
672C22C2×#54711yescharacteristic
673C22C2×#54811yescharacteristic
674C22C2×D4×Q811yescharacteristic
675C22C2×D4×Q811yescharacteristic
676C22C2×#55011yescharacteristic
677C22C2×#55211yescharacteristic
678C22C2×#55211yescharacteristic
679C22C2×#55611yescharacteristic
680C22C2×#55711yescharacteristic
681C22Q8×#4111yescharacteristic
682C22#348011yescharacteristic
683C22#348811yescharacteristic
684C22#348911yescharacteristic
685C22#349711yescharacteristic
686C22#349711yescharacteristic
687C22#350211yescharacteristic
688C22#350311yescharacteristic
689C22#350611yescharacteristic
690C22#350711yescharacteristic
691C22#350711yescharacteristic
692C22#350911yescharacteristic
693C22#351111yescharacteristic
694C22#351211yescharacteristic
695C22#352211yescharacteristic
696C22#352311yescharacteristic
697C22#355011yescharacteristic
698C22#355511yescharacteristic
699C22#355511yescharacteristic
700C22#355611yescharacteristic
701C4—24no—
702C4—24no—
703C4—28no—
704C4—24no—
705C4—44no—
706C4—42no—
707C4—42no—
708C4—44no—
709C4—44no—
710C4—44no—
711C4—84no—
712C4—82no—
713C4—82no—
714C4—84no—
715C4—82no—
716C4#244618yescomplemented
717C4#248618yes—
718C2C2×#244611yescharacteristic
719C2C2×#248611yescharacteristic
720C2#3551711yescharacteristic
721C2#3613411yescharacteristic
722C2#3628511yescharacteristic
723C2Q8×#12111yescharacteristic
724C2#3657611yescharacteristic
725C2#3897611yescharacteristic
726C2#3937911yescharacteristic
727C2#3959711yescharacteristic
728C2#3972411yescharacteristic
729C2#5944311yescharacteristic
730C2#5944511yescharacteristic
731C2#5982111yescharacteristic
732C2#5982811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
151. #389Q8semidirect
716. C4#2446semidirect

Maximal subgroups

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

subgroupindex
C2×#24462
C2×Q8×C4⋊C42
C2×C4×C4⋊Q82
#158882
#153472
#151862
#151522
#151512
#151332
#147122
#140422
#139312
#139262
C4×#3892