Finite Group Databasebeta

← #759766

Subgroups — #759766

Order 512.

2618 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic
2C2×#2446C212yesmaximal
3C2×C4×C4⋊Q8C211yesmaximal, characteristic
4#16529C214yesmaximal
5#15188C212yesmaximal
6#15186C212yesmaximal
7#15139C212yesmaximal
8#15137C212yesmaximal
9#14794C214yesmaximal
10#14155C211yesmaximal, characteristic
11#14133C212yesmaximal
12#14029C212yesmaximal
13#14017C212yesmaximal
14#13926C211yesmaximal, characteristic
15#13910C211yesmaximal, characteristic
16C4×#391C211yesmaximal, characteristic
17C4×#389C211yesmaximal, characteristic
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2212yes—
22#2546—28no—
23#2446—216no—
24C2×#400C2214yes—
25C2×#398C2214yes—
26C2×#398C2214yes—
27C4×C4⋊Q8—28no—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2211yescharacteristic
33C2×#391C2212yes—
34C2×#391C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#389C2211yescharacteristic
43C2×#387C2214yes—
44C2×#387C2214yes—
45C2×#384C2212yes—
46C2×#384C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2212yes—
53C2×C43C2211yescharacteristic
54#1445C2214yes—
55#1445C2214yes—
56#1444C2214yes—
57#1444C2214yes—
58#1443C2214yes—
59#1443C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1442C2214yes—
63#1442C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1436C2214yes—
68#1436C2212yes—
69#1436C2212yes—
70#1436C2214yes—
71#1436C2212yes—
72#1436C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2211yescharacteristic
83C4×#95C2212yes—
84C4×#95C2212yes—
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C23×Q8C2312yes—
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8D418yes—
94C2×C4×Q8—28no—
95C2×C4×Q8—28no—
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8—44no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2314yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#400—216no—
133#398—216no—
134#398—216no—
135#391—28no—
136#391—28no—
137#391—28no—
138#391—28no—
139#391—28no—
140#391—24no—
141#391D418yes—
142#389—28no—
143#389—28no—
144#389—28no—
145#389—28no—
146#389—28no—
147#389—24no—
148#389—44no—
149#389Q818yescomplemented
150#387—216no—
151#387—216no—
152#384—28no—
153#384—24no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—24no—
159C4×C4⋊C4—24no—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C43—24no—
192C22×Q8—22no—
193C22×Q8—24no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—22no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—22no—
202C22×Q8—24no—
203C22×Q8—22no—
204C22×Q8C2×D418yes—
205C22×Q8C2×D414yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C4⋊Q8—232no—
224C4⋊Q8—432no—
225C4×Q8—432no—
226C4×Q8—416no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
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—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4C2×D418yes—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×Q814yes—
283C2×C4⋊C4C2×Q818yes—
284C2×C4⋊C4C2×Q814yes—
285C2×C4⋊C4#4114yes—
286C2×C4⋊C4#4114yes—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—28no—
295C2×C42—28no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—44no—
316C2×C42—41no—
317C2×C42—41no—
318C2×C42C2×D414yes—
319C2×C42C2×D414yes—
320C2×C42C2×Q814yes—
321C2×C42#4114yes—
322C2×C42#4114yes—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—28no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—42no—
346#95—42no—
347#95—42no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95C2×D418yes—
352#95C2×D418yes—
353#95C2×Q818yes—
354C24C2511yescharacteristic
355C2×Q8—216no—
356C2×Q8—28no—
357C2×Q8—216no—
358C2×Q8—28no—
359C2×Q8—216no—
360C2×Q8—28no—
361C2×Q8—44no—
362C2×Q8—48no—
363C2×Q8—44no—
364C2×Q8—48no—
365C2×Q8—44no—
366C2×Q8—48no—
367C2×Q8—44no—
368C2×Q8—48no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4C22×D414yes—
459C22×C4C22×D414yes—
460C22×C4C22×D418yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×Q812yes—
468C22×C4C22×Q814yes—
469C22×C4C22×Q814yes—
470C22×C4C22×Q812yes—
471C22×C4C22×Q812yes—
472C22×C4C2×#4114yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C42−1+412yes—
484C22×C42−1+412yes—
485C22×C42−1+412yes—
486C22×C42−1+412yes—
487C22×C42−1+412yes—
488C4⋊C4—216no—
489C4⋊C4—216no—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—216no—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—48no—
497C4⋊C4—416no—
498C4⋊C4—416no—
499C4⋊C4—416no—
500C4⋊C4—416no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—88no—
508C42—28no—
509C42—28no—
510C42—28no—
511C42—416no—
512C42—44no—
513C42—48no—
514C42—48no—
515C42—48no—
516C42—44no—
517C42—44no—
518C42—48no—
519C42#123116yes—
520C23C23×D411yescharacteristic
521C23C23×D411yescharacteristic
522C23C23×Q811yescharacteristic
523C23C22×#4111yescharacteristic
524C23C22×#4111yescharacteristic
525C23C22×#4111yescharacteristic
526C23C22×#4111yescharacteristic
527C23C2×H(5,𝔽2)11yescharacteristic
528C23C2×2−1+411yescharacteristic
529C23C2×2−1+411yescharacteristic
530C23C2×2−1+411yescharacteristic
531C23C2×2−1+411yescharacteristic
532C23C2×2−1+411yescharacteristic
533C23#58511yescharacteristic
534C23#58511yescharacteristic
535Q8—432no—
536Q8—416no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
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—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4C2×#12314yes—
641C2×C4C2×#12314yes—
642C2×C4C2×#12314yes—
643C2×C4#53214yes—
644C2×C4#53314yes—
645C2×C4#54114yes—
646C2×C4#54214yes—
647C2×C4D4214yes—
648C2×C4#54714yes—
649C2×C4#54714yes—
650C2×C4#54814yes—
651C2×C4D4×Q814yes—
652C2×C4D4×Q814yes—
653C2×C4#55314yes—
654C2×C4#55614yes—
655C2×C4#55714yes—
656C22C22×#12311yescharacteristic
657C22C2×#53211yescharacteristic
658C22C2×#53311yescharacteristic
659C22C2×#54111yescharacteristic
660C22C2×#54211yescharacteristic
661C22#346111yescharacteristic
662C22#346111yescharacteristic
663C22C2×D4211yescharacteristic
664C22C2×#54711yescharacteristic
665C22C2×#54711yescharacteristic
666C22C2×#54811yescharacteristic
667C22C2×D4×Q811yescharacteristic
668C22C2×D4×Q811yescharacteristic
669C22D4×#4111yescharacteristic
670C22C2×#55311yescharacteristic
671C22C2×#55611yescharacteristic
672C22C2×#55711yescharacteristic
673C22Q8×#4111yescharacteristic
674C22#348211yescharacteristic
675C22#348611yescharacteristic
676C22#348711yescharacteristic
677C22#349711yescharacteristic
678C22#349911yescharacteristic
679C22#350211yescharacteristic
680C22#350311yescharacteristic
681C22#350711yescharacteristic
682C22#350711yescharacteristic
683C22#351411yescharacteristic
684C22#351511yescharacteristic
685C22#352311yescharacteristic
686C22#352311yescharacteristic
687C22#355211yescharacteristic
688C22#355411yescharacteristic
689C22#355511yescharacteristic
690C22#355611yescharacteristic
691C4—24no—
692C4—24no—
693C4—44no—
694C4—44no—
695C4—44no—
696C4—48no—
697C4—44no—
698C4—42no—
699C4—42no—
700C4—42no—
701C4—44no—
702C4—44no—
703C4—82no—
704C4—82no—
705C4—84no—
706C4#242218yes—
707C4#244618yescomplemented
708C2C2×#242211yescharacteristic
709C2C2×#244611yescharacteristic
710C2#3492911yescharacteristic
711C2#3518211yescharacteristic
712C2#3519911yescharacteristic
713C2#3520911yescharacteristic
714C2#3763011yescharacteristic
715C2#5906711yescharacteristic
716C2#5907111yescharacteristic
717C2#5908011yescharacteristic
718C2#5912711yescharacteristic
719C2#5942811yescharacteristic
720C2#5943711yescharacteristic
721C2#5968111yescharacteristic
722C2#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
149. #389Q8semidirect
707. C4#2446semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#24462
C2×C4×C4⋊Q82
#165292
#151882
#151862
#151392
#151372
#147942
#141552
#141332
#140292
#140172
#139262
#139102
C4×#3912
C4×#3892