Finite Group Databasebeta

← #525325

Subgroups — #525325

Order 512.

2626 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic
3C2×#2295C211yesmaximal, characteristic
4#14712C211yesmaximal, characteristic
5#14550C214yesmaximal
6#14548C214yesmaximal
7#14266C212yesmaximal
8#14142C212yesmaximal
9#14132C212yesmaximal
10#14046C212yesmaximal
11#14017C212yesmaximal
12#14005C212yesmaximal
13#14000C212yesmaximal
14#13926C211yesmaximal, characteristic
15C4⋊C42C211yesmaximal, characteristic
16#13786C212yesmaximal
17C4×#391C211yesmaximal, characteristic
18C22×C4×Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#2582C4116yescomplemented
22C2×#398C2214yes—
23Q8×C4⋊C4—28no—
24#2295—28no—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2214yes—
35C2×#389C2211yescharacteristic
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2214yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#387C2212yes—
43C2×#387C2212yes—
44C2×#384C2211yescharacteristic
45C2×#384C2212yes—
46C2×#384C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×#376C2212yes—
54C2×C43C2211yescharacteristic
55#1445C2214yes—
56#1444C2214yes—
57#1443C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2212yes—
70#1436C2212yes—
71#1436C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81C4⋊#95C2211yescharacteristic
82#1434C2214yes—
83#1434C2214yes—
84C4×#95C2212yes—
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C23×Q8C2311yescharacteristic
88C2×C4×Q8—28no—
89C2×C4×Q8—24no—
90C2×C4×Q8—24no—
91C2×C4×Q8—28no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—44no—
95C2×C4×Q8C2×C418yes—
96C2×C4×Q8C2×C418yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2314yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2314yes—
129C22×C42C2311yescharacteristic
130#398C2×C4132yes—
131#391—24no—
132#391—24no—
133#391—24no—
134#391—44no—
135#391C2×C4116yes—
136#391C2×C4116yes—
137#391Q818yescomplemented
138#389—24no—
139#389—24no—
140#389—28no—
141#389—216no—
142#389—48no—
143#389—44no—
144#389—44no—
145#389—44no—
146#389C2×C4116yes—
147#389C2×C4116yes—
148#387—28no—
149#387—28no—
150#384—28no—
151#384—24no—
152#384C2×C418yes—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—44no—
157C4×C4⋊C4—44no—
158C4×C4⋊C4D418yes—
159#376—28no—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C43—24no—
191C22×Q8—22no—
192C22×Q8—22no—
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×Q8C22×C414yes—
202C22×Q8C22×C414yes—
203C22×Q8C2×Q814yes—
204C22×Q8#4114yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C4×Q8—232no—
223C4×Q8—232no—
224C4×Q8—432no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C22×C414yes—
275C2×C4⋊C4C22×C414yes—
276C2×C4⋊C4C22×C418yes—
277C2×C4⋊C4C22×C418yes—
278C2×C4⋊C4C22×C418yes—
279C2×C4⋊C4C22×C418yes—
280C2×C4⋊C4C22×C418yes—
281C2×C4⋊C4C22×C4116yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×Q814yes—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—28no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—28no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—44no—
312C2×C42—41no—
313C2×C42—41no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42—44no—
317C2×C42C22×C418yes—
318C2×C42C22×C418yes—
319C2×C42C22×C414yes—
320C2×C42C2×D418yes—
321C2×C42C2×D414yes—
322C2×C42C2×Q814yes—
323C2×C42#4114yes—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—42no—
337#95—42no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95C22×C4116yes—
349#95C22×C4116yes—
350#95C22×C4116yes—
351#95C2×D418yes—
352#95C2×Q818yes—
353#95C2×Q818yes—
354C24C2511yescharacteristic
355C2×Q8—28no—
356C2×Q8—28no—
357C2×Q8—28no—
358C2×Q8—28no—
359C2×Q8—28no—
360C2×Q8—28no—
361C2×Q8—28no—
362C2×Q8—28no—
363C2×Q8—28no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8C4×Q8116yes—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4C23×C414yes—
458C22×C4C23×C418yes—
459C22×C4C23×C414yes—
460C22×C4C23×C414yes—
461C22×C4C23×C412yes—
462C22×C4C23×C414yes—
463C22×C4C23×C412yes—
464C22×C4C23×C412yes—
465C22×C4C22×D414yes—
466C22×C4C22×D414yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×Q812yes—
471C22×C4C22×Q812yes—
472C22×C4C22×Q814yes—
473C22×C4C22×Q814yes—
474C22×C4C22×Q812yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4114yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C4H(5,𝔽2)12yes—
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—28no—
490C4⋊C4—28no—
491C4⋊C4—216no—
492C4⋊C4—216no—
493C4⋊C4—216no—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—48no—
498C4⋊C4—416no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—416no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—84no—
506C4⋊C4—88no—
507C4⋊C4—88no—
508C42—216no—
509C42—216no—
510C42—216no—
511C42—28no—
512C42—44no—
513C42—44no—
514C42—416no—
515C42—44no—
516C42—44no—
517C42—48no—
518C42—44no—
519C42—88no—
520C42C4×D4116yes—
521C23C24×C411yescharacteristic
522C23C23×D411yescharacteristic
523C23C23×Q811yescharacteristic
524C23C22×#4111yescharacteristic
525C23C22×#4111yescharacteristic
526C23C2×H(5,𝔽2)11yescharacteristic
527C23C2×H(5,𝔽2)11yescharacteristic
528C23C2×H(5,𝔽2)11yescharacteristic
529C23C2×2−1+411yescharacteristic
530C23C2×2−1+411yescharacteristic
531C23C2×2−1+411yescharacteristic
532C23C2×2−1+411yescharacteristic
533C23C2×2−1+411yescharacteristic
534C23#58511yescharacteristic
535C23#58511yescharacteristic
536Q8—232no—
537Q8—416no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4C2×C4×D414yes—
642C2×C4C2×C4×D414yes—
643C2×C4C2×C4×D414yes—
644C2×C4C2×C4×Q818yes—
645C2×C4C2×C4×Q814yes—
646C2×C4#51914yes—
647C2×C4#52014yes—
648C2×C4#52014yes—
649C2×C4#53614yes—
650C2×C4#53714yes—
651C2×C4#54114yes—
652C2×C4#54214yes—
653C2×C4#54614yes—
654C2×C4D4×Q814yes—
655C2×C4#55214yes—
656C2×C4#55414yes—
657C22C22×C4×D411yescharacteristic
658C22C22×C4×Q811yescharacteristic
659C22C2×#51911yescharacteristic
660C22C2×#52011yescharacteristic
661C22C2×#52011yescharacteristic
662C22C4×H(5,𝔽2)11yescharacteristic
663C22C4×2−1+411yescharacteristic
664C22C2×#53611yescharacteristic
665C22C2×#53711yescharacteristic
666C22C2×#54111yescharacteristic
667C22C2×#54211yescharacteristic
668C22C2×#54611yescharacteristic
669C22C2×D4×Q811yescharacteristic
670C22C2×#55211yescharacteristic
671C22C2×#55411yescharacteristic
672C22#348811yescharacteristic
673C22#348911yescharacteristic
674C22#350211yescharacteristic
675C22#350211yescharacteristic
676C22#350311yescharacteristic
677C22#350311yescharacteristic
678C22#351111yescharacteristic
679C22#351211yescharacteristic
680C22#351711yescharacteristic
681C22#351811yescharacteristic
682C22#352211yescharacteristic
683C22#352311yescharacteristic
684C22#354311yescharacteristic
685C22#355011yescharacteristic
686C22#355211yescharacteristic
687C22#355411yescharacteristic
688C22#355411yescharacteristic
689C22#355411yescharacteristic
690C22#355511yescharacteristic
691C22#355611yescharacteristic
692C4—28no—
693C4—24no—
694C4—44no—
695C4—42no—
696C4—42no—
697C4—48no—
698C4—44no—
699C4—44no—
700C4—44no—
701C4—44no—
702C4—42no—
703C4—44no—
704C4—42no—
705C4—82no—
706C4—84no—
707C4#232318yes—
708C4#236018yes—
709C2C2×#232311yescharacteristic
710C2C2×#236011yescharacteristic
711C2#3426711yescharacteristic
712C2#3433511yescharacteristic
713C2#3433711yescharacteristic
714C2#3448811yescharacteristic
715C2#3449811yescharacteristic
716C2#3890911yescharacteristic
717C2#3891111yescharacteristic
718C2#3897611yescharacteristic
719C2#3897811yescharacteristic
720C2#3959711yescharacteristic
721C2#3959911yescharacteristic
722C2#3963311yescharacteristic
723C2#3963511yescharacteristic
subgroupG/H
center, Frattini, socle, prime radical, cosocleC24C25
commutatorC23C24×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #2582C4semidirect
137. #391Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
C2×Q8×C4⋊C42
C2×#22952
#147122
#145502
#145482
#142662
#141422
#141322
#140462
#140172
#140052
#140002
#139262
C4⋊C422
#137862
C4×#3912