Finite Group Databasebeta

← #520338

Subgroups — #520338

Order 512.

2674 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2859C211yesmaximal, characteristic
2C2×#2370C212yesmaximal
3#14699C212yesmaximal
4#14160C212yesmaximal
5#14151C211yesmaximal, characteristic
6#14151C211yesmaximal, characteristic
7#14150C212yesmaximal
8#14092C214yesmaximal
9#14083C214yesmaximal
10#14005C212yesmaximal
11#13981C212yesmaximal
12#13804C212yesmaximal
13C4⋊C42C211yesmaximal, characteristic
14C4⋊C42C211yesmaximal, characteristic
15C4×#384C211yesmaximal, characteristic
16C4×#389C212yesmaximal
17C42×C4⋊C4C211yesmaximal, characteristic
18C22×#125C2212yes—
19#2859C4116yescomplemented
20#2370—216no—
21C2×#389C2214yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#387C2214yes—
31C2×#387C2214yes—
32C2×#384C2211yescharacteristic
33C2×#384C2211yescharacteristic
34C2×#384C2211yescharacteristic
35C2×#384C2211yescharacteristic
36C2×#382C2212yes—
37C2×C4×C4⋊C4C2214yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C43C2212yes—
53C2×C43C2211yescharacteristic
54#1444C2214yes—
55#1442C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2211yescharacteristic
80#1434C2214yes—
81C4×#95C2212yes—
82C4×#95C2212yes—
83C4×#95C2214yes—
84C4×#95C2212yes—
85C2×#125—28no—
86C2×#125—28no—
87C2×#125C2×C4116yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2314yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2314yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#389—28no—
128#389—28no—
129#389—28no—
130#389—216no—
131#389—44no—
132#389C2×C4116yes—
133#389C2×C4116yes—
134#389C2×C4116yes—
135#389C2×C4116yes—
136#387—216no—
137#387—216no—
138#384—24no—
139#384—24no—
140#384C2×C418yes—
141#384C2×C418yes—
142#382C2×C4116yes—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—28no—
145C4×C4⋊C4—28no—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—216no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—44no—
155C4×C4⋊C4D418yes—
156C4×C4⋊C4Q818yes—
157C4×C4⋊C4Q818yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C43—44no—
185C43C2×C418yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2414yes—
189C23×C4C2414yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203#125—264no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C22×C418yes—
255C2×C4⋊C4C22×C418yes—
256C2×C4⋊C4C22×C4116yes—
257C2×C4⋊C4C22×C4116yes—
258C2×C4⋊C4C22×C418yes—
259C2×C4⋊C4C22×C418yes—
260C2×C4⋊C4C22×C418yes—
261C2×C4⋊C4C22×C418yes—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2×Q814yes—
265C2×C4⋊C4C2×Q814yes—
266C2×C4⋊C4C2×Q814yes—
267C2×C4⋊C4C2×Q814yes—
268C2×C4⋊C4#4114yes—
269C2×C4⋊C4#4114yes—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—28no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—28no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—41no—
295C2×C42—41no—
296C2×C42—41no—
297C2×C42—41no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—44no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42C22×C414yes—
306C2×C42C22×C414yes—
307C2×C42C22×C414yes—
308C2×C42C22×C414yes—
309C2×C42C22×C414yes—
310C2×C42C22×C414yes—
311C2×C42C22×C414yes—
312C2×C42C2×D418yes—
313C2×C42C2×D414yes—
314C2×C42C2×Q818yes—
315C2×C42C2×Q818yes—
316C2×C42C2×Q814yes—
317C2×C42C2×Q814yes—
318C2×C42#4114yes—
319C2×C42#4114yes—
320C2×C42#4114yes—
321#95—24no—
322#95—28no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—28no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95C22×C4116yes—
343#95C22×C4116yes—
344#95C2×D418yes—
345#95C2×Q818yes—
346#95C2×Q818yes—
347C24C2511yescharacteristic
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4C23×C418yes—
429C22×C4C23×C418yes—
430C22×C4C23×C412yes—
431C22×C4C23×C412yes—
432C22×C4C23×C412yes—
433C22×C4C23×C412yes—
434C22×C4C23×C412yes—
435C22×C4C23×C412yes—
436C22×C4C23×C412yes—
437C22×C4C22×D414yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D414yes—
441C22×C4C22×D412yes—
442C22×C4C22×Q814yes—
443C22×C4C22×Q814yes—
444C22×C4C22×Q812yes—
445C22×C4C22×Q812yes—
446C22×C4C22×Q812yes—
447C22×C4C22×Q814yes—
448C22×C4C22×Q814yes—
449C22×C4C22×Q812yes—
450C22×C4C22×Q812yes—
451C22×C4C22×Q812yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4H(5,𝔽2)12yes—
466C22×C42−1+412yes—
467C4⋊C4—232no—
468C4⋊C4—232no—
469C4⋊C4—216no—
470C4⋊C4—216no—
471C4⋊C4—216no—
472C4⋊C4—216no—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—28no—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—416no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—88no—
488C4⋊C4C4⋊Q8116yes—
489C42—28no—
490C42—28no—
491C42—216no—
492C42—44no—
493C42—44no—
494C42—48no—
495C42—48no—
496C42—416no—
497C42—48no—
498C42—48no—
499C42—48no—
500C42—48no—
501C42—48no—
502C42—88no—
503C42C4×D4116yes—
504C42C4×Q8116yes—
505C42C4×Q8116yes—
506C23C24×C411yescharacteristic
507C23C23×D411yescharacteristic
508C23C23×Q811yescharacteristic
509C23C23×Q811yescharacteristic
510C23C22×#4111yescharacteristic
511C23C22×#4111yescharacteristic
512C23C22×#4111yescharacteristic
513C23C22×#4111yescharacteristic
514C23C22×#4111yescharacteristic
515C23C2×H(5,𝔽2)11yescharacteristic
516C23C2×2−1+411yescharacteristic
517C23#58511yescharacteristic
518C23#58511yescharacteristic
519C23#58511yescharacteristic
520C23#58511yescharacteristic
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
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—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
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—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4C2×C4×D414yes—
621C2×C4C2×C4×D414yes—
622C2×C4C2×C4×D414yes—
623C2×C4C2×C4×Q814yes—
624C2×C4C2×C4×Q814yes—
625C2×C4C2×C4×Q814yes—
626C2×C4C2×C4×Q814yes—
627C2×C4C2×C4×Q814yes—
628C2×C4C2×C4×Q814yes—
629C2×C4C4×#4114yes—
630C2×C4#52014yes—
631C2×C4C2×C4⋊Q818yes—
632C2×C4C2×C4⋊Q814yes—
633C2×C4#53214yes—
634C2×C4#53314yes—
635C2×C4#53314yes—
636C2×C4#55214yes—
637C2×C4#55314yes—
638C2×C4#55414yes—
639C2×C4#55714yes—
640C22C22×C4×D411yescharacteristic
641C22C22×C4×Q811yescharacteristic
642C22C22×C4×Q811yescharacteristic
643C22C2×C4×#4111yescharacteristic
644C22C2×#52011yescharacteristic
645C22#342811yescharacteristic
646C22#342811yescharacteristic
647C22C22×C4⋊Q811yescharacteristic
648C22C2×#53211yescharacteristic
649C22C2×#53311yescharacteristic
650C22C2×#53311yescharacteristic
651C22D4×#4111yescharacteristic
652C22D4×#4111yescharacteristic
653C22C2×#55211yescharacteristic
654C22C2×#55311yescharacteristic
655C22C2×#55411yescharacteristic
656C22#347511yescharacteristic
657C22#347511yescharacteristic
658C22C2×#55711yescharacteristic
659C22Q8×#4111yescharacteristic
660C22Q8×#4111yescharacteristic
661C22#348011yescharacteristic
662C22#348011yescharacteristic
663C22#349611yescharacteristic
664C22#349711yescharacteristic
665C22#350111yescharacteristic
666C22#350211yescharacteristic
667C22#350411yescharacteristic
668C22#350411yescharacteristic
669C22#350611yescharacteristic
670C22#350611yescharacteristic
671C22#350711yescharacteristic
672C22#350711yescharacteristic
673C22#351211yescharacteristic
674C22#351211yescharacteristic
675C4—24no—
676C4—24no—
677C4—24no—
678C4—24no—
679C4—28no—
680C4—24no—
681C4—44no—
682C4—48no—
683C4—44no—
684C4—44no—
685C4—44no—
686C4—48no—
687C4—44no—
688C4—82no—
689C4—82no—
690C4C4×C4⋊Q818yes—
691C4#237018yescomplemented
692C2C2×C4×C4⋊Q811yescharacteristic
693C2#3414111yescharacteristic
694C2C2×#237011yescharacteristic
695C2#3414511yescharacteristic
696C2#3420611yescharacteristic
697C2#3421311yescharacteristic
698C2#3421711yescharacteristic
699C2#3626111yescharacteristic
700C2#3627011yescharacteristic
701C2#3628511yescharacteristic
702C2#3641611yescharacteristic
703C2#3646611yescharacteristic
704C2#3646811yescharacteristic
705C2#3647711yescharacteristic
706C2#3653511yescharacteristic
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
19. #2859C4semidirect
691. C4#2370semidirect

Maximal subgroups

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

subgroupindex
C2×#28592
C2×#23702
#146992
#141602
#141512
#141502
#140922
#140832
#140052
#139812
#138042
C4⋊C422
C4×#3842
C4×#3892
C42×C4⋊C42