Finite Group Databasebeta

← #674914

Subgroups — #674914

Order 512.

2538 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2859C211yesmaximal, characteristic
2C2×#2552C212yesmaximal
3C2×#2552C211yesmaximal, characteristic
4#15766C214yesmaximal
5#15520C214yesmaximal
6#15228C212yesmaximal
7#15218C212yesmaximal
8#15137C212yesmaximal
9#15133C212yesmaximal
10#14006C212yesmaximal
11#14005C212yesmaximal
12#13910C211yesmaximal, characteristic
13#13910C211yesmaximal, characteristic
14#13910C212yesmaximal
15C4⋊C42C211yesmaximal, characteristic
16C4×#389C211yesmaximal, characteristic
17C4×#389C211yesmaximal, characteristic
18C22×#125C2212yes—
19C22×#125C2212yes—
20#2859—28no—
21#2552—216no—
22#2552—28no—
23C2×#398C2214yes—
24C2×#398C2214yes—
25C2×#398C2214yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2214yes—
36C2×#389C2211yescharacteristic
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#384C2211yescharacteristic
42C2×#384C2212yes—
43C2×#384C2212yes—
44C2×#384C2214yes—
45C2×#384C2212yes—
46C2×#384C2211yescharacteristic
47C2×#382C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C43C2211yescharacteristic
57#1444C2214yes—
58#1444C2214yes—
59#1444C2214yes—
60#1444C2214yes—
61#1444C2214yes—
62#1444C2214yes—
63#1442C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1436C2212yes—
70#1436C2212yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2212yes—
74#1436C2212yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2214yes—
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2212yes—
83C4⋊#95C2211yescharacteristic
84C4×#95C2212yes—
85C4×#95C2212yes—
86C2×#125—28no—
87C2×#125—28no—
88C2×#125—28no—
89C2×#125—28no—
90C2×#125—44no—
91C2×#125—44no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2314yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#398—216no—
132#398—216no—
133#398—216no—
134#389—28no—
135#389—28no—
136#389—28no—
137#389—28no—
138#389—28no—
139#389—28no—
140#389—28no—
141#389—24no—
142#389—28no—
143#389—24no—
144#389—216no—
145#389—44no—
146#389—44no—
147#389D418yes—
148#389Q818yes—
149#384—28no—
150#384—216no—
151#384—28no—
152#384—24no—
153#384—44no—
154#384D418yes—
155#382—28no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4—24no—
160C4×C4⋊C4—24no—
161C4×C4⋊C4—44no—
162C4×C4⋊C4Q818yes—
163C4×C4⋊C4Q818yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C43—24no—
191C23×C4C2414yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2414yes—
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208#125—432no—
209#125—432no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4C2×D418yes—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×D418yes—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4C2×D418yes—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×Q814yes—
283C2×C4⋊C4C2×Q814yes—
284C2×C4⋊C4C2×Q814yes—
285C2×C4⋊C4C2×Q818yes—
286C2×C4⋊C4C2×Q814yes—
287C2×C4⋊C4C2×Q814yes—
288C2×C4⋊C4C2×Q814yes—
289C2×C4⋊C4#4114yes—
290C2×C4⋊C4#4114yes—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—28no—
306C2×C42—24no—
307C2×C42—28no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—44no—
316C2×C42—41no—
317C2×C42—41no—
318C2×C42—41no—
319C2×C42—41no—
320C2×C42C2×D414yes—
321C2×C42C2×D414yes—
322C2×C42C2×Q818yes—
323C2×C42C2×Q818yes—
324C2×C42C2×Q814yes—
325C2×C42C2×Q814yes—
326C2×C42C2×Q814yes—
327#95—28no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—44no—
341#95—42no—
342#95—42no—
343#95—42no—
344#95—42no—
345#95—42no—
346#95—42no—
347#95—44no—
348#95—44no—
349#95C2×D418yes—
350#95C2×Q818yes—
351#95C2×Q818yes—
352#95C2×Q818yes—
353C24C2511yescharacteristic
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—22no—
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—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4C22×D418yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D414yes—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×Q812yes—
451C22×C4C22×Q814yes—
452C22×C4C22×Q814yes—
453C22×C4C22×Q814yes—
454C22×C4C22×Q812yes—
455C22×C4C22×Q814yes—
456C22×C4C22×Q812yes—
457C22×C4C22×Q812yes—
458C22×C4C22×Q814yes—
459C22×C4C22×Q814yes—
460C22×C4C22×Q812yes—
461C22×C4C22×Q812yes—
462C22×C4C22×Q812yes—
463C22×C4C22×Q812yes—
464C22×C4C22×Q812yes—
465C22×C4C2×#4114yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4H(5,𝔽2)12yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C42−1+412yes—
472C22×C42−1+412yes—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—216no—
476C4⋊C4—28no—
477C4⋊C4—216no—
478C4⋊C4—28no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—416no—
485C4⋊C4—416no—
486C4⋊C4—416no—
487C4⋊C4—416no—
488C4⋊C4—416no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—88no—
495C4⋊C4—88no—
496C4⋊C4—88no—
497C4⋊C4—88no—
498C4⋊C4#125116yescomplemented
499C4⋊C4C4⋊Q8116yes—
500C42—28no—
501C42—216no—
502C42—28no—
503C42—216no—
504C42—416no—
505C42—44no—
506C42—44no—
507C42—48no—
508C42—44no—
509C42—44no—
510C42—48no—
511C42C4⋊Q8116yes—
512C23C23×D411yescharacteristic
513C23C23×D411yescharacteristic
514C23C23×Q811yescharacteristic
515C23C23×Q811yescharacteristic
516C23C23×Q811yescharacteristic
517C23C22×#4111yescharacteristic
518C23C22×#4111yescharacteristic
519C23C2×H(5,𝔽2)11yescharacteristic
520C23C2×H(5,𝔽2)11yescharacteristic
521C23C2×2−1+411yescharacteristic
522C23C2×2−1+411yescharacteristic
523C23#58511yescharacteristic
524C23#58511yescharacteristic
525C23#58511yescharacteristic
526C23#58511yescharacteristic
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—28no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4C2×#12518yes—
630C2×C4C2×#12514yes—
631C2×C4C2×C4⋊Q814yes—
632C2×C4C2×C4⋊Q818yes—
633C2×C4C2×C4⋊Q814yes—
634C2×C4C2×C4⋊Q814yes—
635C2×C4C2×C4⋊Q814yes—
636C2×C4#54414yes—
637C2×C4#54414yes—
638C2×C4D4214yes—
639C2×C4#54714yes—
640C2×C4D4×Q814yes—
641C2×C4D4×Q814yes—
642C2×C4#55014yes—
643C2×C4#55414yes—
644C2×C4#55714yes—
645C2×C4Q8214yes—
646C22C22×#12511yescharacteristic
647C22C22×C4⋊Q811yescharacteristic
648C22C22×C4⋊Q811yescharacteristic
649C22C2×#54411yescharacteristic
650C22C2×#54411yescharacteristic
651C22C2×D4211yescharacteristic
652C22C2×#54711yescharacteristic
653C22C2×D4×Q811yescharacteristic
654C22C2×D4×Q811yescharacteristic
655C22C2×#55011yescharacteristic
656C22D4×#4111yescharacteristic
657C22D4×#4111yescharacteristic
658C22C2×#55411yescharacteristic
659C22C2×#55711yescharacteristic
660C22C2×Q8211yescharacteristic
661C22Q8×#4111yescharacteristic
662C22Q8×#4111yescharacteristic
663C22#348511yescharacteristic
664C22#348711yescharacteristic
665C22#349611yescharacteristic
666C22#349711yescharacteristic
667C22#350111yescharacteristic
668C22#350211yescharacteristic
669C22#350411yescharacteristic
670C22#350411yescharacteristic
671C22#350411yescharacteristic
672C22#350411yescharacteristic
673C22#351511yescharacteristic
674C22#351511yescharacteristic
675C22#351711yescharacteristic
676C22#351711yescharacteristic
677C22#354611yescharacteristic
678C22#354711yescharacteristic
679C22#355411yescharacteristic
680C22#355511yescharacteristic
681C4—28no—
682C4—24no—
683C4—28no—
684C4—44no—
685C4—42no—
686C4—44no—
687C4—42no—
688C4—44no—
689C4—42no—
690C4—48no—
691C4—42no—
692C4—44no—
693C4—82no—
694C4—84no—
695C4—82no—
696C4#255118yes—
697C4#255218yescomplemented
698C2C2×#255111yescharacteristic
699C2C2×#255211yescharacteristic
700C2#3564011yescharacteristic
701C2D4×#12511yescharacteristic
702C2#3626111yescharacteristic
703C2#3646611yescharacteristic
704C2Q8×#12511yescharacteristic
705C2#3896611yescharacteristic
706C2#3918311yescharacteristic
707C2#3937011yescharacteristic
708C2#3962611yescharacteristic
709C2#5905111yescharacteristic
710C2#5905211yescharacteristic
711C2#5906611yescharacteristic
712C2#5906711yescharacteristic
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
498. C4⋊C4#125semidirect
697. C4#2552semidirect

Maximal subgroups

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

subgroupindex
C2×#28592
C2×#25522
#157662
#155202
#152282
#152182
#151372
#151332
#140062
#140052
#139102
C4⋊C422
C4×#3892