Finite Group Databasebeta

← #532537

Subgroups — #532537

Order 512.

2582 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2C2×#2370C212yesmaximal
3#14532C214yesmaximal
4#14191C214yesmaximal
5#14151C211yesmaximal, characteristic
6#14133C212yesmaximal
7#14133C212yesmaximal
8#14054C212yesmaximal
9#14049C212yesmaximal
10#14046C212yesmaximal
11#14041C212yesmaximal
12C4⋊C42C212yesmaximal
13C4⋊C42C211yesmaximal, characteristic
14C4⋊C42C211yesmaximal, characteristic
15C4×#384C211yesmaximal, characteristic
16C4×#389C211yesmaximal, characteristic
17C4×#389C211yesmaximal, characteristic
18C22×#125C2212yes—
19#2552C4116yescomplemented
20C2×#398C2214yes—
21#2370—216no—
22C2×#389C2211yescharacteristic
23C2×#389C2212yes—
24C2×#389C2214yes—
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2211yescharacteristic
31C2×#387C2214yes—
32C2×#387C2214yes—
33C2×#387C2214yes—
34C2×#384C2211yescharacteristic
35C2×#384C2212yes—
36C2×#384C2211yescharacteristic
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2214yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
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×C43C2211yescharacteristic
52#1445C2214yes—
53#1445C2214yes—
54#1443C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2212yes—
62#1436C2212yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2212yes—
68#1436C2212yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2214yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4×#95C2212yes—
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⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2314yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2314yes—
122C22×C42C2314yes—
123C22×C42C2311yescharacteristic
124#398C2×C4132yes—
125#389—28no—
126#389—216no—
127#389—24no—
128#389—24no—
129#389—24no—
130#389—44no—
131#389C2×C4116yes—
132#389C2×C4116yes—
133#389D418yes—
134#387—216no—
135#387—216no—
136#387—48no—
137#384C2×C4116yes—
138#384C2×C418yes—
139#384Q818yes—
140C4×C4⋊C4—216no—
141C4×C4⋊C4—28no—
142C4×C4⋊C4—28no—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—28no—
145C4×C4⋊C4—28no—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—44no—
150C4×C4⋊C4—44no—
151C4×C4⋊C4—44no—
152C4×C4⋊C4C2×C418yes—
153C4×C4⋊C4C2×C418yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C43—24no—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2414yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C23×C4C2414yes—
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201#125—264no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
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—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4C22×C418yes—
251C2×C4⋊C4C22×C4116yes—
252C2×C4⋊C4C22×C4116yes—
253C2×C4⋊C4C22×C414yes—
254C2×C4⋊C4C22×C414yes—
255C2×C4⋊C4C22×C414yes—
256C2×C4⋊C4C22×C418yes—
257C2×C4⋊C4C22×C4116yes—
258C2×C4⋊C4C22×C418yes—
259C2×C4⋊C4C22×C414yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D418yes—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×Q818yes—
264C2×C4⋊C4C2×Q818yes—
265C2×C4⋊C4C2×Q818yes—
266C2×C4⋊C4#4114yes—
267C2×C4⋊C4#4114yes—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—28no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—28no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—28no—
295C2×C42—28no—
296C2×C42—44no—
297C2×C42—41no—
298C2×C42—41no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42C22×C418yes—
304C2×C42C22×C418yes—
305C2×C42C22×C414yes—
306C2×C42C2×D414yes—
307C2×C42C2×Q814yes—
308C2×C42#4114yes—
309C2×C42#4114yes—
310#95—24no—
311#95—28no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—24no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—24no—
326#95—24no—
327#95—42no—
328#95—44no—
329#95—42no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95C22×C418yes—
336#95C22×C418yes—
337#95C22×C4116yes—
338#95C2×D418yes—
339C24C2511yescharacteristic
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
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×C4C23×C414yes—
427C22×C4C23×C414yes—
428C22×C4C23×C414yes—
429C22×C4C23×C418yes—
430C22×C4C23×C414yes—
431C22×C4C23×C412yes—
432C22×C4C23×C412yes—
433C22×C4C23×C412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D414yes—
436C22×C4C22×D414yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×Q818yes—
440C22×C4C22×Q812yes—
441C22×C4C22×Q812yes—
442C22×C4C22×Q812yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4114yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C4H(5,𝔽2)12yes—
455C22×C42−1+412yes—
456C22×C42−1+412yes—
457C22×C42−1+412yes—
458C22×C42−1+412yes—
459C4⋊C4—232no—
460C4⋊C4—232no—
461C4⋊C4—216no—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—28no—
465C4⋊C4—216no—
466C4⋊C4—216no—
467C4⋊C4—28no—
468C4⋊C4—216no—
469C4⋊C4—28no—
470C4⋊C4—216no—
471C4⋊C4—48no—
472C4⋊C4—416no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—88no—
480C4⋊C4C4×D4116yes—
481C42—28no—
482C42—216no—
483C42—216no—
484C42—28no—
485C42—28no—
486C42—416no—
487C42—44no—
488C42—48no—
489C42—44no—
490C42—48no—
491C42—44no—
492C42—88no—
493C42—88no—
494C42C4×Q8116yes—
495C23C24×C411yescharacteristic
496C23C23×D411yescharacteristic
497C23C23×Q811yescharacteristic
498C23C22×#4111yescharacteristic
499C23C22×#4111yescharacteristic
500C23C22×#4111yescharacteristic
501C23C22×#4111yescharacteristic
502C23C2×H(5,𝔽2)11yescharacteristic
503C23C2×H(5,𝔽2)11yescharacteristic
504C23C2×2−1+411yescharacteristic
505C23C2×2−1+411yescharacteristic
506C23C2×2−1+411yescharacteristic
507C23C2×2−1+411yescharacteristic
508C23#58511yescharacteristic
509C23#58511yescharacteristic
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—28no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—41no—
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—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4C2×C4×D418yes—
613C2×C4C2×C4×D414yes—
614C2×C4C2×C4×Q814yes—
615C2×C4C2×C4×Q814yes—
616C2×C4C2×C4×Q814yes—
617C2×C4C4×#4114yes—
618C2×C4#51914yes—
619C2×C4#52014yes—
620C2×C4#54714yes—
621C2×C4#54714yes—
622C2×C4D4×Q814yes—
623C2×C4#55214yes—
624C2×C4#55314yes—
625C2×C4#55414yes—
626C2×C4#55614yes—
627C2×C4#55714yes—
628C2×C4#55714yes—
629C22C22×C4×D411yescharacteristic
630C22C22×C4×Q811yescharacteristic
631C22C2×C4×#4111yescharacteristic
632C22C2×#51911yescharacteristic
633C22C2×#52011yescharacteristic
634C22C4×H(5,𝔽2)11yescharacteristic
635C22C4×2−1+411yescharacteristic
636C22C2×#54711yescharacteristic
637C22C2×#54711yescharacteristic
638C22C2×D4×Q811yescharacteristic
639C22D4×#4111yescharacteristic
640C22C2×#55211yescharacteristic
641C22C2×#55311yescharacteristic
642C22C2×#55411yescharacteristic
643C22C2×#55611yescharacteristic
644C22#347511yescharacteristic
645C22C2×#55711yescharacteristic
646C22C2×#55711yescharacteristic
647C22#348211yescharacteristic
648C22#349211yescharacteristic
649C22#349311yescharacteristic
650C22#349711yescharacteristic
651C22#349811yescharacteristic
652C22#350111yescharacteristic
653C22#350211yescharacteristic
654C22#350711yescharacteristic
655C22#351111yescharacteristic
656C22#351511yescharacteristic
657C22#351711yescharacteristic
658C22#352211yescharacteristic
659C22#352211yescharacteristic
660C22#355211yescharacteristic
661C22#355411yescharacteristic
662C22#355511yescharacteristic
663C22#355611yescharacteristic
664C4—28no—
665C4—24no—
666C4—24no—
667C4—44no—
668C4—48no—
669C4—44no—
670C4—48no—
671C4—42no—
672C4—44no—
673C4—42no—
674C4—42no—
675C4—44no—
676C4—44no—
677C4—82no—
678C4—82no—
679C4#236518yes—
680C4#237018yescomplemented
681C2C2×#236511yescharacteristic
682C2C2×#237011yescharacteristic
683C2C4×#54711yescharacteristic
684C2#3427111yescharacteristic
685C2#3431311yescharacteristic
686C2#3431711yescharacteristic
687C2#3448811yescharacteristic
688C2#5924311yescharacteristic
689C2#5924911yescharacteristic
690C2#5925811yescharacteristic
691C2#5930411yescharacteristic
692C2#5945811yescharacteristic
693C2#5946711yescharacteristic
694C2#5966511yescharacteristic
695C2#5982811yescharacteristic
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. #2552C4semidirect
680. C4#2370semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#23702
#145322
#141912
#141512
#141332
#140542
#140492
#140462
#140412
C4⋊C422
C4×#3842
C4×#3892