Finite Group Databasebeta

← #516840

Subgroups — #516840

Order 512.

2638 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2C2×#2330C212yesmaximal
3#14712C212yesmaximal
4#14699C212yesmaximal
5#14328C214yesmaximal
6#14211C214yesmaximal
7#14151C211yesmaximal, characteristic
8#14054C212yesmaximal
9#14029C212yesmaximal
10#14005C212yesmaximal
11#14005C212yesmaximal
12#13910C211yesmaximal, characteristic
13#13910C211yesmaximal, characteristic
14#13783C212yesmaximal
15C4×#384C211yesmaximal, characteristic
16#13755C211yesmaximal, characteristic
17C42×C4⋊C4C211yesmaximal, characteristic
18C22×#125C2212yes—
19#2552C4116yescomplemented
20C2×#398C2214yes—
21#2330—216no—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2214yes—
25C2×#389C2214yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2214yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#387C2214yes—
34C2×#387C2214yes—
35C2×#384C2212yes—
36C2×#384C2211yescharacteristic
37C2×#384C2212yes—
38C2×#384C2211yescharacteristic
39C2×#384C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×#376C2212yes—
52C2×C43C2212yes—
53C2×C43C2211yescharacteristic
54#1445C2214yes—
55#1444C2214yes—
56#1444C2214yes—
57#1444C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2214yes—
73C4⋊#95C2214yes—
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2212yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2212yes—
78#1434C2214yes—
79#1434C2212yes—
80#1434C2212yes—
81C4×#95C2212yes—
82C4×#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C2×#125—28no—
86C2×#125—28no—
87C2×#125C2×C4116yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2314yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#398C2×C4132yes—
127#389—28no—
128#389—28no—
129#389—28no—
130#389—216no—
131#389—216no—
132#389—48no—
133#389—44no—
134#389—44no—
135#389—44no—
136#389C2×C4116yes—
137#389C2×C4116yes—
138#387—216no—
139#387—216no—
140#384—28no—
141#384—24no—
142#384—44no—
143#384C2×C4116yes—
144#384C2×C418yes—
145C4×C4⋊C4—24no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—44no—
151C4×C4⋊C4—44no—
152C4×C4⋊C4C2×C418yes—
153C4×C4⋊C4C2×C418yes—
154C4×C4⋊C4D418yes—
155C4×C4⋊C4Q818yes—
156#376—28no—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C43—44no—
185C43Q818yes—
186C23×C4C2412yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2414yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203#125—264no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C22×C418yes—
255C2×C4⋊C4C22×C4116yes—
256C2×C4⋊C4C22×C4116yes—
257C2×C4⋊C4C22×C414yes—
258C2×C4⋊C4C22×C414yes—
259C2×C4⋊C4C22×C418yes—
260C2×C4⋊C4C22×C4116yes—
261C2×C4⋊C4C22×C418yes—
262C2×C4⋊C4C22×C414yes—
263C2×C4⋊C4C22×C414yes—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×Q814yes—
267C2×C4⋊C4C2×Q814yes—
268C2×C4⋊C4#4114yes—
269C2×C4⋊C4#4114yes—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—28no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—41no—
299C2×C42—41no—
300C2×C42—42no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42C22×C418yes—
304C2×C42C22×C418yes—
305C2×C42C22×C414yes—
306C2×C42C2×D418yes—
307C2×C42C2×D414yes—
308C2×C42C2×Q814yes—
309C2×C42C2×Q814yes—
310C2×C42C2×Q814yes—
311C2×C42C2×Q814yes—
312C2×C42C2×Q814yes—
313C2×C42C2×Q818yes—
314C2×C42C2×Q814yes—
315C2×C42C2×Q814yes—
316C2×C42C2×Q814yes—
317C2×C42#4114yes—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—24no—
332#95—24no—
333#95—42no—
334#95—44no—
335#95—42no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95C22×C418yes—
341#95C22×C418yes—
342#95C22×C4116yes—
343#95C2×D418yes—
344#95C2×Q818yes—
345C24C2511yescharacteristic
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4C23×C414yes—
428C22×C4C23×C414yes—
429C22×C4C23×C418yes—
430C22×C4C23×C414yes—
431C22×C4C23×C412yes—
432C22×C4C23×C414yes—
433C22×C4C23×C412yes—
434C22×C4C23×C412yes—
435C22×C4C22×D414yes—
436C22×C4C22×D414yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×Q814yes—
441C22×C4C22×Q812yes—
442C22×C4C22×Q814yes—
443C22×C4C22×Q812yes—
444C22×C4C22×Q812yes—
445C22×C4C22×Q812yes—
446C22×C4C22×Q812yes—
447C22×C4C22×Q812yes—
448C22×C4C22×Q812yes—
449C22×C4C22×Q812yes—
450C22×C4C22×Q812yes—
451C22×C4C22×Q812yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C42−1+412yes—
463C22×C42−1+412yes—
464C22×C42−1+412yes—
465C4⋊C4—232no—
466C4⋊C4—232no—
467C4⋊C4—216no—
468C4⋊C4—28no—
469C4⋊C4—216no—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—48no—
474C4⋊C4—416no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—88no—
481C4⋊C4—88no—
482C4⋊C4—88no—
483C4⋊C4—88no—
484C4⋊C4—88no—
485C4⋊C4C4×D4116yes—
486C4⋊C4C4×Q8116yes—
487C42—216no—
488C42—216no—
489C42—28no—
490C42—28no—
491C42—28no—
492C42—28no—
493C42—48no—
494C42—44no—
495C42—48no—
496C42—48no—
497C42—48no—
498C42—48no—
499C42—48no—
500C42—88no—
501C42C4×Q8116yes—
502C42C4⋊Q8116yes—
503C23C24×C411yescharacteristic
504C23C23×D411yescharacteristic
505C23C23×Q811yescharacteristic
506C23C23×Q811yescharacteristic
507C23C22×#4111yescharacteristic
508C23C22×#4111yescharacteristic
509C23C22×#4111yescharacteristic
510C23C2×H(5,𝔽2)11yescharacteristic
511C23C2×H(5,𝔽2)11yescharacteristic
512C23C2×H(5,𝔽2)11yescharacteristic
513C23C2×2−1+411yescharacteristic
514C23C2×2−1+411yescharacteristic
515C23C2×2−1+411yescharacteristic
516C23#58511yescharacteristic
517C23#58511yescharacteristic
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
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—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4C2×C4×D418yes—
618C2×C4C2×C4×D414yes—
619C2×C4C2×C4×Q814yes—
620C2×C4C2×C4×Q818yes—
621C2×C4C2×C4×Q814yes—
622C2×C4C2×C4×Q814yes—
623C2×C4C2×C4×Q814yes—
624C2×C4#52014yes—
625C2×C4#52014yes—
626C2×C4C2×C4⋊Q814yes—
627C2×C4C2×C4⋊Q814yes—
628C2×C4C2×C4⋊Q814yes—
629C2×C4#53214yes—
630C2×C4#53314yes—
631C2×C4#53314yes—
632C2×C4#54014yes—
633C2×C4#54114yes—
634C2×C4#54414yes—
635C2×C4#54414yes—
636C2×C4#54414yes—
637C22C22×C4×D411yescharacteristic
638C22C22×C4×Q811yescharacteristic
639C22C22×C4×Q811yescharacteristic
640C22C2×#52011yescharacteristic
641C22C2×#52011yescharacteristic
642C22C4×H(5,𝔽2)11yescharacteristic
643C22C4×2−1+411yescharacteristic
644C22C22×C4⋊Q811yescharacteristic
645C22C2×#53211yescharacteristic
646C22C2×#53311yescharacteristic
647C22C2×#53311yescharacteristic
648C22C2×#54011yescharacteristic
649C22C2×#54111yescharacteristic
650C22C2×#54411yescharacteristic
651C22C2×#54411yescharacteristic
652C22C2×#54411yescharacteristic
653C22#345811yescharacteristic
654C22#346011yescharacteristic
655C22#346111yescharacteristic
656C22#349611yescharacteristic
657C22#349711yescharacteristic
658C22#349811yescharacteristic
659C22#349911yescharacteristic
660C22#350111yescharacteristic
661C22#350211yescharacteristic
662C22#350311yescharacteristic
663C22#350311yescharacteristic
664C22#351511yescharacteristic
665C22#351511yescharacteristic
666C22#351711yescharacteristic
667C22#351711yescharacteristic
668C22#351911yescharacteristic
669C22#352011yescharacteristic
670C22#352211yescharacteristic
671C22#352311yescharacteristic
672C4—28no—
673C4—28no—
674C4—24no—
675C4—24no—
676C4—24no—
677C4—24no—
678C4—44no—
679C4—44no—
680C4—48no—
681C4—44no—
682C4—44no—
683C4—42no—
684C4—82no—
685C4—84no—
686C4—82no—
687C4C4×C4⋊Q818yes—
688C4#233018yescomplemented
689C2C2×C4×C4⋊Q811yescharacteristic
690C2C2×#233011yescharacteristic
691C2C4×#54411yescharacteristic
692C2#3430811yescharacteristic
693C2#3431311yescharacteristic
694C2#3433211yescharacteristic
695C2#3433711yescharacteristic
696C2#3516711yescharacteristic
697C2#3517111yescharacteristic
698C2#3517611yescharacteristic
699C2#3520211yescharacteristic
700C2#3520811yescharacteristic
701C2#3520911yescharacteristic
702C2#3523911yescharacteristic
703C2#3527511yescharacteristic
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
688. C4#2330semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#23302
#147122
#146992
#143282
#142112
#141512
#140542
#140292
#140052
#139102
#137832
C4×#3842
#137552
C42×C4⋊C42