Finite Group Databasebeta

← #524629

Subgroups — #524629

Order 512.

3182 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×#2344C211yesmaximal, characteristic
2C2×#2344C211yesmaximal, characteristic
3C22×#389C211yesmaximal, characteristic
4#14697C211yesmaximal, characteristic
5#14502C214yesmaximal, complemented
6#14468C214yesmaximal, complemented
7#14263C212yesmaximal
8#14130C211yesmaximal, characteristic
9#14128C212yesmaximal
10#14045C212yesmaximal, complemented
11#14006C212yesmaximal, complemented
12#13962C212yesmaximal
13#13954C212yesmaximal, complemented
14#13933C212yesmaximal, complemented
15#13881C211yesmaximal, characteristic
16#13795C212yesmaximal
17C4×#385C211yesmaximal, characteristic
18C22×#117C2212yes—
19C22×#117C2212yes—
20C23×C4⋊C4C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2344—28no—
25#2344C4116yescomplemented
26C2×#389—28no—
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2214yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#387C2212yes—
36C2×#387C2212yes—
37C2×#387C2214yes—
38C2×#385C2211yescharacteristic
39C2×#385C2211yescharacteristic
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#382C2212yes—
43C2×#381C2212yes—
44C2×#381C2212yes—
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53C22×#95C2212yes—
54#1444C2214yes—
55#1444C2212yes—
56#1444C2214yes—
57#1444C2214yes—
58#1442C2214yes—
59#1442C2212yes—
60#1442C2214yes—
61#1442C2214yes—
62#1442C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2212yes—
69#1436C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2214yes—
74C4⋊#95C2214yes—
75C4⋊#95C2214yes—
76C4⋊#95C2214yes—
77C4⋊#95C2214yes—
78#1434C2214yes—
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2211yescharacteristic
84C22⋊#95C2211yescharacteristic
85C4×#95C2212yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C2×#117—28no—
92C2×#117—28no—
93C2×#117—44no—
94C2×#117—44no—
95C2×#117—44no—
96C2×#117C2×C4116yes—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—28no—
100C22×C4⋊C4—44no—
101C22×C4⋊C4—44no—
102C22×C4⋊C4C2×C418yes—
103C22×C4⋊C4D418yes—
104C22×C4⋊C4Q818yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2311yescharacteristic
128C22×C42—24no—
129C22×C42—44no—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2314yes—
138#389—24no—
139#389—24no—
140#389—28no—
141#389—216no—
142#389—28no—
143#389—44no—
144#389—44no—
145#389C2×C4116yescomplemented
146#387—28no—
147#387—28no—
148#387C2×C4132yescomplemented
149#385—24no—
150#385—24no—
151#385—44no—
152#385—44no—
153#382—44no—
154#381—28no—
155#381C2×C4116yes—
156#380—24no—
157#380C2×C418yes—
158C4×C4⋊C4—24no—
159C4×C4⋊C4—24no—
160C4×C4⋊C4—44no—
161C4×C4⋊C4C2×C4116yescomplemented
162C4×C22⋊C4—24no—
163C4×C22⋊C4C2×C418yes—
164C2×#95—28no—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2311yescharacteristic
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C25C2411yescharacteristic
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4C22×C418yes—
216C23×C4C22×C414yes—
217C23×C4C2×D418yes—
218C23×C4C2×D414yes—
219C23×C4C2×Q818yes—
220C23×C4C2×Q814yes—
221C23×C4#4114yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238#117—432no—
239#117—432no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—432no—
273C2×C4⋊C4—416no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4C4⋊C4164yescomplemented
280C2×C4⋊C4C22×C418yes—
281C2×C4⋊C4C22×C4116yes—
282C2×C4⋊C4C22×C414yes—
283C2×C4⋊C4C22×C418yes—
284C2×C4⋊C4C22×C414yes—
285C2×C4⋊C4C2×D414yes—
286C2×C4⋊C4C2×D418yes—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2×Q818yes—
289C2×C4⋊C4C2×Q814yes—
290C2×C4⋊C4C2×Q814yes—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4C22×C414yes—
306C2×C22⋊C4C22×C418yes—
307C2×C22⋊C4C22×C414yes—
308C2×C22⋊C4#4114yes—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42—42no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42—416no—
330C2×C42C22×C418yes—
331C2×C42C22×C4116yes—
332#95—28no—
333#95—28no—
334#95—22no—
335#95—22no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—42no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—42no—
350#95—42no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95C22×C418yes—
358#95C22×C4116yes—
359#95C22×C418yes—
360#95C22×C4116yes—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24C23×C412yes—
366C24C22×D412yes—
367C24C22×Q812yes—
368C24C2×#4112yes—
369C24C2×#4112yes—
370C24H(5,𝔽2)12yes—
371C24H(5,𝔽2)12yes—
372C24C2511yescharacteristic
373C22×C4—216no—
374C22×C4—216no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—216no—
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—216no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—216no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—48no—
467C22×C4—44no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4C2×C4⋊C4132yes—
474C22×C4C2×C4⋊C4116yes—
475C22×C4C23×C414yes—
476C22×C4C23×C418yes—
477C22×C4C23×C414yes—
478C22×C4C23×C412yes—
479C22×C4C23×C414yes—
480C22×C4C23×C414yes—
481C22×C4C23×C412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D414yes—
484C22×C4C22×D414yes—
485C22×C4C22×D412yes—
486C22×C4C22×Q812yes—
487C22×C4C22×Q814yes—
488C22×C4C22×Q814yes—
489C22×C4C22×Q812yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4114yes—
492C22×C4C2×#4112yes—
493C22×C4H(5,𝔽2)14yes—
494C22×C42−1+414yes—
495C22×C42−1+414yes—
496C4⋊C4—216no—
497C4⋊C4—416no—
498C4⋊C4—48no—
499C4⋊C4—416no—
500C4⋊C4—416no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—88no—
509C4⋊C4—88no—
510C4⋊C4C2×C4⋊C4132yes—
511C4⋊C4C2×C4⋊C4116yes—
512C4⋊C4C2×C4⋊C4116yes—
513C22⋊C4—28no—
514C22⋊C4—216no—
515C22⋊C4—28no—
516C22⋊C4—28no—
517C22⋊C4—44no—
518C22⋊C4—48no—
519C42—216no—
520C42—416no—
521C42—48no—
522C42—48no—
523C42—48no—
524C42—44no—
525C42—48no—
526C42—44no—
527C23—24no—
528C23—24no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23C22×C4⋊C414yes—
544C23C4×#4114yes—
545C23#51814yes—
546C23#54614yes—
547C23#54614yes—
548C23#55414yes—
549C23#55414yes—
550C23C24×C411yescharacteristic
551C23C23×D411yescharacteristic
552C23C23×Q811yescharacteristic
553C23C22×#4111yescharacteristic
554C23C22×#4111yescharacteristic
555C23C2×H(5,𝔽2)11yescharacteristic
556C23C2×H(5,𝔽2)11yescharacteristic
557C23C2×H(5,𝔽2)12yes—
558C23C2×2−1+412yes—
559C23C2×2−1+412yes—
560C23#58512yes—
561C2×C4—264no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
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—232no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—416no—
599C2×C4—416no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—44no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—816no—
654C2×C4C22×C4⋊C414yes—
655C2×C4C22×C4⋊C418yes—
656C2×C4C22×C4⋊C418yes—
657C2×C4C22×C4⋊C414yes—
658C2×C4#52018yes—
659C2×C4#53514yes—
660C2×C4#53614yes—
661C2×C4#54418yes—
662C22—24no—
663C22—28no—
664C22—24no—
665C22—24no—
666C22—28no—
667C22—28no—
668C22—28no—
669C22—24no—
670C22—24no—
671C22—24no—
672C22#234918yes—
673C22C23×C4⋊C411yescharacteristic
674C22C2×C4×#4111yescharacteristic
675C22C2×#51811yescharacteristic
676C22C2×#52012yes—
677C22C4×2−1+412yes—
678C22C2×#53511yescharacteristic
679C22C2×#53611yescharacteristic
680C22C2×#54412yes—
681C22C2×#54611yescharacteristic
682C22C2×#54611yescharacteristic
683C22C2×#55411yescharacteristic
684C22C2×#55411yescharacteristic
685C22#348611yescharacteristic
686C22#348711yescharacteristic
687C22#350212yes—
688C22#350512yes—
689C22#350912yes—
690C22#351512yes—
691C22#352012yes—
692C22#353612yes—
693C22#354212yes—
694C22#354412yes—
695C22#355912yes—
696C4—24no—
697C4—28no—
698C4—28no—
699C4—44no—
700C4—44no—
701C4—44no—
702C4—48no—
703C4—42no—
704C4—44no—
705C4—42no—
706C4—81no—
707C4—81no—
708C4—84no—
709C4—82no—
710C4—82no—
711C4#231718yes—
712C2—28no—
713C2C2×#231711yescharacteristic
714C2C2×#234911yescharacteristic
715C2#3424911yescharacteristic
716C2#3431412yes—
717C2#3446012yes—
718C2#3890511yescharacteristic
719C2#3890711yescharacteristic
720C2#3896312yes—
721C2#3946511yescharacteristic
722C2#3946711yescharacteristic
723C2#3949612yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #14502C2semidirect
6. #14468C2semidirect
10. #14045C2semidirect
11. #14006C2semidirect
13. #13954C2semidirect
14. #13933C2semidirect
25. #2344C4semidirect
145. #389C2×C4semidirect
148. #387C2×C4semidirect
161. C4×C4⋊C4C2×C4semidirect
279. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23442
C22×#3892
#146972
#145022
#144682
#142632
#141302
#141282
#140452
#140062
#139622
#139542
#139332
#138812
#137952
C4×#3852