Finite Group Databasebeta

← #737676

Subgroups — #737676

Order 512.

2354 conjugacy classes of subgroups.

The subgroup lattice has 739 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2357C211yesmaximal, characteristic
3#16302C214yesmaximal, complemented
4#16085C214yesmaximal, complemented
5#15396C214yesmaximal, complemented
6#15229C211yesmaximal, characteristic
7#15229C211yesmaximal, characteristic
8#15222C211yesmaximal, characteristic
9#15134C211yesmaximal, characteristic
10#14775C214yesmaximal, complemented
11#14348C211yesmaximal, characteristic, complemented
12#14160C211yesmaximal, characteristic, complemented
13#14160C211yesmaximal, characteristic, complemented
14#14051C211yesmaximal, characteristic, complemented
15#14049C211yesmaximal, characteristic, complemented
16#13909C211yesmaximal, characteristic, complemented
17#13906C211yesmaximal, characteristic
18#13848C211yesmaximal, characteristic, complemented
19#13785C211yesmaximal, characteristic
20C22×#125C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23#2552—28no—
24C2×#398C2212yes—
25C2×#398C2212yes—
26C2×#396C2214yes—
27#2357—28no—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#389C2212yes—
37C2×#388C2212yes—
38C2×#388C2212yes—
39C2×#387C2214yes—
40C2×#384C2211yescharacteristic
41C2×#384C2211yescharacteristic
42C2×#384C2212yes—
43C2×#382C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×#381C2212yes—
46C2×#380C2214yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2212yes—
55C2×#376C2211yescharacteristic
56#1445C2214yes—
57#1444C2212yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2212yes—
61#1442C2214yes—
62#1441C2212yes—
63#1441C2212yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1437C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2211yescharacteristic
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2211yescharacteristic
87C4⋊#95C2211yescharacteristic
88C4⋊#95C2211yescharacteristic
89#1434C2214yes—
90C22⋊#95C2214yes—
91C4×#95C2211yescharacteristic
92C4×#95C2211yescharacteristic
93C23×D4C2311yescharacteristic
94C24×C4C2312yes—
95C2×#125—24no—
96C2×#125—24no—
97C2×#125—24no—
98C2×#125—24no—
99C2×#125—42no—
100C2×#125—42no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×C4×D4Q818yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2312yes—
141C22×C42C2312yes—
142C22×C42C2312yes—
143C22×C42C2311yescharacteristic
144#398—28no—
145#398—28no—
146#396—48no—
147#390—24no—
148#390—24no—
149#390—24no—
150#390—24no—
151#390—42no—
152#390—42no—
153#389—28no—
154#389—24no—
155#389—24no—
156#388—28no—
157#388—28no—
158#387—48no—
159#384—28no—
160#384—24no—
161#384—24no—
162#382—24no—
163#382—24no—
164#381—28no—
165#380—48no—
166C4×C4⋊C4—24no—
167C4×C4⋊C4—24no—
168C4×C4⋊C4—24no—
169C4×C4⋊C4—24no—
170C4×C4⋊C4—42no—
171C4×C4⋊C4—42no—
172C4×C4⋊C4D418yes—
173C4×C22⋊C4—28no—
174#376—24no—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2312yes—
201C2×#95C2312yes—
202C2×#95C2311yescharacteristic
203C2×#95C2311yescharacteristic
204C25C2412yes—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4C2×Q814yes—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4C2×Q818yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235#125—416no—
236#125—416no—
237C4×D4—232no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4C2×D414yes—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×Q814yes—
293C2×C4⋊C4#4114yes—
294C2×C4⋊C4#4114yes—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4C2×Q818yes—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—44no—
338C2×C42—44no—
339C2×C42—44no—
340C2×C42C2×D414yes—
341C2×C42C2×D414yes—
342C2×C42C2×D414yes—
343C2×C42C2×Q814yes—
344C2×C42#4114yes—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—28no—
356#95—28no—
357#95—24no—
358#95—44no—
359#95—44no—
360#95—42no—
361#95—44no—
362#95—42no—
363#95—42no—
364#95—44no—
365#95—44no—
366#95—42no—
367#95—42no—
368#95—44no—
369#95—44no—
370#95—44no—
371#95—44no—
372#95C2×D414yes—
373#95C2×D414yes—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—24no—
379C24—22no—
380C24—22no—
381C24—24no—
382C24—24no—
383C24—24no—
384C24C22×Q814yes—
385C24C2511yescharacteristic
386C2×D4—28no—
387C2×D4—28no—
388C2×D4—28no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—216no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—416no—
474C22×C4—88no—
475C22×C4—84no—
476C22×C4—84no—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×Q814yes—
485C22×C4C22×Q812yes—
486C22×C4C22×Q812yes—
487C22×C4C22×Q812yes—
488C22×C4C2×#4114yes—
489C22×C4C2×#4114yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4H(5,𝔽2)14yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C42−1+412yes—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—416no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C4⋊C4—84no—
519C22⋊C4—216no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4—88no—
525C22⋊C4—88no—
526C22⋊C4—88no—
527C22⋊C4—88no—
528C42—28no—
529C42—28no—
530C42—48no—
531C42—44no—
532C42—44no—
533C42—44no—
534C42—44no—
535C42—84no—
536C42—84no—
537C42C22⋊Q8116yes—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—28no—
542C23—28no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—44no—
550C23—42no—
551C23—42no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23C23×D411yescharacteristic
562C23C23×Q811yescharacteristic
563C23C22×#4111yescharacteristic
564C23C22×#4111yescharacteristic
565C23C22×#4111yescharacteristic
566C23C2×H(5,𝔽2)11yescharacteristic
567C23C2×H(5,𝔽2)12yes—
568C23C2×2−1+411yescharacteristic
569C23#58512yes—
570C23#58512yes—
571C23#58512yes—
572D4—416no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
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—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—816no—
665C2×C4C2×C22⋊Q814yes—
666C2×C4C2×C22⋊Q814yes—
667C2×C4C2×C22⋊Q814yes—
668C2×C4#53314yes—
669C2×C4#54414yes—
670C2×C4#54614yes—
671C2×C4#54814yes—
672C2×C4#55514yes—
673C2×C4#55614yes—
674C22—44no—
675C22—48no—
676C22—44no—
677C22—44no—
678C22—48no—
679C22—84no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—84no—
685C22C22×C22⋊Q811yescharacteristic
686C22C2×#53311yescharacteristic
687C22#344911yescharacteristic
688C22#344911yescharacteristic
689C22C2×#54411yescharacteristic
690C22C2×#54611yescharacteristic
691C22C2×#54811yescharacteristic
692C22C2×#55511yescharacteristic
693C22C2×#55611yescharacteristic
694C22#349412yes—
695C22#349811yescharacteristic
696C22#350011yescharacteristic
697C22#350112yes—
698C22#350412yes—
699C22#351212yes—
700C22#351312yes—
701C22#351712yes—
702C22#352112yes—
703C22#352412yes—
704C22#353912yes—
705C22#354212yes—
706C22#355412yes—
707C22#355712yes—
708C4—24no—
709C4—24no—
710C4—44no—
711C4—42no—
712C4—42no—
713C4—42no—
714C4—42no—
715C4—44no—
716C4—44no—
717C4—44no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4#243918yes—
726C2—84no—
727C2C2×#243911yescharacteristic
728C2#3533311yescharacteristic
729C2#3552911yescharacteristic
730C2#3596312yes—
731C2#3808911yescharacteristic
732C2#3809211yescharacteristic
733C2#3916111yescharacteristic
734C2#3916411yescharacteristic
735C2#4249512yes—
736C2#4505212yes—
737C2#4519712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2552C2semidirect
3. #16302C2semidirect
4. #16085C2semidirect
5. #15396C2semidirect
10. #14775C2semidirect
11. #14348C2semidirect
12. #14160C2semidirect
13. #14160C2semidirect
14. #14051C2semidirect
15. #14049C2semidirect
16. #13909C2semidirect
18. #13848C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#23572
#163022
#160852
#153962
#152292
#152222
#151342
#147752
#143482
#141602
#140512
#140492
#139092
#139062
#138482
#137852