Finite Group Databasebeta

← #737675

Subgroups — #737675

Order 512.

2278 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic
2C2×#2356C211yesmaximal, characteristic
3#16669C214yesmaximal
4#16101C214yesmaximal
5#16093C214yesmaximal
6#15237C211yesmaximal, characteristic
7#15191C211yesmaximal, characteristic
8#15191C211yesmaximal, characteristic
9#15184C211yesmaximal, characteristic
10#14901C214yesmaximal
11#14348C211yesmaximal, characteristic
12#14161C211yesmaximal, characteristic
13#14155C211yesmaximal, characteristic
14#14051C211yesmaximal, characteristic
15#14049C211yesmaximal, characteristic
16#13926C211yesmaximal, characteristic
17#13909C211yesmaximal, characteristic
18#13848C211yesmaximal, characteristic
19#13786C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×C4⋊Q8C2212yes—
23C22×C4×Q8C2211yescharacteristic
24#2551—28no—
25C2×#395C2212yes—
26C2×#395C2212yes—
27#2356—28no—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#391C2211yescharacteristic
32C2×#391C2214yes—
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2212yes—
38C2×#387C2212yes—
39C2×#387C2212yes—
40C2×#387C2214yes—
41C2×#384C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C2×#383C2212yes—
44C2×#382C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
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×#376C2211yescharacteristic
54C2×#376C2212yes—
55#1446C2214yes—
56#1446C2214yes—
57#1446C2214yes—
58#1445C2214yes—
59#1445C2214yes—
60#1444C2212yes—
61#1443C2214yes—
62#1442C2214yes—
63#1442C2214yes—
64#1442C2212yes—
65#1442C2214yes—
66#1442C2214yes—
67#1442C2214yes—
68#1441C2212yes—
69#1441C2212yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2214yes—
82C4⋊#95C2214yes—
83C4⋊#95C2211yescharacteristic
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2211yescharacteristic
87C4⋊#95C2211yescharacteristic
88C4⋊#95C2211yescharacteristic
89#1434C2214yes—
90#1434C2214yes—
91C4×#95C2211yescharacteristic
92C4×#95C2211yescharacteristic
93C23×Q8C2311yescharacteristic
94C23×Q8C2312yes—
95C23×Q8C2312yes—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—42no—
102C2×C4⋊Q8—42no—
103C2×C4⋊Q8—48no—
104C2×C4×Q8—24no—
105C2×C4×Q8—24no—
106C2×C4×Q8Q818yescomplemented
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138#395—28no—
139#395—28no—
140#391—24no—
141#391—24no—
142#391—24no—
143#391—24no—
144#391—42no—
145#391—42no—
146#391—48no—
147#389—28no—
148#389—24no—
149#389—24no—
150#387—28no—
151#387—28no—
152#387—48no—
153#384—24no—
154#384—24no—
155#383—28no—
156#382—24no—
157#382—24no—
158C4×C4⋊C4—24no—
159C4×C4⋊C4—24no—
160C4×C4⋊C4—24no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4—42no—
163C4×C4⋊C4—42no—
164C4×C4⋊C4D418yes—
165#376—24no—
166#376—28no—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2312yes—
201C2×#95C2312yes—
202C2×#95C2311yescharacteristic
203C2×#95C2311yescharacteristic
204C22×Q8—22no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—24no—
211C22×Q8—24no—
212C22×Q8—22no—
213C22×Q8—28no—
214C22×Q8—22no—
215C22×Q8—42no—
216C22×Q8—42no—
217C22×Q8—44no—
218C22×Q8—44no—
219C22×Q8—44no—
220C22×Q8C2×Q814yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C4⋊Q8—416no—
241C4⋊Q8—416no—
242C4⋊Q8—432no—
243C4×Q8—232no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D414yes—
294C2×C4⋊C4C2×Q814yes—
295C2×C4⋊C4C2×Q818yes—
296C2×C4⋊C4#4114yes—
297C2×C4⋊C4#4114yes—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—28no—
319C2×C42—28no—
320C2×C42—22no—
321C2×C42—44no—
322C2×C42C2×D414yes—
323C2×C42C2×D414yes—
324C2×C42C2×D414yes—
325C2×C42C2×Q818yes—
326C2×C42C2×Q814yes—
327C2×C42#4114yes—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—44no—
349#95—42no—
350#95—44no—
351#95—44no—
352#95—42no—
353#95—42no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95C2×D414yes—
364#95C2×D414yes—
365C24C2511yescharacteristic
366C2×Q8—28no—
367C2×Q8—28no—
368C2×Q8—28no—
369C2×Q8—48no—
370C2×Q8—48no—
371C2×Q8—416no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—84no—
379C2×Q8—88no—
380C2×Q8—84no—
381C2×Q8—88no—
382C2×Q8—88no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
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—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
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—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×Q814yes—
474C22×C4C22×Q814yes—
475C22×C4C22×Q812yes—
476C22×C4C22×Q812yes—
477C22×C4C22×Q812yes—
478C22×C4C2×#4114yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4H(5,𝔽2)14yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C42−1+414yes—
488C22×C42−1+414yes—
489C22×C42−1+412yes—
490C4⋊C4—28no—
491C4⋊C4—216no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—416no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—84no—
510C42—216no—
511C42—28no—
512C42—28no—
513C42—48no—
514C42—48no—
515C42—48no—
516C42—44no—
517C42—44no—
518C42—44no—
519C42—44no—
520C42C22⋊Q8116yes—
521C23C23×D411yescharacteristic
522C23C23×Q811yescharacteristic
523C23C22×#4111yescharacteristic
524C23C22×#4111yescharacteristic
525C23C22×#4111yescharacteristic
526C23C2×H(5,𝔽2)12yes—
527C23C2×H(5,𝔽2)11yescharacteristic
528C23C2×2−1+412yes—
529C23C2×2−1+411yescharacteristic
530C23C2×2−1+412yes—
531C23#58512yes—
532Q8—416no—
533Q8—816no—
534Q8—816no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
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×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4C2×C22⋊Q814yes—
632C2×C4C2×C22⋊Q814yes—
633C2×C4C2×C22⋊Q814yes—
634C2×C4#53314yes—
635C2×C4#53514yes—
636C2×C4#53614yes—
637C2×C4#54414yes—
638C2×C4#54614yes—
639C2×C4#54814yes—
640C2×C4#55514yes—
641C2×C4#55614yes—
642C22C22×C22⋊Q811yescharacteristic
643C22C2×#53311yescharacteristic
644C22C2×#53511yescharacteristic
645C22C2×#53611yescharacteristic
646C22C2×#54411yescharacteristic
647C22C2×#54611yescharacteristic
648C22C2×#54811yescharacteristic
649C22C2×#55511yescharacteristic
650C22C2×#55611yescharacteristic
651C22#349512yes—
652C22#349911yescharacteristic
653C22#350011yescharacteristic
654C22#350212yes—
655C22#350312yes—
656C22#351112yes—
657C22#351312yes—
658C22#351812yes—
659C22#352012yes—
660C22#352312yes—
661C22#353812yes—
662C22#354212yes—
663C22#355012yes—
664C22#355512yes—
665C4—24no—
666C4—24no—
667C4—24no—
668C4—24no—
669C4—44no—
670C4—44no—
671C4—42no—
672C4—42no—
673C4—44no—
674C4—44no—
675C4—44no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4#243918yes—
684C2C2×#243911yescharacteristic
685C2#3527911yescharacteristic
686C2#3566811yescharacteristic
687C2#3586712yes—
688C2#3808111yescharacteristic
689C2#3808411yescharacteristic
690C2#3916711yescharacteristic
691C2#3917211yescharacteristic
692C2#4282012yes—
693C2#4506512yes—
694C2#4507812yes—
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
106. C2×C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#23562
#166692
#161012
#160932
#152372
#151912
#151842
#149012
#143482
#141612
#141552
#140512
#140492
#139262
#139092
#138482
#137862