Finite Group Databasebeta

← #874354

Subgroups — #874354

Order 512.

2238 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×#2511C211yesmaximal, characteristic
3#16524C214yesmaximal
4#15807C214yesmaximal
5#15627C214yesmaximal
6#15199C211yesmaximal, characteristic
7#15191C211yesmaximal, characteristic
8#15185C211yesmaximal, characteristic
9#15185C211yesmaximal, characteristic
10#15180C211yesmaximal, characteristic
11#15143C211yesmaximal, characteristic
12#14714C211yesmaximal, characteristic
13#14687C211yesmaximal, characteristic
14#14404C214yesmaximal
15#14024C211yesmaximal, characteristic
16#14000C211yesmaximal, characteristic
17#13976C211yesmaximal, characteristic
18#13910C211yesmaximal, characteristic
19C4×#387C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23#2567—28no—
24#2511—28no—
25C2×#400C2212yes—
26C2×#400C2212yes—
27C2×#395C2212yes—
28C2×#395C2212yes—
29C2×#395C2214yes—
30C2×#391C2211yescharacteristic
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#391C2211yescharacteristic
36C2×#391C2211yescharacteristic
37C2×#391C2211yescharacteristic
38C2×#391C2211yescharacteristic
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#389C2214yes—
45C2×#387C2212yes—
46C2×#387C2212yes—
47C2×#387C2212yes—
48C2×#384C2211yescharacteristic
49C2×#384C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C2×#384C2211yescharacteristic
52C2×#384C2211yescharacteristic
53C2×#384C2211yescharacteristic
54C2×#382C2214yes—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C43C2211yescharacteristic
59#1446C2214yes—
60#1445C2214yes—
61#1445C2214yes—
62#1444C2214yes—
63#1444C2214yes—
64#1444C2214yes—
65#1444C2214yes—
66#1444C2214yes—
67#1442C2214yes—
68#1442C2214yes—
69#1442C2214yes—
70#1442C2214yes—
71#1442C2214yes—
72#1441C2214yes—
73#1441C2214yes—
74#1436C2212yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78#1436C2212yes—
79C4⋊#95C2214yes—
80C4⋊#95C2214yes—
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2214yes—
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2211yescharacteristic
87C4⋊#95C2211yescharacteristic
88#1434C2212yes—
89C4×#95C2211yescharacteristic
90C4×#95C2211yescharacteristic
91C4×#95C2211yescharacteristic
92C4×#95C2211yescharacteristic
93C23×Q8C2311yescharacteristic
94C23×Q8C2311yescharacteristic
95C23×Q8C2312yes—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—42no—
100C2×C4⋊Q8—42no—
101C2×C4⋊Q8—42no—
102C2×C4×Q8—24no—
103C2×C4×Q8—42no—
104C2×C4×Q8—42no—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137C22×C42C2312yes—
138#400—28no—
139#400—28no—
140#395—28no—
141#395—28no—
142#395—48no—
143#391—24no—
144#391—24no—
145#391—24no—
146#391—24no—
147#391—24no—
148#391—24no—
149#391—24no—
150#391—42no—
151#391—42no—
152#391—42no—
153#391—42no—
154#389—28no—
155#389—28no—
156#389—24no—
157#389—48no—
158#387—28no—
159#387—28no—
160#387Q8116yescomplemented
161#384—24no—
162#384—24no—
163#384—24no—
164#384—42no—
165#384—42no—
166#384—42no—
167#382—48no—
168C4×C4⋊C4—24no—
169C4×C4⋊C4—24no—
170C4×C4⋊C4D418yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2314yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2311yescharacteristic
201C2×#95C2311yescharacteristic
202C2×#95C2312yes—
203C2×#95C2312yes—
204C2×#95C2312yes—
205C2×#95C2312yes—
206C43—24no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—24no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—42no—
222C22×Q8—42no—
223C22×Q8—44no—
224C22×Q8—44no—
225C22×Q8#4114yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C4⋊Q8—416no—
246C4⋊Q8—416no—
247C4×Q8—416no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4C2×D414yes—
302C2×C4⋊C4C2×D414yes—
303C2×C4⋊C4C2×Q818yes—
304C2×C4⋊C4C2×Q818yes—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42—42no—
331C2×C42—44no—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42C2×D414yes—
335C2×C42C2×Q818yes—
336C2×C42#4114yes—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—24no—
355#95—24no—
356#95—42no—
357#95—42no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95C2×D414yes—
367#95C2×D414yes—
368#95C2×Q818yes—
369#95C2×Q818yes—
370#95C2×Q818yes—
371#95C2×Q818yes—
372C24C2511yescharacteristic
373C2×Q8—28no—
374C2×Q8—28no—
375C2×Q8—28no—
376C2×Q8—48no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—84no—
389C2×Q8—84no—
390C2×Q8—88no—
391C2×Q8—88no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
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—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
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—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
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—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
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×Q814yes—
486C22×C4C22×Q814yes—
487C22×C4C22×Q814yes—
488C22×C4C22×Q814yes—
489C22×C4C22×Q814yes—
490C22×C4C22×Q814yes—
491C22×C4C2×#4114yes—
492C22×C4C2×#4114yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4H(5,𝔽2)14yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C4H(5,𝔽2)12yes—
498C22×C42−1+414yes—
499C22×C42−1+412yes—
500C22×C42−1+412yes—
501C4⋊C4—28no—
502C4⋊C4—216no—
503C4⋊C4—416no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—48no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4—84no—
517C4⋊C4—88no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C4⋊C4—88no—
521C4⋊C4—88no—
522C4⋊C4C4⋊Q8116yescomplemented
523C42—28no—
524C42—28no—
525C42—28no—
526C42—28no—
527C42—216no—
528C42—48no—
529C42—48no—
530C42—44no—
531C42—44no—
532C42—44no—
533C42—84no—
534C23C23×D411yescharacteristic
535C23C23×Q812yes—
536C23C22×#4111yescharacteristic
537C23C22×#4111yescharacteristic
538C23C2×H(5,𝔽2)11yescharacteristic
539C23C2×H(5,𝔽2)11yescharacteristic
540C23C2×H(5,𝔽2)12yes—
541C23C2×2−1+412yes—
542C23C2×2−1+411yescharacteristic
543C23C2×2−1+411yescharacteristic
544C23#58512yes—
545Q8—416no—
546Q8—816no—
547Q8—88no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—42no—
589C2×C4—42no—
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—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
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—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4C2×C4⋊Q814yes—
646C2×C4C2×C4⋊Q814yes—
647C2×C4C2×C4⋊Q814yes—
648C2×C4#53514yes—
649C2×C4#53614yes—
650C2×C4#53914yes—
651C2×C4#54214yes—
652C2×C4#54714yes—
653C2×C4#54814yes—
654C2×C4#55114yes—
655C2×C4#55418yes—
656C2×C4#56314yes—
657C22C22×C4⋊Q811yescharacteristic
658C22C2×#53511yescharacteristic
659C22C2×#53611yescharacteristic
660C22C2×#53911yescharacteristic
661C22C2×#54211yescharacteristic
662C22C2×#54711yescharacteristic
663C22C2×#54811yescharacteristic
664C22C2×#55111yescharacteristic
665C22C2×#55412yes—
666C22Q8×#4112yes—
667C22#349512yes—
668C22#349811yescharacteristic
669C22#350011yescharacteristic
670C22#350212yes—
671C22#350312yes—
672C22#350312yes—
673C22#350312yes—
674C22#352212yes—
675C22C2×#56311yescharacteristic
676C22#354612yes—
677C22#355012yes—
678C22#355812yes—
679C22#356312yes—
680C4—24no—
681C4—24no—
682C4—28no—
683C4—24no—
684C4—24no—
685C4—44no—
686C4—44no—
687C4—42no—
688C4—44no—
689C4—42no—
690C4—44no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4#272318yes—
699C2#3528111yescharacteristic
700C2#3618011yescharacteristic
701C2#3647411yescharacteristic
702C2#3730511yescharacteristic
703C2#3786511yescharacteristic
704C2#3916811yescharacteristic
705C2#3960912yes—
706C2#3966212yes—
707C2#4600612yes—
708C2C2×#272311yescharacteristic
709C2#5927412yes—
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
160. #387Q8semidirect
522. C4⋊C4C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#25112
#165242
#158072
#156272
#151992
#151912
#151852
#151802
#151432
#147142
#146872
#144042
#140242
#140002
#139762
#139102
C4×#3872