Finite Group Databasebeta

← #523519

Subgroups — #523519

Order 512.

2490 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic
3C2×#2351C211yesmaximal, characteristic
4#14707C211yesmaximal, characteristic
5#14552C214yesmaximal
6#14378C214yesmaximal
7#14282C211yesmaximal, characteristic
8#14277C211yesmaximal, characteristic
9#14192C214yesmaximal
10#14059C214yesmaximal
11#14051C211yesmaximal, characteristic
12#14050C211yesmaximal, characteristic
13#13978C211yesmaximal, characteristic
14#13976C211yesmaximal, characteristic
15#13909C211yesmaximal, characteristic
16#13786C211yesmaximal, characteristic
17#13786C211yesmaximal, characteristic
18C4×#391C211yesmaximal, characteristic
19#13759C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24#2568C4116yescomplemented
25C2×#400C2214yes—
26#2369—28no—
27#2351—28no—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2211yescharacteristic
31C2×#391C2214yes—
32C2×#391C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#389C2212yes—
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2212yes—
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2212yes—
42C2×#384C2211yescharacteristic
43C2×#382C2212yes—
44C2×#382C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#382C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×#376C2211yescharacteristic
56C2×#376C2211yescharacteristic
57C2×C43C2211yescharacteristic
58#1446C2214yes—
59#1445C2214yes—
60#1444C2214yes—
61#1444C2214yes—
62#1443C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1441C2214yes—
67#1436C2212yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2212yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78#1436C2212yes—
79#1436C2212yes—
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2211yescharacteristic
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2211yescharacteristic
87C4⋊#95C2211yescharacteristic
88#1434C2214yes—
89#1434C2214yes—
90#1434C2214yes—
91C4×#95C2214yes—
92C4×#95C2211yescharacteristic
93C4×#95C2211yescharacteristic
94C23×Q8C2311yescharacteristic
95C23×Q8C2312yes—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8C2×C418yes—
99C2×C4×Q8—24no—
100C2×C4×Q8—24no—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C2×C4×Q8—24no—
104C2×C4×Q8—24no—
105C2×C4×Q8—42no—
106C2×C4×Q8—42no—
107C2×C4×Q8C2×C418yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2312yes—
140C22×C42C2312yes—
141C22×C42C2311yescharacteristic
142#400C2×C4132yes—
143#391—24no—
144#391—28no—
145#391—28no—
146#391C2×C4132yes—
147#391C2×C418yes—
148#391C2×C418yes—
149#391Q818yescomplemented
150#389—28no—
151#389—28no—
152#389—28no—
153#389—24no—
154#389—24no—
155#389—24no—
156#389—24no—
157#384—42no—
158#382—24no—
159#382—28no—
160#382—42no—
161#382C2×C418yes—
162#382C2×C418yes—
163C4×C4⋊C4—28no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4—24no—
166C4×C4⋊C4—24no—
167C4×C4⋊C4—28no—
168C4×C4⋊C4C2×C418yes—
169C4×C4⋊C4D418yes—
170#376—24no—
171#376—42no—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2314yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2312yes—
199C2×#95C2312yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2311yescharacteristic
205C2×#95C2311yescharacteristic
206C2×#95C2314yes—
207C43—42no—
208C22×Q8—22no—
209C22×Q8—24no—
210C22×Q8—22no—
211C22×Q8—24no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—44no—
215C22×Q8—44no—
216C22×Q8C22×C414yes—
217C22×Q8C22×C418yes—
218C22×Q8C2×Q814yes—
219C22×Q8#4114yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C4⋊Q8—232no—
240C4×Q8—232no—
241C4×Q8—416no—
242C4×Q8—416no—
243C2×C4⋊C4—24no—
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—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
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—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4C22×C414yes—
286C2×C4⋊C4C22×C418yes—
287C2×C4⋊C4C22×C414yes—
288C2×C4⋊C4C22×C414yes—
289C2×C4⋊C4C22×C418yes—
290C2×C4⋊C4C22×C418yes—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×Q814yes—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42—44no—
323C2×C42—42no—
324C2×C42—42no—
325C2×C42—44no—
326C2×C42—44no—
327C2×C42C22×C418yes—
328C2×C42C22×C414yes—
329C2×C42C22×C414yes—
330C2×C42C22×C414yes—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42C2×Q814yes—
335C2×C42#4114yes—
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—24no—
345#95—24no—
346#95—24no—
347#95—28no—
348#95—24no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—42no—
356#95—42no—
357#95—42no—
358#95—44no—
359#95—44no—
360#95—42no—
361#95—42no—
362#95—44no—
363#95C22×C418yes—
364#95C22×C4116yes—
365#95C22×C4116yes—
366#95C22×C4116yes—
367#95C22×C4116yes—
368#95C2×D414yes—
369#95C2×D414yes—
370#95C2×Q8116yes—
371C24C2511yescharacteristic
372C2×Q8—28no—
373C2×Q8—216no—
374C2×Q8—28no—
375C2×Q8—28no—
376C2×Q8—28no—
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8—416no—
380C2×Q8—84no—
381C2×Q8—84no—
382C2×Q8—88no—
383C2×Q8C4×Q8116yes—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
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—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
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—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4C23×C414yes—
464C22×C4C23×C414yes—
465C22×C4C23×C414yes—
466C22×C4C23×C414yes—
467C22×C4C23×C414yes—
468C22×C4C23×C412yes—
469C22×C4C23×C412yes—
470C22×C4C23×C414yes—
471C22×C4C23×C412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×Q814yes—
480C22×C4C22×Q814yes—
481C22×C4C22×Q814yes—
482C22×C4C22×Q812yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4114yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C42−1+414yes—
490C22×C42−1+412yes—
491C4⋊C4—216no—
492C4⋊C4—28no—
493C4⋊C4—216no—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C4⋊C4—88no—
510C42—216no—
511C42—28no—
512C42—28no—
513C42—28no—
514C42—28no—
515C42—28no—
516C42—48no—
517C42—48no—
518C42—48no—
519C42—48no—
520C42—48no—
521C42—44no—
522C42—84no—
523C42—84no—
524C42C4×D4116yes—
525C23C24×C411yescharacteristic
526C23C23×D411yescharacteristic
527C23C23×Q811yescharacteristic
528C23C22×#4111yescharacteristic
529C23C22×#4111yescharacteristic
530C23C2×H(5,𝔽2)11yescharacteristic
531C23C2×2−1+411yescharacteristic
532C23C2×2−1+412yes—
533C23#58512yes—
534C23#58512yes—
535C23#58512yes—
536Q8—232no—
537Q8—816no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
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—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
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—44no—
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—44no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4C2×C4×D414yes—
635C2×C4C2×C4×D414yes—
636C2×C4C2×C4×D414yes—
637C2×C4C2×C4×Q818yes—
638C2×C4C2×C4×Q814yes—
639C2×C4#52014yes—
640C2×C4#53614yes—
641C2×C4#54614yes—
642C2×C4D4×Q814yes—
643C2×C4#55214yes—
644C2×C4#55414yes—
645C22C22×C4×D411yescharacteristic
646C22C22×C4×Q811yescharacteristic
647C22C2×#52011yescharacteristic
648C22#342812yes—
649C22C4×2−1+412yes—
650C22C2×#53611yescharacteristic
651C22#344911yescharacteristic
652C22#346112yes—
653C22C2×#54611yescharacteristic
654C22C2×D4×Q811yescharacteristic
655C22C2×#55211yescharacteristic
656C22C2×#55411yescharacteristic
657C22#350011yescharacteristic
658C22#350011yescharacteristic
659C22#350212yes—
660C22#350412yes—
661C22#351012yes—
662C22#351612yes—
663C22#352412yes—
664C22#354012yes—
665C22#354112yes—
666C22#355512yes—
667C22#356512yes—
668C4—24no—
669C4—28no—
670C4—24no—
671C4—24no—
672C4—44no—
673C4—44no—
674C4—44no—
675C4—44no—
676C4—44no—
677C4—44no—
678C4—44no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4#236018yes—
687C2C4×#53611yescharacteristic
688C2C2×#236011yescharacteristic
689C2#3421111yescharacteristic
690C2#3433612yes—
691C2#3446312yes—
692C2#3808311yescharacteristic
693C2#3809511yescharacteristic
694C2#3827712yes—
695C2#3961311yescharacteristic
696C2#3961911yescharacteristic
697C2#3969512yes—
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
24. #2568C4semidirect
149. #391Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
C2×#23692
C2×#23512
#147072
#145522
#143782
#142822
#142772
#141922
#140592
#140512
#140502
#139782
#139762
#139092
#137862
C4×#3912
#137592