Finite Group Databasebeta

← #6374584

Subgroups — #6374584

Order 512.

3038 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×#3455C211yesmaximal, characteristic
2#34179C211yesmaximal, characteristic, complemented
3#34177C211yesmaximal, characteristic, complemented
4#21877C214yesmaximal, complemented
5#21876C214yesmaximal, complemented
6#21826C212yesmaximal, complemented
7#21794C212yesmaximal, complemented
8#21748C214yesmaximal, complemented
9#21423C214yesmaximal, complemented
10#21422C214yesmaximal, complemented
11#20723C214yesmaximal, complemented
12#3455—24no—
13#3455—24no—
14#3455—24no—
15#3455—24no—
16C2×#542C412yescomplemented
17C2×#542C412yescomplemented
18C2×#541C412yescomplemented
19C2×#540C412yescomplemented
20C2×#539C2212yes—
21C2×#539C2212yes—
22C2×#538C2212yes—
23C2×#538C2212yes—
24C2×#533C2211yescharacteristic
25C2×#532C2211yescharacteristic
26C2×#529C414yescomplemented
27C2×#529C414yescomplemented
28C2×#525C2212yes—
29C2×C4×#41C2212yes—
30C22×#117C2211yescharacteristic
31#2347—24no—
32#2346—24no—
33#2343—24no—
34#2343—24no—
35#2316C2212yes—
36#2315C2212yes—
37#2313C2212yes—
38#2313C2212yes—
39#2312C2212yes—
40#2312C2212yes—
41#2283C2212yes—
42#2283C2212yes—
43#2283C2212yes—
44#2283C2212yes—
45#2280C2212yes—
46C2×#381C2212yes—
47#2274C2212yes—
48#2273C2212yes—
49#1996C2218yes—
50#1996C2218yes—
51#1964C2218yes—
52#1964C2218yes—
53#1899C22116yes—
54#1898C22116yes—
55#1820C2218yes—
56#1799C22116yes—
57#1782C2218yes—
58C4×#99C22116yes—
59C22×#41C2312yes—
60C24×C4C2311yescharacteristic
61#542—28no—
62#542—28no—
63#542—28no—
64#542—28no—
65#541—28no—
66#541—28no—
67#540—28no—
68#540—28no—
69#539—28no—
70#539—28no—
71#539—28no—
72#539—28no—
73#538—28no—
74#538—28no—
75#538—28no—
76#538—28no—
77#533—22no—
78#533—22no—
79#533—22no—
80#533—22no—
81#532—22no—
82#532—22no—
83#532—22no—
84#532—22no—
85C2×C4⋊Q8C2×C411yescharacteristic, complemented
86C2×C4⋊Q8C2×C411yescharacteristic, complemented
87C2×C4⋊Q8C2×C411yescharacteristic, complemented
88C2×#127C2×C411yescharacteristic, complemented
89#529—216no—
90#529—216no—
91#529—216no—
92#529—216no—
93C2×#126C2×C414yes—
94C2×#126C2×C414yes—
95C2×#126C2×C414yescomplemented
96C2×#126C2×C414yescomplemented
97C2×#125C2×C414yescomplemented
98C2×#125C2×C414yescomplemented
99C2×#125C2×C412yescomplemented
100C2×#124C2×C414yes—
101C2×#124C2×C414yes—
102C2×#124C2×C412yescomplemented
103#525—24no—
104#525—24no—
105#525—24no—
106#525—24no—
107C2×#123C2×C414yes—
108C2×#123C2×C414yes—
109C2×#123C2×C414yescomplemented
110C2×#123C2×C414yescomplemented
111C2×#123C2314yes—
112C2×C22⋊Q8C2×C412yes—
113C2×C22⋊Q8C2×C412yes—
114C2×C22⋊Q8C2×C414yescomplemented
115C2×C22⋊Q8C2×C414yescomplemented
116C2×C22⋊Q8C2312yes—
117C2×C22⋊Q8C2312yes—
118C2×#121C2×C412yes—
119C2×#121C2×C412yes—
120C2×#121C2×C414yes—
121C2×#121C2×C414yes—
122C2×#121C2312yes—
123C2×#121C2312yes—
124C2×#120C2314yes—
125C4×#41—24no—
126C4×#41—24no—
127C4×#41—24no—
128C4×#41—24no—
129C2×C4×Q8C2×C412yescomplemented
130C2×C4×Q8C2×C412yescomplemented
131C2×C4×Q8C2×C412yescomplemented
132C2×C4×Q8C2×C412yescomplemented
133C2×C4×D4C2×C414yes—
134C2×C4×D4C2×C414yes—
135C2×C4×D4C2×C412yescomplemented
136C2×C4×D4C2×C412yescomplemented
137C2×C4×D4C2×C412yescomplemented
138C2×C4×D4C2×C412yescomplemented
139C2×C4×D4C2314yes—
140C2×C4×D4C2314yes—
141C2×#117C2×C412yes—
142C2×#117C2×C412yes—
143C2×#117C2×C412yes—
144C2×#117C2×C412yes—
145C2×#117C2×C414yescomplemented
146C2×#117D414yes—
147C2×#117D414yes—
148C2×#117D414yes—
149C2×#117D414yes—
150C2×#117C2314yes—
151C2×#117C2312yes—
152C22×C4⋊C4C2311yescharacteristic
153C22×C4⋊C4C2311yescharacteristic
154C22×C22⋊C4C2314yes—
155C22×C22⋊C4C2311yescharacteristic
156C22×C22⋊C4C2311yescharacteristic
157C22×C42C2×C414yescomplemented
158C22×C42C2311yescharacteristic
159C22×C42C2311yescharacteristic
160C2×#99C23132yes—
161#389—28no—
162#388—28no—
163#388—28no—
164#387—28no—
165#387—28no—
166#385—28no—
167#383—22no—
168#383—22no—
169#383—22no—
170#383—22no—
171#382—22no—
172#382—22no—
173#382—22no—
174#382—22no—
175#381—22no—
176#381—22no—
177#381C2314yes—
178#381C2314yes—
179#381C2314yes—
180#380—22no—
181#380—22no—
182#380C2314yes—
183#380C2314yes—
184#380C2314yes—
185#380C2314yes—
186#379C2314yes—
187C4×C4⋊C4—28no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—28no—
195C4×C22⋊C4C2314yes—
196C4×C22⋊C4C2314yes—
197C4×C22⋊C4C2314yes—
198C4×C22⋊C4C2314yes—
199#376—22no—
200#376—22no—
201#376—22no—
202#376—22no—
203C2×#95—22no—
204C2×#95—22no—
205#342C23132yes—
206C25C2411yescharacteristic
207C2×#41—22no—
208C2×#41—22no—
209C2×#41—22no—
210C2×#41—22no—
211C2×#41—48no—
212C22×Q8C22×C412yes—
213C22×Q8C22×C412yes—
214C22×Q8C22×C412yes—
215C22×D4C22×C412yes—
216C22×D4C22×C414yes—
217C22×D4C22×C414yes—
218C22×D4C22×C412yes—
219C22×D4C22×C412yes—
220C22×D4C2414yes—
221C23×C4—24no—
222C23×C4—44no—
223C23×C4C22×C414yes—
224C23×C4C22×C414yes—
225C23×C4C22×C414yes—
226C23×C4C22×C412yes—
227C23×C4C22×C412yes—
228C23×C4C2×D412yes—
229C23×C4C2×D412yes—
230C23×C4C2×D412yes—
231C23×C4C2×D412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C4⋊Q8—22no—
239C4⋊Q8—22no—
240C4⋊Q8—22no—
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243C4⋊Q8—22no—
244#127—22no—
245#127—22no—
246#126—216no—
247#126—216no—
248#126—216no—
249#126—216no—
250#125—216no—
251#125—216no—
252#125—24no—
253#125—24no—
254#124—216no—
255#124—216no—
256#124—24no—
257#124—24no—
258#123—28no—
259#123—28no—
260#123—216no—
261#123—216no—
262#123—216no—
263#123—216no—
264C22⋊Q8—24no—
265C22⋊Q8—24no—
266C22⋊Q8—24no—
267C22⋊Q8—24no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—24no—
271C22⋊Q8—24no—
272C22⋊Q8—216no—
273C22⋊Q8—216no—
274#121—24no—
275#121—24no—
276#121—24no—
277#121—24no—
278#121—24no—
279#121—24no—
280#121—24no—
281#121—24no—
282#121—216no—
283#121—216no—
284#120—28no—
285#120—28no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—24no—
294C4×D4—28no—
295C4×D4—28no—
296C4×D4—28no—
297C4×D4—28no—
298C4×D4—28no—
299C4×D4—28no—
300C4×D4—28no—
301C4×D4—28no—
302C4×D4—24no—
303C4×D4—24no—
304C4×D4—24no—
305C4×D4—24no—
306C4×D4—24no—
307C4×D4—24no—
308C4×D4—24no—
309C4×D4—24no—
310#117—22no—
311#117—22no—
312#117—22no—
313#117—22no—
314#117—28no—
315#117—28no—
316#117—48no—
317#117C22⋊C418yes—
318#117C22⋊C418yes—
319#117C22⋊C418yes—
320#117C22⋊C418yes—
321#117C22⋊C418yes—
322#117C22⋊C418yes—
323#117C22⋊C418yes—
324#117C22⋊C418yes—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4C22×C414yes—
331C2×C4⋊C4C22×C414yes—
332C2×C4⋊C4C22×C412yes—
333C2×C4⋊C4C22×C412yes—
334C2×C4⋊C4C22×C414yes—
335C2×C4⋊C4C22×C414yes—
336C2×C4⋊C4C22×C412yes—
337C2×C4⋊C4C22×C412yes—
338C2×C4⋊C4C22×C414yes—
339C2×C4⋊C4C22×C414yes—
340C2×C4⋊C4C22×C412yes—
341C2×C4⋊C4C22×C412yes—
342C2×C4⋊C4C22×C412yes—
343C2×C4⋊C4C22×C412yes—
344C2×C4⋊C4C22×C412yes—
345C2×C4⋊C4C22×C412yes—
346C2×C4⋊C4C2×D414yes—
347C2×C4⋊C4C2×D414yes—
348C2×C4⋊C4C2×D414yes—
349C2×C4⋊C4C2×D414yes—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—48no—
354C2×C22⋊C4C22×C414yes—
355C2×C22⋊C4C22×C414yes—
356C2×C22⋊C4C22×C414yes—
357C2×C22⋊C4C22×C414yes—
358C2×C22⋊C4C22×C414yes—
359C2×C22⋊C4C22×C414yes—
360C2×C22⋊C4C22×C414yes—
361C2×C22⋊C4C22×C414yes—
362C2×C22⋊C4C22×C412yes—
363C2×C22⋊C4C22×C412yes—
364C2×C22⋊C4C22×C412yes—
365C2×C22⋊C4C22×C412yes—
366C2×C22⋊C4C2×D414yes—
367C2×C22⋊C4C2×D414yes—
368C2×C22⋊C4C2×D414yes—
369C2×C22⋊C4C2×D414yes—
370C2×C22⋊C4C2418yes—
371C2×C22⋊C4C2418yes—
372C2×C22⋊C4C2414yes—
373C2×C42—28no—
374C2×C42—28no—
375C2×C42—24no—
376C2×C42—24no—
377C2×C42—24no—
378C2×C42—24no—
379C2×C42—24no—
380C2×C42C22×C414yes—
381C2×C42C22×C414yes—
382C2×C42C22×C414yes—
383C2×C42C22×C412yes—
384C2×C42C22×C411yescharacteristic
385C2×C42C22×C411yescharacteristic
386C2×C42C22×C412yes—
387C2×C42C22×C411yescharacteristic
388C2×C42C22×C411yescharacteristic
389C2×C42C2×D412yes—
390C2×C42C2×D412yes—
391C2×C42C2×D412yes—
392C2×C42C2×D412yes—
393C2×C42C2×D412yes—
394C2×C42C2×D412yes—
395C2×C42C2×D412yes—
396C2×C42C2×D412yes—
397#99—432no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95—24no—
410#95—24no—
411#95—24no—
412C24—24no—
413C24—81no—
414C24—81no—
415C24C23×C414yes—
416C24C23×C412yes—
417C24C22×D412yes—
418C24C22×D412yes—
419C24C2511yescharacteristic
420#41—48no—
421#41—48no—
422#41—48no—
423#41—48no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—24no—
427C2×Q8—24no—
428C2×Q8—44no—
429C2×Q8—82no—
430C2×Q8—82no—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—216no—
434C2×D4—28no—
435C2×D4—28no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—44no—
441C2×D4—82no—
442C2×D4—82no—
443C2×D4—84no—
444C2×D4—84no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—28no—
450C22×C4—28no—
451C22×C4—28no—
452C22×C4—28no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—81no—
462C22×C4—81no—
463C22×C4—81no—
464C22×C4—81no—
465C22×C4—84no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4C2×C22⋊C414yes—
470C22×C4C2×C22⋊C414yes—
471C22×C4C2×C22⋊C414yes—
472C22×C4C2×C22⋊C414yes—
473C22×C4C23×C414yes—
474C22×C4C23×C411yescharacteristic
475C22×C4C23×C411yescharacteristic
476C22×C4C23×C412yes—
477C22×C4C23×C414yes—
478C22×C4C23×C414yes—
479C22×C4C23×C412yes—
480C22×C4C23×C412yes—
481C22×C4C23×C411yescharacteristic
482C22×C4C23×C411yescharacteristic
483C22×C4C23×C411yescharacteristic
484C22×C4C23×C411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—28no—
504C4⋊C4—28no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—28no—
510C4⋊C4—28no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—48no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C4⋊C4C2×C22⋊C418yes—
519C4⋊C4C2×C22⋊C418yes—
520C4⋊C4C2×C22⋊C418yes—
521C4⋊C4C2×C22⋊C418yes—
522C22⋊C4—28no—
523C22⋊C4—28no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—28no—
527C22⋊C4—28no—
528C22⋊C4—28no—
529C22⋊C4—28no—
530C22⋊C4—28no—
531C22⋊C4—28no—
532C22⋊C4—216no—
533C22⋊C4—216no—
534C22⋊C4—48no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C22⋊C4C2×C22⋊C418yes—
538C22⋊C4C2×C22⋊C418yes—
539C22⋊C4C2×C22⋊C418yes—
540C22⋊C4C2×C22⋊C418yes—
541C42—24no—
542C42—24no—
543C42—28no—
544C42—28no—
545C42—48no—
546C42—84no—
547C42—84no—
548C42—84no—
549C42—84no—
550C42C2×C22⋊C414yes—
551C42C2×C22⋊C414yes—
552C42C2×C22⋊C412yes—
553C42C2×C22⋊C412yes—
554C42C2×C22⋊C412yes—
555C42C2×C22⋊C412yes—
556C42C2×C22⋊C414yes—
557C42C2×C22⋊C414yes—
558C42C2×C22⋊C412yes—
559C42C2×C22⋊C412yes—
560C42C2×C22⋊C412yes—
561C42C2×C22⋊C412yes—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—28no—
567C23—42no—
568C23—42no—
569C23—41no—
570C23—41no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—44no—
576C23—44no—
577C23—88no—
578C23C22×C22⋊C414yes—
579C23C24×C411yescharacteristic
580C23C23×D411yescharacteristic
581C23C23×D411yescharacteristic
582C23#58514yes—
583Q8—44no—
584Q8—44no—
585Q8—84no—
586Q8—84no—
587D4—44no—
588D4—44no—
589D4—48no—
590D4—48no—
591D4—84no—
592D4—84no—
593D4—88no—
594D4—88no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—28no—
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—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—48no—
627C2×C4—48no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—48no—
631C2×C4—48no—
632C2×C4—48no—
633C2×C4—48no—
634C2×C4—48no—
635C2×C4—48no—
636C2×C4—88no—
637C2×C4—88no—
638C2×C4—168no—
639C2×C4C22×C22⋊C411yescharacteristic
640C2×C4C22×C22⋊C411yescharacteristic
641C2×C4C22×C22⋊C414yes—
642C2×C4C22×C22⋊C414yes—
643C2×C4C22×C22⋊C411yescharacteristic
644C2×C4C22×C22⋊C411yescharacteristic
645C2×C4C22×C22⋊C412yes—
646C2×C4C22×C22⋊C412yes—
647C2×C4C22×C22⋊C412yes—
648C2×C4C22×C22⋊C412yes—
649C2×C4C22×C22⋊C414yes—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—44no—
660C22—48no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—84no—
665C22—84no—
666C22—88no—
667C22C23×C22⋊C411yescharacteristic
668C22#342812yes—
669C22#344912yes—
670C22#344912yes—
671C4—24no—
672C4—24no—
673C4—22no—
674C4—22no—
675C4—22no—
676C4—22no—
677C4—21no—
678C4—21no—
679C4—24no—
680C4—21no—
681C4—21no—
682C4—44no—
683C4—44no—
684C4—44no—
685C4—88no—
686C4—88no—
687C2—22no—
688C2—22no—
689C2—22no—
690C2—44no—
691C2—84no—
692C2#3404411yescharacteristic
693C2#6007011yescharacteristic
694C2#6007011yescharacteristic
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC23C24×C4
Frattini, cosocleC24C25
prime radicalC2×#3455C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×#117 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34179C2semidirect
3. #34177C2semidirect
4. #21877C2semidirect
5. #21876C2semidirect
6. #21826C2semidirect
7. #21794C2semidirect
8. #21748C2semidirect
9. #21423C2semidirect
10. #21422C2semidirect
11. #20723C2semidirect
16. C2×#542C4semidirect
17. C2×#542C4semidirect
18. C2×#541C4semidirect
19. C2×#540C4semidirect
26. C2×#529C4semidirect
27. C2×#529C4semidirect
85. C2×C4⋊Q8C2×C4semidirect
86. C2×C4⋊Q8C2×C4semidirect
87. C2×C4⋊Q8C2×C4semidirect
88. C2×#127C2×C4semidirect
95. C2×#126C2×C4semidirect
96. C2×#126C2×C4semidirect
97. C2×#125C2×C4semidirect
98. C2×#125C2×C4semidirect
99. C2×#125C2×C4semidirect
102. C2×#124C2×C4semidirect
109. C2×#123C2×C4semidirect
110. C2×#123C2×C4semidirect
114. C2×C22⋊Q8C2×C4semidirect
115. C2×C22⋊Q8C2×C4semidirect
129. C2×C4×Q8C2×C4semidirect
130. C2×C4×Q8C2×C4semidirect
131. C2×C4×Q8C2×C4semidirect
132. C2×C4×Q8C2×C4semidirect
135. C2×C4×D4C2×C4semidirect
136. C2×C4×D4C2×C4semidirect
137. C2×C4×D4C2×C4semidirect
138. C2×C4×D4C2×C4semidirect
145. C2×#117C2×C4semidirect
157. C22×C42C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#34552
#341792
#341772
#218772
#218762
#218262
#217942
#217482
#214232
#214222
#207232