Finite Group Databasebeta

← #6343501

Subgroups — #6343501

Order 512.

3145 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3501C211yesmaximal, characteristic
2#60230C211yesmaximal, characteristic
3#34126C211yesmaximal, characteristic
4#17627C214yesmaximal, complemented
5#17621C218yesmaximal, complemented
6#17614C214yesmaximal, complemented
7#17584C212yesmaximal, complemented
8#17578C214yesmaximal, complemented
9#17545C212yesmaximal, complemented
10C2×#1508C214yesmaximal, complemented
11#3501—24no—
12#3501—21no—
13#3501—22no—
14#3501—21no—
15#3501C418yescomplemented
16#3501C2212yes—
17#3501C2214yes—
18#3501C2212yes—
19C2×#554—24no—
20C2×#554C414yescomplemented
21C2×D4×Q8C414yescomplemented
22C2×#544—21no—
23C2×#543—22no—
24C2×#533C2212yes—
25C22×C22⋊Q8C2214yes—
26C2×#518—21no—
27C22×C4×D4C2211yescharacteristic
28#2975C2212yes—
29#2974—24no—
30#2973C2214yes—
31#2972C2212yes—
32D4×M16—24no—
33#2920C2212yes—
34C2×#407C2214yes—
35#2357C2212yes—
36#2353C2214yes—
37#2349C2212yes—
38#2345—21no—
39#2342—22no—
40D4×C22⋊C4—21no—
41#2293—21no—
42#2290—22no—
43#2283C2211yescharacteristic
44#2280C2212yes—
45C42×D4—21no—
46C4×#117C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48#1520C22116yescomplemented
49#1518C2218yescomplemented
50#1515C2218yescomplemented
51#1511C22116yescomplemented
52#1508C2214yes—
53#1508C2218yes—
54#1508C2214yes—
55#1508C22116yescomplemented
56#1506C2218yescomplemented
57#1503C2218yescomplemented
58C2×#328C22116yescomplemented
59C23×Q8C2×C414yescomplemented
60C23×D4D411yescharacteristic
61C24×C4C2312yes—
62C22×M16—24no—
63#554—28no—
64#554—216no—
65#554—28no—
66#554—24no—
67#554—28no—
68#554—24no—
69#554—28no—
70#554—216no—
71#554—28no—
72#554C2×C418yes—
73#554C2×C4116yescomplemented
74#554C2×C418yescomplemented
75D4×Q8—24no—
76D4×Q8—28no—
77D4×Q8—24no—
78D4×Q8C2×C418yes—
79D4×Q8C2×C418yescomplemented
80D4×Q8C2×C4116yescomplemented
81#544—28no—
82#544—22no—
83#544—24no—
84#544—22no—
85#543—216no—
86#543—24no—
87#543—24no—
88#543—24no—
89#543—24no—
90#533—22no—
91#533—22no—
92#533—22no—
93#533—22no—
94#533C2×C414yescomplemented
95#533C2×C414yescomplemented
96#533C2314yes—
97#533C2314yes—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8C2×C412yescomplemented
100C2×C4⋊Q8C2×C412yescomplemented
101C2×#125—24no—
102C2×#125—22no—
103C2×C22⋊Q8—24no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—28no—
111C2×C22⋊Q8—28no—
112C2×C22⋊Q8C2×C418yescomplemented
113C2×C22⋊Q8C2×C418yescomplemented
114C2×C22⋊Q8C2×C418yescomplemented
115C2×C22⋊Q8C2314yes—
116C2×C22⋊Q8C2318yes—
117C2×C22⋊Q8C2318yes—
118#518—28no—
119#518—22no—
120#518—24no—
121#518—22no—
122C2×C4×Q8—22no—
123C2×C4×Q8C2×C414yescomplemented
124C2×C4×D4—24no—
125C2×C4×D4—24no—
126C2×C4×D4—22no—
127C2×C4×D4C2×C414yes—
128C2×C4×D4C2×C414yes—
129C2×C4×D4D414yes—
130C2×C4×D4D414yes—
131C2×C4×D4D411yescharacteristic
132C2×C4×D4D412yes—
133C2×C4×D4D411yescharacteristic
134C2×C4×D4C2311yescharacteristic
135C2×C4×D4C2312yes—
136C2×C4×D4C2311yescharacteristic
137C2×#117—22no—
138C2×#117—21no—
139C2×#117—22no—
140C22×C4⋊C4—24no—
141C22×C4⋊C4C2×C414yescomplemented
142C22×C4⋊C4D411yescharacteristic
143C22×C22⋊C4—24no—
144C22×C22⋊C4—22no—
145C22×C22⋊C4D412yes—
146C22×C42—22no—
147C22×C42C2311yescharacteristic
148#436—28no—
149#435—216no—
150C8×D4—28no—
151#432—22no—
152#432C2314yes—
153#431—22no—
154#431C2314yes—
155#423—22no—
156#407—24no—
157#407—24no—
158#407C2314yes—
159#407C2318yes—
160C2×C22⋊C8—24no—
161C2×C22⋊C8C2318yes—
162C4×M16—22no—
163#390—22no—
164#389—22no—
165#388—24no—
166#388—24no—
167#387—24no—
168#386—24no—
169#385—24no—
170#383C2312yes—
171#382C2312yes—
172#381—21no—
173#381C2314yes—
174#380—21no—
175#380—22no—
176#380C2314yes—
177#379—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4C2312yes—
182C4×C22⋊C4—21no—
183C4×C22⋊C4—22no—
184C4×C22⋊C4—22no—
185C4×C22⋊C4—22no—
186C4×C22⋊C4—24no—
187C4×C22⋊C4C2314yes—
188C4×C22⋊C4C2311yescharacteristic
189C4×C22⋊C4C2312yes—
190C4×C22⋊C4C2311yescharacteristic
191#376—21no—
192#376C2312yes—
193C2×#95C2314yes—
194C43—21no—
195#328C23116yes—
196#328C23132yes—
197#328C23116yes—
198C25C2×D412yes—
199C22×Q8—22no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—88no—
204C22×Q8C22×C414yes—
205C22×Q8C22×C418yes—
206C22×Q8C22×C418yes—
207C22×D4—28no—
208C22×D4C22⋊C414yes—
209C22×D4C22⋊C414yes—
210C22×D4C2×D412yes—
211C22×D4C2×D411yescharacteristic
212C22×D4C2×D411yescharacteristic
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—84no—
217C23×C4C22×C414yes—
218C23×C4C22×C414yes—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D412yes—
222C23×C4C2×D412yes—
223C23×C4C2×D414yes—
224C23×C4C2×D412yes—
225C23×C4C2×D414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2412yes—
229C2×M16—24no—
230C2×M16—28no—
231C2×M16—88no—
232C22×C8—28no—
233C4⋊Q8—28no—
234C4⋊Q8—216no—
235C4⋊Q8—28no—
236C4⋊Q8—22no—
237C4⋊Q8—22no—
238C4⋊Q8—22no—
239C4⋊Q8—22no—
240C4⋊Q8C22×C414yes—
241C4⋊Q8C22×C414yes—
242C4⋊Q8C22×C414yes—
243C4⋊Q8C22×C414yes—
244#125—28no—
245#125—216no—
246#125—28no—
247#125—24no—
248#125—24no—
249#125—24no—
250#125—24no—
251C22⋊Q8—28no—
252C22⋊Q8—28no—
253C22⋊Q8—28no—
254C22⋊Q8—28no—
255C22⋊Q8—28no—
256C22⋊Q8—28no—
257C22⋊Q8—24no—
258C22⋊Q8—24no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C22⋊Q8—28no—
262C22⋊Q8—28no—
263C22⋊Q8—28no—
264C22⋊Q8—28no—
265C22⋊Q8—24no—
266C22⋊Q8—24no—
267C22⋊Q8—216no—
268C22⋊Q8—216no—
269C22⋊Q8—216no—
270C22⋊Q8—216no—
271C22⋊Q8—216no—
272C22⋊Q8—216no—
273C22⋊Q8—216no—
274C22⋊Q8—216no—
275C22⋊Q8C22×C418yes—
276C22⋊Q8C22×C4116yes—
277C22⋊Q8C22×C4116yes—
278C22⋊Q8C2418yes—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8C22×C418yes—
286C4×Q8C22×C418yes—
287C4×D4—28no—
288C4×D4—216no—
289C4×D4—28no—
290C4×D4—28no—
291C4×D4—216no—
292C4×D4—28no—
293C4×D4—48no—
294C4×D4C22⋊C418yes—
295C4×D4C22⋊C414yes—
296C4×D4C22⋊C414yes—
297C4×D4C22⋊C418yes—
298C4×D4C22⋊C414yes—
299C4×D4C22⋊C414yes—
300C4×D4C22×C414yes—
301C4×D4C22×C418yes—
302C4×D4C22×C414yes—
303C4×D4C2×D414yes—
304C4×D4C2×D418yes—
305C4×D4C2×D414yes—
306#117—24no—
307#117—24no—
308#117—24no—
309#117—24no—
310#117—44no—
311#117—84no—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—28no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4C22⋊C414yes—
329C2×C4⋊C4C22⋊C414yes—
330C2×C4⋊C4C22×C414yes—
331C2×C4⋊C4C22×C418yes—
332C2×C4⋊C4C22×C418yes—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C22⋊C4—28no—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—28no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—28no—
342C2×C22⋊C4—28no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—48no—
349C2×C22⋊C4C22⋊C418yes—
350C2×C22⋊C4C22⋊C418yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C42—24no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—21no—
361C2×C42—24no—
362C2×C42—84no—
363C2×C42C22×C412yes—
364C2×C42C22×C412yes—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D412yes—
368C2×C42C2×D412yes—
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371C4⋊C8—24no—
372C4⋊C8—24no—
373C22⋊C8—24no—
374C22⋊C8—28no—
375C22⋊C8—28no—
376C22⋊C8C2418yes—
377#97—24no—
378C4×C8—24no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—24no—
384#95—22no—
385#95C2412yes—
386#95C2414yes—
387#95C2412yes—
388C24—28no—
389C24—44no—
390C24—82no—
391C24C2×C22⋊C414yes—
392C24C2×C22⋊C414yes—
393C24C22×D411yescharacteristic
394C24C22×D412yes—
395C24C22×D412yes—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—28no—
400C2×Q8—28no—
401C2×Q8—48no—
402C2×Q8—84no—
403C2×Q8—88no—
404C2×Q8—88no—
405C2×Q8—88no—
406C2×Q8—88no—
407C2×Q8—88no—
408C2×Q8C23×C418yes—
409C2×D4—28no—
410C2×D4—216no—
411C2×D4—216no—
412C2×D4—24no—
413C2×D4C2×C22⋊C418yes—
414C2×D4C2×C22⋊C414yes—
415C2×D4C2×C22⋊C414yes—
416C2×D4#13718yes—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—48no—
430C22×C4—42no—
431C22×C4—48no—
432C22×C4—42no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—84no—
436C22×C4—82no—
437C22×C4—84no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4—88no—
441C22×C4—84no—
442C22×C4—88no—
443C22×C4C2×C22⋊C412yes—
444C22×C4C2×C22⋊C412yes—
445C22×C4C2×C22⋊C412yes—
446C22×C4C2×C22⋊C414yes—
447C22×C4C2×C22⋊C414yes—
448C22×C4C2×C22⋊C412yes—
449C22×C4C2×C22⋊C414yes—
450C22×C4C2×C22⋊C414yes—
451C22×C4C2×C22⋊C414yes—
452C22×C4C2×C22⋊C414yes—
453C22×C4C23×C411yescharacteristic
454C22×C4C23×C411yescharacteristic
455C22×C4C23×C414yes—
456C22×C4C23×C414yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D414yes—
466C22×C4C22×D414yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C2511yescharacteristic
469M16—48no—
470M16—88no—
471C2×C8—28no—
472C2×C8—88no—
473C4⋊C4—28no—
474C4⋊C4—24no—
475C4⋊C4—216no—
476C4⋊C4—216no—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—28no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—84no—
491C4⋊C4C2×C22⋊C418yes—
492C4⋊C4C2×C22⋊C414yes—
493C4⋊C4C2×C22⋊C414yes—
494C4⋊C4C23×C418yes—
495C22⋊C4—24no—
496C22⋊C4—28no—
497C22⋊C4—216no—
498C22⋊C4—216no—
499C22⋊C4—28no—
500C22⋊C4—48no—
501C22⋊C4—44no—
502C22⋊C4—42no—
503C22⋊C4—48no—
504C22⋊C4—84no—
505C22⋊C4C2×C22⋊C418yes—
506C22⋊C4C2×C22⋊C418yes—
507C22⋊C4C2×C22⋊C418yes—
508C22⋊C4C2×C22⋊C418yes—
509C42—28no—
510C42—28no—
511C42—42no—
512C42—44no—
513C42—44no—
514C42—42no—
515C42—44no—
516C42—84no—
517C42C2×C22⋊C412yes—
518C42C2×C22⋊C412yes—
519C42C2×C22⋊C412yes—
520C42C2×C22⋊C412yes—
521C42C23×C412yes—
522C42C23×C412yes—
523C42C22×D412yes—
524C42C22×D412yes—
525C23—28no—
526C23—22no—
527C23—28no—
528C23—22no—
529C23—28no—
530C23—24no—
531C23—42no—
532C23—44no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—816no—
540C23C22×C22⋊C411yescharacteristic
541C23C22×C22⋊C414yes—
542C23C22×C22⋊C414yes—
543C23C22×C22⋊C411yescharacteristic
544C23C2×#13718yes—
545C23C23×D411yescharacteristic
546Q8—48no—
547Q8—48no—
548Q8—84no—
549Q8—84no—
550Q8—88no—
551Q8—88no—
552D4—216no—
553D4—28no—
554D4#419116yes—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—88no—
594C2×C4—88no—
595C2×C4—88no—
596C2×C4—88no—
597C2×C4—816no—
598C2×C4C22×C22⋊C412yes—
599C2×C4C22×C22⋊C411yescharacteristic
600C2×C4C22×C22⋊C412yes—
601C2×C4C22×C22⋊C412yes—
602C2×C4C22×C22⋊C414yes—
603C2×C4C22×C22⋊C414yes—
604C2×C4C22×C22⋊C411yescharacteristic
605C2×C4C22×C22⋊C412yes—
606C2×C4C2×#13714yes—
607C2×C4C24×C411yescharacteristic
608C2×C4C23×D411yescharacteristic
609C8—48no—
610C22—21no—
611C22—21no—
612C22—24no—
613C22—21no—
614C22—21no—
615C22—24no—
616C22—21no—
617C22—24no—
618C22—21no—
619C22—24no—
620C22—21no—
621C22—24no—
622C22—48no—
623C22—416no—
624C22—44no—
625C22—84no—
626C22—88no—
627C22C2×#41918yes—
628C22#290112yes—
629C22C23×C22⋊C411yescharacteristic
630C22C22×#13712yes—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—24no—
635C4—24no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—44no—
640C4—48no—
641C4—48no—
642C4—82no—
643C4—84no—
644C4#288312yes—
645C4C2×#41914yes—
646C2—22no—
647C2—22no—
648C2—28no—
649C2—44no—
650C2—41no—
651C2C2×#288311yescharacteristic
652C2C22×#41911yescharacteristic
653C2C2×#290111yescharacteristic
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17627C2semidirect
5. #17621C2semidirect
6. #17614C2semidirect
7. #17584C2semidirect
8. #17578C2semidirect
9. #17545C2semidirect
10. C2×#1508C2semidirect
15. #3501C4semidirect
20. C2×#554C4semidirect
21. C2×D4×Q8C4semidirect
48. #1520C22semidirect
49. #1518C22semidirect
50. #1515C22semidirect
51. #1511C22semidirect
55. #1508C22semidirect
56. #1506C22semidirect
57. #1503C22semidirect
58. C2×#328C22semidirect
59. C23×Q8C2×C4semidirect
73. #554C2×C4semidirect
74. #554C2×C4semidirect
79. D4×Q8C2×C4semidirect
80. D4×Q8C2×C4semidirect
94. #533C2×C4semidirect
95. #533C2×C4semidirect
99. C2×C4⋊Q8C2×C4semidirect
100. C2×C4⋊Q8C2×C4semidirect
112. C2×C22⋊Q8C2×C4semidirect
113. C2×C22⋊Q8C2×C4semidirect
114. C2×C22⋊Q8C2×C4semidirect
123. C2×C4×Q8C2×C4semidirect
141. C22×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#35012
#602302
#341262
#176272
#176212
#176142
#175842
#175782
#175452
C2×#15082