Finite Group Databasebeta

← #6343603

Subgroups — #6343603

Order 512.

3088 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3501C211yesmaximal, characteristic
2#60230C211yesmaximal, characteristic
3#34190C211yesmaximal, characteristic
4#17627C214yesmaximal, complemented
5#17621C218yesmaximal, complemented
6#17614C214yesmaximal, complemented
7#17590C212yesmaximal, complemented
8#17578C214yesmaximal, complemented
9#17562C212yesmaximal, complemented
10C2×#1511C214yesmaximal, complemented
11#3501—24no—
12#3501—22no—
13#3501—21no—
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#2344—21no—
39#2342—22no—
40D4×C22⋊C4—21no—
41#2319—21no—
42#2318—22no—
43#2283C2211yescharacteristic
44#2283C2211yescharacteristic
45#2280C2212yes—
46C2×#381C2212yes—
47#2275—21no—
48#1520C22116yescomplemented
49#1518C2218yescomplemented
50#1515C2218yescomplemented
51#1511C2214yes—
52#1511C2218yes—
53#1511C2214yes—
54#1511C22116yescomplemented
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×D4D412yes—
132C2×C4×D4D411yescharacteristic
133C2×C4×D4D411yescharacteristic
134C2×C4×D4C2311yescharacteristic
135C2×C4×D4C2312yes—
136C2×C4×D4C2311yescharacteristic
137C2×#117—22no—
138C2×#117—22no—
139C22×C4⋊C4—24no—
140C22×C4⋊C4C2×C414yescomplemented
141C22×C4⋊C4D411yescharacteristic
142C22×C22⋊C4—24no—
143C22×C22⋊C4—22no—
144C22×C22⋊C4D412yes—
145C22×C42C2311yescharacteristic
146#436—28no—
147#435—216no—
148C8×D4—28no—
149#432—22no—
150#432C2314yes—
151#431—22no—
152#431C2314yes—
153#423—22no—
154#407—24no—
155#407—24no—
156#407C2314yes—
157#407C2318yes—
158C2×C22⋊C8—24no—
159C2×C22⋊C8C2318yes—
160C4×M16—22no—
161#390—22no—
162#389—22no—
163#388—24no—
164#388—24no—
165#387—24no—
166#386—24no—
167#385—24no—
168#383—21no—
169#383C2312yes—
170#382—21no—
171#382C2312yes—
172#381—22no—
173#381—22no—
174#381C2314yes—
175#381C2311yescharacteristic
176#381C2312yes—
177#381C2311yescharacteristic
178#380—22no—
179#380—21no—
180#380—21no—
181#380C2314yes—
182#379—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4C2312yes—
186C4×C22⋊C4—22no—
187C4×C22⋊C4—21no—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—24no—
190C4×C22⋊C4C2314yes—
191#376—21no—
192#376C2312yes—
193C2×#95—22no—
194C2×#95C2314yes—
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×D412yes—
221C23×C4C2×D414yes—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D412yes—
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—84no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—28no—
323C2×C4⋊C4—28no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4C22⋊C414yes—
328C2×C4⋊C4C22⋊C414yes—
329C2×C4⋊C4C22×C414yes—
330C2×C4⋊C4C22×C418yes—
331C2×C4⋊C4C22×C418yes—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C22⋊C4—28no—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—28no—
340C2×C22⋊C4—28no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—84no—
348C2×C22⋊C4C22⋊C418yes—
349C2×C22⋊C4C22⋊C418yes—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C42—24no—
355C2×C42—22no—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42C22×C412yes—
359C2×C42C22×C412yes—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D412yes—
363C2×C42C2×D412yes—
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366C4⋊C8—24no—
367C4⋊C8—24no—
368C22⋊C8—24no—
369C22⋊C8—28no—
370C22⋊C8—28no—
371C22⋊C8C2418yes—
372#97—24no—
373C4×C8—24no—
374#95—21no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95C2412yes—
384#95C2414yes—
385#95C2412yes—
386C24—28no—
387C24—44no—
388C24—82no—
389C24C2×C22⋊C414yes—
390C24C2×C22⋊C414yes—
391C24C22×D411yescharacteristic
392C24C22×D412yes—
393C24C22×D412yes—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—28no—
398C2×Q8—28no—
399C2×Q8—48no—
400C2×Q8—88no—
401C2×Q8—84no—
402C2×Q8—88no—
403C2×Q8—88no—
404C2×Q8—88no—
405C2×Q8—88no—
406C2×Q8C23×C418yes—
407C2×D4—28no—
408C2×D4—216no—
409C2×D4—216no—
410C2×D4—24no—
411C2×D4C2×C22⋊C418yes—
412C2×D4C2×C22⋊C414yes—
413C2×D4C2×C22⋊C414yes—
414C2×D4#13718yes—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—44no—
426C22×C4—48no—
427C22×C4—42no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—84no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4—84no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4—81no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4—84no—
440C22×C4—84no—
441C22×C4C2×C22⋊C412yes—
442C22×C4C2×C22⋊C412yes—
443C22×C4C2×C22⋊C414yes—
444C22×C4C2×C22⋊C412yes—
445C22×C4C2×C22⋊C412yes—
446C22×C4C2×C22⋊C414yes—
447C22×C4C2×C22⋊C414yes—
448C22×C4C2×C22⋊C414yes—
449C22×C4C2×C22⋊C414yes—
450C22×C4C2×C22⋊C414yes—
451C22×C4C23×C411yescharacteristic
452C22×C4C23×C411yescharacteristic
453C22×C4C23×C414yes—
454C22×C4C23×C414yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C2511yescharacteristic
467M16—48no—
468M16—88no—
469C2×C8—28no—
470C2×C8—88no—
471C4⋊C4—28no—
472C4⋊C4—24no—
473C4⋊C4—216no—
474C4⋊C4—216no—
475C4⋊C4—28no—
476C4⋊C4—28no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—28no—
484C4⋊C4—28no—
485C4⋊C4—28no—
486C4⋊C4—44no—
487C4⋊C4—82no—
488C4⋊C4—84no—
489C4⋊C4C2×C22⋊C418yes—
490C4⋊C4C2×C22⋊C414yes—
491C4⋊C4C2×C22⋊C414yes—
492C4⋊C4C23×C418yes—
493C22⋊C4—28no—
494C22⋊C4—216no—
495C22⋊C4—216no—
496C22⋊C4—28no—
497C22⋊C4—42no—
498C22⋊C4—44no—
499C22⋊C4—42no—
500C22⋊C4—48no—
501C22⋊C4—84no—
502C22⋊C4—84no—
503C22⋊C4C2×C22⋊C418yes—
504C22⋊C4C2×C22⋊C418yes—
505C22⋊C4C2×C22⋊C418yes—
506C22⋊C4C2×C22⋊C418yes—
507C42—28no—
508C42—28no—
509C42—44no—
510C42—82no—
511C42—84no—
512C42C2×C22⋊C412yes—
513C42C2×C22⋊C412yes—
514C42C2×C22⋊C412yes—
515C42C2×C22⋊C412yes—
516C42C23×C412yes—
517C42C23×C412yes—
518C42C22×D412yes—
519C42C22×D412yes—
520C23—22no—
521C23—28no—
522C23—28no—
523C23—22no—
524C23—28no—
525C23—24no—
526C23—42no—
527C23—44no—
528C23—42no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23—816no—
535C23C22×C22⋊C411yescharacteristic
536C23C22×C22⋊C414yes—
537C23C22×C22⋊C411yescharacteristic
538C23C22×C22⋊C414yes—
539C23C2×#13718yes—
540C23C23×D411yescharacteristic
541Q8—48no—
542Q8—48no—
543Q8—84no—
544Q8—84no—
545Q8—88no—
546Q8—88no—
547D4—216no—
548D4—28no—
549D4#419116yes—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—88no—
589C2×C4—88no—
590C2×C4—88no—
591C2×C4—88no—
592C2×C4—168no—
593C2×C4C22×C22⋊C412yes—
594C2×C4C22×C22⋊C411yescharacteristic
595C2×C4C22×C22⋊C411yescharacteristic
596C2×C4C22×C22⋊C412yes—
597C2×C4C22×C22⋊C412yes—
598C2×C4C22×C22⋊C414yes—
599C2×C4C22×C22⋊C414yes—
600C2×C4C22×C22⋊C412yes—
601C2×C4C2×#13714yes—
602C2×C4C24×C411yescharacteristic
603C2×C4C23×D411yescharacteristic
604C8—48no—
605C22—21no—
606C22—24no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—24no—
611C22—21no—
612C22—24no—
613C22—21no—
614C22—24no—
615C22—21no—
616C22—24no—
617C22—48no—
618C22—416no—
619C22—44no—
620C22—84no—
621C22—88no—
622C22#288312yes—
623C22C2×#41918yes—
624C22C23×C22⋊C411yescharacteristic
625C22C22×#13712yes—
626C4—21no—
627C4—21no—
628C4—21no—
629C4—21no—
630C4—24no—
631C4—24no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—48no—
636C4—48no—
637C4—82no—
638C4—84no—
639C4—82no—
640C4C2×#41914yes—
641C2—22no—
642C2—22no—
643C2—28no—
644C2—44no—
645C2—41no—
646C2C2×#288311yescharacteristic
647C2C22×#41911yescharacteristic
648C2#6009311yescharacteristic
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. #17590C2semidirect
8. #17578C2semidirect
9. #17562C2semidirect
10. C2×#1511C2semidirect
15. #3501C4semidirect
20. C2×#554C4semidirect
21. C2×D4×Q8C4semidirect
48. #1520C22semidirect
49. #1518C22semidirect
50. #1515C22semidirect
54. #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
140. 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
#341902
#176272
#176212
#176142
#175902
#175782
#175622
C2×#15112