Finite Group Databasebeta

← #6343767

Subgroups — #6343767

Order 512.

3303 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3514C211yesmaximal, characteristic
2#60282C211yesmaximal, characteristic, complemented
3C4×#568C211yesmaximal, characteristic, complemented
4#17652C2116yesmaximal, complemented
5#17650C214yesmaximal, complemented
6#17647C212yesmaximal, complemented
7#17642C212yesmaximal, complemented
8C2×#1527C214yesmaximal, complemented
9C2×#568—21no—
10#3514—24no—
11#3514—24no—
12#3514C418yes—
13#3514C2218yes—
14C2×#559C411yescharacteristic
15C2×#556C411yescharacteristic
16C2×#555—21no—
17C2×#551—22no—
18C2×D4×Q8C411yescharacteristic
19C2×#548C412yes—
20C2×#546—21no—
21C2×#546C412yes—
22C2×D42C411yescharacteristic
23C2×#542—21no—
24C2×#538—22no—
25C2×#532C2211yescharacteristic
26C2×#529C2211yescharacteristic
27C2×#529C2212yes—
28C22×#124C2212yes—
29C2×#518—21no—
30C2×C4×#41C2211yescharacteristic
31#2990—24no—
32#2983—21no—
33#2973—22no—
34#2972—21no—
35#2970C2218yescomplemented
36#2920C2212yescomplemented
37C2×#432C2214yescomplemented
38#2875C2218yescomplemented
39C4×#131—24no—
40#2873—22no—
41#2873—21no—
42C2×C4×M16C2214yescomplemented
43C2×C4×M16C2212yescomplemented
44#1532C22116yescomplemented
45#1530C2218yescomplemented
46#1529C2218yescomplemented
47#1528C2214yescomplemented
48#1527C2214yes—
49#1527C22116yescomplemented
50#1527C2218yescomplemented
51#1525C2218yescomplemented
52#1524C2218yescomplemented
53#1524C22116yescomplemented
54C2×#329C22116yescomplemented
55C22×#41C2×C412yes—
56C22×#41D411yescharacteristic
57C23×Q8C2×C412yes—
58C23×D4—21no—
59C23×D4C2×C412yes—
60#568—216no—
61C2×#131—28no—
62C22×M16—24no—
63C22×M16—22no—
64#559—24no—
65#559C2×C418yes—
66#556—24no—
67#556C2×C418yes—
68#555—28no—
69#555—28no—
70#551—216no—
71#551—216no—
72D4×Q8—24no—
73D4×Q8C2×C418yes—
74#548—28no—
75#548C2×C4116yes—
76#546—28no—
77#546—28no—
78#546—28no—
79#546C2×C4116yes—
80D42—24no—
81D42C2×C418yes—
82#542—28no—
83#542—28no—
84#538—216no—
85#538—216no—
86#532—22no—
87#532—22no—
88#532C2×C414yes—
89#532C2314yes—
90C2×C4⋊Q8—21no—
91C2×C4⋊Q8C2×C411yescharacteristic
92C2×#127C2×C411yescharacteristic
93#529—24no—
94#529—24no—
95#529—22no—
96#529—22no—
97#529C2×C418yes—
98#529C2×C414yes—
99#529C2314yes—
100#529C2318yes—
101C2×#126—22no—
102C2×#126—22no—
103C2×#126—21no—
104C2×#125C2311yescharacteristic
105C2×#124—21no—
106C2×#124—22no—
107C2×#124—22no—
108C2×#124—22no—
109C2×#124—24no—
110C2×#124—24no—
111C2×#124C2×C412yes—
112C2×#124C2×C412yes—
113C2×#124C2×C418yes—
114C2×#124C2311yescharacteristic
115C2×#124C2312yes—
116C2×#124C2318yes—
117C2×#123—21no—
118C2×#123—22no—
119C2×#123—22no—
120C2×#123—22no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8—22no—
124C2×C22⋊Q8C2×C412yes—
125C2×C22⋊Q8C2×C411yescharacteristic
126C2×#121—22no—
127C2×#121—21no—
128C2×#121—21no—
129C2×#121—21no—
130C2×#121—22no—
131C2×#121C2×C412yes—
132C2×#121C2×C411yescharacteristic
133C2×#120—22no—
134C2×#120—22no—
135#518—28no—
136#518—28no—
137C4×#41C2×C414yes—
138C4×#41D414yes—
139C4×#41D414yes—
140C4×#41C2314yes—
141C2×C4×Q8C2×C412yes—
142C2×C4×Q8C2×C411yescharacteristic
143C2×C4×Q8C2×C411yescharacteristic
144C2×C4×Q8D411yescharacteristic
145C2×C4×D4—21no—
146C2×C4×D4—22no—
147C2×C4×D4—21no—
148C2×C4×D4—21no—
149C2×C4×D4—22no—
150C2×C4×D4—21no—
151C2×C4×D4—21no—
152C2×C4×D4C2×C411yescharacteristic
153C2×C4×D4C2×C412yes—
154C2×C4×D4C2×C411yescharacteristic
155C2×C4×D4C2×C412yes—
156C2×C4×D4D411yescharacteristic
157C2×C4×D4D412yes—
158C2×#117—21no—
159C2×#117—21no—
160C2×#117—22no—
161C2×#117D412yes—
162C2×#117D411yescharacteristic
163C22×C22⋊C4—22no—
164C22×C22⋊C4—22no—
165C22×C22⋊C4—22no—
166C22×C42C2312yes—
167C22×C42C2311yescharacteristic
168#446—28no—
169#435—216no—
170C8×D4—28no—
171#432—24no—
172#432—24no—
173#432C2314yes—
174#432C2318yescomplemented
175#432C2314yescomplemented
176#431—22no—
177#431—22no—
178#423—22no—
179C2×C4⋊C8—24no—
180#407—22no—
181C2×C22⋊C8—24no—
182#405—216no—
183#405—28no—
184C4×M16—24no—
185C4×M16—22no—
186C4×M16—24no—
187C4×M16—22no—
188C4×M16C2314yes—
189C4×M16C2318yescomplemented
190C4×M16C2314yescomplemented
191C4×M16C2314yescomplemented
192C2×#97C2318yescomplemented
193C2×#97C2314yescomplemented
194C2×#97C2318yescomplemented
195C2×C4×C8—24no—
196C2×C4×C8—22no—
197#330C23116yescomplemented
198#329C23116yes—
199#329C23132yescomplemented
200C25—22no—
201C2×#41—28no—
202C2×#41—24no—
203C2×#41—84no—
204C2×#41C22⋊C414yes—
205C2×#41C22×C418yes—
206C2×#41C2×D414yes—
207C22×Q8—22no—
208C22×Q8—24no—
209C22×Q8—84no—
210C22×Q8C22⋊C411yescharacteristic
211C22×Q8C22×C411yescharacteristic
212C22×Q8C22×C412yes—
213C22×Q8C22×C418yes—
214C22×D4—22no—
215C22×D4—21no—
216C22×D4—22no—
217C22×D4—28no—
218C22×D4—21no—
219C22×D4—21no—
220C22×D4—22no—
221C22×D4—24no—
222C22×D4—48no—
223C22×D4—84no—
224C22×D4C22⋊C412yes—
225C22×D4C22⋊C411yescharacteristic
226C22×D4C22×C411yescharacteristic
227C22×D4C22×C412yes—
228C22×D4C22×C418yes—
229C23×C4—22no—
230C23×C4—21no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4C22×C412yes—
234C23×C4C22×C411yescharacteristic
235C23×C4C2×D412yes—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D412yes—
238C23×C4C2×D411yescharacteristic
239#131—264no—
240C2×M16—232no—
241C2×M16—216no—
242C2×M16—24no—
243C2×M16—28no—
244C2×M16—24no—
245C2×M16—24no—
246C22×C8—28no—
247C22×C8—24no—
248C22×C8—28no—
249C4⋊Q8—28no—
250C4⋊Q8—22no—
251C4⋊Q8C22×C414yes—
252#127—22no—
253#127C22×C414yes—
254#126—216no—
255#126—28no—
256#126—24no—
257#126—28no—
258#126—24no—
259#125—21no—
260#125—21no—
261#125C22×C412yes—
262#125C2412yes—
263#124—28no—
264#124—216no—
265#124—28no—
266#124—24no—
267#124—24no—
268#124—28no—
269#124—21no—
270#124—21no—
271#124—22no—
272#124—22no—
273#124—28no—
274#124—28no—
275#124C22×C412yes—
276#124C22×C414yes—
277#124C22×C418yes—
278#124C22×C418yes—
279#124C2412yes—
280#124C2414yes—
281#123—24no—
282#123—28no—
283#123—24no—
284#123—28no—
285#123—216no—
286#123—216no—
287C22⋊Q8—28no—
288C22⋊Q8—24no—
289C22⋊Q8—22no—
290C22⋊Q8—216no—
291C22⋊Q8—216no—
292C22⋊Q8C22×C418yes—
293C22⋊Q8C22×C414yes—
294#121—216no—
295#121—24no—
296#121—24no—
297#121—24no—
298#121—22no—
299#121—24no—
300#121—24no—
301#121—28no—
302#121—216no—
303#121C22×C418yes—
304#121C22×C414yes—
305#120—216no—
306#120—216no—
307C4×Q8—24no—
308C4×Q8—22no—
309C4×Q8C22⋊C414yes—
310C4×Q8C22⋊C414yes—
311C4×Q8C22×C418yes—
312C4×Q8C22×C414yes—
313C4×Q8C22×C414yes—
314C4×Q8C2×D414yes—
315C4×D4—28no—
316C4×D4—28no—
317C4×D4—24no—
318C4×D4—28no—
319C4×D4—24no—
320C4×D4—28no—
321C4×D4—216no—
322C4×D4—24no—
323C4×D4—24no—
324C4×D4—22no—
325C4×D4—24no—
326C4×D4—24no—
327C4×D4—24no—
328C4×D4C22⋊C414yes—
329C4×D4C22⋊C418yes—
330C4×D4C22⋊C414yes—
331C4×D4C22⋊C418yes—
332C4×D4C22×C414yes—
333C4×D4C22×C418yes—
334C4×D4C22×C414yes—
335C4×D4C22×C418yes—
336C4×D4C2×D414yes—
337C4×D4C2×D418yes—
338#117—28no—
339#117—24no—
340#117—28no—
341#117—24no—
342#117—28no—
343#117C22⋊C418yes—
344#117C22⋊C414yes—
345#117C2×D418yes—
346#117C2×D414yes—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4C22⋊C411yescharacteristic
356C2×C4⋊C4C22⋊C411yescharacteristic
357C2×C4⋊C4C22⋊C412yes—
358C2×C4⋊C4C22⋊C412yes—
359C2×C4⋊C4C22×C411yescharacteristic
360C2×C4⋊C4C22×C411yescharacteristic
361C2×C22⋊C4—22no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—28no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—28no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—28no—
378C2×C22⋊C4—28no—
379C2×C22⋊C4—216no—
380C2×C22⋊C4C22⋊C412yes—
381C2×C22⋊C4C22⋊C412yes—
382C2×C22⋊C4C22⋊C412yes—
383C2×C42—22no—
384C2×C42—21no—
385C2×C42C22×C412yes—
386C2×C42C22×C411yescharacteristic
387C2×C42C22×C414yes—
388C2×C42C22×C412yes—
389C2×C42C2×D412yes—
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D412yes—
393C2×C42C2×D414yes—
394C2×C42C2×D414yes—
395C2×C42C2×D412yes—
396C2×C42C2×D412yes—
397C2×C42C2414yes—
398C2×C42C2412yes—
399C2×C42C2411yescharacteristic
400C4⋊C8—24no—
401C4⋊C8—28no—
402C4⋊C8—24no—
403C22⋊C8—24no—
404C22⋊C8—28no—
405C22⋊C8—24no—
406#97—28no—
407#97—24no—
408#97—28no—
409#97C2418yes—
410#97C2418yes—
411C4×C8—24no—
412C4×C8—28no—
413C4×C8—24no—
414C4×C8—24no—
415C24—21no—
416C24—22no—
417C24—28no—
418C24—48no—
419C24—84no—
420C24—84no—
421C24C2×C22⋊C411yescharacteristic
422C24C2×C22⋊C412yes—
423#41—216no—
424#41—216no—
425#41—88no—
426#41—88no—
427C2×Q8—21no—
428C2×Q8—22no—
429C2×Q8—28no—
430C2×Q8—24no—
431C2×Q8—42no—
432C2×Q8—44no—
433C2×Q8—88no—
434C2×Q8—84no—
435C2×Q8—88no—
436C2×Q8C2×C22⋊C414yes—
437C2×Q8C23×C412yes—
438C2×Q8C23×C414yes—
439C2×D4—28no—
440C2×D4—28no—
441C2×D4—24no—
442C2×D4—28no—
443C2×D4—24no—
444C2×D4—21no—
445C2×D4—22no—
446C2×D4—24no—
447C2×D4—28no—
448C2×D4—42no—
449C2×D4—44no—
450C2×D4—48no—
451C2×D4—432no—
452C2×D4—44no—
453C2×D4—88no—
454C2×D4—88no—
455C2×D4—82no—
456C2×D4—82no—
457C2×D4—84no—
458C2×D4—84no—
459C2×D4#10418yes—
460C2×D4C2×C22⋊C418yes—
461C2×D4C2×C22⋊C414yes—
462C2×D4C23×C412yes—
463C2×D4C23×C414yes—
464C22×C4—28no—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—28no—
471C22×C4—24no—
472C22×C4—28no—
473C22×C4—28no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—82no—
478C22×C4—81no—
479C22×C4—84no—
480C22×C4—84no—
481C22×C4—84no—
482C22×C4C2×C22⋊C412yes—
483C22×C4C2×C22⋊C414yes—
484C22×C4C2×C22⋊C412yes—
485C22×C4C2×C22⋊C414yes—
486C22×C4C2×C22⋊C412yes—
487C22×C4C2×C22⋊C412yes—
488C22×C4C2×C22⋊C414yes—
489C22×C4C2×C22⋊C411yescharacteristic
490C22×C4C2×C22⋊C412yes—
491C22×C4C2×C22⋊C411yescharacteristic
492C22×C4C2×C22⋊C412yes—
493C22×C4C2×C22⋊C411yescharacteristic
494C22×C4C23×C411yescharacteristic
495C22×C4C23×C414yes—
496C22×C4C23×C412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D414yes—
499C22×C4C22×D412yes—
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D414yes—
502C22×C4C22×D412yes—
503M16—216no—
504M16—232no—
505M16—216no—
506M16—84no—
507C2×C8—232no—
508C2×C8—216no—
509C2×C8—28no—
510C2×C8—28no—
511C2×C8—88no—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—28no—
517C4⋊C4—24no—
518C4⋊C4—28no—
519C4⋊C4—24no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4C2×C22⋊C414yes—
525C4⋊C4C2×C22⋊C414yes—
526C4⋊C4C2×C22⋊C418yes—
527C4⋊C4C2×C22⋊C418yes—
528C4⋊C4C23×C412yes—
529C4⋊C4C23×C412yes—
530C22⋊C4—28no—
531C22⋊C4—24no—
532C22⋊C4—28no—
533C22⋊C4—24no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—28no—
539C22⋊C4—28no—
540C22⋊C4—28no—
541C22⋊C4—28no—
542C22⋊C4—24no—
543C22⋊C4—28no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—24no—
547C22⋊C4—24no—
548C22⋊C4C2×C22⋊C418yes—
549C22⋊C4C2×C22⋊C418yes—
550C22⋊C4C2×C22⋊C418yes—
551C42—28no—
552C42—24no—
553C42C2×C22⋊C414yes—
554C42C2×C22⋊C412yes—
555C42C2×C22⋊C412yes—
556C42C2×C22⋊C414yes—
557C42C2×C22⋊C412yes—
558C42C2×C22⋊C414yes—
559C42C23×C411yescharacteristic
560C42C23×C414yes—
561C42C23×C412yes—
562C42C22×D411yescharacteristic
563C42C22×D414yes—
564C42C22×D412yes—
565C42C22×D412yes—
566C42C22×D414yes—
567C42C22×D411yescharacteristic
568C42C2511yescharacteristic
569C23—22no—
570C23—22no—
571C23—24no—
572C23—42no—
573C23—42no—
574C23—41no—
575C23—41no—
576C23—44no—
577C23—44no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—44no—
582C23—416no—
583C23—88no—
584C23—88no—
585C23—1616no—
586C23C2×#10418yes—
587C23C22×C22⋊C412yes—
588C23C22×C22⋊C414yes—
589C23C22×C22⋊C411yescharacteristic
590Q8—28no—
591Q8—42no—
592Q8—44no—
593Q8—42no—
594Q8—44no—
595Q8—88no—
596D4—216no—
597D4—42no—
598D4—44no—
599D4—416no—
600D4—42no—
601D4—44no—
602D4—48no—
603D4—84no—
604D4—88no—
605D4—84no—
606D4—88no—
607D4C2×#104116yes—
608C2×C4—28no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—22no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—48no—
637C2×C4—88no—
638C2×C4—88no—
639C2×C4—88no—
640C2×C4C2×#10414yes—
641C2×C4C22×C22⋊C414yes—
642C2×C4C22×C22⋊C411yescharacteristic
643C2×C4C22×C22⋊C411yescharacteristic
644C2×C4C22×C22⋊C414yes—
645C2×C4C22×C22⋊C412yes—
646C2×C4C22×C22⋊C412yes—
647C2×C4C22×C22⋊C414yes—
648C2×C4C22×C22⋊C411yescharacteristic
649C2×C4C22×C22⋊C412yes—
650C2×C4C24×C411yescharacteristic
651C2×C4C23×D411yescharacteristic
652C2×C4C23×D411yescharacteristic
653C8—216no—
654C8—48no—
655C22—21no—
656C22—24no—
657C22—21no—
658C22—24no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—48no—
663C22—44no—
664C22—42no—
665C22—42no—
666C22—44no—
667C22—44no—
668C22—42no—
669C22—82no—
670C22—88no—
671C22—84no—
672C22—1616no—
673C22C22×#10412yes—
674C22C22×#10418yes—
675C22C23×C22⋊C411yescharacteristic
676C4—21no—
677C4—22no—
678C4—24no—
679C4—21no—
680C4—21no—
681C4—24no—
682C4—21no—
683C4—21no—
684C4—24no—
685C4—22no—
686C4—44no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4#288812yes—
691C4C22×#10414yes—
692C2—22no—
693C2—28no—
694C2—42no—
695C2—84no—
696C2—82no—
697C2C2×#288811yescharacteristic
698C2C23×#10411yescharacteristic
699C2#6009311yescharacteristic
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
prime radicalC2×#3514C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60282C2semidirect
3. C4×#568C2semidirect
4. #17652C2semidirect
5. #17650C2semidirect
6. #17647C2semidirect
7. #17642C2semidirect
8. C2×#1527C2semidirect
35. #2970C22semidirect
36. #2920C22semidirect
37. C2×#432C22semidirect
38. #2875C22semidirect
42. C2×C4×M16C22semidirect
43. C2×C4×M16C22semidirect
44. #1532C22semidirect
45. #1530C22semidirect
46. #1529C22semidirect
47. #1528C22semidirect
49. #1527C22semidirect
50. #1527C22semidirect
51. #1525C22semidirect
52. #1524C22semidirect
53. #1524C22semidirect
54. C2×#329C22semidirect
174. #432C23semidirect
175. #432C23semidirect
189. C4×M16C23semidirect
190. C4×M16C23semidirect
191. C4×M16C23semidirect
192. C2×#97C23semidirect
193. C2×#97C23semidirect
194. C2×#97C23semidirect
197. #330C23semidirect
199. #329C23semidirect

Maximal subgroups

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

subgroupindex
C2×#35142
#602822
C4×#5682
#176522
#176502
#176472
#176422
C2×#15272