Finite Group Databasebeta

← #6510798

Subgroups — #6510798

Order 512.

2270 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3195C211yesmaximal, characteristic, complemented
2C2×#3193C211yesmaximal, characteristic
3C2×#3095C211yesmaximal, characteristic, complemented
4C2×#3094C211yesmaximal, characteristic
5#35019C211yesmaximal, characteristic
6C2×#2499C211yesmaximal, characteristic, complemented
7#28361C214yesmaximal, complemented
8#28359C214yesmaximal, complemented
9#28315C212yesmaximal, complemented
10#28306C212yesmaximal, complemented
11#27947C214yesmaximal, complemented
12#27943C214yesmaximal, complemented
13#27914C212yesmaximal, complemented
14#27912C212yesmaximal, complemented
15#20524C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×#117C2211yescharacteristic
20#3195—28no—
21#3193—28no—
22#3095—28no—
23#3094—28no—
24C2×#484C2212yes—
25C2×#484C2212yescomplemented
26C2×#481C2212yes—
27C2×#485C2214yes—
28C2×#482C2214yes—
29C2×#483C2212yescomplemented
30C2×#483C2212yescomplemented
31C2×#483C2212yescomplemented
32C2×#480C2212yes—
33C2×#480C2212yes—
34C2×#470C2214yes—
35C2×#468C2214yes—
36C2×#451C2212yescomplemented
37C2×#451C2212yescomplemented
38C2×#448C2212yes—
39C2×#450C2212yes—
40C2×#428C2212yes—
41C2×#428C2212yescomplemented
42C2×#419C2212yescomplemented
43C2×#418C2212yes—
44C2×#407C2211yescharacteristic
45C2×#407C2211yescharacteristic
46#2850C2212yes—
47#2850C2212yes—
48#2639—22no—
49#2591—22no—
50#2561—22no—
51#2504C2212yes—
52#2503C2212yes—
53#2499—22no—
54#2499—22no—
55#2499C2214yes—
56#2499C2214yescomplemented
57#2436C2212yes—
58#2435C2212yes—
59C2×#397—22no—
60#2308—22no—
61C2×#391C2211yescharacteristic, complemented
62C2×#391C2211yescharacteristic, complemented
63C2×#387—22no—
64C2×#384C2211yescharacteristic, complemented
65C2×#382C2211yescharacteristic, complemented
66C2×C4×C22⋊C4C2211yescharacteristic
67#2097C2214yescomplemented
68#2097C2214yescomplemented
69#2097C2214yescomplemented
70#2095C2214yescomplemented
71#2094C2214yescomplemented
72#2092C2214yescomplemented
73#2015C2214yescomplemented
74#2015C2214yescomplemented
75#1973C2214yescomplemented
76#1971C2214yescomplemented
77#1969C2214yescomplemented
78#1968C2214yescomplemented
79#1959C2214yescomplemented
80#1958C2214yescomplemented
81#1847C2214yescomplemented
82#1846C2214yescomplemented
83#1753C2214yes—
84#1753C2214yescomplemented
85#1749C2212yescomplemented
86C4×C22⋊C8C2212yescomplemented
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×D4—21no—
90C23×D4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C22×Q16C2314yes—
93C22×SD16C2314yes—
94C22×M16C2312yes—
95C22×M16C2312yes—
96C2×#124C2312yes—
97C2×#124C2312yes—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—24no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2312yes—
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2312yes—
110C2×#121—21no—
111C2×#121—21no—
112C2×#121—24no—
113C2×#121—24no—
114C2×#121C2312yes—
115C2×#121C2312yes—
116C2×#117—21no—
117C2×#117—21no—
118C2×#117—24no—
119C2×#117—24no—
120C2×#117C2312yes—
121C2×#117C2312yes—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4D412yes—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#485—216no—
139#484—28no—
140#484—28no—
141#483—28no—
142#483—28no—
143#483—28no—
144#482—216no—
145#481—28no—
146#480—28no—
147#480—28no—
148#470—216no—
149#468—216no—
150#451—28no—
151#451—28no—
152#450—28no—
153#448—28no—
154#428—28no—
155#428—28no—
156C2×#107C2314yes—
157C2×#107C2314yes—
158C2×#107C2314yes—
159C2×#106C2314yes—
160C2×#106C2314yes—
161C2×#106C2314yes—
162#419—28no—
163#418—28no—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#102C2314yes—
170C2×#102C2314yes—
171C2×#102C2314yes—
172#407D418yes—
173#407D418yes—
174C2×C22⋊C8C2312yes—
175C2×C22⋊C8C2312yes—
176C2×C22⋊C8C2312yes—
177C2×C22⋊C8C2312yes—
178C2×#97C2314yes—
179C2×C4×C8C2314yes—
180#400—24no—
181#397—28no—
182#397—28no—
183#397—24no—
184#394—24no—
185#393—24no—
186#391—22no—
187#391—22no—
188#391C2312yes—
189#391C2312yes—
190#391C2314yes—
191#391C2314yes—
192#390C2312yes—
193#390C2312yes—
194#388—24no—
195#388—24no—
196#387—28no—
197#387—28no—
198#386—24no—
199#386—24no—
200#386—24no—
201#386—24no—
202#384—22no—
203#384C2314yes—
204#382—22no—
205#382C2312yes—
206#382C2312yes—
207#382C2314yes—
208#381—24no—
209#379—24no—
210C4×C22⋊C4D412yes—
211C4×C22⋊C4D412yes—
212C4×C22⋊C4C2312yes—
213C4×C22⋊C4C2312yes—
214C2×#95—22no—
215C2×#95—22no—
216C2×#95—22no—
217C2×#95D412yes—
218#336C2314yes—
219#336C2314yes—
220C25—21no—
221C25C2×D411yescharacteristic
222C22×Q8—21no—
223C22×Q8—21no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—42no—
229C22×Q8—42no—
230C22×Q8—42no—
231C22×Q8—42no—
232C22×Q8C2412yes—
233C22×Q8C2412yes—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—21no—
239C22×D4—42no—
240C22×D4—42no—
241C22×D4—48no—
242C22×D4—44no—
243C22×D4C2412yes—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—42no—
252C23×C4—42no—
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D414yes—
259C23×C4C2×D414yes—
260C23×C4C2411yescharacteristic
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C2×Q16—28no—
264C2×Q16—44no—
265C2×Q16—44no—
266C2×SD16—28no—
267C2×SD16—44no—
268C2×SD16—44no—
269C2×M16—42no—
270C2×M16—42no—
271C2×M16—42no—
272C2×M16—42no—
273C2×M16C2×D418yes—
274C2×M16C2×D418yes—
275C22×C8C2414yes—
276C22×C8C2414yes—
277#124—44no—
278#124—44no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283C22⋊Q8—24no—
284C22⋊Q8—24no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8—44no—
288C22⋊Q8—44no—
289C22⋊Q8—44no—
290C22⋊Q8—44no—
291#121—24no—
292#121—24no—
293#121—24no—
294#121—24no—
295#121—44no—
296#121—44no—
297#117—24no—
298#117—24no—
299#117—24no—
300#117—24no—
301#117—44no—
302#117—44no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—28no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—48no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4C2×D414yes—
349C2×C22⋊C4C2×D414yes—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—42no—
353C2×C42—42no—
354C2×C42—44no—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2412yes—
358C2×C42C2412yes—
359#107—28no—
360#107—28no—
361#107—28no—
362#106—28no—
363#106—28no—
364#106—28no—
365#103—28no—
366#103—28no—
367#103—28no—
368#103—28no—
369#103—28no—
370#102—28no—
371#102—28no—
372#102—28no—
373C22⋊C8C2×D418yes—
374C22⋊C8C2×D418yes—
375C22⋊C8C2×D418yes—
376C22⋊C8C2×D418yes—
377#97—44no—
378C4×C8—44no—
379#95—28no—
380#95—28no—
381#95—28no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95C2×D414yes—
387#95C2×D414yes—
388C24—24no—
389C24—22no—
390C24—41no—
391C24—41no—
392C24—42no—
393C24—44no—
394C24—42no—
395C24—81no—
396C24—81no—
397C24—82no—
398C24#12714yes—
399C24C22×D411yescharacteristic
400C24C22×D411yescharacteristic
401C24C22×D411yescharacteristic
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—48no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—81no—
416C2×Q8—81no—
417C2×Q8—84no—
418C2×Q8—84no—
419C2×Q8—84no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—48no—
429C2×D4—42no—
430C2×D4—84no—
431C2×D4—81no—
432C2×D4—81no—
433C2×D4—816no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—42no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—82no—
475C22×C4—81no—
476C22×C4—81no—
477C22×C4#12714yes—
478C22×C4#12714yes—
479C22×C4#12714yes—
480C22×C4C22×D414yes—
481C22×C4C22×D414yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C2511yescharacteristic
492Q16—88no—
493SD16—88no—
494M16—48no—
495M16—48no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8C22×D418yes—
501C2×C8C22×D418yes—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—416no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C42—24no—
537C42—24no—
538C42—42no—
539C42—42no—
540C42—44no—
541C42—44no—
542C23—22no—
543C23—21no—
544C23—21no—
545C23—22no—
546C23—22no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—22no—
553C23—28no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—44no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23—164no—
567C23—164no—
568C23C2×#12712yes—
569C23C2×#12712yes—
570C23C2×#12712yes—
571C23#57514yes—
572C23C23×D411yescharacteristic
573Q8—44no—
574Q8—44no—
575Q8—44no—
576Q8—44no—
577Q8—84no—
578Q8—84no—
579D4—44no—
580D4—44no—
581D4—84no—
582D4—88no—
583D4—88no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—44no—
607C2×C4—44no—
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—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4C2×#12712yes—
631C2×C4C2×#12712yes—
632C2×C4C2×#12712yes—
633C2×C4C23×D411yescharacteristic
634C2×C4C23×D411yescharacteristic
635C2×C4#58512yes—
636C2×C4#58512yes—
637C8—28no—
638C8—28no—
639C22—24no—
640C22—24no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—88no—
654C22—84no—
655C22—84no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—168no—
660C22#315314yes—
661C22C22×#12711yescharacteristic
662C22#348511yescharacteristic
663C22#348511yescharacteristic
664C22C2×#57512yes—
665C22#358612yes—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—44no—
670C4—44no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4#348512yes—
679C4#348512yes—
680C2—24no—
681C2—42no—
682C2—42no—
683C2—162no—
684C2#3546311yescharacteristic
685C2C2×#315311yescharacteristic
686C2#6064511yescharacteristic
687C2#6171212yes—
688C2#6171612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3195C2semidirect
3. C2×#3095C2semidirect
6. C2×#2499C2semidirect
7. #28361C2semidirect
8. #28359C2semidirect
9. #28315C2semidirect
10. #28306C2semidirect
11. #27947C2semidirect
12. #27943C2semidirect
13. #27914C2semidirect
14. #27912C2semidirect
15. #20524C2semidirect
25. C2×#484C22semidirect
29. C2×#483C22semidirect
30. C2×#483C22semidirect
31. C2×#483C22semidirect
36. C2×#451C22semidirect
37. C2×#451C22semidirect
41. C2×#428C22semidirect
42. C2×#419C22semidirect
56. #2499C22semidirect
61. C2×#391C22semidirect
62. C2×#391C22semidirect
64. C2×#384C22semidirect
65. C2×#382C22semidirect
67. #2097C22semidirect
68. #2097C22semidirect
69. #2097C22semidirect
70. #2095C22semidirect
71. #2094C22semidirect
72. #2092C22semidirect
73. #2015C22semidirect
74. #2015C22semidirect
75. #1973C22semidirect
76. #1971C22semidirect
77. #1969C22semidirect
78. #1968C22semidirect
79. #1959C22semidirect
80. #1958C22semidirect
81. #1847C22semidirect
82. #1846C22semidirect
84. #1753C22semidirect
85. #1749C22semidirect
86. C4×C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31952
C2×#31932
C2×#30952
C2×#30942
#350192
C2×#24992
#283612
#283592
#283152
#283062
#279472
#279432
#279142
#279122
#205242