Finite Group Databasebeta

← #6345196

Subgroups — #6345196

Order 512.

1582 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60952C211yesmaximal, characteristic
2#60204C211yesmaximal, characteristic
3C2×#2960C211yesmaximal, characteristic, complemented
4#17984C214yesmaximal, complemented
5#17965C214yesmaximal, complemented
6#17941C214yesmaximal, complemented
7#17936C212yesmaximal, complemented
8#17934C212yesmaximal, complemented
9#17913C214yesmaximal, complemented
10#17903C212yesmaximal, complemented
11#17901C212yesmaximal, complemented
12#17820C211yesmaximal, characteristic, complemented
13#17498C211yesmaximal, characteristic
14#17491C211yesmaximal, characteristic, complemented
15#17323C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17C2×#533C2211yescharacteristic
18#3255C412yes—
19#3254C412yes—
20#3239C412yes—
21#3237C412yes—
22#3218C412yes—
23#3216C412yes—
24#3196C414yes—
25#3172C2214yes—
26#3107C2214yes—
27#3104C2212yes—
28#3103C2212yes—
29#2966—21no—
30#2966—21no—
31#2966C2212yes—
32#2966C2212yes—
33#2965—21no—
34#2965C2212yes—
35C8×#41—21no—
36C8×#41C2212yes—
37#2960—21no—
38#2960—21no—
39#2960—21no—
40#2960—21no—
41#2960C2211yescharacteristic
42#2960C2212yes—
43#2960C2212yes—
44#2960C2212yes—
45#2960C2212yes—
46#2960C2211yescharacteristic
47C2×#446—21no—
48C2×#446C2212yes—
49C2×C8×Q8—21no—
50C2×C8×Q8C2212yes—
51#2931C2211yescharacteristic
52#2931C2211yescharacteristic
53#2929—21no—
54#2929—21no—
55#2922—21no—
56#2920C2211yescharacteristic
57#2920C2211yescharacteristic
58#2920C2211yescharacteristic
59C2×#424—21no—
60C2×#423C2211yescharacteristic
61C4×#131—21no—
62C2×#405—21no—
63C22×C4×C8C2211yescharacteristic
64#1593C2218yescomplemented
65#1592C2218yescomplemented
66#1587C2218yescomplemented
67#1585C2218yescomplemented
68#1584C2218yescomplemented
69#1583C2218yescomplemented
70C8×Q16C2218yescomplemented
71C8×SD16C2218yescomplemented
72#1574C2214yes—
73#1571C2212yes—
74#1570C2212yes—
75#1569C2214yes—
76#1497C2214yes—
77#1495C2214yes—
78#1489C2214yes—
79#1487C2212yes—
80#1487C2212yes—
81#1485C2214yes—
82#1481C2212yes—
83#1480C2212yes—
84#1452C2212yes—
85#1451C2212yes—
86#1451C2212yes—
87#1450C2212yes—
88C8×M16C2212yes—
89C8×M16C2212yes—
90C2×#131—21no—
91C2×#131—21no—
92C22×M16—21no—
93C23×C8D411yescharacteristic
94C23×C8D411yescharacteristic
95#533—21no—
96#533—21no—
97#533—21no—
98#533—21no—
99#533C2×C412yes—
100#533C2×C412yes—
101#533C2×C411yescharacteristic
102#533C2×C411yescharacteristic
103#533C2311yescharacteristic
104#533C2312yes—
105#533C2312yes—
106#532C2×C411yescharacteristic
107#532C2×C411yescharacteristic
108#532C2311yescharacteristic
109C2×C4⋊Q8C2×C411yescharacteristic
110C2×C4⋊Q8C2×C411yescharacteristic
111#529—21no—
112#529—21no—
113#529—21no—
114#529—21no—
115#529C2×C412yes—
116#529C2312yes—
117C2×#125—21no—
118#525—21no—
119#525—21no—
120#525C2×C412yes—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8C2×C412yes—
123C4×#41—21no—
124C4×#41—21no—
125C4×#41C2312yes—
126C2×C4×Q8—21no—
127C2×C4×Q8C2312yes—
128C2×#117—21no—
129C22×C42C2311yescharacteristic
130#501C2×C414yes—
131#497C2×C414yes—
132#491C2×C414yes—
133#489C2×C414yes—
134#485C2×C418yes—
135#484C2×C414yes—
136#481C2×C414yes—
137#477C2×C414yes—
138#475C2×C414yes—
139#475C2×C414yes—
140#474C2×C414yes—
141#470C2×C418yes—
142#468C2×C418yes—
143#462C2×C414yes—
144#462C2×C414yes—
145#461C2×C414yes—
146#460C2×C414yes—
147#452C2×C418yes—
148#451C2×C414yes—
149#448C2×C414yes—
150#446—22no—
151#446—22no—
152#446—22no—
153#446—22no—
154#446—22no—
155#446—22no—
156#446—21no—
157#446—21no—
158#446C2312yes—
159#446C2314yes—
160#446C2314yes—
161#446C2312yes—
162C8×Q8—22no—
163C8×Q8—22no—
164C8×Q8—22no—
165C8×Q8—22no—
166C8×Q8—22no—
167C8×Q8—22no—
168C8×Q8—21no—
169C8×Q8—21no—
170C8×Q8C2314yes—
171C8×Q8C2312yes—
172C8×Q8C2314yes—
173C8×Q8C2312yes—
174C4×Q16C2318yes—
175C4×SD16C2318yes—
176#436—21no—
177#436—21no—
178#436C2312yes—
179#436C2312yes—
180#435—22no—
181#435—22no—
182#435—22no—
183#435—22no—
184C8×D4—22no—
185C8×D4—22no—
186C8×D4—22no—
187C8×D4—22no—
188C8×D4—21no—
189C8×D4—21no—
190C8×D4C2312yes—
191C8×D4C2312yes—
192#433—22no—
193#433—22no—
194#432—21no—
195#432—21no—
196#432—21no—
197#432C2×C412yes—
198#432C2×C412yes—
199#432C2312yes—
200#431—21no—
201#431—22no—
202#431—22no—
203#431C2312yes—
204#431C2312yes—
205#431C2312yes—
206#431C2312yes—
207#431C2312yes—
208#424—22no—
209#424—22no—
210#424—22no—
211#424—22no—
212#424—21no—
213#424—21no—
214#423—21no—
215#423—21no—
216#423—21no—
217#423C2×C412yes—
218#423C2312yes—
219#423C2311yescharacteristic
220#423C2311yescharacteristic
221#423C2312yes—
222C2×C4⋊C8—22no—
223C2×C4⋊C8—22no—
224C2×C4⋊C8—21no—
225C2×C4⋊C8C2312yes—
226#408—21no—
227#408—21no—
228#407—21no—
229#407—21no—
230#407C2×C412yes—
231#405—22no—
232#405—22no—
233#405—22no—
234#405—22no—
235#405—21no—
236#405—21no—
237#405—22no—
238#405—22no—
239C4×M16—21no—
240C4×M16—22no—
241C4×M16C2312yes—
242C4×M16C2312yes—
243C2×#97—22no—
244C2×C4×C8—22no—
245C2×C4×C8—22no—
246C2×C4×C8D411yescharacteristic
247C2×C4×C8D411yescharacteristic
248C2×C4×C8D411yescharacteristic
249C2×C4×C8D411yescharacteristic
250C2×C4×C8D411yescharacteristic
251C2×C4×C8D411yescharacteristic
252C2×C4×C8C2311yescharacteristic
253C2×C4×C8C2311yescharacteristic
254C2×C4×C8C2311yescharacteristic
255C2×C4×C8C2311yescharacteristic
256C2×C4×C8C2311yescharacteristic
257C2×C4×C8C2311yescharacteristic
258#335C2318yes—
259#334C2318yes—
260Q8⋊C8C2318yes—
261Q8⋊C8C2318yes—
262Q8⋊C8C2318yes—
263D4⋊C8C2318yes—
264#322C2314yes—
265#322C2314yes—
266#322C2314yes—
267C82C2314yes—
268C2×#41—21no—
269C2×#41—21no—
270C2×#41C22×C412yes—
271C22×Q8—21no—
272C22×Q8C22×C412yes—
273C23×C4C22×C411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4#4111yescharacteristic
276C2×Q16C22×C418yes—
277C2×SD16C22×C418yes—
278#131—44no—
279#131—44no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—21no—
283C2×M16—21no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—44no—
288C2×M16C22×C414yes—
289C22×C8—24no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—42no—
294C22×C8C2×D411yescharacteristic
295C22×C8C2×D411yescharacteristic
296C22×C8C2×D411yescharacteristic
297C22×C8C2×D411yescharacteristic
298C22×C8C2×D411yescharacteristic
299C22×C8C2×D411yescharacteristic
300C22×C8C2×D411yescharacteristic
301C22×C8C2×D411yescharacteristic
302C22×C8C2×D411yescharacteristic
303C22×C8C2×D411yescharacteristic
304C22×C8C2×D411yescharacteristic
305C22×C8C2×D411yescharacteristic
306C4⋊Q8—21no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8C22×C412yes—
311C4⋊Q8C22×C412yes—
312C4⋊Q8C22×C411yescharacteristic
313C4⋊Q8C22×C412yes—
314C4⋊Q8C22×C412yes—
315C4⋊Q8C22×C411yescharacteristic
316C4⋊Q8C22×C411yescharacteristic
317#127C22×C411yescharacteristic
318#126—22no—
319#126—22no—
320#125—22no—
321#125—22no—
322#125—21no—
323#125—22no—
324#125—22no—
325#125—21no—
326#125C22×C412yes—
327#124—21no—
328#124—21no—
329#124C22×C412yes—
330#123—22no—
331#123—22no—
332#123—22no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—21no—
338C22⋊Q8—21no—
339C22⋊Q8—21no—
340C22⋊Q8—22no—
341C22⋊Q8—22no—
342C22⋊Q8—21no—
343C22⋊Q8C22×C414yes—
344C22⋊Q8C22×C414yes—
345C22⋊Q8C22×C412yes—
346C22⋊Q8C22×C412yes—
347#121—21no—
348#121—21no—
349#121—21no—
350#121—21no—
351#121C22×C412yes—
352#121C22×C412yes—
353#120—22no—
354C4×Q8—22no—
355C4×Q8—22no—
356C4×Q8—21no—
357C4×Q8—21no—
358C4×Q8—21no—
359C4×Q8—22no—
360C4×Q8—22no—
361C4×Q8—22no—
362C4×Q8—21no—
363C4×Q8—22no—
364C4×Q8C22×C414yes—
365C4×Q8C22×C412yes—
366C4×Q8C2414yes—
367C4×Q8C2412yes—
368C4×D4—22no—
369C4×D4—22no—
370C4×D4—22no—
371C4×D4—22no—
372C4×D4—22no—
373C4×D4—22no—
374C4×D4—21no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4C22×C412yes—
379C4×D4C2412yes—
380#117—22no—
381#117—22no—
382#117—21no—
383#117—22no—
384#117—22no—
385#117—21no—
386#117—22no—
387#117—22no—
388#117—22no—
389#117C22×C412yes—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4C22×C412yes—
396C2×C22⋊C4—22no—
397C2×C42—22no—
398C2×C42—22no—
399C2×C42C22×C411yescharacteristic
400C2×C42C22×C411yescharacteristic
401C2×C42C22×C411yescharacteristic
402C2×C42C2×D411yescharacteristic
403C2×C42C2×D411yescharacteristic
404C2×C42C2×D411yescharacteristic
405C2×C42#4111yescharacteristic
406C2×C42#4111yescharacteristic
407C2×C42#4111yescharacteristic
408C2×C42C2411yescharacteristic
409C2×C42C2411yescharacteristic
410C2×C42C2411yescharacteristic
411#107C22×C418yes—
412#106C22×C418yes—
413C4⋊C8—24no—
414C4⋊C8—24no—
415C4⋊C8—24no—
416C4⋊C8—24no—
417C4⋊C8—22no—
418C4⋊C8—22no—
419C4⋊C8—22no—
420C4⋊C8—22no—
421C4⋊C8—22no—
422C4⋊C8—22no—
423C4⋊C8—22no—
424C4⋊C8—21no—
425C4⋊C8—21no—
426C4⋊C8—22no—
427C4⋊C8—22no—
428C4⋊C8C22×C414yes—
429C4⋊C8C22×C414yes—
430C4⋊C8C2414yes—
431C4⋊C8C2412yes—
432C4⋊C8C2412yes—
433#103C22×C418yes—
434#103C22×C418yes—
435#103C22×C418yes—
436#102C22×C418yes—
437C22⋊C8—22no—
438C22⋊C8—22no—
439C22⋊C8—22no—
440C22⋊C8—22no—
441C22⋊C8—22no—
442C22⋊C8—22no—
443C22⋊C8—22no—
444C22⋊C8—22no—
445C22⋊C8C22×C414yes—
446C22⋊C8C22×C414yes—
447#97—24no—
448#97—24no—
449#97—22no—
450#97—22no—
451#97—22no—
452#97—22no—
453#97C22×C414yes—
454C4×C8—22no—
455C4×C8—24no—
456C4×C8—22no—
457C4×C8—24no—
458C4×C8—22no—
459C4×C8C2×D411yescharacteristic
460C4×C8C2×D411yescharacteristic
461C4×C8C2×D411yescharacteristic
462C4×C8C2×D411yescharacteristic
463C4×C8C2×D411yescharacteristic
464C4×C8C2×D411yescharacteristic
465C4×C8C2×D411yescharacteristic
466C4×C8C2×D411yescharacteristic
467C4×C8C2×D411yescharacteristic
468C4×C8C2×D411yescharacteristic
469C4×C8C2×D411yescharacteristic
470C4×C8C2×D411yescharacteristic
471C4×C8C2414yes—
472C4×C8C2414yes—
473C4×C8C2411yescharacteristic
474C4×C8C2411yescharacteristic
475C4×C8C2411yescharacteristic
476C4×C8C2411yescharacteristic
477C24C4×D411yescharacteristic
478#41—44no—
479#41—44no—
480#41—42no—
481#41—42no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—22no—
485C2×Q8—22no—
486C2×Q8—22no—
487C2×Q8—44no—
488C2×Q8—44no—
489C2×Q8C23×C412yes—
490C2×Q8C23×C414yes—
491C2×D4—22no—
492C2×D4—22no—
493C2×D4—22no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4C23×C412yes—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4C4×D411yescharacteristic
507C22×C4C4×D411yescharacteristic
508C22×C4C4×D411yescharacteristic
509C22×C4C4×D411yescharacteristic
510C22×C4C4×D411yescharacteristic
511C22×C4C4×D411yescharacteristic
512C22×C4C4×D411yescharacteristic
513C22×C4C4×D411yescharacteristic
514C22×C4C4×D411yescharacteristic
515C22×C4C23×C411yescharacteristic
516C22×C4C23×C411yescharacteristic
517C22×C4C23×C411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C2×#4111yescharacteristic
522C22×C4C2×#4111yescharacteristic
523C22×C4C2×#4111yescharacteristic
524Q16—48no—
525SD16—48no—
526M16—44no—
527M16—44no—
528M16—44no—
529M16—44no—
530M16—44no—
531M16—42no—
532M16—42no—
533M16—44no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—24no—
544C2×C8—24no—
545C2×C8—41no—
546C2×C8—41no—
547C2×C8—41no—
548C2×C8—41no—
549C2×C8—44no—
550C2×C8—44no—
551C2×C8—44no—
552C2×C8#13714yes—
553C2×C8C23×C418yes—
554C2×C8C22×D411yescharacteristic
555C2×C8C22×D411yescharacteristic
556C2×C8C22×D411yescharacteristic
557C2×C8C22×D411yescharacteristic
558C2×C8C22×D411yescharacteristic
559C2×C8C22×D411yescharacteristic
560C2×C8C22×D411yescharacteristic
561C2×C8C22×D411yescharacteristic
562C4⋊C4—22no—
563C4⋊C4—22no—
564C4⋊C4—22no—
565C4⋊C4—22no—
566C4⋊C4—22no—
567C4⋊C4—22no—
568C4⋊C4—24no—
569C4⋊C4—24no—
570C4⋊C4—24no—
571C4⋊C4—22no—
572C4⋊C4—21no—
573C4⋊C4—21no—
574C4⋊C4—21no—
575C4⋊C4—21no—
576C4⋊C4—22no—
577C4⋊C4—22no—
578C4⋊C4—22no—
579C4⋊C4—22no—
580C4⋊C4—22no—
581C4⋊C4—24no—
582C4⋊C4—22no—
583C4⋊C4C23×C414yes—
584C4⋊C4C23×C412yes—
585C4⋊C4C23×C412yes—
586C22⋊C4—24no—
587C22⋊C4—24no—
588C22⋊C4—22no—
589C22⋊C4—22no—
590C22⋊C4—22no—
591C22⋊C4—22no—
592C22⋊C4—22no—
593C22⋊C4—22no—
594C22⋊C4—22no—
595C22⋊C4—22no—
596C22⋊C4—22no—
597C42—22no—
598C42—24no—
599C42—22no—
600C42—22no—
601C42—24no—
602C42C4×D411yescharacteristic
603C42C4×D411yescharacteristic
604C42C4×D411yescharacteristic
605C42C4×D411yescharacteristic
606C42C4×D411yescharacteristic
607C42C4×D411yescharacteristic
608C42C23×C411yescharacteristic
609C42C23×C411yescharacteristic
610C42C23×C411yescharacteristic
611C42C22×D411yescharacteristic
612C42C22×D411yescharacteristic
613C42C22×D411yescharacteristic
614C42C2×#4111yescharacteristic
615C42C2×#4111yescharacteristic
616C42C2×#4111yescharacteristic
617C42C2511yescharacteristic
618C23—22no—
619C23—22no—
620C23—41no—
621C23—41no—
622C23C2×C4×D411yescharacteristic
623C23C2×C4×D411yescharacteristic
624C23C2×C4×D411yescharacteristic
625Q8—24no—
626Q8—24no—
627Q8—24no—
628Q8—44no—
629Q8—44no—
630Q8—42no—
631Q8—42no—
632Q8—44no—
633D4—24no—
634D4—42no—
635D4—44no—
636D4—44no—
637D4—44no—
638D4—42no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—24no—
656C2×C4—24no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4C2×C4×D411yescharacteristic
670C2×C4C2×C4×D411yescharacteristic
671C2×C4C2×C4×D411yescharacteristic
672C2×C4C2×C4×D411yescharacteristic
673C2×C4C2×C4×D411yescharacteristic
674C2×C4C2×C4×D411yescharacteristic
675C2×C4C2×C4×D411yescharacteristic
676C2×C4C2×C4×D411yescharacteristic
677C2×C4C2×C4×D411yescharacteristic
678C2×C4C2×#13712yes—
679C2×C4#57512yes—
680C2×C4C24×C411yescharacteristic
681C2×C4C23×D411yescharacteristic
682C2×C4C22×#4111yescharacteristic
683C8—28no—
684C8—21no—
685C8—21no—
686C8—21no—
687C8—21no—
688C8—21no—
689C8—21no—
690C8—44no—
691C8—42no—
692C8—42no—
693C8—44no—
694C8C2×#13712yes—
695C8C2×#13712yes—
696C22—21no—
697C22—21no—
698C22—21no—
699C22—21no—
700C22—21no—
701C22—21no—
702C22—21no—
703C22—21no—
704C22—44no—
705C22—42no—
706C22—42no—
707C22—42no—
708C22C4×#13712yes—
709C22C22×C4×D411yescharacteristic
710C4—21no—
711C4—21no—
712C4—21no—
713C4—21no—
714C4—21no—
715C4—21no—
716C4—21no—
717C4—21no—
718C4—42no—
719C4—44no—
720C4—42no—
721C4—44no—
722C4—42no—
723C4C4×#13712yes—
724C4#295512yes—
725C4#357312yes—
726C4C22×#13711yescharacteristic
727C4C2×#57511yescharacteristic
728C2—22no—
729C2—41no—
730C2—41no—
731C2—82no—
732C2#6014011yescharacteristic
733C2C2×C4×#13711yescharacteristic
734C2C2×#295511yescharacteristic
subgroupG/H
centerC2×C8C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radical#3444C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C22×C4×C8 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2960C2semidirect
4. #17984C2semidirect
5. #17965C2semidirect
6. #17941C2semidirect
7. #17936C2semidirect
8. #17934C2semidirect
9. #17913C2semidirect
10. #17903C2semidirect
11. #17901C2semidirect
12. #17820C2semidirect
14. #17491C2semidirect
15. #17323C2semidirect
64. #1593C22semidirect
65. #1592C22semidirect
66. #1587C22semidirect
67. #1585C22semidirect
68. #1584C22semidirect
69. #1583C22semidirect
70. C8×Q16C22semidirect
71. C8×SD16C22semidirect

Maximal subgroups

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

subgroupindex
#609522
#602042
C2×#29602
#179842
#179652
#179412
#179362
#179342
#179132
#179032
#179012
#178202
#174982
#174912
#173232