Finite Group Databasebeta

← #368673

Subgroups — #368673

Order 512.

1138 conjugacy classes of subgroups.

The subgroup lattice has 764 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21257C211yesmaximal, characteristic
2C2×#1980C211yesmaximal, characteristic
3#21188C211yesmaximal, characteristic
4C2×#1938C211yesmaximal, characteristic, complemented
5C2×#1937C211yesmaximal, characteristic, complemented
6C2×#1771C211yesmaximal, characteristic, complemented
7#20427C211yesmaximal, characteristic, complemented
8#11191C211yesmaximal, characteristic, complemented
9#11191C211yesmaximal, characteristic, complemented
10#11068C211yesmaximal, characteristic, complemented
11#11068C211yesmaximal, characteristic, complemented
12#10930C211yesmaximal, characteristic, complemented
13#10929C211yesmaximal, characteristic, complemented
14#10853C211yesmaximal, characteristic, complemented
15#10852C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C22×#104C2211yescharacteristic
24C22×#104C2211yescharacteristic
25C2×#416C2211yescharacteristic
26C2×#416C2211yescharacteristic
27C2×#416C2211yescharacteristic
28C2×#416C2211yescharacteristic
29C2×#408C2211yescharacteristic
30C2×#408C2211yescharacteristic
31C2×#408C2211yescharacteristic
32C2×#408C2211yescharacteristic
33C2×C4×M16C2211yescharacteristic
34C22×#97C2211yescharacteristic
35C22×#97C2211yescharacteristic
36C22×C4×C8C2211yescharacteristic
37C4×#117C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39#1983C412yescomplemented
40#1983C412yescomplemented
41#1982C412yescomplemented
42#1981C412yescomplemented
43#1980—24no—
44#1955—22no—
45#1955—22no—
46#1938—24no—
47#1937—24no—
48#1771—24no—
49#1768—22no—
50#1767—22no—
51C2×#341C2211yescharacteristic
52C2×#341C2211yescharacteristic
53C2×#341C2211yescharacteristic
54C2×#341C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59#1729C2211yescharacteristic
60#1729C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C22×#41C2311yescharacteristic
64C22×#41C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73#532—22no—
74#532—22no—
75#532—22no—
76#532—22no—
77C2×C4⋊Q8C2×C411yescharacteristic
78C2×#127C2×C411yescharacteristic
79C2×#124C2×C411yescharacteristic
80C2×#124C2×C411yescharacteristic
81C2×#121—21no—
82C2×#121—21no—
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×#117C2311yescharacteristic
86C2×#117C2311yescharacteristic
87C22×C4⋊C4—21no—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91#424—22no—
92#424—22no—
93#424—22no—
94#424D414yescomplemented
95#423—22no—
96#423—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8C2×C412yes—
104C2×C4⋊C8C2×C412yes—
105C2×#104—22no—
106C2×#104—22no—
107C2×#104—22no—
108C2×#104—22no—
109C2×#104—22no—
110C2×#104—22no—
111C2×#104—22no—
112C2×#104—22no—
113C2×#104—41no—
114C2×#104—41no—
115C2×#104—41no—
116C2×#104—41no—
117#416—22no—
118#416—22no—
119#416—22no—
120#416—22no—
121#416—41no—
122#416—41no—
123#416—41no—
124#416—41no—
125C2×#103C2×C412yes—
126C2×#103C2×C412yes—
127C2×#103C2×C412yes—
128C2×#103C2×C412yes—
129C2×#102C2×C412yes—
130C2×#102C2×C412yes—
131C2×#102C2×C412yes—
132C2×#102C2×C412yes—
133#408—22no—
134#408—22no—
135#408—22no—
136#408—22no—
137#408—41no—
138#408—41no—
139#408—41no—
140#408—41no—
141C2×C22⋊C8—22no—
142C2×C22⋊C8—22no—
143C2×C22⋊C8—22no—
144C2×C22⋊C8—22no—
145C4×M16—22no—
146C4×M16—22no—
147C2×#97—22no—
148C2×#97—22no—
149C2×#97—22no—
150C2×#97—22no—
151C2×#97—21no—
152C2×#97—41no—
153C2×#97D412yes—
154C2×#97D412yes—
155C2×C4×C8—22no—
156C2×C4×C8—22no—
157C2×C4×C8—21no—
158C2×C4×C8—41no—
159C4×C4⋊C4—22no—
160C4×C4⋊C4C2×C412yes—
161C4×C4⋊C4C2×C412yes—
162C4×C4⋊C4D414yes—
163C4×C22⋊C4—21no—
164C4×C22⋊C4—21no—
165#376—21no—
166C2×#95—21no—
167C43—21no—
168#341—22no—
169#341—22no—
170#341—41no—
171#341—41no—
172#339—22no—
173#339—22no—
174#339—41no—
175#339—41no—
176#336—22no—
177#336—22no—
178#336—22no—
179#336—22no—
180C2×#41—21no—
181C2×#41—21no—
182C2×#41—21no—
183C2×#41—21no—
184C2×#41—21no—
185C2×#41—21no—
186C2×#41—21no—
187C2×#41—21no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41—21no—
196C2×#41—21no—
197C2×#41—21no—
198C2×#41—21no—
199C2×#41—21no—
200C2×#41—21no—
201C2×#41—21no—
202C2×#41—21no—
203C2×#41—21no—
204C22×Q8C22×C411yescharacteristic
205C22×Q8C22×C411yescharacteristic
206C22×D4—21no—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4C22×C411yescharacteristic
211C22×D4C22×C411yescharacteristic
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4#4111yescharacteristic
217C23×C4#4111yescharacteristic
218C23×C4C2411yescharacteristic
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—21no—
235C2×M16—21no—
236C2×M16—21no—
237C2×M16—21no—
238C2×M16—21no—
239C2×M16—21no—
240C2×M16—21no—
241C2×M16C2×D412yes—
242C2×M16#4112yes—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—21no—
261C22×C8—21no—
262C22×C8—21no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—21no—
267C22×C8—21no—
268C22×C8—21no—
269C22×C8C22×C412yes—
270C22×C8C22×C412yes—
271C22×C8C2×D412yes—
272C22×C8C2×D412yes—
273C22×C8C2×D412yes—
274C22×C8#4112yes—
275C4⋊Q8—22no—
276C4⋊Q8—22no—
277#127—22no—
278#127—22no—
279#124—22no—
280#124—22no—
281#124—22no—
282#124—22no—
283#121—42no—
284#121—42no—
285#121—42no—
286#121—42no—
287C4×D4—42no—
288C4×D4—42no—
289C4×D4—42no—
290C4×D4—42no—
291#117—21no—
292#117—21no—
293#117—21no—
294#117—21no—
295#117—21no—
296#117—21no—
297#117C2×D412yes—
298#117#4112yes—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4C22×C412yes—
304C2×C4⋊C4C22×C412yes—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42C22×C411yescharacteristic
326C2×C42C22×C411yescharacteristic
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42#4112yes—
333C2×C42#4112yes—
334C2×C42#4112yes—
335C4⋊C8—28no—
336C4⋊C8—28no—
337C4⋊C8—24no—
338C4⋊C8—24no—
339C4⋊C8—42no—
340C4⋊C8—42no—
341#104—44no—
342#104—44no—
343#104—44no—
344#104—44no—
345#103—24no—
346#103—24no—
347#103—42no—
348#103—42no—
349#102—24no—
350#102—24no—
351#102—42no—
352#102—42no—
353C22⋊C8—44no—
354C22⋊C8—44no—
355C22⋊C8—44no—
356C22⋊C8—44no—
357#97—24no—
358#97—24no—
359#97—44no—
360#97—42no—
361C4×C8—44no—
362C4×C8—42no—
363#95—22no—
364#95—22no—
365#95—24no—
366C24—21no—
367C24—21no—
368C24C2×#4111yescharacteristic
369#41—41no—
370#41—41no—
371#41—41no—
372#41—41no—
373#41—41no—
374#41—41no—
375#41—41no—
376#41—41no—
377#41—41no—
378#41—41no—
379#41—41no—
380#41—41no—
381#41—41no—
382#41—41no—
383#41—41no—
384#41—41no—
385#41—41no—
386#41—41no—
387#41—41no—
388#41—41no—
389#41—41no—
390#41—41no—
391#41—41no—
392#41—41no—
393#41—41no—
394#41—41no—
395#41—41no—
396#41—41no—
397#41—41no—
398#41—41no—
399#41—41no—
400#41—41no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—41no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4—41no—
431C2×D4—41no—
432C2×D4—41no—
433C2×D4—41no—
434C2×D4—41no—
435C2×D4—41no—
436C2×D4—41no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—82no—
440C2×D4—82no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4C4×D411yescharacteristic
470C22×C4C4×D411yescharacteristic
471C22×C4C4×D411yescharacteristic
472C22×C4C4×D411yescharacteristic
473C22×C4#12312yes—
474C22×C4#12312yes—
475C22×C4#12612yes—
476C22×C4#12612yes—
477C22×C4C23×C411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C2×#4111yescharacteristic
482C22×C4C2×#4111yescharacteristic
483C22×C4C2×#4111yescharacteristic
484C22×C4C2×#4111yescharacteristic
485C22×C4C2×#4111yescharacteristic
486C22×C4C2×#4111yescharacteristic
487C22×C4H(5,𝔽2)11yescharacteristic
488C22×C4H(5,𝔽2)11yescharacteristic
489C22×C4H(5,𝔽2)11yescharacteristic
490C22×C42−1+411yescharacteristic
491M16—22no—
492M16—22no—
493M16—22no—
494M16—22no—
495M16—41no—
496M16—41no—
497M16—41no—
498M16—41no—
499M16—41no—
500M16—41no—
501M16—41no—
502M16—41no—
503M16—41no—
504M16—41no—
505M16—41no—
506M16—41no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—41no—
525C2×C8—41no—
526C2×C8—41no—
527C2×C8—41no—
528C2×C8—41no—
529C2×C8—41no—
530C2×C8—41no—
531C2×C8—41no—
532C2×C8—41no—
533C2×C8—41no—
534C2×C8—41no—
535C2×C8—41no—
536C2×C8—41no—
537C2×C8—41no—
538C2×C8—41no—
539C2×C8—41no—
540C2×C8—41no—
541C2×C8—41no—
542C2×C8—41no—
543C2×C8—41no—
544C2×C8—41no—
545C2×C8—41no—
546C2×C8#12114yes—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—48no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4C4×D414yes—
554C22⋊C4—22no—
555C22⋊C4—22no—
556C22⋊C4—41no—
557C22⋊C4—41no—
558C22⋊C4—42no—
559C22⋊C4—42no—
560C22⋊C4—42no—
561C22⋊C4—42no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—41no—
569C42—41no—
570C42—41no—
571C42—41no—
572C42—41no—
573C42—41no—
574C42—82no—
575C42—82no—
576C42#11712yes—
577C42#11712yes—
578C42C4×D412yes—
579C42C4×D412yes—
580C42#12112yes—
581C42#12112yes—
582C42#12412yes—
583C42#12412yes—
584C23—22no—
585C23—22no—
586C23—41no—
587C23—41no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23C2×C4×D411yescharacteristic
597C23C2×#12311yescharacteristic
598C23C2×#12611yescharacteristic
599C23#52911yescharacteristic
600C23#53211yescharacteristic
601C23#54011yescharacteristic
602C23#54211yescharacteristic
603Q8—41no—
604Q8—41no—
605Q8—41no—
606Q8—41no—
607Q8—41no—
608Q8—41no—
609Q8—41no—
610Q8—41no—
611D4—41no—
612D4—41no—
613D4—41no—
614D4—41no—
615D4—41no—
616D4—41no—
617D4—41no—
618D4—41no—
619D4—84no—
620D4—84no—
621D4—81no—
622D4—81no—
623D4—81no—
624D4—81no—
625D4—81no—
626D4—81no—
627D4—81no—
628D4—81no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—82no—
658C2×C4—82no—
659C2×C4—81no—
660C2×C4—81no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4—81no—
664C2×C4—81no—
665C2×C4—81no—
666C2×C4—81no—
667C2×C4#38812yes—
668C2×C4#38812yes—
669C2×C4#38812yes—
670C2×C4#38812yes—
671C2×C4C2×#11711yescharacteristic
672C2×C4C2×C4×D411yescharacteristic
673C2×C4C2×C4×D411yescharacteristic
674C2×C4C4×#4111yescharacteristic
675C2×C4#51811yescharacteristic
676C2×C4#52011yescharacteristic
677C2×C4C2×#12111yescharacteristic
678C2×C4C2×#12111yescharacteristic
679C2×C4#52511yescharacteristic
680C2×C4C2×#12411yescharacteristic
681C2×C4#53511yescharacteristic
682C2×C4#53711yescharacteristic
683C2×C4D4211yescharacteristic
684C2×C4#54611yescharacteristic
685C2×C4#54711yescharacteristic
686C2×C4#54811yescharacteristic
687C2×C4#55011yescharacteristic
688C2×C4#55011yescharacteristic
689C2×C4#55011yescharacteristic
690C2×C4#55311yescharacteristic
691C2×C4#55311yescharacteristic
692C2×C4#55511yescharacteristic
693C2×C4#55611yescharacteristic
694C2×C4#55611yescharacteristic
695C2×C4#55911yescharacteristic
696C2×C4#55911yescharacteristic
697C2×C4#55911yescharacteristic
698C2×C4#55911yescharacteristic
699C8—24no—
700C8—42no—
701C8—42no—
702C8—42no—
703C8—42no—
704C8—42no—
705C8—81no—
706C8—81no—
707C22—22no—
708C22—22no—
709C22—21no—
710C22—21no—
711C22—21no—
712C22—21no—
713C22—22no—
714C22—22no—
715C22—82no—
716C22—82no—
717C22—81no—
718C22—81no—
719C22—81no—
720C22—81no—
721C22—82no—
722C22—82no—
723C22—81no—
724C22—81no—
725C22C2×#38811yescharacteristic
726C22C4×#12111yescharacteristic
727C22#232411yescharacteristic
728C22#244511yescharacteristic
729C22#244811yescharacteristic
730C22#251211yescharacteristic
731C22#251311yescharacteristic
732C22#295612yes—
733C22#295712yes—
734C4—22no—
735C4—22no—
736C4—22no—
737C4—22no—
738C4—44no—
739C4—44no—
740C4—82no—
741C4—82no—
742C4—81no—
743C4—81no—
744C4#235411yescharacteristic
745C4#235711yescharacteristic
746C4#236611yescharacteristic
747C4#236711yescharacteristic
748C4#241411yescharacteristic
749C4#241611yescharacteristic
750C4#254811yescharacteristic
751C4#259211yescharacteristic
752C2—21no—
753C2—21no—
754C2—21no—
755C2—21no—
756C2—161no—
757C2—161no—
758C2#1402211yescharacteristic
759C2#2109811yescharacteristic
760C2#2109911yescharacteristic
761C2#2151912yes—
762C2#2216712yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×#97 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1938C2semidirect
5. C2×#1937C2semidirect
6. C2×#1771C2semidirect
7. #20427C2semidirect
8. #11191C2semidirect
9. #11191C2semidirect
10. #11068C2semidirect
11. #11068C2semidirect
12. #10930C2semidirect
13. #10929C2semidirect
14. #10853C2semidirect
15. #10852C2semidirect
39. #1983C4semidirect
40. #1983C4semidirect
41. #1982C4semidirect
42. #1981C4semidirect
94. #424D4semidirect

Maximal subgroups

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

subgroupindex
#212572
C2×#19802
#211882
C2×#19382
C2×#19372
C2×#17712
#204272
#111912
#110682
#109302
#109292
#108532
#108522