Finite Group Databasebeta

← #109648

Subgroups — #109648

Order 512.

1042 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1853C211yesmaximal, characteristic
2C22⋊C4×C4⋊C4C211yesmaximal, characteristic
3#10854C211yesmaximal, characteristic
4#10730C211yesmaximal, characteristic
5#8023C211yesmaximal, characteristic
6C2×#1289C211yesmaximal, characteristic
7C2×#1282C211yesmaximal, characteristic
8C23×C4⋊C4C2211yescharacteristic
9C22×#106C412yescomplemented
10C22×C22⋊C8C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C2×#380C2211yescharacteristic
13C2×#380C2211yescharacteristic
14C2×#380C2211yescharacteristic
15C2×C4×C4⋊C4—21no—
16C2×C4×C22⋊C4—21no—
17C2×C4×C22⋊C4C2211yescharacteristic
18#1853—21no—
19#1853—22no—
20#1853—21no—
21C2×#342—21no—
22C2×#342—21no—
23C2×#340—22no—
24C2×#340C412yescomplemented
25#1436—22no—
26C4⋊#95—21no—
27C22⋊#95—21no—
28#1294—22no—
29#1289C414yescomplemented
30#1282C414yescomplemented
31C24×C4—21no—
32C24×C4C2311yescharacteristic
33C23×C8—21no—
34C23×C8C2×C412yes—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—44no—
41C22×C4⋊C4C2×C412yes—
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C4⋊C4D412yes—
46C22×C4⋊C4D411yescharacteristic
47C22×C22⋊C4—21no—
48C22×C22⋊C4—21no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4D411yescharacteristic
51C22×C22⋊C4D411yescharacteristic
52C22×C42—21no—
53C22×C42—22no—
54C2×#106—22no—
55C2×#106—22no—
56C2×#106—44no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8C2×C412yes—
67C2×C22⋊C8C2×C412yes—
68C2×C22⋊C8C2×C412yes—
69#380—21no—
70#380—21no—
71#380—21no—
72#380—21no—
73#380—21no—
74#380—21no—
75#380—21no—
76#380—21no—
77#380—21no—
78#380C2×C412yes—
79#380C2×C412yes—
80#380Q812yes—
81C4×C4⋊C4—44no—
82C4×C22⋊C4—21no—
83C4×C22⋊C4—21no—
84C4×C22⋊C4—42no—
85C4×C22⋊C4—42no—
86C4×C22⋊C4C2×C412yes—
87C4×C22⋊C4Q812yes—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95D412yes—
95#342—42no—
96#342—42no—
97#340—22no—
98#340—22no—
99#340—44no—
100C25C2×D411yescharacteristic
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—42no—
111C23×C4—42no—
112C23×C4—44no—
113C23×C4—41no—
114C23×C4—41no—
115C23×C4—41no—
116C23×C4—41no—
117C23×C4—41no—
118C23×C4—82no—
119C23×C4C22×C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C22×C411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×Q811yescharacteristic
126C23×C4C2×Q811yescharacteristic
127C23×C4#4111yescharacteristic
128C23×C4#4111yescharacteristic
129C23×C4#4111yescharacteristic
130C23×C4#4111yescharacteristic
131C23×C4#4111yescharacteristic
132C23×C4#4111yescharacteristic
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—42no—
137C22×C8—42no—
138C22×C8—42no—
139C22×C8—42no—
140C22×C8—41no—
141C22×C8—41no—
142C22×C8C4214yescomplemented
143C2×C4⋊C4—21no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22⋊C412yes—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4C4⋊C412yes—
199C2×C22⋊C4C4⋊C412yes—
200C2×C22⋊C4SD1612yes—
201C2×C22⋊C4SD1612yes—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—44no—
207C2×C42—44no—
208C2×C42—82no—
209#106—44no—
210#106—44no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—41no—
218C22⋊C8—41no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—44no—
224#95—44no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—82no—
229#95C4⋊C414yes—
230C24—41no—
231C24—41no—
232C24—41no—
233C24—41no—
234C24C2×C4⋊C411yescharacteristic
235C24C4×D411yescharacteristic
236C24C4×D411yescharacteristic
237C24#12011yescharacteristic
238C24#12111yescharacteristic
239C24C22⋊Q811yescharacteristic
240C24#12311yescharacteristic
241C24C2×SD1611yescharacteristic
242C24C2×SD1611yescharacteristic
243C24#13511yescharacteristic
244C24#13511yescharacteristic
245C24AGL(1,ℤ/8)11yescharacteristic
246C24AGL(1,ℤ/8)11yescharacteristic
247C24#13711yescharacteristic
248C24#13711yescharacteristic
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—84no—
316C22×C4—84no—
317C22×C4—84no—
318C22×C4—84no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4—81no—
322C22×C4—81no—
323C22×C4—81no—
324C22×C4—81no—
325C22×C4—81no—
326C22×C4—81no—
327C22×C4—81no—
328C22×C4—81no—
329C22×C4C2×C4211yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C4⋊C411yescharacteristic
332C22×C4#11711yescharacteristic
333C22×C4#11711yescharacteristic
334C22×C4#11711yescharacteristic
335C22×C4C4×D411yescharacteristic
336C22×C4C4×D411yescharacteristic
337C22×C4C4×D411yescharacteristic
338C22×C4C4×D411yescharacteristic
339C22×C4C4×D411yescharacteristic
340C22×C4C4×D411yescharacteristic
341C22×C4C4×Q811yescharacteristic
342C22×C4C4×Q811yescharacteristic
343C22×C4C4×Q811yescharacteristic
344C22×C4C4×Q811yescharacteristic
345C22×C4#12111yescharacteristic
346C22×C4C22⋊Q811yescharacteristic
347C22×C4C22⋊Q811yescharacteristic
348C22×C4C22⋊Q811yescharacteristic
349C22×C4#12311yescharacteristic
350C22×C4#12311yescharacteristic
351C22×C4#12411yescharacteristic
352C22×C4#12511yescharacteristic
353C22×C4#12511yescharacteristic
354C22×C4#12611yescharacteristic
355C22×C4#12611yescharacteristic
356C22×C4C4⋊Q811yescharacteristic
357C2×C8—24no—
358C2×C8—44no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—44no—
362C2×C8—44no—
363C2×C8—42no—
364C2×C8—81no—
365C2×C8—81no—
366C2×C8—82no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—81no—
380C4⋊C4—81no—
381C4⋊C4—82no—
382C4⋊C4#9914yes—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—81no—
397C22⋊C4—81no—
398C22⋊C4—81no—
399C22⋊C4—81no—
400C22⋊C4—81no—
401C22⋊C4—81no—
402C22⋊C4#10614yescomplemented
403C42—44no—
404C42—44no—
405C42—88no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23C4×C4⋊C411yescharacteristic
443C23#38511yescharacteristic
444C23#38511yescharacteristic
445C23#38611yescharacteristic
446C23#38711yescharacteristic
447C23#38811yescharacteristic
448C23#38911yescharacteristic
449C23C2×#10611yescharacteristic
450C23#42711yescharacteristic
451C23#42811yescharacteristic
452C23#42811yescharacteristic
453C23C4×SD1611yescharacteristic
454C23C4×SD1611yescharacteristic
455C23C4×SD1611yescharacteristic
456C23C4×SD1611yescharacteristic
457C23#44011yescharacteristic
458C23#44011yescharacteristic
459C23#44111yescharacteristic
460C23#44211yescharacteristic
461C23#44811yescharacteristic
462C23#44911yescharacteristic
463C23#45011yescharacteristic
464C23#45211yescharacteristic
465C23#46011yescharacteristic
466C23#46111yescharacteristic
467C23#46311yescharacteristic
468C23#46411yescharacteristic
469C23#47511yescharacteristic
470C23#47611yescharacteristic
471C23#47811yescharacteristic
472C23#47911yescharacteristic
473C23#48111yescharacteristic
474C23#48211yescharacteristic
475C23#48311yescharacteristic
476C23#48511yescharacteristic
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4—84no—
531C2×C4—84no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—82no—
538C2×C4—81no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—82no—
551C2×C4—161no—
552C2×C4—161no—
553C2×C4—161no—
554C2×C4—161no—
555C2×C4—161no—
556C2×C4—161no—
557C2×C4—161no—
558C2×C4—161no—
559C2×C4C4×C22⋊C411yescharacteristic
560C2×C4C4×C4⋊C411yescharacteristic
561C2×C4#38711yescharacteristic
562C2×C4#38711yescharacteristic
563C2×C4#38711yescharacteristic
564C2×C4#38811yescharacteristic
565C2×C4#38911yescharacteristic
566C2×C4#39111yescharacteristic
567C2×C4C2×#9912yes—
568C2×C4#41012yes—
569C8—44no—
570C8—82no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22#143611yescharacteristic
624C22C4×#10611yescharacteristic
625C22#177611yescharacteristic
626C22#185311yescharacteristic
627C22#185411yescharacteristic
628C22#186411yescharacteristic
629C22#186511yescharacteristic
630C22#187711yescharacteristic
631C22#188011yescharacteristic
632C22#189811yescharacteristic
633C22#189911yescharacteristic
634C22#190411yescharacteristic
635C22#190511yescharacteristic
636C22#191911yescharacteristic
637C22#192011yescharacteristic
638C22#192611yescharacteristic
639C22#192711yescharacteristic
640C22#194811yescharacteristic
641C22#194911yescharacteristic
642C4—21no—
643C4—21no—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—44no—
649C4—44no—
650C4—84no—
651C4—82no—
652C4—82no—
653C4—162no—
654C4—161no—
655C4—161no—
656C4C4×#9912yes—
657C4#199612yes—
658C2—21no—
659C2—21no—
660C2—21no—
661C2—21no—
662C2—21no—
663C2—21no—
664C2—21no—
665C2—21no—
666C2—21no—
667C2—21no—
668C2—21no—
669C2—21no—
670C2#925511yescharacteristic
671C2#925611yescharacteristic
672C2#926111yescharacteristic
673C2#926211yescharacteristic
674C2#1090311yescharacteristic
675C2#1090811yescharacteristic
676C2#1091211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
9. C22×#106C4semidirect
24. C2×#340C4semidirect
29. #1289C4semidirect
30. #1282C4semidirect
142. C22×C8C42semidirect
402. C22⋊C4#106semidirect

Maximal subgroups

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

subgroupindex
C2×#18532
C22⋊C4×C4⋊C42
#108542
#107302
#80232
C2×#12892
C2×#12822