Finite Group Databasebeta

← #188974

Subgroups — #188974

Order 512.

973 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60233C211yesmaximal, characteristic
2#21948C211yesmaximal, characteristic, complemented
3#21939C211yesmaximal, characteristic, complemented
4#17582C211yesmaximal, characteristic
5#17559C211yesmaximal, characteristic
6#8629C212yesmaximal, complemented
7#8628C212yesmaximal, complemented
8#8627C212yesmaximal, complemented
9#8626C212yesmaximal, complemented
10#8108C211yesmaximal, characteristic, complemented
11#8104C211yesmaximal, characteristic, complemented
12C2×#525C2211yescharacteristic
13#2977—22no—
14#2976—21no—
15#2976—21no—
16#2974—22no—
17#2973—21no—
18#2972—21no—
19#2933C412yescomplemented
20#2933C412yescomplemented
21#2929C2212yes—
22#2929C2212yes—
23#2924C412yescomplemented
24#2880C412yescomplemented
25C2×#407C2211yescharacteristic
26C2×#407C2211yescharacteristic
27C2×#407C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29#2280C2211yescharacteristic
30#2280C2211yescharacteristic
31#1855—22no—
32#1855—22no—
33#1854—21no—
34#1854—21no—
35#1854—21no—
36#1854—21no—
37#1810C2212yescomplemented
38#1810C2212yescomplemented
39#1808C2211yescharacteristic, complemented
40#1808C2211yescharacteristic, complemented
41#1807C2211yescharacteristic, complemented
42#1807C2211yescharacteristic, complemented
43#1516—22no—
44#1513C2212yes—
45#1513C2212yes—
46#1513C2212yes—
47#1513C2212yes—
48#1511—21no—
49#1510—21no—
50#1508—21no—
51#1507—21no—
52#1501—22no—
53#1305C2212yescomplemented
54#1305C2212yescomplemented
55#1305C2212yescomplemented
56#1305C2212yescomplemented
57C22×#41C2×C411yescharacteristic
58C24×C4C2311yescharacteristic
59C2×#131C2×C412yes—
60C2×#131C2×C412yes—
61C22×M16—21no—
62C22×M16—21no—
63C22×M16C2×C411yescharacteristic
64C22×M16C2×C411yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67#525—21no—
68#525—21no—
69#525—21no—
70#525—21no—
71#525C2×C412yes—
72#525D412yes—
73#525Q812yes—
74#525C2312yes—
75C2×#123—21no—
76C2×C22⋊Q8D411yescharacteristic
77C2×C22⋊Q8Q811yescharacteristic
78C2×#121D411yescharacteristic
79C2×#121Q811yescharacteristic
80C2×#120—21no—
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×#117—21no—
84C2×#117—21no—
85C2×#117C2×C411yescharacteristic
86#436—22no—
87#436—22no—
88#435—22no—
89#435—22no—
90#435—22no—
91#435—22no—
92#435—22no—
93#435—22no—
94#435—22no—
95#435—22no—
96C8×D4—22no—
97C8×D4—22no—
98C8×D4—22no—
99C8×D4—22no—
100C8×D4—22no—
101C8×D4—22no—
102#433—22no—
103#433—22no—
104#432—22no—
105#431—22no—
106C2×#108—22no—
107C2×#108—22no—
108C2×#107—21no—
109C2×#107—21no—
110C2×#106—21no—
111C2×#106—21no—
112#424C2×C412yes—
113#424C2×C412yes—
114#408—22no—
115#408—22no—
116#408C2×C412yes—
117#408C2×C412yes—
118#407—21no—
119#407—21no—
120#407—22no—
121#407—21no—
122#407—21no—
123#407—21no—
124#407—21no—
125#407—21no—
126#407—21no—
127#407C2×C412yes—
128#407C2312yes—
129#407C2312yes—
130#407C2312yes—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—22no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8—21no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8C2×C412yes—
139C2×C22⋊C8C2×C412yes—
140C2×C22⋊C8D411yescharacteristic
141C2×C22⋊C8D411yescharacteristic
142C2×C22⋊C8D411yescharacteristic
143C2×C22⋊C8D411yescharacteristic
144C2×C22⋊C8D412yes—
145C2×C22⋊C8D412yes—
146#405C2×C412yes—
147#405C2×C412yes—
148#381—21no—
149#381—21no—
150#380—21no—
151#380—21no—
152#380C2311yescharacteristic
153#380C2311yescharacteristic
154#380C2311yescharacteristic
155#380C2311yescharacteristic
156#379—21no—
157#379—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—21no—
161C4×C22⋊C4—21no—
162#341—22no—
163#341—22no—
164#340—21no—
165#340—21no—
166#340—21no—
167#340—21no—
168#328—22no—
169#328—22no—
170#328—22no—
171#328—22no—
172#327—22no—
173#327—22no—
174#327—22no—
175#327—22no—
176C25#4111yescharacteristic
177C2×#41—21no—
178C2×#41—21no—
179C2×#41—81no—
180C2×#41—81no—
181C2×#41C4⋊C412yes—
182C2×#41C22×C412yes—
183C22×Q8C4⋊C411yescharacteristic
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4C4⋊C411yescharacteristic
188C23×C4—21no—
189C23×C4—81no—
190C23×C4—81no—
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×Q811yescharacteristic
200C23×C4C2×Q811yescharacteristic
201C23×C4#4111yescharacteristic
202C23×C4#4111yescharacteristic
203C23×C4#4111yescharacteristic
204C23×C4C2411yescharacteristic
205#131—82no—
206#131—82no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—22no—
210C2×M16—22no—
211C2×M16—22no—
212C2×M16—22no—
213C2×M16—41no—
214C2×M16—41no—
215C2×M16—81no—
216C2×M16—81no—
217C2×M16—81no—
218C2×M16—81no—
219C2×M16—81no—
220C2×M16—81no—
221C2×M16C22×C412yes—
222C2×M16C22×C412yes—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—41no—
228C22×C8—41no—
229C22×C8—81no—
230C22×C8—81no—
231C22×C8C22×C412yes—
232C22×C8C22×C412yes—
233C22×C8C2×D411yescharacteristic
234C22×C8C2×D411yescharacteristic
235C22×C8C2×D411yescharacteristic
236C22×C8C2×D411yescharacteristic
237C22×C8C2×D412yes—
238C22×C8C2×D412yes—
239#123—22no—
240#123—22no—
241#123—22no—
242#123—22no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245C22⋊Q8—21no—
246C22⋊Q8—21no—
247C22⋊Q8C4⋊C412yes—
248C22⋊Q8C4⋊C412yes—
249C22⋊Q8C2×D412yes—
250C22⋊Q8C2×Q812yes—
251#121—21no—
252#121—21no—
253#121—21no—
254#121—21no—
255#121C4⋊C412yes—
256#121C4⋊C412yes—
257#121C2×D412yes—
258#121C2×Q812yes—
259#120—22no—
260#120—22no—
261#120—22no—
262#120—22no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—22no—
271#117—21no—
272#117—21no—
273#117—81no—
274#117—81no—
275#117—81no—
276#117—81no—
277#117C4⋊C412yes—
278#117C22×C412yes—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4C4⋊C411yescharacteristic
286C2×C4⋊C4C4⋊C411yescharacteristic
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—41no—
297C2×C22⋊C4—41no—
298C2×C22⋊C4—41no—
299C2×C22⋊C4—41no—
300C2×C42—21no—
301C2×C42—41no—
302C2×C42—41no—
303#108—82no—
304#108—82no—
305#107—81no—
306#107—81no—
307#106—81no—
308#106—81no—
309C4⋊C8—22no—
310C4⋊C8—22no—
311C4⋊C8—22no—
312C4⋊C8—22no—
313C22⋊C8—22no—
314C22⋊C8—22no—
315C22⋊C8—22no—
316C22⋊C8—22no—
317C22⋊C8—22no—
318C22⋊C8—22no—
319C22⋊C8—22no—
320C22⋊C8—22no—
321C22⋊C8—22no—
322C22⋊C8—22no—
323C22⋊C8—22no—
324C22⋊C8—22no—
325C22⋊C8—41no—
326C22⋊C8—41no—
327C22⋊C8—41no—
328C22⋊C8—41no—
329C22⋊C8C22⋊C412yes—
330C22⋊C8C22⋊C412yes—
331C22⋊C8C22⋊C412yes—
332C22⋊C8C22⋊C412yes—
333C22⋊C8C22⋊C412yes—
334C22⋊C8C22⋊C412yes—
335C22⋊C8C22⋊C412yes—
336C22⋊C8C22⋊C412yes—
337#97—22no—
338#97—22no—
339C4×C8—22no—
340C4×C8—22no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—41no—
346#95—41no—
347C24—21no—
348C24—41no—
349C24—41no—
350C24—81no—
351C24—81no—
352C24#11711yescharacteristic
353C24#11711yescharacteristic
354C24#12111yescharacteristic
355C24#12111yescharacteristic
356C24C22⋊Q811yescharacteristic
357C24C22⋊Q811yescharacteristic
358C24C2×#4111yescharacteristic
359#41—42no—
360#41—42no—
361#41—81no—
362#41—81no—
363#41—82no—
364#41—82no—
365C2×Q8—21no—
366C2×Q8—81no—
367C2×Q8C2×C4⋊C412yes—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—21no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—81no—
378C2×D4—82no—
379C2×D4—82no—
380C2×D4C2×C4⋊C412yes—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—81no—
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C4⋊C411yescharacteristic
418C22×C4C2×C4⋊C411yescharacteristic
419C22×C4C2×C4⋊C411yescharacteristic
420C22×C4C2×C4⋊C411yescharacteristic
421C22×C4#11711yescharacteristic
422C22×C4#11711yescharacteristic
423C22×C4#12111yescharacteristic
424C22×C4#12111yescharacteristic
425C22×C4#12111yescharacteristic
426C22×C4#12111yescharacteristic
427C22×C4#12111yescharacteristic
428C22×C4#12111yescharacteristic
429C22×C4C22⋊Q811yescharacteristic
430C22×C4C22⋊Q811yescharacteristic
431C22×C4C22⋊Q811yescharacteristic
432C22×C4C22⋊Q811yescharacteristic
433C22×C4C22⋊Q811yescharacteristic
434C22×C4C22⋊Q811yescharacteristic
435C22×C4C23×C411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×Q811yescharacteristic
440C22×C4C2×#4111yescharacteristic
441M16—42no—
442M16—42no—
443M16—42no—
444M16—42no—
445M16—42no—
446M16—42no—
447M16—82no—
448M16—82no—
449M16—81no—
450M16—81no—
451M16—82no—
452M16—82no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—82no—
466C2×C8—82no—
467C2×C8—81no—
468C2×C8—81no—
469C2×C8—82no—
470C2×C8—82no—
471C2×C8C2×C22⋊C412yes—
472C2×C8C2×C22⋊C412yes—
473C2×C8C2×C22⋊C412yes—
474C2×C8C2×C22⋊C412yes—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4C2×C4⋊C412yes—
490C4⋊C4C2×C4⋊C412yes—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—81no—
506C22⋊C4—81no—
507C22⋊C4—81no—
508C22⋊C4—81no—
509C42—22no—
510C42—22no—
511C42—82no—
512C42—82no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—81no—
540C23—81no—
541C23—84no—
542C23—84no—
543C23#38011yescharacteristic
544C23#38011yescharacteristic
545C23#38011yescharacteristic
546C23#38011yescharacteristic
547C23C2×#11711yescharacteristic
548C23C2×#12111yescharacteristic
549C23C2×C22⋊Q811yescharacteristic
550Q8—42no—
551Q8—81no—
552D4—42no—
553D4—44no—
554D4—44no—
555D4—44no—
556D4—42no—
557D4—42no—
558D4—81no—
559D4—82no—
560D4—82no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—82no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4#38011yescharacteristic
618C2×C4#38011yescharacteristic
619C2×C4#38011yescharacteristic
620C2×C4#38011yescharacteristic
621C2×C4C22×C22⋊C411yescharacteristic
622C2×C4C22×C4⋊C411yescharacteristic
623C2×C4C2×#12111yescharacteristic
624C2×C4C2×C22⋊Q811yescharacteristic
625C8—22no—
626C8—22no—
627C8—22no—
628C8—22no—
629C8—42no—
630C8—42no—
631C8—82no—
632C8—82no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—162no—
667C22C2×#38011yescharacteristic
668C22#306711yescharacteristic
669C22#306711yescharacteristic
670C4—21no—
671C4—21no—
672C4—21no—
673C4—21no—
674C4—21no—
675C4—21no—
676C4—21no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—161no—
681C4—161no—
682C4—161no—
683C4—161no—
684C4#288311yescharacteristic
685C4#288311yescharacteristic
686C2—21no—
687C2—21no—
688C2—41no—
689C2—41no—
690C2—41no—
691C2—41no—
692C2—41no—
693C2—41no—
694C2—82no—
695C2#2063311yescharacteristic
696C2#2067511yescharacteristic
697C2#2067811yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21948C2semidirect
3. #21939C2semidirect
6. #8629C2semidirect
7. #8628C2semidirect
8. #8627C2semidirect
9. #8626C2semidirect
10. #8108C2semidirect
11. #8104C2semidirect
19. #2933C4semidirect
20. #2933C4semidirect
23. #2924C4semidirect
24. #2880C4semidirect
37. #1810C22semidirect
38. #1810C22semidirect
39. #1808C22semidirect
40. #1808C22semidirect
41. #1807C22semidirect
42. #1807C22semidirect
53. #1305C22semidirect
54. #1305C22semidirect
55. #1305C22semidirect
56. #1305C22semidirect

Maximal subgroups

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

subgroupindex
#602332
#219482
#219392
#175822
#175592
#86292
#86282
#86272
#86262
#81082
#81042