Finite Group Databasebeta

← #377758

Subgroups — #377758

Order 512.

1323 conjugacy classes of subgroups.

The subgroup lattice has 773 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23479C211yesmaximal, characteristic
2#21191C211yesmaximal, characteristic
3C2×#1938C211yesmaximal, characteristic, complemented
4#21053C211yesmaximal, characteristic, complemented
5C2×#1896C211yesmaximal, characteristic
6C2×#1773C211yesmaximal, characteristic, complemented
7C2×#1749C211yesmaximal, characteristic, complemented
8#11221C211yesmaximal, characteristic, complemented
9#11125C211yesmaximal, characteristic, complemented
10#11097C211yesmaximal, characteristic, complemented
11#10934C211yesmaximal, characteristic, complemented
12#10930C211yesmaximal, characteristic, complemented
13#10887C211yesmaximal, characteristic, complemented
14#10883C211yesmaximal, characteristic, complemented
15#10719C211yesmaximal, characteristic, complemented
16C2×#525C2211yescharacteristic
17C23×C42C2211yescharacteristic
18C22×#108C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C2×#424C2211yescharacteristic
21C2×#424C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C22×#104C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#416C2211yescharacteristic
26C2×#408C2211yescharacteristic
27C2×#408C2211yescharacteristic
28C2×#407C2211yescharacteristic
29C2×#407C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C2×#405C2211yescharacteristic
33C2×#405C2211yescharacteristic
34C22×#97C2211yescharacteristic
35C22×#97C2211yescharacteristic
36#2280C2211yescharacteristic
37#2036C412yescomplemented
38#2035C412yescomplemented
39#2034C412yescomplemented
40#2033C412yescomplemented
41#1955—22no—
42#1954—22no—
43#1938—24no—
44#1924—22no—
45#1923—22no—
46#1896—24no—
47#1773—24no—
48#1749—24no—
49C2×#341C2211yescharacteristic
50C2×#341C2211yescharacteristic
51C2×#341C2211yescharacteristic
52#1735C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59#1729C2211yescharacteristic
60#1729C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C22×#41C2311yescharacteristic
64C24×C4—21no—
65C24×C4C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74#525—22no—
75#525—22no—
76#525—41no—
77#525—41no—
78C2×#123—21no—
79C2×C22⋊Q8C2×C412yes—
80C2×#121C2×C412yes—
81C2×#120—21no—
82C2×C4×D4—22no—
83C2×#117C2311yescharacteristic
84C2×#117C2311yescharacteristic
85C2×#117C2311yescharacteristic
86C22×C42—21no—
87C22×C42—22no—
88C22×C42—22no—
89C22×C42—22no—
90C22×C42—22no—
91C22×C42—41no—
92C22×C42—41no—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C2×#108—22no—
96C2×#108—22no—
97C2×#108—22no—
98#427—22no—
99#427—22no—
100C2×#107C2×C412yescomplemented
101C2×#106C2×C412yescomplemented
102#424—22no—
103#424—22no—
104#424—22no—
105#424—41no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—22no—
109C2×#104—22no—
110C2×#104—22no—
111C2×#104—22no—
112C2×#104—22no—
113C2×#104—22no—
114C2×#104—41no—
115#416—22no—
116#416—22no—
117#416—41no—
118#416—41no—
119C2×#103C2×C411yescharacteristic, complemented
120C2×#103C2×C411yescharacteristic, complemented
121C2×#103C2×C412yescomplemented
122C2×#102C2×C411yescharacteristic, complemented
123C2×#102C2×C411yescharacteristic, complemented
124C2×#102C2×C412yescomplemented
125#408—22no—
126#408—22no—
127#408—22no—
128#408—41no—
129#407—22no—
130#407—22no—
131#407—41no—
132#407—41no—
133C2×C22⋊C8—22no—
134C2×C22⋊C8—22no—
135C2×C22⋊C8—22no—
136C2×C22⋊C8—22no—
137C2×C22⋊C8—22no—
138C2×C22⋊C8—22no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8—22no—
141C2×C22⋊C8—21no—
142C2×C22⋊C8C2×C412yes—
143C2×C22⋊C8C2×C412yes—
144C2×C22⋊C8D412yes—
145C2×C22⋊C8D412yes—
146#405—22no—
147#405—22no—
148#405—41no—
149#405—41no—
150C2×#97—22no—
151C2×#97—22no—
152C2×#97—21no—
153C2×#97—21no—
154C2×#97—22no—
155C2×#97—41no—
156C2×#97—41no—
157C2×#97D414yescomplemented
158C2×C4×C8—21no—
159C2×C4×C8—21no—
160#381—21no—
161#380C2×C412yes—
162#380C2×C412yes—
163#379—21no—
164C4×C22⋊C4—22no—
165#341—22no—
166#341—41no—
167#341—41no—
168#340C2×C412yescomplemented
169#340C2×C412yescomplemented
170#339—22no—
171#339—22no—
172#339—22no—
173#339—22no—
174#339—41no—
175#339D414yes—
176#336—22no—
177#336—22no—
178#336—22no—
179#336—22no—
180C25#4111yescharacteristic
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—
193C22×Q8C22×C411yescharacteristic
194C22×D4—21no—
195C22×D4—22no—
196C22×D4C22×C411yescharacteristic
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—24no—
212C23×C4—21no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4C22×C412yes—
218C23×C4C2×D412yes—
219C23×C4#4111yescharacteristic
220C23×C4C2411yescharacteristic
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×M16—21no—
242C2×M16—21no—
243C2×M16C2×D412yes—
244C2×M16#4112yes—
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×C411yescharacteristic
271C22×C8C22×C411yescharacteristic
272C22×C8C2×D412yes—
273C22×C8C2×D412yes—
274C22×C8C2×D412yes—
275C22×C8C2×D411yescharacteristic
276C22×C8C2×D411yescharacteristic
277C22×C8#4112yes—
278C22×C8#4112yes—
279#123—42no—
280#123—42no—
281C22⋊Q8—24no—
282C22⋊Q8—42no—
283#121—24no—
284#121—42no—
285#120—42no—
286#120—42no—
287C4×D4—44no—
288C4×D4—44no—
289#117—21no—
290#117—21no—
291#117—21no—
292#117—21no—
293#117—21no—
294#117—21no—
295#117—21no—
296#117—21no—
297#117—21no—
298#117—21no—
299#117C2×D412yes—
300#117#4112yes—
301C2×C4⋊C4—22no—
302C2×C4⋊C4C22×C411yescharacteristic
303C2×C4⋊C4C22×C411yescharacteristic
304C2×C4⋊C4C22×C412yes—
305C2×C4⋊C4C22×C412yes—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
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—24no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—44no—
326C2×C42—44no—
327C2×C42—44no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42—44no—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42#4112yes—
334C2×C42#4112yes—
335#108—28no—
336#107—24no—
337#106—24no—
338C4⋊C8—44no—
339C4⋊C8—42no—
340C4⋊C8—42no—
341#104—44no—
342#104—44no—
343#103—22no—
344#103—22no—
345#103—42no—
346#102—22no—
347#102—22no—
348#102—42no—
349C22⋊C8—24no—
350C22⋊C8—24no—
351C22⋊C8—24no—
352C22⋊C8—24no—
353C22⋊C8—44no—
354C22⋊C8—44no—
355C22⋊C8—44no—
356C22⋊C8—42no—
357C22⋊C8—42no—
358#97—28no—
359#97—42no—
360#97—44no—
361#97—42no—
362C4×C8—42no—
363C4×C8—42no—
364#95—22no—
365#95—22no—
366C24—21no—
367C24—21no—
368C24—24no—
369C24—21no—
370C24—21no—
371C24—21no—
372C24—21no—
373C24—21no—
374C24—42no—
375C24C4×D412yes—
376C24C2×#4111yescharacteristic
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—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—44no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—44no—
413C2×D4—82no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—48no—
448C22×C4—416no—
449C22×C4—416no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
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×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4C4×D412yes—
475C22×C4C23×C411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C2×#4111yescharacteristic
480C22×C4C2×#4111yescharacteristic
481C22×C4C2×#4111yescharacteristic
482C22×C4C2×#4111yescharacteristic
483C22×C4C2×#4111yescharacteristic
484C22×C4C2×#4111yescharacteristic
485C22×C4H(5,𝔽2)11yescharacteristic
486C22×C4H(5,𝔽2)11yescharacteristic
487C22×C4H(5,𝔽2)11yescharacteristic
488C22×C42−1+411yescharacteristic
489M16—22no—
490M16—22no—
491M16—22no—
492M16—22no—
493M16—41no—
494M16—41no—
495M16—41no—
496M16—41no—
497M16—41no—
498M16—41no—
499M16—41no—
500M16—41no—
501M16—41no—
502M16—41no—
503M16—41no—
504M16—41no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
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—41no—
547C2×C8C4×D412yes—
548C2×C8C4×D412yes—
549C2×C8#12112yes—
550C2×C8#12112yes—
551C4⋊C4—22no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—22no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—22no—
558C4⋊C4—44no—
559C4⋊C4C4×D414yes—
560C22⋊C4—22no—
561C22⋊C4—22no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—41no—
565C22⋊C4—41no—
566C22⋊C4—41no—
567C22⋊C4—41no—
568C22⋊C4—44no—
569C22⋊C4—82no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—48no—
578C42—41no—
579C42—41no—
580C42—41no—
581C42—41no—
582C42—41no—
583C42—41no—
584C42#12114yes—
585C23—22no—
586C23—22no—
587C23—24no—
588C23—24no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—42no—
600C23—42no—
601C23—41no—
602C23—41no—
603C23—41no—
604C23—41no—
605C23—41no—
606C23—41no—
607C23—42no—
608C23—42no—
609C23—41no—
610C23#44414yes—
611C23C2×C4×D411yescharacteristic
612C23#52511yescharacteristic
613C23#52911yescharacteristic
614C23#54611yescharacteristic
615C23#55111yescharacteristic
616C23#55111yescharacteristic
617C23#55211yescharacteristic
618Q8—41no—
619Q8—41no—
620Q8—41no—
621Q8—41no—
622D4—41no—
623D4—41no—
624D4—41no—
625D4—41no—
626D4—88no—
627D4—81no—
628D4—81no—
629D4—81no—
630D4—81no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—24no—
636C2×C4—22no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—432no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—81no—
671C2×C4—81no—
672C2×C4—81no—
673C2×C4—81no—
674C2×C4C2×C4×D411yescharacteristic
675C2×C4C2×C4×D411yescharacteristic
676C2×C4C4×#4111yescharacteristic
677C2×C4C4×#4111yescharacteristic
678C2×C4#51811yescharacteristic
679C2×C4#52011yescharacteristic
680C2×C4C2×#12111yescharacteristic
681C2×C4C2×#12111yescharacteristic
682C2×C4#52511yescharacteristic
683C2×C4#52911yescharacteristic
684C2×C4#53211yescharacteristic
685C2×C4#53211yescharacteristic
686C2×C4#53511yescharacteristic
687C2×C4#53711yescharacteristic
688C2×C4#54011yescharacteristic
689C2×C4#54211yescharacteristic
690C2×C4D4211yescharacteristic
691C2×C4#54611yescharacteristic
692C2×C4#54611yescharacteristic
693C2×C4#54611yescharacteristic
694C2×C4#54711yescharacteristic
695C2×C4#54811yescharacteristic
696C2×C4#55111yescharacteristic
697C2×C4#55311yescharacteristic
698C2×C4#55311yescharacteristic
699C2×C4#55611yescharacteristic
700C2×C4#55911yescharacteristic
701C2×C4#55911yescharacteristic
702C8—22no—
703C8—22no—
704C8—22no—
705C8—22no—
706C8—42no—
707C8—42no—
708C8—42no—
709C8—42no—
710C8—81no—
711C8—81no—
712C22—21no—
713C22—21no—
714C22—21no—
715C22—21no—
716C22—22no—
717C22—22no—
718C22—22no—
719C22—22no—
720C22—24no—
721C22—24no—
722C22—24no—
723C22—44no—
724C22—44no—
725C22—44no—
726C22—44no—
727C22—42no—
728C22—42no—
729C22—42no—
730C22—42no—
731C22—81no—
732C22—81no—
733C22—82no—
734C22—81no—
735C22—82no—
736C22C4×#12111yescharacteristic
737C22#235811yescharacteristic
738C22#235911yescharacteristic
739C22#241111yescharacteristic
740C22#245811yescharacteristic
741C22#251611yescharacteristic
742C22#260111yescharacteristic
743C22C2×#44412yes—
744C22#295612yes—
745C4—44no—
746C4—44no—
747C4—44no—
748C4—42no—
749C4—81no—
750C4—81no—
751C4—82no—
752C4—81no—
753C4C4×#12111yescharacteristic
754C4#232411yescharacteristic
755C4#236711yescharacteristic
756C4#237411yescharacteristic
757C4#241611yescharacteristic
758C4#246411yescharacteristic
759C4#252111yescharacteristic
760C4#254411yescharacteristic
761C2—21no—
762C2—21no—
763C2—21no—
764C2—21no—
765C2—44no—
766C2—161no—
767C2#1420111yescharacteristic
768C2#2154611yescharacteristic
769C2#2155111yescharacteristic
770C2#2288612yes—
771C2#2295012yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1938C2semidirect
4. #21053C2semidirect
6. C2×#1773C2semidirect
7. C2×#1749C2semidirect
8. #11221C2semidirect
9. #11125C2semidirect
10. #11097C2semidirect
11. #10934C2semidirect
12. #10930C2semidirect
13. #10887C2semidirect
14. #10883C2semidirect
15. #10719C2semidirect
37. #2036C4semidirect
38. #2035C4semidirect
39. #2034C4semidirect
40. #2033C4semidirect
100. C2×#107C2×C4semidirect
101. C2×#106C2×C4semidirect
119. C2×#103C2×C4semidirect
120. C2×#103C2×C4semidirect
121. C2×#103C2×C4semidirect
122. C2×#102C2×C4semidirect
123. C2×#102C2×C4semidirect
124. C2×#102C2×C4semidirect
157. C2×#97D4semidirect
168. #340C2×C4semidirect
169. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
#234792
#211912
C2×#19382
#210532
C2×#18962
C2×#17732
C2×#17492
#112212
#111252
#110972
#109342
#109302
#108872
#108832
#107192