Finite Group Databasebeta

← #361451

Subgroups — #361451

Order 512.

1524 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2933C211yesmaximal, characteristic
2#20983C212yesmaximal
3#20843C212yesmaximal, complemented
4C2×#1812C211yesmaximal, characteristic, complemented
5C22×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
6#11229C212yesmaximal, complemented
7#11098C212yesmaximal, complemented
8#10866C211yesmaximal, characteristic, complemented
9#10814C211yesmaximal, characteristic, complemented
10#10759C211yesmaximal, characteristic, complemented
11#10717C211yesmaximal, characteristic, complemented
12C22×#131C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C2×#525C2211yescharacteristic
15C22×#117C2212yes—
16#2933—22no—
17#2933—22no—
18#2933—22no—
19#2933—22no—
20C2×#436C412yescomplemented
21C2×#435C412yescomplemented
22C2×#435C412yescomplemented
23C2×C8×D4C412yescomplemented
24C2×#427C2212yes—
25C2×#427C2212yes—
26C2×#424C2211yescharacteristic
27C2×#410C2212yes—
28C2×#410C2212yes—
29C2×#408C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31C2×#405C2211yescharacteristic
32#2280C2212yes—
33#1899—24no—
34#1898—24no—
35#1854—22no—
36#1854—22no—
37#1853—22no—
38#1852—22no—
39#1812—24no—
40#1740C2212yes—
41#1740C2212yes—
42#1735C2212yes—
43#1735C2212yes—
44C2×H(3,ℤ/4)—24no—
45C2×H(3,ℤ/4)C2211yescharacteristic
46C2×H(3,ℤ/4)C2211yescharacteristic
47C2×H(3,ℤ/4)C2212yes—
48C2×H(3,ℤ/4)C2211yescharacteristic
49C2×H(3,ℤ/4)C2212yes—
50C2×H(3,ℤ/4)C2211yescharacteristic
51C22×#41C2311yescharacteristic
52C24×C4C2311yescharacteristic
53C2×#131—22no—
54C2×#131—22no—
55C2×#131—22no—
56C2×#131—22no—
57C2×#131—42no—
58C22×M16—22no—
59C22×M16—22no—
60C22×M16—22no—
61C22×M16—22no—
62C22×M16—42no—
63C22×M16C2×C412yes—
64C22×M16C2×C412yescomplemented
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C23×C8C2×C412yes—
68C23×C8C2×C412yescomplemented
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71#525—22no—
72#525—22no—
73#525—22no—
74#525—22no—
75C2×#123D412yes—
76C2×C22⋊Q8Q811yescharacteristic
77C2×C22⋊Q8Q811yescharacteristic
78C2×#121Q811yescharacteristic
79C2×#121Q811yescharacteristic
80C2×#120D412yes—
81C2×C4×D4C2×C412yes—
82C2×C4×D4C2×C412yes—
83C2×#117—22no—
84C2×#117—24no—
85C2×#117—24no—
86C2×#117—22no—
87C2×#117—22no—
88C2×#117—42no—
89C2×#117—42no—
90C2×#117C2312yes—
91C2×#117C2312yes—
92C2×#117C2312yes—
93C2×#117C2312yes—
94C2×#117C2311yescharacteristic
95C22×C4⋊C4—22no—
96C22×C22⋊C4—22no—
97C22×C42—22no—
98#436—24no—
99#436—24no—
100#435—24no—
101#435—24no—
102#435—24no—
103#435—24no—
104C8×D4—22no—
105C8×D4—22no—
106C8×D4—22no—
107C8×D4—22no—
108#427—24no—
109#427—24no—
110#427—42no—
111#427—42no—
112C2×#107—22no—
113C2×#107—22no—
114C2×#106—22no—
115C2×#106—22no—
116#424—22no—
117#424—22no—
118C2×C4⋊C8C2×C412yescomplemented
119C2×C4⋊C8C2×C412yescomplemented
120#410—24no—
121#410—24no—
122#410—42no—
123#410—42no—
124C2×#99—24no—
125C2×#99—24no—
126#408—22no—
127#408—22no—
128C2×C22⋊C8C2×C411yescharacteristic
129C2×C22⋊C8C2×C411yescharacteristic
130C2×C22⋊C8C2×C411yescharacteristic
131C2×C22⋊C8C2×C411yescharacteristic
132C2×C22⋊C8C2×C412yescomplemented
133C2×C22⋊C8C2×C412yescomplemented
134C2×C22⋊C8D412yes—
135C2×C22⋊C8D412yes—
136C2×C22⋊C8D412yes—
137C2×C22⋊C8D412yes—
138#405—22no—
139#405—22no—
140C2×#97C2×C412yescomplemented
141C2×C4×C8C2×C412yescomplemented
142#381—22no—
143#380—22no—
144#380—22no—
145#379—22no—
146C4×C22⋊C4—22no—
147C4×C22⋊C4—22no—
148#342—24no—
149#342—24no—
150#340—22no—
151#340—22no—
152#340—22no—
153#340—22no—
154H(3,ℤ/4)—22no—
155H(3,ℤ/4)—22no—
156H(3,ℤ/4)—22no—
157H(3,ℤ/4)—24no—
158H(3,ℤ/4)—42no—
159H(3,ℤ/4)D414yescomplemented
160C25C2×D411yescharacteristic
161C2×#41—22no—
162C2×#41—21no—
163C2×#41—22no—
164C2×#41—21no—
165C2×#41—22no—
166C2×#41—22no—
167C2×#41—21no—
168C2×#41—21no—
169C22×Q8C2×Q811yescharacteristic
170C22×D4C4⋊C412yes—
171C22×D4C4⋊C412yes—
172C22×D4C2×D412yes—
173C22×D4C2×Q811yescharacteristic
174C23×C4—21no—
175C23×C4—22no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4—21no—
180C23×C4—42no—
181C23×C4C22×C412yes—
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×Q811yescharacteristic
188C23×C4C2×Q811yescharacteristic
189C23×C4#4112yes—
190C23×C4#4112yes—
191C23×C4C2411yescharacteristic
192#131—44no—
193#131—44no—
194#131—44no—
195#131—44no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—24no—
199C2×M16—21no—
200C2×M16—22no—
201C2×M16—21no—
202C2×M16—22no—
203C2×M16—24no—
204C2×M16—24no—
205C2×M16—24no—
206C2×M16—42no—
207C2×M16—42no—
208C2×M16—44no—
209C2×M16—44no—
210C2×M16—44no—
211C2×M16—44no—
212C2×M16C2×D412yes—
213C2×M16#4112yes—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—24no—
225C22×C8—24no—
226C22×C8—42no—
227C22×C8—42no—
228C22×C8C22×C412yes—
229C22×C8C22×C411yescharacteristic
230C22×C8C22×C411yescharacteristic
231C22×C8C22×C411yescharacteristic
232C22×C8C22×C411yescharacteristic
233C22×C8C22×C412yes—
234C22×C8C2×D412yes—
235C22×C8C2×D412yes—
236C22×C8C2×D412yes—
237C22×C8C2×D412yes—
238#123—24no—
239#123—24no—
240C22⋊Q8—22no—
241C22⋊Q8—22no—
242C22⋊Q8—22no—
243C22⋊Q8—22no—
244#121—22no—
245#121—22no—
246#121—22no—
247#121—22no—
248#120—24no—
249#120—24no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—24no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—24no—
256#117—22no—
257#117—22no—
258#117—28no—
259#117—22no—
260#117—22no—
261#117—22no—
262#117—22no—
263#117—22no—
264#117—22no—
265#117—22no—
266#117—22no—
267#117—22no—
268#117—22no—
269#117—22no—
270#117—22no—
271#117—22no—
272#117—22no—
273#117—44no—
274#117—44no—
275#117—44no—
276#117—44no—
277#117—44no—
278#117C2×D414yes—
279#117#4114yes—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4C4⋊C412yes—
288C2×C4⋊C4C4⋊C412yes—
289C2×C4⋊C4C2×Q811yescharacteristic
290C2×C4⋊C4C2×Q811yescharacteristic
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4C4⋊C412yes—
301C2×C22⋊C4C4⋊C412yes—
302C2×C22⋊C4C4⋊C412yes—
303C2×C22⋊C4C4⋊C412yes—
304C2×C22⋊C4C2×D412yes—
305C2×C42—28no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—44no—
311C2×C42—42no—
312C2×C42C22×C412yes—
313#107—44no—
314#107—44no—
315#106—44no—
316#106—44no—
317C4⋊C8—22no—
318C4⋊C8—22no—
319C4⋊C8—24no—
320#99—48no—
321#99—48no—
322C22⋊C8—22no—
323C22⋊C8—22no—
324C22⋊C8—24no—
325C22⋊C8C22⋊C411yescharacteristic
326C22⋊C8C22⋊C411yescharacteristic
327C22⋊C8C22⋊C411yescharacteristic
328C22⋊C8C22⋊C411yescharacteristic
329C22⋊C8C22⋊C412yes—
330C22⋊C8C22⋊C412yes—
331C22⋊C8C22⋊C412yes—
332C22⋊C8C22⋊C412yes—
333C22⋊C8C22⋊C412yes—
334C22⋊C8C22⋊C412yes—
335#97—24no—
336C4×C8—22no—
337C4×C8—22no—
338#95—42no—
339#95—42no—
340C24—21no—
341C24—22no—
342C24—21no—
343C24—42no—
344C24—81no—
345C24C2×C4⋊C411yescharacteristic
346C24C2×C4⋊C412yes—
347C24C2×C4⋊C411yescharacteristic
348C24#12112yes—
349C24#12112yes—
350C24C22×D411yescharacteristic
351#41—42no—
352#41—42no—
353#41—42no—
354#41—42no—
355#41—42no—
356#41—42no—
357#41—42no—
358#41—42no—
359C2×Q8—22no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—22no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—24no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—24no—
370C2×D4—22no—
371C2×D4—21no—
372C2×D4—22no—
373C2×D4—21no—
374C2×D4—44no—
375C2×D4—82no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—48no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C412yes—
425C22×C4C2×C4⋊C412yes—
426C22×C4C2×C4⋊C411yescharacteristic
427C22×C4C2×C4⋊C411yescharacteristic
428C22×C4C2×C4⋊C412yes—
429C22×C4C2×C4⋊C412yes—
430C22×C4#11711yescharacteristic
431C22×C4#11712yes—
432C22×C4#11711yescharacteristic
433C22×C4#12112yes—
434C22×C4#12112yes—
435C22×C4C22⋊Q812yes—
436C22×C4C22⋊Q812yes—
437C22×C4C22⋊Q812yes—
438C22×C4C22⋊Q812yes—
439C22×C4C23×C411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×Q811yescharacteristic
443C22×C4C2×#4112yes—
444C22×C4C2×#4111yescharacteristic
445C22×C4C2×#4112yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C4H(5,𝔽2)11yescharacteristic
448C22×C42−1+411yescharacteristic
449M16—24no—
450M16—24no—
451M16—48no—
452M16—48no—
453M16—42no—
454M16—42no—
455C2×C8—22no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—22no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—44no—
477C2×C8—44no—
478C2×C8—44no—
479C2×C8—44no—
480C2×C8C2×C22⋊C411yescharacteristic
481C2×C8C2×C22⋊C411yescharacteristic
482C2×C8C2×C22⋊C412yes—
483C2×C8C2×C22⋊C411yescharacteristic
484C2×C8C2×C22⋊C411yescharacteristic
485C2×C8C2×C22⋊C412yes—
486C2×C8C4×D414yes—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—48no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C22⋊C4—24no—
502C22⋊C4—22no—
503C22⋊C4—24no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—48no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C42—22no—
520C42—22no—
521C42—24no—
522C42—24no—
523C42—48no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—42no—
529C42—42no—
530C23—24no—
531C23—22no—
532C23—21no—
533C23—21no—
534C23—22no—
535C23—22no—
536C23—21no—
537C23—21no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—42no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23—41no—
549C23—41no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—82no—
554C23—84no—
555C23—82no—
556C23#38011yescharacteristic
557C23#38012yes—
558C23#38011yescharacteristic
559C23C22×C4⋊C411yescharacteristic
560C23C2×#12112yes—
561C23D4211yescharacteristic
562C23#54612yes—
563C23#54711yescharacteristic
564Q8—42no—
565Q8—42no—
566D4—44no—
567D4—44no—
568D4—44no—
569D4—44no—
570D4—42no—
571D4—42no—
572D4—84no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—816no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4#38011yescharacteristic
626C2×C4#38011yescharacteristic
627C2×C4#38012yes—
628C2×C4C22×C22⋊C411yescharacteristic
629C2×C4C2×#11711yescharacteristic
630C2×C4C2×C4×D411yescharacteristic
631C2×C4C4×#4111yescharacteristic
632C2×C4#51811yescharacteristic
633C2×C4#52011yescharacteristic
634C2×C4C2×C22⋊Q812yes—
635C2×C4#52512yes—
636C2×C4#52912yes—
637C2×C4#53812yes—
638C2×C4#53912yes—
639C2×C4D4212yes—
640C2×C4#54612yes—
641C2×C4#54612yes—
642C2×C4#54712yes—
643C2×C4D4×Q811yescharacteristic
644C2×C4#55412yes—
645C2×C4#55411yescharacteristic
646C8—21no—
647C8—21no—
648C8—22no—
649C8—24no—
650C8—21no—
651C8—22no—
652C8—21no—
653C8—42no—
654C22—21no—
655C22—21no—
656C22—22no—
657C22—21no—
658C22—22no—
659C22—21no—
660C22—21no—
661C22—22no—
662C22—21no—
663C22—22no—
664C22—22no—
665C22—21no—
666C22—41no—
667C22—42no—
668C22—41no—
669C22—44no—
670C22—44no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—82no—
677C22—82no—
678C22—81no—
679C22—81no—
680C22—82no—
681C22—82no—
682C22#181212yescomplemented
683C22C2×#38011yescharacteristic
684C22D4×C4⋊C411yescharacteristic
685C22#234911yescharacteristic
686C22#242412yes—
687C22#242512yes—
688C4—21no—
689C4—22no—
690C4—21no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—84no—
695C4—84no—
696C4—162no—
697C4—162no—
698C4#229011yescharacteristic
699C4#231811yescharacteristic
700C4D4×C22⋊C411yescharacteristic
701C4#233911yescharacteristic
702C4#244912yes—
703C4#245012yes—
704C2—22no—
705C2—22no—
706C2—22no—
707C2—21no—
708C2—21no—
709C2—41no—
710C2—42no—
711C2—82no—
712C2#1389911yescharacteristic
713C2C2×#181211yescharacteristic
714C2#2067111yescharacteristic
subgroupG/H
centerC2×C4C2×#117
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#380
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×M16 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20843C2semidirect
4. C2×#1812C2semidirect
5. C22×H(3,ℤ/4)C2semidirect
6. #11229C2semidirect
7. #11098C2semidirect
8. #10866C2semidirect
9. #10814C2semidirect
10. #10759C2semidirect
11. #10717C2semidirect
20. C2×#436C4semidirect
21. C2×#435C4semidirect
22. C2×#435C4semidirect
23. C2×C8×D4C4semidirect
64. C22×M16C2×C4semidirect
68. C23×C8C2×C4semidirect
118. C2×C4⋊C8C2×C4semidirect
119. C2×C4⋊C8C2×C4semidirect
132. C2×C22⋊C8C2×C4semidirect
133. C2×C22⋊C8C2×C4semidirect
140. C2×#97C2×C4semidirect
141. C2×C4×C8C2×C4semidirect
159. H(3,ℤ/4)D4semidirect
682. C22#1812semidirect

Maximal subgroups

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

subgroupindex
C2×#29332
#209832
#208432
C2×#18122
C22×H(3,ℤ/4)2
#112292
#110982
#108662
#108142
#107592
#107172