Finite Group Databasebeta

← #363741

Subgroups — #363741

Order 512.

1402 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34121C211yesmaximal, characteristic
2#21146C211yesmaximal, characteristic, complemented
3C2×#1806C211yesmaximal, characteristic
4C2×C4×#104C211yesmaximal, characteristic
5C4×#389C211yesmaximal, characteristic, complemented
6#11075C212yesmaximal
7#11053C212yesmaximal
8#10852C212yesmaximal, complemented
9#10851C211yesmaximal, characteristic, complemented
10#10839C211yesmaximal, characteristic, complemented
11#10808C212yesmaximal, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×#423C2212yes—
15C22×#104C2211yescharacteristic
16C22×#104C2211yescharacteristic
17C22×#104C2212yes—
18C2×C4×M16C2211yescharacteristic
19#2357C412yescomplemented
20#2356C412yescomplemented
21#2355C411yescharacteristic, complemented
22#2355C411yescharacteristic, complemented
23#2354C411yescharacteristic, complemented
24#2354C411yescharacteristic, complemented
25#2353—22no—
26#2352—21no—
27#2352—21no—
28#2298—22no—
29#2297C2212yes—
30C2×#389—21no—
31#2285C2211yescharacteristic
32#2283—21no—
33C4×#117—21no—
34C4×#117C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C43C2211yescharacteristic
39#1940—22no—
40#1939—21no—
41#1939—21no—
42#1806—24no—
43C4×#104—24no—
44C2×#339C2212yes—
45C2×#339C2211yescharacteristic
46C2×#339C2212yes—
47C2×#339C2212yes—
48C2×#339C2211yescharacteristic
49C2×#339C2212yes—
50C2×#339C2212yes—
51#1728C2212yes—
52#1436—22no—
53C4⋊#95—21no—
54C4×#95C2212yes—
55C22×#41C2311yescharacteristic
56C22×M16C2312yes—
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C4×#41—22no—
60C4×#41—22no—
61C4×#41D414yes—
62C4×#41Q814yes—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4C2×C411yescharacteristic
69C2×#117—21no—
70C2×#117—21no—
71C2×#117—21no—
72C2×#117C2312yes—
73C2×#117C2311yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C42—21no—
79C22×C42—21no—
80C22×C42—21no—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2312yes—
85C22×C42C2311yescharacteristic
86#423—24no—
87#423—22no—
88#423—22no—
89#423—42no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—21no—
94C2×#104—22no—
95C2×#104—22no—
96C2×#104—22no—
97C2×#104—22no—
98C2×#104—22no—
99C2×#104—22no—
100C2×#104—24no—
101C2×#104—42no—
102C2×#104—42no—
103C4×M16—22no—
104C4×M16—22no—
105C2×#97—21no—
106C2×C4×C8—21no—
107#391C2×C412yes—
108#391C2×C412yes—
109#390C2×C412yes—
110#390C2×C412yes—
111#389—28no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#387—22no—
117#387—22no—
118#387—22no—
119#387—22no—
120#386—22no—
121#386—22no—
122#384C2×C411yescharacteristic
123#384C2×C411yescharacteristic
124#383—22no—
125#382—21no—
126#382—21no—
127#382C2×C412yes—
128#381—22no—
129#380—21no—
130#380—21no—
131#380—21no—
132#380—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—24no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4—42no—
141C4×C4⋊C4C2×C412yes—
142C4×C4⋊C4C2×C412yes—
143C4×C4⋊C4C2×C412yes—
144C4×C4⋊C4C2×C412yes—
145C4×C4⋊C4C2×C411yescharacteristic
146C4×C4⋊C4C2×C411yescharacteristic
147C4×C4⋊C4C2×C412yes—
148C4×C4⋊C4D411yescharacteristic
149C4×C4⋊C4D411yescharacteristic
150C4×C4⋊C4D412yes—
151C4×C22⋊C4—22no—
152C4×C22⋊C4—22no—
153C4×C22⋊C4—22no—
154C4×C22⋊C4—22no—
155C4×C22⋊C4—22no—
156#376—22no—
157#376—22no—
158C2×#95—22no—
159C2×#95—22no—
160C2×#95—22no—
161C2×#95—21no—
162C2×#95D412yes—
163C43—22no—
164C43—22no—
165C43—22no—
166#339—24no—
167#339—24no—
168#339—22no—
169#339—22no—
170#339—24no—
171#339—42no—
172#339—42no—
173#336—22no—
174#336—22no—
175C2×#41—21no—
176C2×#41—21no—
177C2×#41—21no—
178C2×#41—21no—
179C2×#41—21no—
180C2×#41—21no—
181C2×#41—21no—
182C2×#41—21no—
183C2×#41—21no—
184C2×#41—21no—
185C2×#41C2×D412yes—
186C2×#41C2×Q812yes—
187C22×Q8C22×C411yescharacteristic
188C22×D4—21no—
189C22×D4C22×C411yescharacteristic
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—22no—
196C23×C4C2×D412yes—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D412yes—
199C23×C4C2×D411yescharacteristic
200C23×C4C2411yescharacteristic
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—22no—
212C2×M16—22no—
213C2×M16—22no—
214C2×M16—22no—
215C2×M16—22no—
216C2×M16—22no—
217C2×M16#4112yes—
218C2×M16#4112yes—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—22no—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8C4⋊C414yescomplemented
225C4×Q8C4⋊C414yescomplemented
226C4×D4—24no—
227C4×D4—24no—
228C4×D4—24no—
229C4×D4—24no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4C4⋊C414yescomplemented
233C4×D4C4⋊C414yescomplemented
234#117—21no—
235#117—21no—
236#117—24no—
237#117—24no—
238#117—22no—
239#117—22no—
240#117—22no—
241#117—22no—
242#117—44no—
243#117—44no—
244#117C2×D412yes—
245#117C2×Q812yes—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4C22⋊C411yescharacteristic
261C2×C4⋊C4C22⋊C411yescharacteristic
262C2×C4⋊C4C22⋊C411yescharacteristic
263C2×C4⋊C4C22⋊C412yes—
264C2×C4⋊C4C22⋊C412yes—
265C2×C4⋊C4C22⋊C411yescharacteristic
266C2×C4⋊C4C22×C412yes—
267C2×C4⋊C4C22×C412yes—
268C2×C4⋊C4C22×C411yescharacteristic
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42C22×C412yes—
315C2×C42C22×C412yes—
316C2×C42C22×C411yescharacteristic
317C2×C42C22×C411yescharacteristic
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×Q812yes—
324C2×C42#4112yes—
325C2×C42#4112yes—
326C2×C42#4112yes—
327C2×C42#4112yes—
328C4⋊C8—44no—
329C4⋊C8—44no—
330C4⋊C8—42no—
331C4⋊C8—42no—
332#104—28no—
333#104—28no—
334#104—48no—
335#97—42no—
336C4×C8—42no—
337#95—28no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—44no—
349#95—44no—
350#95—42no—
351#95C22⋊C412yes—
352#95C22⋊C412yes—
353#95C22⋊C412yes—
354#95C22⋊C412yes—
355C24—21no—
356C24C22×D411yescharacteristic
357#41—22no—
358#41—22no—
359#41—22no—
360#41—22no—
361#41—22no—
362#41—22no—
363#41—22no—
364#41—22no—
365#41—41no—
366#41—41no—
367#41—41no—
368#41—41no—
369#41—41no—
370#41—41no—
371#41—41no—
372#41—41no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8C2×C4⋊C412yes—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4C2×C4⋊C412yes—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—81no—
440C22×C4—81no—
441C22×C4C2×C22⋊C412yes—
442C22×C4C2×C22⋊C412yes—
443C22×C4C2×C22⋊C412yes—
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C412yes—
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4#12112yes—
450C22×C4#12112yes—
451C22×C4C22⋊Q812yes—
452C22×C4C22⋊Q812yes—
453C22×C4C23×C411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×Q811yescharacteristic
457C22×C4C2×#4111yescharacteristic
458C22×C4C2×#4111yescharacteristic
459C22×C4C2×#4111yescharacteristic
460C22×C4C2×#4111yescharacteristic
461C22×C4C2×#4111yescharacteristic
462C22×C4C2×#4111yescharacteristic
463C22×C4H(5,𝔽2)11yescharacteristic
464C22×C4H(5,𝔽2)11yescharacteristic
465C22×C42−1+411yescharacteristic
466C22×C42−1+411yescharacteristic
467M16—22no—
468M16—22no—
469M16—22no—
470M16—22no—
471M16—41no—
472M16—41no—
473M16—41no—
474M16—41no—
475M16—42no—
476M16—42no—
477M16—42no—
478M16—42no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—42no—
486C2×C8—42no—
487C2×C8—42no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—42no—
501C4⋊C4—48no—
502C4⋊C4—416no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—81no—
506C4⋊C4—81no—
507C4⋊C4#10414yescomplemented
508C4⋊C4C2×C4⋊C412yes—
509C4⋊C4C2×C4⋊C412yes—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—44no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—24no—
531C42—24no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—48no—
537C42—44no—
538C42—48no—
539C42—41no—
540C42—41no—
541C42—42no—
542C42—42no—
543C42—41no—
544C42—41no—
545C42—44no—
546C42—42no—
547C42—42no—
548C42—42no—
549C42—42no—
550C42—82no—
551C42—82no—
552C42C2×C4⋊C412yes—
553C42C2×C4⋊C412yes—
554C42#11712yes—
555C42#11712yes—
556C42#12112yes—
557C42#12112yes—
558C42C22⋊Q812yes—
559C42C22⋊Q812yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—22no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23C22×C22⋊C411yescharacteristic
572C23C2×#12111yescharacteristic
573C23C2×C22⋊Q811yescharacteristic
574C23#54611yescharacteristic
575C23#54711yescharacteristic
576C23#54811yescharacteristic
577C23#55011yescharacteristic
578Q8—22no—
579Q8—22no—
580Q8—41no—
581Q8—41no—
582D4—22no—
583D4—22no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—41no—
589D4—41no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—48no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—84no—
651C2×C4—82no—
652C2×C4—82no—
653C2×C4—82no—
654C2×C4—82no—
655C2×C4—82no—
656C2×C4—82no—
657C2×C4#38012yes—
658C2×C4#38012yes—
659C2×C4#38012yes—
660C2×C4#38012yes—
661C2×C4C2×#10414yes—
662C2×C4C2×#10412yes—
663C2×C4#42112yes—
664C2×C4C22×C4⋊C411yescharacteristic
665C2×C4C2×#11711yescharacteristic
666C2×C4C4×#4111yescharacteristic
667C2×C4C4×#4111yescharacteristic
668C2×C4#52011yescharacteristic
669C2×C4#52011yescharacteristic
670C2×C4C2×#12111yescharacteristic
671C2×C4C2×C22⋊Q811yescharacteristic
672C2×C4#52511yescharacteristic
673C2×C4#52511yescharacteristic
674C2×C4#53311yescharacteristic
675C2×C4#53311yescharacteristic
676C2×C4#53511yescharacteristic
677C2×C4#53611yescharacteristic
678C2×C4#54411yescharacteristic
679C2×C4#54411yescharacteristic
680C2×C4#54611yescharacteristic
681C2×C4#54711yescharacteristic
682C2×C4#54811yescharacteristic
683C2×C4#55011yescharacteristic
684C2×C4#55211yescharacteristic
685C2×C4#55211yescharacteristic
686C2×C4#55311yescharacteristic
687C2×C4#55311yescharacteristic
688C2×C4#55511yescharacteristic
689C2×C4#55511yescharacteristic
690C2×C4#55611yescharacteristic
691C2×C4#55611yescharacteristic
692C8—42no—
693C8—42no—
694C8—82no—
695C22—21no—
696C22—21no—
697C22—21no—
698C22—21no—
699C22—21no—
700C22—21no—
701C22—21no—
702C22—21no—
703C22—21no—
704C22—21no—
705C22—21no—
706C22—21no—
707C22—42no—
708C22—42no—
709C22—42no—
710C22—81no—
711C22—81no—
712C22—81no—
713C22—81no—
714C22C2×#38011yescharacteristic
715C22#233911yescharacteristic
716C22#234111yescharacteristic
717C22#241411yescharacteristic
718C22#241511yescharacteristic
719C22#243911yescharacteristic
720C22#244111yescharacteristic
721C22C22×#10411yescharacteristic
722C22C2×#42111yescharacteristic
723C22#290111yescharacteristic
724C22#290111yescharacteristic
725C4—22no—
726C4—22no—
727C4—22no—
728C4—24no—
729C4—44no—
730C4—44no—
731C4—42no—
732C4—42no—
733C4—42no—
734C4—42no—
735C4—42no—
736C4—42no—
737C4—84no—
738C4—82no—
739C4—82no—
740C4#180614yes—
741C4#229611yescharacteristic
742C4#231711yescharacteristic
743C4#234611yescharacteristic
744C4#234711yescharacteristic
745C4#241411yescharacteristic
746C4#241511yescharacteristic
747C4#243911yescharacteristic
748C4#244111yescharacteristic
749C2—21no—
750C2—21no—
751C2—21no—
752C2—21no—
753C2—82no—
754C2#1395111yescharacteristic
755C2C2×#180611yescharacteristic
756C2#2065511yescharacteristic
757C2#2250511yescharacteristic
758C2#2250711yescharacteristic
759C2#2255011yescharacteristic
760C2#2255111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21146C2semidirect
5. C4×#389C2semidirect
8. #10852C2semidirect
9. #10851C2semidirect
10. #10839C2semidirect
11. #10808C2semidirect
19. #2357C4semidirect
20. #2356C4semidirect
21. #2355C4semidirect
22. #2355C4semidirect
23. #2354C4semidirect
24. #2354C4semidirect
224. C4×Q8C4⋊C4semidirect
225. C4×Q8C4⋊C4semidirect
232. C4×D4C4⋊C4semidirect
233. C4×D4C4⋊C4semidirect
507. C4⋊C4#104semidirect

Maximal subgroups

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

subgroupindex
#341212
#211462
C2×#18062
C2×C4×#1042
C4×#3892
#110752
#110532
#108522
#108512
#108392
#108082