Finite Group Databasebeta

← #363745

Subgroups — #363745

Order 512.

1142 conjugacy classes of subgroups.

The subgroup lattice has 765 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2925C211yesmaximal, characteristic
2C2×#1950C211yesmaximal, characteristic, complemented
3C2×#1949C211yesmaximal, characteristic, complemented
4#20649C211yesmaximal, characteristic
5C4×#417C211yesmaximal, characteristic
6#11107C211yesmaximal, characteristic
7#11106C211yesmaximal, characteristic
8#11105C211yesmaximal, characteristic
9#11105C211yesmaximal, characteristic
10#10870C211yesmaximal, characteristic, complemented
11#10869C211yesmaximal, characteristic, complemented
12#10858C211yesmaximal, characteristic, complemented
13#10847C211yesmaximal, characteristic, complemented
14#10827C211yesmaximal, characteristic, complemented
15#10827C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17#2925—24no—
18#2925C418yescomplemented
19C2×#433—21no—
20C2×#433—21no—
21C2×#432—21no—
22C2×#431—21no—
23C22×#108C2211yescharacteristic
24C22×#108C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#423C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C2×#417C2211yescharacteristic
31C2×#417C2211yescharacteristic
32C2×#417C2211yescharacteristic
33C2×#417C2211yescharacteristic
34C2×#408C2211yescharacteristic
35C2×#408C2211yescharacteristic
36C2×#405C2211yescharacteristic
37C2×#405C2211yescharacteristic
38C2×C4×M16C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41#1950—24no—
42#1949—21no—
43#1949—21no—
44#1949—21no—
45#1949—21no—
46#1805—21no—
47#1805—21no—
48#1804—21no—
49#1804—21no—
50#1763—21no—
51#1762—21no—
52C4×#103—21no—
53C4×#102—21no—
54C2×#343C2211yescharacteristic
55C2×#343C2211yescharacteristic
56C2×#343C2211yescharacteristic
57C2×#343C2211yescharacteristic
58C2×#343C2211yescharacteristic
59C2×#343C2211yescharacteristic
60C2×#343C2211yescharacteristic
61C2×#343C2211yescharacteristic
62C2×#341C2211yescharacteristic
63C2×#341C2211yescharacteristic
64#1736C2211yescharacteristic
65#1736C2211yescharacteristic
66#1736C2211yescharacteristic
67#1736C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C22×#41C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C22×M16C2311yescharacteristic
76C22×M16C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C23×C8C2311yescharacteristic
79C4×#41—22no—
80C4×#41—22no—
81C4×#41C2×C414yes—
82C4×#41Q814yes—
83C2×C4×Q8D411yescharacteristic
84C2×C4×Q8D411yescharacteristic
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4D411yescharacteristic
88C2×C4×D4D411yescharacteristic
89C2×#117—21no—
90C2×#117C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C42—21no—
96C22×C42—21no—
97C22×C42C2311yescharacteristic
98#433—28no—
99#433—28no—
100#432—28no—
101#431—28no—
102C2×#108—22no—
103C2×#108—22no—
104C2×#108—22no—
105C2×#108—22no—
106C2×#108—41no—
107C2×#108—41no—
108C2×#107—21no—
109C2×#107—21no—
110C2×#107—22no—
111C2×#107—41no—
112C2×#106—21no—
113C2×#106—21no—
114C2×#106—22no—
115C2×#106—41no—
116#423—22no—
117#423—22no—
118#423—22no—
119#423C2×C414yes—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8C2×C414yes—
125C2×C4⋊C8D411yescharacteristic
126C2×C4⋊C8D411yescharacteristic
127C2×C4⋊C8D411yescharacteristic
128C2×C4⋊C8D411yescharacteristic
129C2×C4⋊C8D414yes—
130#417—22no—
131#417—22no—
132#417—22no—
133#417—22no—
134#417—41no—
135#417—41no—
136#417—41no—
137#417—41no—
138C2×#103—21no—
139C2×#103—21no—
140C2×#103—21no—
141C2×#103—21no—
142C2×#102—21no—
143C2×#102—21no—
144C2×#102—21no—
145C2×#102—21no—
146#408—22no—
147#408—22no—
148#408C2×C414yes—
149#408C2×C414yes—
150C2×C22⋊C8—21no—
151C2×C22⋊C8—21no—
152C2×C22⋊C8—21no—
153C2×C22⋊C8—21no—
154#405—22no—
155#405—22no—
156#405C2×C414yes—
157#405C2×C414yes—
158C4×M16—22no—
159C4×M16—22no—
160C2×#97—21no—
161C2×#97—21no—
162C2×#97—21no—
163C2×C4×C8—21no—
164C2×C4×C8—21no—
165C2×C4×C8—21no—
166#382—21no—
167#382—21no—
168#382—21no—
169#382—21no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177#343—22no—
178#343—22no—
179#343—22no—
180#343—22no—
181#343—41no—
182#343—41no—
183#343—41no—
184#343—41no—
185#341—22no—
186#341—22no—
187#340—21no—
188#340—21no—
189#340—21no—
190#340—21no—
191#340—21no—
192#340—21no—
193#340—21no—
194#340—21no—
195#340—21no—
196#340—21no—
197#340—21no—
198#340—21no—
199C2×#41—21no—
200C2×#41—21no—
201C2×#41—21no—
202C2×#41—21no—
203C2×#41—21no—
204C2×#41—21no—
205C2×#41—21no—
206C2×#41—21no—
207C2×#41—21no—
208C2×#41—21no—
209C2×#41C22×C412yes—
210C2×#41C2×Q812yes—
211C22×Q8C2×D411yescharacteristic
212C22×D4—21no—
213C22×D4C2×D411yescharacteristic
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4#4111yescharacteristic
222C23×C4#4111yescharacteristic
223C23×C4C2411yescharacteristic
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×M16—21no—
244C2×M16—21no—
245C2×M16—21no—
246C2×M16—21no—
247C2×M16—21no—
248C2×M16—21no—
249C2×M16—21no—
250C2×M16—21no—
251C2×M16—21no—
252C2×M16—21no—
253C2×M16—21no—
254C2×M16—21no—
255C2×M16—21no—
256C2×M16C22×C412yes—
257C2×M16C22×C412yes—
258C2×M16#4112yes—
259C2×M16#4112yes—
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×C8—21no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—21no—
273C22×C8—21no—
274C22×C8C22×C412yes—
275C22×C8C22×C412yes—
276C22×C8C2×D411yescharacteristic
277C22×C8C2×D411yescharacteristic
278C22×C8C2×D411yescharacteristic
279C22×C8C2×D411yescharacteristic
280C22×C8C2×D412yes—
281C22×C8C2×D412yes—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8C4⋊C414yes—
285C4×Q8C4⋊C414yes—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4C4⋊C414yes—
293C4×D4C4⋊C414yes—
294#117—21no—
295#117—21no—
296#117—24no—
297#117—24no—
298#117C22×C412yes—
299#117C2×Q812yes—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4#4112yes—
327C2×C4⋊C4#4112yes—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—41no—
337C2×C42—41no—
338C2×C42—82no—
339C2×C42C22×C412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×Q812yes—
346C2×C42#4112yes—
347#108—44no—
348#108—44no—
349#107—42no—
350#107—42no—
351#106—42no—
352#106—42no—
353C4⋊C8—24no—
354C4⋊C8—24no—
355C4⋊C8—42no—
356C4⋊C8—42no—
357C4⋊C8C22⋊C414yes—
358C4⋊C8C22⋊C414yes—
359C4⋊C8C22⋊C414yes—
360C4⋊C8C22⋊C414yes—
361#103—24no—
362#103—24no—
363#103—42no—
364#103—42no—
365#102—24no—
366#102—24no—
367#102—42no—
368#102—42no—
369C22⋊C8—24no—
370C22⋊C8—24no—
371C22⋊C8—24no—
372C22⋊C8—24no—
373#97—24no—
374#97—24no—
375#97—42no—
376C4×C8—24no—
377C4×C8—24no—
378C4×C8—42no—
379#95—41no—
380#95—41no—
381#95—41no—
382#95—41no—
383#95—41no—
384#95—41no—
385C24—21no—
386C24C2×#4111yescharacteristic
387#41—22no—
388#41—22no—
389#41—22no—
390#41—22no—
391#41—22no—
392#41—22no—
393#41—22no—
394#41—22no—
395#41—41no—
396#41—41no—
397#41—41no—
398#41—41no—
399#41—41no—
400#41—41no—
401#41—41no—
402#41—41no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8C2×C4⋊C412yes—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—41no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4C2×C4⋊C412yes—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—81no—
451C22×C4—81no—
452C22×C4—81no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4—81no—
456C22×C4—81no—
457C22×C4—81no—
458C22×C4—81no—
459C22×C4—81no—
460C22×C4#11712yes—
461C22×C4#11712yes—
462C22×C4#12111yescharacteristic
463C22×C4#12111yescharacteristic
464C22×C4#12111yescharacteristic
465C22×C4#12111yescharacteristic
466C22×C4C22⋊Q812yes—
467C22×C4C22⋊Q812yes—
468C22×C4C23×C411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×Q811yescharacteristic
473C22×C4C2×#4111yescharacteristic
474C22×C4C2×#4111yescharacteristic
475C22×C4C2×#4111yescharacteristic
476C22×C4C2×#4111yescharacteristic
477C22×C4C2×#4111yescharacteristic
478C22×C4H(5,𝔽2)11yescharacteristic
479C22×C4H(5,𝔽2)11yescharacteristic
480C22×C42−1+411yescharacteristic
481C22×C42−1+411yescharacteristic
482M16—22no—
483M16—22no—
484M16—22no—
485M16—22no—
486M16—22no—
487M16—22no—
488M16—22no—
489M16—22no—
490M16—41no—
491M16—41no—
492M16—41no—
493M16—41no—
494M16—41no—
495M16—41no—
496M16—41no—
497M16—41no—
498M16—41no—
499M16—41no—
500M16—41no—
501M16—41no—
502M16—41no—
503M16—41no—
504M16—41no—
505M16—41no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—41no—
527C2×C8—41no—
528C2×C8—41no—
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×C8C2×C22⋊C412yes—
541C2×C8C2×C22⋊C412yes—
542C2×C8C2×C22⋊C412yes—
543C2×C8C2×C22⋊C412yes—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—24no—
547C4⋊C4—24no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—41no—
553C4⋊C4—41no—
554C4⋊C4—41no—
555C4⋊C4—41no—
556C4⋊C4—41no—
557C4⋊C4—41no—
558C4⋊C4—41no—
559C4⋊C4—41no—
560C4⋊C4—41no—
561C4⋊C4—41no—
562C4⋊C4—41no—
563C4⋊C4—41no—
564C4⋊C4C2×C4⋊C412yes—
565C4⋊C4C2×C4⋊C412yes—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—24no—
569C22⋊C4—24no—
570C42—22no—
571C42—22no—
572C42—24no—
573C42—24no—
574C42—82no—
575C42—82no—
576C42C2×C22⋊C412yes—
577C42C2×C22⋊C412yes—
578C42C2×C4⋊C412yes—
579C42C2×C4⋊C412yes—
580C42#12112yes—
581C42#12112yes—
582C42C22⋊Q812yes—
583C42C22⋊Q812yes—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—22no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23C2×#11711yescharacteristic
596C23C2×#12111yescharacteristic
597C23C2×C22⋊Q811yescharacteristic
598C23#55211yescharacteristic
599C23#55311yescharacteristic
600C23#55511yescharacteristic
601C23#55611yescharacteristic
602Q8—22no—
603Q8—22no—
604Q8—41no—
605Q8—41no—
606D4—22no—
607D4—22no—
608D4—42no—
609D4—42no—
610D4—42no—
611D4—42no—
612D4—41no—
613D4—41no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—48no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—84no—
643C2×C4—82no—
644C2×C4—82no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4—81no—
648C2×C4—81no—
649C2×C4—82no—
650C2×C4—82no—
651C2×C4—81no—
652C2×C4—81no—
653C2×C4—81no—
654C2×C4—81no—
655C2×C4—81no—
656C2×C4—81no—
657C2×C4—161no—
658C2×C4—161no—
659C2×C4—161no—
660C2×C4—161no—
661C2×C4#38012yes—
662C2×C4#38012yes—
663C2×C4#38012yes—
664C2×C4#38012yes—
665C2×C4C22×C22⋊C411yescharacteristic
666C2×C4C22×C4⋊C411yescharacteristic
667C2×C4C4×#4111yescharacteristic
668C2×C4C4×#4111yescharacteristic
669C2×C4#52011yescharacteristic
670C2×C4#52011yescharacteristic
671C2×C4C2×#12111yescharacteristic
672C2×C4C2×C22⋊Q811yescharacteristic
673C2×C4#52511yescharacteristic
674C2×C4#52511yescharacteristic
675C2×C4#53311yescharacteristic
676C2×C4#53311yescharacteristic
677C2×C4#53511yescharacteristic
678C2×C4#53611yescharacteristic
679C2×C4#54411yescharacteristic
680C2×C4#54411yescharacteristic
681C2×C4#54611yescharacteristic
682C2×C4#54611yescharacteristic
683C2×C4#54711yescharacteristic
684C2×C4#54711yescharacteristic
685C2×C4#54811yescharacteristic
686C2×C4#54811yescharacteristic
687C2×C4#55011yescharacteristic
688C2×C4#55011yescharacteristic
689C2×C4#55211yescharacteristic
690C2×C4#55311yescharacteristic
691C2×C4#55511yescharacteristic
692C2×C4#55611yescharacteristic
693C8—22no—
694C8—22no—
695C8—22no—
696C8—22no—
697C8—42no—
698C8—42no—
699C8—42no—
700C8—42no—
701C8—81no—
702C8—81no—
703C22—21no—
704C22—21no—
705C22—21no—
706C22—21no—
707C22—21no—
708C22—21no—
709C22—21no—
710C22—21no—
711C22—21no—
712C22—21no—
713C22—21no—
714C22—21no—
715C22—42no—
716C22—42no—
717C22—42no—
718C22—81no—
719C22—81no—
720C22—81no—
721C22—81no—
722C22C2×#38011yescharacteristic
723C22#234611yescharacteristic
724C22#234711yescharacteristic
725C22#241411yescharacteristic
726C22#241511yescharacteristic
727C22#243911yescharacteristic
728C22#244111yescharacteristic
729C22#306711yescharacteristic
730C22#306711yescharacteristic
731C22#306811yescharacteristic
732C22#306911yescharacteristic
733C4—22no—
734C4—22no—
735C4—22no—
736C4—22no—
737C4—42no—
738C4—42no—
739C4—42no—
740C4—82no—
741C4—82no—
742C4—161no—
743C4—161no—
744C4#229611yescharacteristic
745C4#231711yescharacteristic
746C4#233911yescharacteristic
747C4#234111yescharacteristic
748C4#241411yescharacteristic
749C4#241511yescharacteristic
750C4#243911yescharacteristic
751C4#244111yescharacteristic
752C2—21no—
753C2—21no—
754C2—21no—
755C2—21no—
756C2—82no—
757C2#1395111yescharacteristic
758C2#2067811yescharacteristic
759C2#2067911yescharacteristic
760C2#2467411yescharacteristic
761C2#2467511yescharacteristic
762C2#2472411yescharacteristic
763C2#2472511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1950C2semidirect
3. C2×#1949C2semidirect
10. #10870C2semidirect
11. #10869C2semidirect
12. #10858C2semidirect
13. #10847C2semidirect
14. #10827C2semidirect
15. #10827C2semidirect
18. #2925C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29252
C2×#19502
C2×#19492
#206492
C4×#4172
#111072
#111062
#111052
#108702
#108692
#108582
#108472
#108272