Finite Group Databasebeta

← #376080

Subgroups — #376080

Order 512.

1021 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×#2962C211yesmaximal, characteristic
2#23551C211yesmaximal, characteristic
3C2×#1922C211yesmaximal, characteristic
4#21024C211yesmaximal, characteristic
5C2×#1773C211yesmaximal, characteristic
6#11386C211yesmaximal, characteristic
7#11291C211yesmaximal, characteristic
8#11221C211yesmaximal, characteristic
9#10911C211yesmaximal, characteristic
10#10910C211yesmaximal, characteristic
11#10909C211yesmaximal, characteristic
12#10886C211yesmaximal, characteristic
13#10883C211yesmaximal, characteristic
14#10877C211yesmaximal, characteristic
15#10809C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17#2962—24no—
18#2962—24no—
19C2×#446—21no—
20C2×#446—21no—
21C2×#446—21no—
22C2×C8×Q8—21no—
23C22×#108C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C2×#427C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29C2×#405C2211yescharacteristic
30C2×#405C2211yescharacteristic
31C2×#405C2211yescharacteristic
32C2×C4×M16C2211yescharacteristic
33C22×#97C2211yescharacteristic
34#2296C2211yescharacteristic
35#2059C411yescharacteristic, complemented
36#2059C411yescharacteristic, complemented
37#2059C411yescharacteristic, complemented
38#2059C411yescharacteristic, complemented
39#2058C411yescharacteristic, complemented
40#2058C411yescharacteristic, complemented
41#2057C411yescharacteristic, complemented
42#2057C411yescharacteristic, complemented
43#1922—24no—
44#1917—21no—
45#1917—21no—
46#1916—21no—
47#1916—21no—
48#1773—24no—
49C2×#341C2211yescharacteristic
50C2×#341C2211yescharacteristic
51C2×#341C2211yescharacteristic
52#1735C2211yescharacteristic
53#1735C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60C2×#339C2211yescharacteristic
61C2×#339C2211yescharacteristic
62#1729C2211yescharacteristic
63#1729C2211yescharacteristic
64#1729C2211yescharacteristic
65#1728C2211yescharacteristic
66C2×#336C2211yescharacteristic
67C2×#336C2211yescharacteristic
68C2×#336C2211yescharacteristic
69C2×#336C2211yescharacteristic
70C4×#95C2211yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C23×C8C2311yescharacteristic
79#533—22no—
80#533—22no—
81#533—22no—
82#533—22no—
83C2×C4⋊Q8C2×C411yescharacteristic
84C2×C4⋊Q8C2×C411yescharacteristic
85C2×#125C2×C411yescharacteristic
86C2×#125C2×C411yescharacteristic
87C2×C22⋊Q8—21no—
88C2×C22⋊Q8—21no—
89C2×C4×Q8—21no—
90C2×C4×Q8—21no—
91C2×#117C2311yescharacteristic
92C2×#117C2311yescharacteristic
93C2×#117C2311yescharacteristic
94C2×#117C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98#446—28no—
99#446—28no—
100#446—28no—
101C8×Q8—28no—
102C2×#108—22no—
103C2×#108—22no—
104C2×#108Q814yescomplemented
105#427—22no—
106#427—22no—
107#427—22no—
108#427—22no—
109C2×#107C2×C411yescharacteristic
110C2×#107C2×C411yescharacteristic
111C2×#107C2×C411yescharacteristic
112C2×#107C2×C411yescharacteristic
113C2×#106C2×C411yescharacteristic
114C2×#106C2×C411yescharacteristic
115C2×#106C2×C411yescharacteristic
116C2×#106C2×C411yescharacteristic
117#424—22no—
118#424—22no—
119#424—22no—
120#424D414yes—
121C2×C4⋊C8—22no—
122C2×C4⋊C8—22no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8D411yescharacteristic
128C2×C4⋊C8D411yescharacteristic
129C2×C4⋊C8D411yescharacteristic
130C2×C4⋊C8D411yescharacteristic
131#405—22no—
132#405—22no—
133#405—22no—
134#405—22no—
135#405—22no—
136#405D414yes—
137C4×M16—22no—
138C4×M16—22no—
139C2×#97—22no—
140C2×#97—21no—
141C2×#97—21no—
142C2×#97—22no—
143C2×#97—21no—
144C2×#97—22no—
145C2×#97—21no—
146C2×C4×C8—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8—21no—
150#389C2×C411yescharacteristic
151#389C2×C411yescharacteristic
152#389C2×C411yescharacteristic
153#389C2×C411yescharacteristic
154#385—21no—
155#385—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C2×#95—21no—
159C2×#95—21no—
160#341—22no—
161#341—22no—
162#341—22no—
163#340C2×C411yescharacteristic
164#340C2×C411yescharacteristic
165#340C2×C411yescharacteristic
166#340C2×C411yescharacteristic
167#340C2×C411yescharacteristic
168#340C2×C411yescharacteristic
169#340C2×C411yescharacteristic
170#340C2×C411yescharacteristic
171#339—22no—
172#339—22no—
173#339—22no—
174#339—22no—
175#339—22no—
176#339—22no—
177#339—22no—
178#339—22no—
179#336—21no—
180#336—21no—
181#336—21no—
182#336—21no—
183#336—21no—
184#336—21no—
185#336—22no—
186#336—22no—
187#336—21no—
188#336—21no—
189#336—22no—
190#336C2×C411yescharacteristic
191#336C2×C411yescharacteristic
192#336C2×C411yescharacteristic
193#336C2×C411yescharacteristic
194C22×Q8—21no—
195C22×Q8—21no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4#4111yescharacteristic
199C23×C4#4111yescharacteristic
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—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16C2×D412yes—
221C2×M16C2×D412yes—
222C2×M16C2×Q812yes—
223C2×M16C2×Q812yes—
224C2×M16#4112yes—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8C22×C411yescharacteristic
250C22×C8C22×C411yescharacteristic
251C22×C8C22×C411yescharacteristic
252C22×C8C22×C411yescharacteristic
253C22×C8C2×D411yescharacteristic
254C22×C8C2×D411yescharacteristic
255C22×C8C2×D412yes—
256C22×C8C2×D412yes—
257C22×C8C2×D411yescharacteristic
258C22×C8C2×D411yescharacteristic
259C22×C8C2×Q812yes—
260C22×C8#4112yes—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265#125—22no—
266#125—22no—
267#125—22no—
268#125—22no—
269C22⋊Q8—24no—
270C22⋊Q8—24no—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—24no—
277#117—21no—
278#117—21no—
279#117—21no—
280#117—21no—
281#117—21no—
282#117—21no—
283#117—21no—
284#117—21no—
285#117—21no—
286#117—21no—
287#117—21no—
288#117—21no—
289#117C2×D412yes—
290#117C2×D412yes—
291#117#4112yes—
292#117#4112yes—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4C22×C411yescharacteristic
304C2×C4⋊C4C22×C411yescharacteristic
305C2×C4⋊C4C22×C411yescharacteristic
306C2×C4⋊C4C22×C411yescharacteristic
307C2×C4⋊C4C22×C411yescharacteristic
308C2×C4⋊C4C22×C411yescharacteristic
309C2×C4⋊C4C22×C411yescharacteristic
310C2×C4⋊C4C22×C411yescharacteristic
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42C22×C411yescharacteristic
335C2×C42C22×C411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42#4112yes—
339#108—28no—
340#107—22no—
341#107—22no—
342#107—22no—
343#107—22no—
344#106—22no—
345#106—22no—
346#106—22no—
347#106—22no—
348C4⋊C8—24no—
349C4⋊C8—24no—
350C4⋊C8—22no—
351C4⋊C8—22no—
352C4⋊C8—22no—
353C4⋊C8—22no—
354C4⋊C8—24no—
355C4⋊C8—24no—
356#97—28no—
357#97—24no—
358#97—24no—
359#97—24no—
360#97—42no—
361C4×C8—24no—
362C4×C8—24no—
363C4×C8—24no—
364C4×C8—42no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—42no—
370#95—42no—
371C24C2×#4111yescharacteristic
372C2×Q8—24no—
373C2×Q8—24no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8—81no—
377C2×Q8—81no—
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—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
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—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4C4×D411yescharacteristic
409C22×C4C4×D411yescharacteristic
410C22×C4C4×D411yescharacteristic
411C22×C4C4×D411yescharacteristic
412C22×C4C23×C411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×Q811yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C42−1+411yescharacteristic
423C22×C42−1+411yescharacteristic
424C22×C42−1+411yescharacteristic
425C22×C42−1+411yescharacteristic
426M16—22no—
427M16—22no—
428M16—22no—
429M16—22no—
430M16—22no—
431M16—22no—
432M16—22no—
433M16—41no—
434M16—41no—
435M16—41no—
436M16—41no—
437M16—41no—
438M16—41no—
439M16—41no—
440M16—41no—
441M16C22⋊Q814yes—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—22no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—22no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8C4×Q812yes—
491C2×C8C4×Q812yes—
492C2×C8#12012yes—
493C2×C8#12012yes—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—81no—
515C4⋊C4—81no—
516C4⋊C4C4×D412yes—
517C4⋊C4C4×D412yes—
518C4⋊C4C4×D412yes—
519C4⋊C4C4×D412yes—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—41no—
525C22⋊C4—41no—
526C22⋊C4—41no—
527C22⋊C4—41no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—22no—
534C42—22no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—41no—
546C42—41no—
547C42—41no—
548C42—41no—
549C42—41no—
550C42—41no—
551C42—41no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23C2×C4×D411yescharacteristic
557C23#52511yescharacteristic
558C23#52511yescharacteristic
559C23#55211yescharacteristic
560C23#55611yescharacteristic
561C23#55611yescharacteristic
562C23#55711yescharacteristic
563Q8—48no—
564Q8—84no—
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—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—42no—
581C2×C4—42no—
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—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
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—81no—
618C2×C4#44412yes—
619C2×C4C2×C4×D411yescharacteristic
620C2×C4C2×C4×D411yescharacteristic
621C2×C4C2×C4×Q811yescharacteristic
622C2×C4C4×#4111yescharacteristic
623C2×C4#51911yescharacteristic
624C2×C4#51911yescharacteristic
625C2×C4C2×#12011yescharacteristic
626C2×C4C2×C22⋊Q811yescharacteristic
627C2×C4#52511yescharacteristic
628C2×C4#52511yescharacteristic
629C2×C4#52911yescharacteristic
630C2×C4#53311yescharacteristic
631C2×C4#53611yescharacteristic
632C2×C4#53611yescharacteristic
633C2×C4#54111yescharacteristic
634C2×C4#54111yescharacteristic
635C2×C4#54711yescharacteristic
636C2×C4#54711yescharacteristic
637C2×C4#54711yescharacteristic
638C2×C4#54711yescharacteristic
639C2×C4#54811yescharacteristic
640C2×C4#54811yescharacteristic
641C2×C4D4×Q811yescharacteristic
642C2×C4D4×Q811yescharacteristic
643C2×C4#55211yescharacteristic
644C2×C4#55611yescharacteristic
645C2×C4#55611yescharacteristic
646C2×C4#55711yescharacteristic
647C8—22no—
648C8—22no—
649C8—24no—
650C8—22no—
651C8—22no—
652C8—41no—
653C8—41no—
654C8—41no—
655C8—41no—
656C8—42no—
657C8—42no—
658C8—81no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—21no—
667C22—21no—
668C22—21no—
669C22—21no—
670C22—21no—
671C22C4×#12011yescharacteristic
672C22#236211yescharacteristic
673C22#236511yescharacteristic
674C22#246211yescharacteristic
675C22#246911yescharacteristic
676C22#258111yescharacteristic
677C22#259911yescharacteristic
678C22C2×#44411yescharacteristic
679C22#295511yescharacteristic
680C22#295611yescharacteristic
681C22#295711yescharacteristic
682C4—22no—
683C4—41no—
684C4—42no—
685C4—42no—
686C4—41no—
687C4—41no—
688C4—42no—
689C4—42no—
690C4—82no—
691C4—82no—
692C4—81no—
693C4—81no—
694C4—81no—
695C4—81no—
696C4C4×C22⋊Q811yescharacteristic
697C4#232711yescharacteristic
698C4#236211yescharacteristic
699C4#236511yescharacteristic
700C4#240611yescharacteristic
701C4#240911yescharacteristic
702C4#246211yescharacteristic
703C4#246911yescharacteristic
704C2—21no—
705C2—21no—
706C2—21no—
707C2—21no—
708C2#1417411yescharacteristic
709C2#2146711yescharacteristic
710C2#2146811yescharacteristic
711C2#2308111yescharacteristic
712C2#2308211yescharacteristic
713C2#2317111yescharacteristic
714C2#2317211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
35. #2059C4semidirect
36. #2059C4semidirect
37. #2059C4semidirect
38. #2059C4semidirect
39. #2058C4semidirect
40. #2058C4semidirect
41. #2057C4semidirect
42. #2057C4semidirect
104. C2×#108Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#29622
#235512
C2×#19222
#210242
C2×#17732
#113862
#112912
#112212
#109112
#109102
#109092
#108862
#108832
#108772
#108092