Finite Group Databasebeta

← #378360

Subgroups — #378360

Order 512.

861 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23666C211yesmaximal, characteristic
2C2×#1950C211yesmaximal, characteristic
3#21148C211yesmaximal, characteristic
4C2×#1922C211yesmaximal, characteristic
5#20439C211yesmaximal, characteristic
6#11329C211yesmaximal, characteristic
7#11329C211yesmaximal, characteristic
8#11209C211yesmaximal, characteristic
9#10910C211yesmaximal, characteristic
10#10887C211yesmaximal, characteristic
11#10886C211yesmaximal, characteristic
12#10877C211yesmaximal, characteristic
13#10860C211yesmaximal, characteristic
14#10809C211yesmaximal, characteristic
15#10707C211yesmaximal, characteristic
16C22×#108C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C2×#424C2211yescharacteristic
21C2×#424C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C2×#405C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C2×C4×M16C2211yescharacteristic
28C22×#97C2211yescharacteristic
29#2296C2211yescharacteristic
30#2296C2211yescharacteristic
31#2088C411yescharacteristic, complemented
32#2088C411yescharacteristic, complemented
33#2088C411yescharacteristic, complemented
34#2088C411yescharacteristic, complemented
35#2087C411yescharacteristic, complemented
36#2087C411yescharacteristic, complemented
37#2086C411yescharacteristic, complemented
38#2086C411yescharacteristic, complemented
39#1950—24no—
40#1940—21no—
41#1940—21no—
42#1939—21no—
43#1939—21no—
44#1922—24no—
45#1771—21no—
46#1771—21no—
47#1770—21no—
48C8×C4⋊C4—21no—
49C2×#341C2211yescharacteristic
50C2×#341C2211yescharacteristic
51#1735C2211yescharacteristic
52#1735C2211yescharacteristic
53#1735C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60C2×#339C2211yescharacteristic
61C2×#339C2211yescharacteristic
62#1729C2211yescharacteristic
63#1729C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C2×#336C2211yescharacteristic
67C2×#336C2211yescharacteristic
68C4×#95C2211yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×#108—22no—
85C2×#108—22no—
86C2×#108—22no—
87C2×#108—22no—
88C2×#108—41no—
89C2×#108—41no—
90#427—22no—
91#427—22no—
92C2×#107C2×C411yescharacteristic
93C2×#107C2×C411yescharacteristic
94C2×#106C2×C411yescharacteristic
95C2×#106C2×C411yescharacteristic
96#424—22no—
97#424—22no—
98#424—22no—
99#424—22no—
100#424—41no—
101#424—41no—
102#423—22no—
103#423—41no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—22no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8C2×C411yescharacteristic
115C2×C4⋊C8C2×C411yescharacteristic
116C2×C4⋊C8C2×C411yescharacteristic
117C2×C4⋊C8C2×C411yescharacteristic
118#405—22no—
119#405—22no—
120#405—22no—
121#405—41no—
122#405—41no—
123#405Q814yes—
124C4×M16—22no—
125C4×M16—41no—
126C2×#97—21no—
127C2×#97—21no—
128C2×#97—21no—
129C2×#97—22no—
130C2×#97—22no—
131C2×#97—21no—
132C2×#97—41no—
133C2×C4×C8—21no—
134C2×C4×C8—21no—
135C2×C4×C8—21no—
136C2×C4×C8—21no—
137#389C2×C411yescharacteristic
138#389C2×C411yescharacteristic
139#389C2×C411yescharacteristic
140#389C2×C411yescharacteristic
141#389C2×C411yescharacteristic
142#389C2×C411yescharacteristic
143#389C2×C411yescharacteristic
144#389C2×C411yescharacteristic
145#385—21no—
146#385—21no—
147#385—21no—
148#385—21no—
149C4×C4⋊C4—21no—
150C4×C4⋊C4—21no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4—21no—
153C2×#95—21no—
154C2×#95—21no—
155#341—22no—
156#341D414yes—
157#340C2×C411yescharacteristic
158#340C2×C411yescharacteristic
159#340C2×C411yescharacteristic
160#340C2×C411yescharacteristic
161#340C2×C411yescharacteristic
162#340C2×C411yescharacteristic
163#340C2×C411yescharacteristic
164#340C2×C411yescharacteristic
165#340C2×C411yescharacteristic
166#340C2×C411yescharacteristic
167#340C2×C411yescharacteristic
168#340C2×C411yescharacteristic
169#339—22no—
170#339—22no—
171#339—22no—
172#339—22no—
173#339—22no—
174#339—41no—
175#339—41no—
176#339Q814yes—
177#336—22no—
178#336—21no—
179#336—21no—
180#336—21no—
181#336—21no—
182#336—22no—
183#336—41no—
184#336D411yescharacteristic
185#336D411yescharacteristic
186#336D411yescharacteristic
187#336D411yescharacteristic
188C23×C4—21no—
189C23×C4—21no—
190C23×C4#4111yescharacteristic
191C23×C4#4111yescharacteristic
192C23×C4C2411yescharacteristic
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
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×M16C2×D412yes—
213C2×M16C2×D412yes—
214C2×M16C2×Q812yes—
215C2×M16C2×Q812yes—
216C2×M16#4112yes—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
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×C8C22×C411yescharacteristic
242C22×C8C22×C411yescharacteristic
243C22×C8C22×C411yescharacteristic
244C22×C8C22×C411yescharacteristic
245C22×C8C2×D411yescharacteristic
246C22×C8C2×D411yescharacteristic
247C22×C8C2×D412yes—
248C22×C8C2×D411yescharacteristic
249C22×C8C2×D411yescharacteristic
250C22×C8C2×Q812yes—
251C22×C8#4112yes—
252C22×C8#4112yes—
253#117—21no—
254#117—21no—
255#117—21no—
256#117—21no—
257#117—21no—
258#117—21no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265#117C2×Q812yes—
266#117C2×Q812yes—
267#117#4112yes—
268#117#4112yes—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4C22×C411yescharacteristic
278C2×C4⋊C4C22×C411yescharacteristic
279C2×C4⋊C4C22×C411yescharacteristic
280C2×C4⋊C4C22×C411yescharacteristic
281C2×C4⋊C4C22×C411yescharacteristic
282C2×C4⋊C4C22×C411yescharacteristic
283C2×C4⋊C4C22×C411yescharacteristic
284C2×C4⋊C4C22×C411yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42C22×C411yescharacteristic
309C2×C42C22×C411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×Q812yes—
313#108—44no—
314#108—44no—
315#107—22no—
316#107—22no—
317#106—22no—
318#106—22no—
319C4⋊C8—24no—
320C4⋊C8—24no—
321C4⋊C8—22no—
322C4⋊C8—22no—
323C4⋊C8—22no—
324C4⋊C8—22no—
325C4⋊C8—42no—
326C4⋊C8—42no—
327C4⋊C8—42no—
328C4⋊C8—42no—
329C4⋊C8—42no—
330C4⋊C8—42no—
331#97—24no—
332#97—42no—
333#97—42no—
334#97—44no—
335#97—42no—
336C4×C8—24no—
337C4×C8—42no—
338C4×C8—42no—
339C4×C8—42no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—42no—
349#95—42no—
350C24C2×#4111yescharacteristic
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4C4×D411yescharacteristic
382C22×C4C4×D411yescharacteristic
383C22×C4C4×D411yescharacteristic
384C22×C4C4×D411yescharacteristic
385C22×C4C23×C411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×Q811yescharacteristic
389C22×C4C22×Q811yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4H(5,𝔽2)11yescharacteristic
396C22×C4H(5,𝔽2)11yescharacteristic
397C22×C42−1+411yescharacteristic
398C22×C42−1+411yescharacteristic
399M16—22no—
400M16—22no—
401M16—22no—
402M16—22no—
403M16—22no—
404M16—22no—
405M16—22no—
406M16—41no—
407M16—41no—
408M16—41no—
409M16—41no—
410M16—41no—
411M16—41no—
412M16—41no—
413M16—41no—
414M16C4⋊Q814yes—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—22no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—22no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8C4×D412yes—
464C2×C8C4×D412yes—
465C2×C8C22⋊Q812yes—
466C2×C8C22⋊Q812yes—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4—81no—
487C4⋊C4C4×Q812yes—
488C4⋊C4C4×Q812yes—
489C4⋊C4C4×Q812yes—
490C4⋊C4C4×Q812yes—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—22no—
504C42—22no—
505C42—21no—
506C42—21no—
507C42—22no—
508C42—21no—
509C42—21no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—41no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23C2×C4×D411yescharacteristic
528C23#52911yescharacteristic
529C23#53311yescharacteristic
530C23#54811yescharacteristic
531C23#55211yescharacteristic
532C23#55311yescharacteristic
533C23#55411yescharacteristic
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4#44412yes—
588C2×C4C2×C4×D411yescharacteristic
589C2×C4C2×C4×Q811yescharacteristic
590C2×C4C2×C4×Q811yescharacteristic
591C2×C4C4×#4111yescharacteristic
592C2×C4#52011yescharacteristic
593C2×C4#52011yescharacteristic
594C2×C4C2×C22⋊Q811yescharacteristic
595C2×C4#52511yescharacteristic
596C2×C4C2×C4⋊Q811yescharacteristic
597C2×C4#53211yescharacteristic
598C2×C4#53311yescharacteristic
599C2×C4#53311yescharacteristic
600C2×C4#54011yescharacteristic
601C2×C4#54111yescharacteristic
602C2×C4#54411yescharacteristic
603C2×C4#54411yescharacteristic
604C2×C4D4211yescharacteristic
605C2×C4#54611yescharacteristic
606C2×C4#54611yescharacteristic
607C2×C4#54711yescharacteristic
608C2×C4#54711yescharacteristic
609C2×C4D4×Q811yescharacteristic
610C2×C4#55111yescharacteristic
611C2×C4#55211yescharacteristic
612C2×C4#55211yescharacteristic
613C2×C4#55311yescharacteristic
614C2×C4#55411yescharacteristic
615C2×C4#55711yescharacteristic
616C8—22no—
617C8—22no—
618C8—22no—
619C8—22no—
620C8—24no—
621C8—41no—
622C8—41no—
623C8—41no—
624C8—41no—
625C8—42no—
626C8—42no—
627C8—81no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22C4×C22⋊Q811yescharacteristic
641C22#236011yescharacteristic
642C22#236111yescharacteristic
643C22#251911yescharacteristic
644C22#252211yescharacteristic
645C22#255011yescharacteristic
646C22#260611yescharacteristic
647C22C2×#44411yescharacteristic
648C22#295511yescharacteristic
649C22#295611yescharacteristic
650C22#295711yescharacteristic
651C4—22no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—82no—
660C4—82no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4—81no—
665C4C4×C4⋊Q811yescharacteristic
666C4#233011yescharacteristic
667C4#235911yescharacteristic
668C4#237011yescharacteristic
669C4#243811yescharacteristic
670C4#244011yescharacteristic
671C4#246211yescharacteristic
672C4#246411yescharacteristic
673C2—21no—
674C2—21no—
675C2—21no—
676C2—21no—
677C2#1421111yescharacteristic
678C2#2162811yescharacteristic
679C2#2162911yescharacteristic
680C2#2298311yescharacteristic
681C2#2298411yescharacteristic
682C2#2301911yescharacteristic
683C2#2302011yescharacteristic
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 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
31. #2088C4semidirect
32. #2088C4semidirect
33. #2088C4semidirect
34. #2088C4semidirect
35. #2087C4semidirect
36. #2087C4semidirect
37. #2086C4semidirect
38. #2086C4semidirect

Maximal subgroups

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

subgroupindex
#236662
C2×#19502
#211482
C2×#19222
#204392
#113292
#112092
#109102
#108872
#108862
#108772
#108602
#108092
#107072