Finite Group Databasebeta

← #378467

Subgroups — #378467

Order 512.

909 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23551C211yesmaximal, characteristic
2C2×#1986C211yesmaximal, characteristic
3#21304C211yesmaximal, characteristic
4C2×#1922C211yesmaximal, characteristic
5#20439C211yesmaximal, characteristic
6#11329C211yesmaximal, characteristic
7#11291C211yesmaximal, characteristic
8#11226C211yesmaximal, characteristic
9#10911C211yesmaximal, characteristic
10#10909C211yesmaximal, characteristic
11#10909C211yesmaximal, characteristic
12#10875C211yesmaximal, characteristic
13#10860C211yesmaximal, characteristic
14#10809C211yesmaximal, characteristic
15#10707C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C2×#424C2211yescharacteristic
21C2×#424C2211yescharacteristic
22C2×#405C2211yescharacteristic
23C2×#405C2211yescharacteristic
24C2×#405C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26C22×#97C2211yescharacteristic
27#2296C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#2059C411yescharacteristic, complemented
30#2059C411yescharacteristic, complemented
31#2059C411yescharacteristic, complemented
32#2059C411yescharacteristic, complemented
33#2058C411yescharacteristic, complemented
34#2058C411yescharacteristic, complemented
35#2057C411yescharacteristic, complemented
36#2057C411yescharacteristic, complemented
37#1986—24no—
38#1985—21no—
39#1985—21no—
40#1984—21no—
41#1984—21no—
42#1922—24no—
43#1771—21no—
44#1771—21no—
45#1770—21no—
46C8×C4⋊C4—21no—
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49C2×#341C2211yescharacteristic
50#1735C2211yescharacteristic
51#1735C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60#1729C2211yescharacteristic
61#1729C2211yescharacteristic
62#1729C2211yescharacteristic
63#1728C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C2×#336C2211yescharacteristic
67C2×#336C2211yescharacteristic
68C2×#336C2211yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77#533—22no—
78#533—22no—
79#533—41no—
80#533—41no—
81C2×C4⋊Q8C2×C411yescharacteristic
82C2×C4⋊Q8C2×C411yescharacteristic
83C2×#125C2×C411yescharacteristic
84C2×#125C2×C411yescharacteristic
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8—21no—
87C2×C4×Q8—21no—
88C2×C4×Q8—21no—
89C2×#117C2311yescharacteristic
90C2×#117C2311yescharacteristic
91C2×#117C2311yescharacteristic
92C2×#117C2311yescharacteristic
93C22×C4⋊C4—21no—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97C2×#108—22no—
98C2×#108—22no—
99C2×#108—41no—
100#427—22no—
101#427—22no—
102#427—41no—
103#427—41no—
104C2×#107C2×C411yescharacteristic
105C2×#107C2×C411yescharacteristic
106C2×#107C2×C411yescharacteristic
107C2×#107C2×C411yescharacteristic
108C2×#106C2×C411yescharacteristic
109C2×#106C2×C411yescharacteristic
110C2×#106C2×C411yescharacteristic
111C2×#106C2×C411yescharacteristic
112#424—22no—
113#424—22no—
114#424—22no—
115#424—41no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120#405—22no—
121#405—22no—
122#405—22no—
123#405—41no—
124#405—41no—
125#405Q814yescomplemented
126C4×M16—22no—
127C4×M16—41no—
128C2×#97—21no—
129C2×#97—21no—
130C2×#97—21no—
131C2×#97—22no—
132C2×#97—22no—
133C2×#97—21no—
134C2×#97—41no—
135C2×C4×C8—21no—
136C2×C4×C8—21no—
137C2×C4×C8—21no—
138C2×C4×C8—21no—
139#389C2×C411yescharacteristic
140#389C2×C411yescharacteristic
141#389C2×C411yescharacteristic
142#389C2×C411yescharacteristic
143#385—21no—
144#385—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—22no—
148C4×C4⋊C4—41no—
149C2×#95—21no—
150#341—22no—
151#341—22no—
152#341Q814yescomplemented
153#340C2×C411yescharacteristic
154#340C2×C411yescharacteristic
155#340C2×C411yescharacteristic
156#340C2×C411yescharacteristic
157#340C2×C411yescharacteristic
158#340C2×C411yescharacteristic
159#340C2×C411yescharacteristic
160#340C2×C411yescharacteristic
161#339—22no—
162#339—22no—
163#339—22no—
164#339—22no—
165#339—22no—
166#339—22no—
167#339—41no—
168#339D414yes—
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—22no—
176#336—22no—
177#336—22no—
178#336—21no—
179#336—21no—
180#336C2×C411yescharacteristic
181#336C2×C411yescharacteristic
182#336C2×C411yescharacteristic
183#336C2×C411yescharacteristic
184#336D411yescharacteristic
185#336D411yescharacteristic
186#336D411yescharacteristic
187#336D411yescharacteristic
188C22×Q8—21no—
189C22×Q8—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4#4111yescharacteristic
193C23×C4#4111yescharacteristic
194C23×C4C2411yescharacteristic
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×M16—21no—
213C2×M16—21no—
214C2×M16C2×D412yes—
215C2×M16C2×Q812yes—
216C2×M16C2×Q812yes—
217C2×M16C2×Q812yes—
218C2×M16#4112yes—
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×C8—21no—
242C22×C8—21no—
243C22×C8C22×C411yescharacteristic
244C22×C8C22×C411yescharacteristic
245C22×C8C22×C411yescharacteristic
246C22×C8C22×C411yescharacteristic
247C22×C8C2×D411yescharacteristic
248C22×C8C2×D411yescharacteristic
249C22×C8C2×D411yescharacteristic
250C22×C8C2×D411yescharacteristic
251C22×C8C2×Q812yes—
252C22×C8C2×Q812yes—
253C22×C8#4112yes—
254C22×C8#4112yes—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—41no—
258C4⋊Q8—41no—
259#125—22no—
260#125—22no—
261#125—41no—
262#125—41no—
263C22⋊Q8—42no—
264C22⋊Q8—42no—
265C22⋊Q8—42no—
266C22⋊Q8—42no—
267C4×Q8—42no—
268C4×Q8—42no—
269C4×Q8—42no—
270C4×Q8—42no—
271#117—21no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117—21no—
278#117—21no—
279#117—21no—
280#117—21no—
281#117—21no—
282#117—21no—
283#117C2×D412yes—
284#117C2×Q812yes—
285#117#4112yes—
286#117#4112yes—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4C22×C411yescharacteristic
301C2×C4⋊C4C22×C411yescharacteristic
302C2×C4⋊C4C22×C411yescharacteristic
303C2×C4⋊C4C22×C411yescharacteristic
304C2×C4⋊C4C22×C411yescharacteristic
305C2×C4⋊C4C22×C411yescharacteristic
306C2×C4⋊C4C22×C411yescharacteristic
307C2×C4⋊C4C22×C411yescharacteristic
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—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×C42C22×C411yescharacteristic
332C2×C42C22×C411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336#108—44no—
337#107—22no—
338#107—22no—
339#107—41no—
340#107—41no—
341#106—22no—
342#106—22no—
343#106—41no—
344#106—41no—
345C4⋊C8—24no—
346C4⋊C8—24no—
347C4⋊C8—42no—
348C4⋊C8—42no—
349#97—24no—
350#97—42no—
351#97—42no—
352#97—44no—
353#97—42no—
354C4×C8—24no—
355C4×C8—42no—
356C4×C8—42no—
357C4×C8—42no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—42no—
363C24C2×#4111yescharacteristic
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—81no—
369C2×Q8—81no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4C4×D411yescharacteristic
401C22×C4C4×D411yescharacteristic
402C22×C4C4×D411yescharacteristic
403C22×C4C4×D411yescharacteristic
404C22×C4C23×C411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×Q811yescharacteristic
408C22×C4C22×Q811yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4H(5,𝔽2)11yescharacteristic
415C22×C4H(5,𝔽2)11yescharacteristic
416C22×C42−1+411yescharacteristic
417C22×C42−1+411yescharacteristic
418M16—22no—
419M16—22no—
420M16—22no—
421M16—22no—
422M16—22no—
423M16—22no—
424M16—22no—
425M16—41no—
426M16—41no—
427M16—41no—
428M16—41no—
429M16—41no—
430M16—41no—
431M16—41no—
432M16—41no—
433M16C4⋊Q814yes—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
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—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—22no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—22no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8C4×Q812yes—
483C2×C8C4×Q812yes—
484C2×C8C22⋊Q812yes—
485C2×C8C22⋊Q812yes—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—48no—
506C4⋊C4—42no—
507C4⋊C4—81no—
508C4⋊C4—81no—
509C4⋊C4C4×D412yes—
510C4⋊C4C4×D412yes—
511C4⋊C4C4×Q812yes—
512C4⋊C4C4×Q812yes—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—22no—
526C42—22no—
527C42—21no—
528C42—21no—
529C42—22no—
530C42—22no—
531C42—21no—
532C42—21no—
533C42—22no—
534C42—22no—
535C42—41no—
536C42—41no—
537C42—41no—
538C42—41no—
539C42—41no—
540C42—41no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—41no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23C2×C4×D411yescharacteristic
550C23#52911yescharacteristic
551C23#53311yescharacteristic
552C23#54611yescharacteristic
553C23#55111yescharacteristic
554C23#55611yescharacteristic
555C23#55711yescharacteristic
556Q8—84no—
557Q8—84no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4#44412yes—
612C2×C4C2×C4×D411yescharacteristic
613C2×C4C2×C4×Q811yescharacteristic
614C2×C4C2×C4×Q811yescharacteristic
615C2×C4C4×#4111yescharacteristic
616C2×C4#51811yescharacteristic
617C2×C4#51911yescharacteristic
618C2×C4C2×C22⋊Q811yescharacteristic
619C2×C4#52511yescharacteristic
620C2×C4C2×C4⋊Q811yescharacteristic
621C2×C4#53211yescharacteristic
622C2×C4#53311yescharacteristic
623C2×C4#53311yescharacteristic
624C2×C4#54211yescharacteristic
625C2×C4#54211yescharacteristic
626C2×C4#54411yescharacteristic
627C2×C4#54411yescharacteristic
628C2×C4D4211yescharacteristic
629C2×C4#54611yescharacteristic
630C2×C4#54711yescharacteristic
631C2×C4#54711yescharacteristic
632C2×C4#54811yescharacteristic
633C2×C4D4×Q811yescharacteristic
634C2×C4#55111yescharacteristic
635C2×C4#55111yescharacteristic
636C2×C4#55211yescharacteristic
637C2×C4#55411yescharacteristic
638C2×C4#55411yescharacteristic
639C2×C4#55611yescharacteristic
640C8—22no—
641C8—22no—
642C8—22no—
643C8—22no—
644C8—24no—
645C8—41no—
646C8—41no—
647C8—41no—
648C8—41no—
649C8—42no—
650C8—42no—
651C8—81no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—21no—
664C22C4×C22⋊Q811yescharacteristic
665C22#235811yescharacteristic
666C22#236511yescharacteristic
667C22#252011yescharacteristic
668C22#252111yescharacteristic
669C22#255011yescharacteristic
670C22#260511yescharacteristic
671C22C2×#44411yescharacteristic
672C22#295511yescharacteristic
673C22#295611yescharacteristic
674C22#295711yescharacteristic
675C4—22no—
676C4—41no—
677C4—42no—
678C4—41no—
679C4—41no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—82no—
684C4—82no—
685C4—81no—
686C4—81no—
687C4—81no—
688C4—81no—
689C4C4×C4⋊Q811yescharacteristic
690C4#233111yescharacteristic
691C4#236211yescharacteristic
692C4#236811yescharacteristic
693C4#243811yescharacteristic
694C4#244011yescharacteristic
695C4#246111yescharacteristic
696C4#246511yescharacteristic
697C2—21no—
698C2—21no—
699C2—21no—
700C2—21no—
701C2#1421411yescharacteristic
702C2#2162811yescharacteristic
703C2#2162911yescharacteristic
704C2#2288611yescharacteristic
705C2#2288911yescharacteristic
706C2#2317111yescharacteristic
707C2#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 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
29. #2059C4semidirect
30. #2059C4semidirect
31. #2059C4semidirect
32. #2059C4semidirect
33. #2058C4semidirect
34. #2058C4semidirect
35. #2057C4semidirect
36. #2057C4semidirect
125. #405Q8semidirect
152. #341Q8semidirect

Maximal subgroups

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

subgroupindex
#235512
C2×#19862
#213042
C2×#19222
#204392
#113292
#112912
#112262
#109112
#109092
#108752
#108602
#108092
#107072