Finite Group Databasebeta

← #379162

Subgroups — #379162

Order 512.

1108 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23730C211yesmaximal, characteristic
2C2×#1945C211yesmaximal, characteristic, complemented
3C2×#1922C211yesmaximal, characteristic, complemented
4#21025C211yesmaximal, characteristic, complemented
5C2×#1866C211yesmaximal, characteristic
6#20839C211yesmaximal, characteristic
7C2×#1764C211yesmaximal, characteristic
8#11376C211yesmaximal, characteristic, complemented
9#11326C211yesmaximal, characteristic, complemented
10#11125C211yesmaximal, characteristic
11#10929C211yesmaximal, characteristic
12#10923C211yesmaximal, characteristic
13#10859C211yesmaximal, characteristic, complemented
14#10839C211yesmaximal, characteristic, complemented
15C4×#336C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C22×#108C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C2×#427C2211yescharacteristic
21C2×#424C2211yescharacteristic
22C2×#424C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C22×#104C2211yescharacteristic
25C22×#104C2211yescharacteristic
26C2×#416C2211yescharacteristic
27C2×#416C2211yescharacteristic
28C2×#408C2211yescharacteristic
29C2×#405C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32#2297C2211yescharacteristic
33#2285C2211yescharacteristic
34C2×C43C2211yescharacteristic
35#2097C411yescharacteristic, complemented
36#2097C411yescharacteristic, complemented
37#2096C411yescharacteristic, complemented
38#2095C411yescharacteristic, complemented
39#2094C411yescharacteristic, complemented
40#2093C411yescharacteristic, complemented
41#2092C411yescharacteristic, complemented
42#2092C411yescharacteristic, complemented
43#1945—24no—
44#1922—24no—
45#1917—22no—
46#1916—22no—
47#1866—24no—
48#1851—22no—
49#1850—22no—
50#1764—24no—
51C2×#341C2211yescharacteristic
52C2×#341C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59#1729C2211yescharacteristic
60#1729C2211yescharacteristic
61#1729C2211yescharacteristic
62#1728C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C2×#336C2211yescharacteristic
67C22×#41C2311yescharacteristic
68C2×#131—22no—
69C2×#131—22no—
70C2×#131—22no—
71C2×#131—22no—
72C2×#131—41no—
73C2×#131—41no—
74C22×M16—21no—
75C22×M16C2311yescharacteristic
76C22×M16C2311yescharacteristic
77C22×M16C2311yescharacteristic
78C22×M16C2311yescharacteristic
79C23×C8—21no—
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×#117C2311yescharacteristic
85C2×#117C2311yescharacteristic
86C2×#117C2311yescharacteristic
87C22×C42—22no—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C2×#108—22no—
92C2×#108—22no—
93C2×#108—22no—
94C2×#108—41no—
95C2×#108—41no—
96C2×#108—41no—
97#427—22no—
98#427—22no—
99#427—22no—
100#427—22no—
101C2×#107C2×C411yescharacteristic
102C2×#107C2×C411yescharacteristic
103C2×#107C2×C411yescharacteristic
104C2×#107C2×C411yescharacteristic
105C2×#106C2×C411yescharacteristic
106C2×#106C2×C411yescharacteristic
107C2×#106C2×C411yescharacteristic
108C2×#106C2×C411yescharacteristic
109#424—22no—
110#424—22no—
111#424—22no—
112#424—22no—
113#423—22no—
114#423—41no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—22no—
120C2×#104—22no—
121C2×#104—22no—
122C2×#104—22no—
123C2×#104—22no—
124C2×#104—41no—
125C2×#104Q814yes—
126#416—22no—
127#416—22no—
128#416—22no—
129#416—22no—
130C2×#103C2×C411yescharacteristic
131C2×#103C2×C411yescharacteristic
132C2×#103C2×C411yescharacteristic
133C2×#103C2×C411yescharacteristic
134C2×#102C2×C411yescharacteristic
135C2×#102C2×C411yescharacteristic
136C2×#102C2×C411yescharacteristic
137C2×#102C2×C411yescharacteristic
138#408—22no—
139#408—41no—
140C2×C22⋊C8—22no—
141#405—22no—
142#405—22no—
143C2×#97—21no—
144C2×C4×C8—21no—
145C2×C4×C8—22no—
146C2×C4×C8—22no—
147C2×C4×C8—41no—
148C2×C4×C8—41no—
149C2×C4×C8—41no—
150C2×C4×C8D414yes—
151#391C2×C411yescharacteristic
152#391C2×C411yescharacteristic
153#390C2×C411yescharacteristic
154#390C2×C411yescharacteristic
155#386—22no—
156#384C2×C411yescharacteristic
157#384C2×C411yescharacteristic
158#382C2×C411yescharacteristic
159#382C2×C411yescharacteristic
160#380—22no—
161C4×C4⋊C4—22no—
162C4×C22⋊C4—22no—
163C43—22no—
164C43—41no—
165#341—22no—
166#341—22no—
167#339—22no—
168#339—22no—
169#339—22no—
170#339—22no—
171#339—22no—
172#339—22no—
173#336—22no—
174#336—22no—
175#336—22no—
176#336—22no—
177#336—22no—
178#336C2×C411yescharacteristic
179#336C2×C411yescharacteristic
180#336C2×C411yescharacteristic
181#336C2×C411yescharacteristic
182#336D412yes—
183#336D412yes—
184#336D414yes—
185C2×#41—21no—
186C2×#41—21no—
187C2×#41—21no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41C2×Q812yes—
196C2×#41#4112yes—
197C22×Q8C22×C411yescharacteristic
198C22×D4—21no—
199C22×D4C22×C411yescharacteristic
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4#4112yes—
204C23×C4C2411yescharacteristic
205#131—28no—
206#131—44no—
207#131—44no—
208#131—44no—
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×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—42no—
232C2×M16—42no—
233C2×M16—42no—
234C2×M16C2×Q812yes—
235C2×M16#4112yes—
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×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—21no—
258C22×C8—42no—
259C22×C8—42no—
260C22×C8—42no—
261C22×C8C22×C411yescharacteristic
262C22×C8C22×C411yescharacteristic
263C22×C8C22×C411yescharacteristic
264C22×C8C22×C411yescharacteristic
265C22×C8C2×D412yes—
266C22×C8C2×D412yes—
267C22×C8C2×D412yes—
268C22×C8C2×D412yes—
269C22×C8C2×D412yes—
270C22×C8C2×D411yescharacteristic
271C22×C8C2×D411yescharacteristic
272C22×C8#4112yes—
273C22×C8#4112yes—
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#117—21no—
284#117—21no—
285#117—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4C22×C411yescharacteristic
289C2×C4⋊C4C22×C411yescharacteristic
290C2×C4⋊C4C22×C411yescharacteristic
291C2×C4⋊C4C22×C411yescharacteristic
292C2×C4⋊C4C22×C411yescharacteristic
293C2×C4⋊C4C22×C411yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
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×C42—21no—
309C2×C42—21no—
310C2×C42—28no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—44no—
316C2×C42—44no—
317C2×C42C22×C411yescharacteristic
318C2×C42C22×C411yescharacteristic
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322C2×C42C2×Q812yes—
323C2×C42#4112yes—
324#108—44no—
325#108—44no—
326#107—22no—
327#107—22no—
328#107—22no—
329#107—22no—
330#106—22no—
331#106—22no—
332#106—22no—
333#106—22no—
334C4⋊C8—44no—
335C4⋊C8—42no—
336C4⋊C8—42no—
337C4⋊C8—42no—
338C4⋊C8—42no—
339#104—28no—
340#104—44no—
341#103—22no—
342#103—22no—
343#103—22no—
344#103—22no—
345#102—22no—
346#102—22no—
347#102—22no—
348#102—22no—
349C22⋊C8—44no—
350#97—42no—
351C4×C8—44no—
352C4×C8—44no—
353C4×C8—42no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358C24—21no—
359C24C2×#4111yescharacteristic
360#41—22no—
361#41—22no—
362#41—22no—
363#41—22no—
364#41—22no—
365#41—22no—
366#41—22no—
367#41—22no—
368#41—41no—
369#41—41no—
370#41—41no—
371#41—41no—
372#41—41no—
373#41—41no—
374#41—41no—
375#41—41no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8C4×Q812yes—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4C4×Q812yes—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4C4×D412yes—
422C22×C4C4×D412yes—
423C22×C4C23×C411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×Q811yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4H(5,𝔽2)11yescharacteristic
434C22×C4H(5,𝔽2)11yescharacteristic
435C22×C42−1+411yescharacteristic
436C22×C42−1+411yescharacteristic
437M16—22no—
438M16—22no—
439M16—22no—
440M16—22no—
441M16—48no—
442M16—41no—
443M16—41no—
444M16—41no—
445M16—41no—
446M16—41no—
447M16—41no—
448M16—41no—
449M16—41no—
450M16—41no—
451M16—41no—
452M16—41no—
453M16—41no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—48no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8C4×D412yes—
499C2×C8C4×D412yes—
500C2×C8C4×D412yes—
501C2×C8C4×D412yes—
502C2×C8#12712yes—
503C2×C8#12712yes—
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—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—44no—
517C4⋊C4—82no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—44no—
525C22⋊C4—82no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—416no—
538C42—41no—
539C42—41no—
540C42—41no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—41no—
546C42C22⋊Q814yes—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—41no—
551C23—41no—
552C23—42no—
553C23—42no—
554C23C2×C4×D411yescharacteristic
555C23#53211yescharacteristic
556C23#53211yescharacteristic
557C23#55111yescharacteristic
558C23#55211yescharacteristic
559C23#55211yescharacteristic
560C23#55411yescharacteristic
561Q8—22no—
562Q8—22no—
563Q8—41no—
564Q8—41no—
565D4—22no—
566D4—22no—
567D4—42no—
568D4—42no—
569D4—42no—
570D4—42no—
571D4—41no—
572D4—41no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4#44314yes—
610C2×C4C2×C4×D411yescharacteristic
611C2×C4C2×C4×D411yescharacteristic
612C2×C4C2×C4×Q811yescharacteristic
613C2×C4C4×#4111yescharacteristic
614C2×C4#52011yescharacteristic
615C2×C4#52011yescharacteristic
616C2×C4C2×C22⋊Q811yescharacteristic
617C2×C4#52511yescharacteristic
618C2×C4#52911yescharacteristic
619C2×C4C2×#12711yescharacteristic
620C2×C4#53211yescharacteristic
621C2×C4#53311yescharacteristic
622C2×C4#53411yescharacteristic
623C2×C4#53611yescharacteristic
624C2×C4#53911yescharacteristic
625C2×C4#53911yescharacteristic
626C2×C4#54611yescharacteristic
627C2×C4#54611yescharacteristic
628C2×C4#54611yescharacteristic
629C2×C4#54611yescharacteristic
630C2×C4#54711yescharacteristic
631C2×C4#54711yescharacteristic
632C2×C4D4×Q811yescharacteristic
633C2×C4D4×Q811yescharacteristic
634C2×C4#55111yescharacteristic
635C2×C4#55211yescharacteristic
636C2×C4#55211yescharacteristic
637C2×C4#55411yescharacteristic
638C8—22no—
639C8—22no—
640C8—22no—
641C8—22no—
642C8—22no—
643C8—22no—
644C8—42no—
645C8—42no—
646C8—42no—
647C8—81no—
648C8—81no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—42no—
658C22—44no—
659C22—82no—
660C22—81no—
661C22—81no—
662C22C4×#12711yescharacteristic
663C22#235911yescharacteristic
664C22#236011yescharacteristic
665C22#255311yescharacteristic
666C22#255611yescharacteristic
667C22#256411yescharacteristic
668C22#261111yescharacteristic
669C22C2×#44312yes—
670C22#295612yes—
671C4—28no—
672C4—24no—
673C4—44no—
674C4—42no—
675C4—42no—
676C4—81no—
677C4—81no—
678C4—81no—
679C4—81no—
680C4—81no—
681C4—81no—
682C4C4×C22⋊Q811yescharacteristic
683C4#232211yescharacteristic
684C4#235911yescharacteristic
685C4#236011yescharacteristic
686C4#253911yescharacteristic
687C4#254211yescharacteristic
688C4#255311yescharacteristic
689C4#255611yescharacteristic
690C2—21no—
691C2—21no—
692C2—21no—
693C2—21no—
694C2—82no—
695C2#1423011yescharacteristic
696C2#2170511yescharacteristic
697C2#2171011yescharacteristic
698C2#2294812yes—
699C2#2298212yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
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
2. C2×#1945C2semidirect
3. C2×#1922C2semidirect
4. #21025C2semidirect
8. #11376C2semidirect
9. #11326C2semidirect
13. #10859C2semidirect
14. #10839C2semidirect
15. C4×#336C2semidirect
35. #2097C4semidirect
36. #2097C4semidirect
37. #2096C4semidirect
38. #2095C4semidirect
39. #2094C4semidirect
40. #2093C4semidirect
41. #2092C4semidirect
42. #2092C4semidirect

Maximal subgroups

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

subgroupindex
#237302
C2×#19452
C2×#19222
#210252
C2×#18662
#208392
C2×#17642
#113762
#113262
#111252
#109292
#109232
#108592
#108392
C4×#3362