Finite Group Databasebeta

← #106984

Subgroups — #106984

Order 512.

1189 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#117C211yesmaximal, characteristic
2C2×#1486C211yesmaximal, characteristic
3C2×#1486C211yesmaximal, characteristic
4#10856C211yesmaximal, characteristic, complemented
5#8095C211yesmaximal, characteristic, complemented
6#7957C211yesmaximal, characteristic, complemented
7#7924C211yesmaximal, characteristic, complemented
8C22×C4×D4C2211yescharacteristic
9C22×#117—21no—
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#423C2211yescharacteristic
14#2354—21no—
15#2352—22no—
16D4×C4⋊C4—21no—
17#2344C412yes—
18#2342—22no—
19D4×C22⋊C4C412yes—
20#2293—21no—
21#2290—22no—
22#2285C2211yescharacteristic
23#2280—21no—
24C42×D4—21no—
25C4×#117C2211yescharacteristic
26C2×#344—21no—
27C2×#344—21no—
28#1486—21no—
29#1486—21no—
30#1486—21no—
31#1486—21no—
32#1486—22no—
33#1486C414yescomplemented
34C2×D4⋊C8—22no—
35C2×D4⋊C8C414yescomplemented
36#1303—22no—
37#1286—22no—
38#1282—22no—
39C23×D4C2×C411yescharacteristic
40C24×C4—21no—
41C22×M16—21no—
42C22×M16—21no—
43C2×C4×D4—22no—
44C2×C4×D4—21no—
45C2×C4×D4—21no—
46C2×C4×D4—21no—
47C2×C4×D4—21no—
48C2×C4×D4—21no—
49C2×C4×D4—21no—
50C2×C4×D4C2×C412yes—
51C2×C4×D4C2×C412yes—
52C2×C4×D4D411yescharacteristic
53C2×C4×D4D411yescharacteristic
54C2×C4×D4D411yescharacteristic
55C2×C4×D4D411yescharacteristic
56C2×C4×D4D412yes—
57C2×C4×D4Q812yes—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—21no—
61C2×#117—22no—
62C2×#117—42no—
63C2×#117—42no—
64C2×#117—42no—
65C2×#117—42no—
66C2×#117D411yescharacteristic
67C2×#117D411yescharacteristic
68C22×C4⋊C4—22no—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C42—22no—
73C22×C42C2311yescharacteristic
74#431—21no—
75#431—21no—
76#431—21no—
77#431—21no—
78#431—21no—
79#431—21no—
80#431—21no—
81#431C2×C412yescomplemented
82#423—21no—
83#423—21no—
84#423—21no—
85#423—21no—
86#423—21no—
87#423—21no—
88#423—21no—
89#423C2×C412yescomplemented
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2×C412yescomplemented
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C4×C8—21no—
99C2×C4×C8C2×C412yescomplemented
100#390—22no—
101#389—22no—
102#388—24no—
103#388—24no—
104#387—24no—
105#386—24no—
106#385—24no—
107#384—21no—
108#382D412yes—
109#381—22no—
110#380—22no—
111#380—22no—
112#380—21no—
113#380C2×C412yes—
114#379—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4D412yes—
120C4×C22⋊C4—24no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4C2×C412yes—
124C4×C22⋊C4D412yes—
125#376—21no—
126C43—21no—
127#344—42no—
128#344—42no—
129D4⋊C8—24no—
130D4⋊C8—24no—
131D4⋊C8—24no—
132D4⋊C8—24no—
133C25—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—44no—
138C22×D4C4212yes—
139C22×D4C22⋊C411yescharacteristic
140C22×D4C22⋊C411yescharacteristic
141C22×D4C4⋊C412yes—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—41no—
153C23×C4—41no—
154C23×C4—41no—
155C23×C4—41no—
156C23×C4—41no—
157C23×C4—41no—
158C23×C4—41no—
159C23×C4—41no—
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C2×M16—22no—
164C2×M16—41no—
165C2×M16—41no—
166C2×M16—41no—
167C2×M16—41no—
168C2×M16—41no—
169C2×M16—41no—
170C2×M16—41no—
171C2×M16—41no—
172C2×M16—41no—
173C2×M16—41no—
174C2×M16—41no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—41no—
178C22×C8—41no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—48no—
196C4×D4C22⋊C412yes—
197C4×D4C22⋊C412yes—
198C4×D4C22⋊C412yes—
199C4×D4C22⋊C412yes—
200C4×D4C4⋊C412yes—
201C4×D4C4⋊C412yes—
202C4×D4C4⋊C412yes—
203C4×D4C4⋊C412yes—
204#117—21no—
205#117—21no—
206#117—21no—
207#117—21no—
208#117—21no—
209#117—21no—
210#117—44no—
211#117—44no—
212#117—44no—
213#117—44no—
214#117—44no—
215#117—41no—
216#117—41no—
217#117—41no—
218#117—41no—
219#117—44no—
220#117D812yes—
221#117SD1612yes—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C4212yes—
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C4⋊C412yes—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4C22⋊C412yes—
252C2×C22⋊C4C22⋊C412yes—
253C2×C42—21no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—44no—
260C2×C42—44no—
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×Q811yescharacteristic
269C2×C42#4111yescharacteristic
270C2×C42#4111yescharacteristic
271C2×C42#4111yescharacteristic
272C2×C42#4111yescharacteristic
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—22no—
278C4⋊C8—41no—
279C4⋊C8—41no—
280C4⋊C8—41no—
281C4⋊C8—41no—
282C4⋊C8—41no—
283C4⋊C8—41no—
284C4⋊C8—41no—
285C22⋊C8—22no—
286C22⋊C8—41no—
287C22⋊C8—41no—
288C22⋊C8—41no—
289C4×C8—22no—
290C4×C8—22no—
291C4×C8—22no—
292C4×C8—41no—
293#95—24no—
294#95—24no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95C22⋊C414yes—
299C24—21no—
300C24—21no—
301C24—22no—
302C24—22no—
303C24—44no—
304C24—41no—
305C24—41no—
306C24—41no—
307C24—41no—
308C24—82no—
309C24C2×C22⋊C411yescharacteristic
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4#9512yes—
324C2×D4#9512yes—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—42no—
341C22×C4—48no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—48no—
379C22×C4—81no—
380C22×C4—84no—
381C22×C4C2×C4211yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C4⋊C411yescharacteristic
385C22×C4#11711yescharacteristic
386C22×C4#11711yescharacteristic
387C22×C4C4×D411yescharacteristic
388C22×C4C4×D411yescharacteristic
389C22×C4C4×D411yescharacteristic
390C22×C4C4×D411yescharacteristic
391C22×C4#12011yescharacteristic
392C22×C4#12011yescharacteristic
393C22×C4#12011yescharacteristic
394C22×C4#12011yescharacteristic
395C22×C4#12111yescharacteristic
396C22×C4#12111yescharacteristic
397C22×C4C22⋊Q811yescharacteristic
398C22×C4C22⋊Q811yescharacteristic
399C22×C4C2×D811yescharacteristic
400C22×C4C2×SD1611yescharacteristic
401C22×C4#13511yescharacteristic
402C22×C4#13511yescharacteristic
403C22×C4AGL(1,ℤ/8)11yescharacteristic
404C22×C4AGL(1,ℤ/8)11yescharacteristic
405C22×C4AGL(1,ℤ/8)11yescharacteristic
406C22×C4#13711yescharacteristic
407M16—42no—
408M16—42no—
409M16—81no—
410M16—81no—
411M16—81no—
412M16—81no—
413M16—81no—
414M16—81no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—82no—
422C2×C8—81no—
423C2×C8—81no—
424C2×C8—82no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—48no—
432C4⋊C4—44no—
433C4⋊C4—42no—
434C4⋊C4—41no—
435C4⋊C4—42no—
436C4⋊C4—41no—
437C4⋊C4—82no—
438C4⋊C4#9512yes—
439C4⋊C4#9512yes—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—84no—
456C22⋊C4#10214yes—
457C42—22no—
458C42—22no—
459C42—48no—
460C42—44no—
461C42—42no—
462C42—41no—
463C42—42no—
464C42—41no—
465C42—82no—
466C42C2×C22⋊C411yescharacteristic
467C42C2×C22⋊C411yescharacteristic
468C42C2×C4⋊C411yescharacteristic
469C42C2×C4⋊C411yescharacteristic
470C42C4×D411yescharacteristic
471C42C4×D411yescharacteristic
472C42C4×D411yescharacteristic
473C42C4×D411yescharacteristic
474C42#12111yescharacteristic
475C42#12111yescharacteristic
476C42C22⋊Q811yescharacteristic
477C42C22⋊Q811yescharacteristic
478C42#12311yescharacteristic
479C42#12311yescharacteristic
480C42#12311yescharacteristic
481C42#12311yescharacteristic
482C23—24no—
483C23—22no—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—41no—
493C23—41no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—82no—
501C23—82no—
502C23—84no—
503C23—84no—
504C23—84no—
505C23—84no—
506C23C4×C22⋊C411yescharacteristic
507C23#37911yescharacteristic
508C23#38011yescharacteristic
509C23C2×#10211yescharacteristic
510C23#41611yescharacteristic
511C23#41711yescharacteristic
512C23#41811yescharacteristic
513D4—24no—
514D4—22no—
515D4—22no—
516D4—22no—
517D4—44no—
518D4—44no—
519D4—42no—
520D4#34414yescomplemented
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—816no—
569C2×C4C2×#9511yescharacteristic
570C2×C4C4×C22⋊C411yescharacteristic
571C2×C4#38011yescharacteristic
572C2×C4#38111yescharacteristic
573C2×C4#38511yescharacteristic
574C2×C4#38511yescharacteristic
575C2×C4#38511yescharacteristic
576C2×C4#38511yescharacteristic
577C2×C4#38611yescharacteristic
578C2×C4#38611yescharacteristic
579C2×C4#38611yescharacteristic
580C2×C4#38611yescharacteristic
581C2×C4C4×D811yescharacteristic
582C2×C4C4×D811yescharacteristic
583C2×C4C4×SD1611yescharacteristic
584C2×C4C4×SD1611yescharacteristic
585C2×C4#44011yescharacteristic
586C2×C4#44011yescharacteristic
587C2×C4#44211yescharacteristic
588C2×C4#44211yescharacteristic
589C2×C4#44711yescharacteristic
590C2×C4#44711yescharacteristic
591C2×C4#44911yescharacteristic
592C2×C4#44911yescharacteristic
593C2×C4#45011yescharacteristic
594C2×C4#45011yescharacteristic
595C2×C4#45211yescharacteristic
596C2×C4#45211yescharacteristic
597C2×C4#45911yescharacteristic
598C2×C4#46111yescharacteristic
599C2×C4#46311yescharacteristic
600C2×C4#46311yescharacteristic
601C2×C4#47411yescharacteristic
602C2×C4#47611yescharacteristic
603C2×C4#47811yescharacteristic
604C2×C4#47811yescharacteristic
605C8—82no—
606C8—82no—
607C8—82no—
608C8—82no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—24no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—84no—
628C22—84no—
629C22—84no—
630C22—84no—
631C22—82no—
632C22—82no—
633C22C22⋊#9511yescharacteristic
634C22C4×#10211yescharacteristic
635C22#176211yescharacteristic
636C22#178611yescharacteristic
637C22#178811yescharacteristic
638C22#180211yescharacteristic
639C22#180411yescharacteristic
640C4—21no—
641C4—21no—
642C4—21no—
643C4—21no—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—44no—
648C4—42no—
649C4—42no—
650C4—82no—
651C4—84no—
652C4C2×#34412yes—
653C4#186211yescharacteristic
654C4#186211yescharacteristic
655C4#186411yescharacteristic
656C4#186411yescharacteristic
657C4#187711yescharacteristic
658C4#187711yescharacteristic
659C4#187911yescharacteristic
660C4#187911yescharacteristic
661C2—21no—
662C2—21no—
663C2—21no—
664C2—21no—
665C2—21no—
666C2—21no—
667C2—44no—
668C2—82no—
669C2#1073811yescharacteristic
670C2#1078311yescharacteristic
671C2#1078811yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10856C2semidirect
5. #8095C2semidirect
6. #7957C2semidirect
7. #7924C2semidirect
33. #1486C4semidirect
35. C2×D4⋊C8C4semidirect
81. #431C2×C4semidirect
89. #423C2×C4semidirect
95. C2×C4⋊C8C2×C4semidirect
99. C2×C4×C8C2×C4semidirect
520. D4#344semidirect

Maximal subgroups

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

subgroupindex
D4×#1172
C2×#14862
#108562
#80952
#79572
#79242