Finite Group Databasebeta

← #225110

Subgroups — #225110

Order 512.

870 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626C211yesmaximal, characteristic, complemented
2D4×C22⋊C8C211yesmaximal, characteristic
3#20732C211yesmaximal, characteristic
4#9558C211yesmaximal, characteristic, complemented
5#9554C211yesmaximal, characteristic, complemented
6#9546C211yesmaximal, characteristic, complemented
7#9542C211yesmaximal, characteristic, complemented
8#8845C211yesmaximal, characteristic, complemented
9#8845C211yesmaximal, characteristic, complemented
10#8710C211yesmaximal, characteristic, complemented
11#8598C211yesmaximal, characteristic, complemented
12#8310C211yesmaximal, characteristic
13#8202C211yesmaximal, characteristic, complemented
14C8×#102C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C22×C22⋊C8—21no—
21#2285C2211yescharacteristic
22#1954C2211yescharacteristic
23#1954C2211yescharacteristic
24#1939C2211yescharacteristic, complemented
25#1939C2211yescharacteristic, complemented
26#1917—21no—
27#1916—21no—
28#1869—21no—
29#1869—21no—
30#1823C411yescharacteristic
31#1823C411yescharacteristic
32#1802C411yescharacteristic
33#1802C411yescharacteristic
34#1802C411yescharacteristic
35#1802C411yescharacteristic
36#1798—21no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39#1786C411yescharacteristic
40#1786C411yescharacteristic
41#1779—21no—
42#1770C2211yescharacteristic, complemented
43C8×C4⋊C4C2211yescharacteristic, complemented
44C4×#102C2211yescharacteristic
45C4×#102C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1750—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#334C2211yescharacteristic, complemented
52C2×#334C2211yescharacteristic, complemented
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×#322C2211yescharacteristic, complemented
58C2×C82C2211yescharacteristic, complemented
59#1279C2211yescharacteristic, complemented
60#1279C2211yescharacteristic, complemented
61#1279C2211yescharacteristic, complemented
62#1279C2211yescharacteristic, complemented
63#1277C2211yescharacteristic, complemented
64#1277C2211yescharacteristic, complemented
65#1273C2211yescharacteristic, complemented
66#1273C2211yescharacteristic, complemented
67C23×D4C2×C411yescharacteristic
68C24×C4—21no—
69C23×C8—21no—
70C23×C8—21no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—41no—
76C2×C4×D4—41no—
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C2×C4×D4C2311yescharacteristic
81C2×#117—21no—
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C22⋊C4—21no—
84C22×C42C2311yescharacteristic
85C8×D4—22no—
86C8×D4—22no—
87C8×D4—41no—
88C8×D4—41no—
89#431—41no—
90#431—41no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2×C411yescharacteristic
94C2×C4⋊C8C2×C411yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#102C2×C411yescharacteristic
100C2×#102C2×C411yescharacteristic
101C2×#102C2×C411yescharacteristic
102C2×#102C2×C411yescharacteristic
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—44no—
117C2×C22⋊C8C2×C411yescharacteristic
118C2×C22⋊C8C2×C411yescharacteristic
119C2×C22⋊C8D411yescharacteristic
120C2×C22⋊C8D411yescharacteristic
121C2×#97—21no—
122C2×C4×C8—21no—
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129#384C2×C411yescharacteristic
130#384C2×C411yescharacteristic
131#382—21no—
132#380—21no—
133#380C2×C411yescharacteristic
134#380C2×C411yescharacteristic
135C4×C4⋊C4—21no—
136C4×C4⋊C4C2311yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336D411yescharacteristic
154#336D411yescharacteristic
155#334—41no—
156#334—41no—
157D4⋊C8—22no—
158D4⋊C8—22no—
159D4⋊C8—41no—
160D4⋊C8—41no—
161#322—41no—
162C82—41no—
163C25—21no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—21no—
169C22×D4—41no—
170C22×D4—41no—
171C22×D4C22×C411yescharacteristic
172C22×D4C22×C411yescharacteristic
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—81no—
183C23×C4—81no—
184C23×C4C22×C411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4#4111yescharacteristic
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—42no—
194C22×C8—42no—
195C22×C8—42no—
196C22×C8—42no—
197C22×C8—81no—
198C22×C8—81no—
199C22×C8C22×C411yescharacteristic
200C22×C8C22×C411yescharacteristic
201C22×C8C22×C411yescharacteristic
202C22×C8C22×C411yescharacteristic
203C22×C8C2×D411yescharacteristic
204C22×C8C2×D411yescharacteristic
205C22×C8C2×D411yescharacteristic
206C22×C8C2×D411yescharacteristic
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—21no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—41no—
218C4×D4—41no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4#4112yes—
222C4×D4#4112yes—
223#117—81no—
224#117—81no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42C22×C411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42#4111yescharacteristic
255C2×C42C2411yescharacteristic
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—41no—
265C4⋊C8—41no—
266C4⋊C8—81no—
267C4⋊C8—81no—
268#102—22no—
269#102—22no—
270#102—22no—
271#102—22no—
272#102—22no—
273#102—22no—
274#102—41no—
275#102—41no—
276C22⋊C8—22no—
277C22⋊C8—41no—
278C22⋊C8—41no—
279C22⋊C8—42no—
280C22⋊C8—42no—
281C22⋊C8—42no—
282C22⋊C8—42no—
283C22⋊C8—42no—
284C22⋊C8—42no—
285C22⋊C8—42no—
286C22⋊C8—81no—
287C22⋊C8—81no—
288C22⋊C8SD1614yes—
289#97—81no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—81no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307C24—21no—
308C24—24no—
309C24—21no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—81no—
315C24—81no—
316C24C4×D411yescharacteristic
317C2×D4—24no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4#11712yes—
333C2×D4#13112yes—
334C2×D4#13112yes—
335C2×D4#13112yes—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—81no—
375C22×C4—81no—
376C22×C4—84no—
377C22×C4—84no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4—81no—
381C22×C4C4×D411yescharacteristic
382C22×C4C4×D411yescharacteristic
383C22×C4C4×D411yescharacteristic
384C22×C4C2×SD1612yes—
385C22×C4#13512yes—
386C22×C4C23×C411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—22no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—84no—
430C2×C8—84no—
431C2×C8C2×SD1612yes—
432C2×C8C2×SD1612yes—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—81no—
449C4⋊C4#11712yes—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—41no—
455C22⋊C4—41no—
456C22⋊C4—81no—
457C22⋊C4—81no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—41no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42C2×#4111yescharacteristic
469C42C2×#4111yescharacteristic
470C42H(5,𝔽2)11yescharacteristic
471C42H(5,𝔽2)11yescharacteristic
472C23—22no—
473C23—21no—
474C23—21no—
475C23—22no—
476C23—21no—
477C23—22no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—22no—
482C23—48no—
483C23—48no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—84no—
495C23—84no—
496C23—84no—
497C23—84no—
498C23—84no—
499C23C4×SD1612yes—
500C23C2×C4×D411yescharacteristic
501D4—22no—
502D4—22no—
503D4—22no—
504D4—22no—
505D4—22no—
506D4—48no—
507D4—41no—
508D4—41no—
509D4#43314yes—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4C4×SD1612yes—
560C2×C4C2×#11711yescharacteristic
561C2×C4#51811yescharacteristic
562C2×C4#54611yescharacteristic
563C2×C4#54611yescharacteristic
564C2×C4#55111yescharacteristic
565C2×C4#55111yescharacteristic
566C2×C4C2×#13111yescharacteristic
567C2×C4C2×#13111yescharacteristic
568C2×C4C2×#13111yescharacteristic
569C2×C4#56811yescharacteristic
570C2×C4C22×SD1611yescharacteristic
571C2×C4C2×#13511yescharacteristic
572C2×C4#57611yescharacteristic
573C2×C4#57611yescharacteristic
574C8—22no—
575C8—22no—
576C8—42no—
577C8—41no—
578C8—41no—
579C8—41no—
580C8—41no—
581C8—41no—
582C8—41no—
583C8—81no—
584C8—81no—
585C8—81no—
586C8—81no—
587C8—81no—
588C22—24no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—41no—
593C22—42no—
594C22—41no—
595C22—42no—
596C22—41no—
597C22—42no—
598C22—42no—
599C22—41no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—816no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22#234211yescharacteristic
611C22#292911yescharacteristic
612C22#293311yescharacteristic
613C22C2×C4×SD1611yescharacteristic
614C22#294811yescharacteristic
615C22#295511yescharacteristic
616C22#295611yescharacteristic
617C4—22no—
618C4—21no—
619C4—21no—
620C4—41no—
621C4—42no—
622C4—41no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4C2×#43311yescharacteristic
629C4#292411yescharacteristic
630C4#297611yescharacteristic
631C4#297711yescharacteristic
632C4#329511yescharacteristic
633C4#329511yescharacteristic
634C4#329711yescharacteristic
635C4#329711yescharacteristic
636C2—42no—
637C2—44no—
638C2—82no—
639C2#1790211yescharacteristic
640C2#1792211yescharacteristic
641C2#1799511yescharacteristic
642C2#1802611yescharacteristic
643C2#2255811yescharacteristic
644C2#2257411yescharacteristic
645C2#2258811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22626C2semidirect
4. #9558C2semidirect
5. #9554C2semidirect
6. #9546C2semidirect
7. #9542C2semidirect
8. #8845C2semidirect
9. #8845C2semidirect
10. #8710C2semidirect
11. #8598C2semidirect
13. #8202C2semidirect
14. C8×#102C2semidirect
15. #8148C2semidirect
24. #1939C22semidirect
25. #1939C22semidirect
42. #1770C22semidirect
43. C8×C4⋊C4C22semidirect
51. C2×#334C22semidirect
52. C2×#334C22semidirect
57. C2×#322C22semidirect
58. C2×C82C22semidirect
59. #1279C22semidirect
60. #1279C22semidirect
61. #1279C22semidirect
62. #1279C22semidirect
63. #1277C22semidirect
64. #1277C22semidirect
65. #1273C22semidirect
66. #1273C22semidirect

Maximal subgroups

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

subgroupindex
#226262
D4×C22⋊C82
#207322
#95582
#95542
#95462
#95422
#88452
#87102
#85982
#83102
#82022
C8×#1022
#81482