Finite Group Databasebeta

← #380233

Subgroups — #380233

Order 512.

2898 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#380C211yesmaximal, characteristic
2C2×#1853C211yesmaximal, characteristic, complemented
3C2×#1852C211yesmaximal, characteristic, complemented
4C2×#1779C211yesmaximal, characteristic
5#14680C211yesmaximal, characteristic
6#11096C214yesmaximal, complemented
7#11095C214yesmaximal, complemented
8#10730C211yesmaximal, characteristic, complemented
9#10730C211yesmaximal, characteristic, complemented
10C25×C4C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C22⋊C4—21no—
14C22×#107C2212yescomplemented
15C22×#106C2212yescomplemented
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2214yes—
19C2×#380—22no—
20C2×#380—24no—
21C2×#380—24no—
22C2×#380C2214yes—
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#379—21no—
28C22×#95—21no—
29#1853—24no—
30#1852—24no—
31#1779C418yescomplemented
32C2×#340C2212yescomplemented
33C2×#340C2212yescomplemented
34C2×#340C2214yescomplemented
35C2×#340C2214yescomplemented
36C22⋊#95—21no—
37C22⋊#95—22no—
38C26D411yescharacteristic
39C24×C4—22no—
40C24×C4—22no—
41C24×C4—22no—
42C24×C4—22no—
43C24×C4—21no—
44C24×C4—21no—
45C24×C4—44no—
46C24×C4—44no—
47C24×C4C2×C414yes—
48C24×C4D411yescharacteristic
49C24×C4D412yes—
50C24×C4Q814yes—
51C24×C4C2312yes—
52C24×C4C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2312yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—44no—
60C22×C4⋊C4—44no—
61C22×C4⋊C4D414yes—
62C22×C4⋊C4D414yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—28no—
69C22×C22⋊C4—24no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—48no—
75C2×#107—24no—
76C2×#107—44no—
77C2×#106—24no—
78C2×#106—44no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—44no—
84C2×C22⋊C8—44no—
85C2×C22⋊C8C2×C414yes—
86C2×C22⋊C8C2×C414yes—
87C2×C22⋊C8C2×C4116yescomplemented
88#380—216no—
89#380—28no—
90#380—28no—
91#380—22no—
92#380—22no—
93#380—22no—
94#380—22no—
95#380—44no—
96#380—44no—
97#379—44no—
98C2×#95—28no—
99C2×#95—24no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—21no—
104#340—24no—
105#340—24no—
106#340—44no—
107#340—44no—
108C25—22no—
109C25—22no—
110C25—24no—
111C25—24no—
112C25—22no—
113C25—22no—
114C25—22no—
115C25—22no—
116C25—42no—
117C25—42no—
118C25C4⋊C414yes—
119C25C2×D412yes—
120C25C2×D411yescharacteristic
121C23×C4—24no—
122C23×C4—24no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—28no—
126C23×C4—28no—
127C23×C4—24no—
128C23×C4—28no—
129C23×C4—28no—
130C23×C4—24no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—416no—
147C23×C4—416no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—48no—
155C23×C4—48no—
156C23×C4—48no—
157C23×C4—48no—
158C23×C4—416no—
159C23×C4—416no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4C4⋊C414yes—
169C23×C4C4⋊C418yes—
170C23×C4C22×C414yes—
171C23×C4C22×C412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×Q814yes—
177C23×C4C2×Q812yes—
178C23×C4#4112yes—
179C23×C4#4112yes—
180C23×C4C2411yescharacteristic
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—24no—
186C22×C8—24no—
187C22×C8—24no—
188C22×C8—24no—
189C22×C8C22×C414yes—
190C22×C8C22×C414yes—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4—48no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4C2×D414yes—
217C2×C22⋊C4—216no—
218C2×C22⋊C4—432no—
219C2×C22⋊C4—48no—
220C2×C22⋊C4—48no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—44no—
233#107—48no—
234#106—48no—
235C22⋊C8—28no—
236C22⋊C8—28no—
237C22⋊C8—416no—
238#95—216no—
239#95—44no—
240#95—44no—
241#95—42no—
242#95—42no—
243C24—22no—
244C24—22no—
245C24—22no—
246C24—22no—
247C24—28no—
248C24—216no—
249C24—28no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—28no—
263C24—216no—
264C24—28no—
265C24—28no—
266C24—48no—
267C24—48no—
268C24—48no—
269C24—48no—
270C24—48no—
271C24—48no—
272C24—48no—
273C24—48no—
274C24—48no—
275C24—44no—
276C24—44no—
277C24—48no—
278C24—44no—
279C24—44no—
280C24—44no—
281C24—44no—
282C24C2×C4⋊C414yes—
283C24C2×C4⋊C412yes—
284C24#12012yes—
285C24#12312yes—
286C24#13512yes—
287C24#13512yes—
288C24AGL(1,ℤ/8)14yes—
289C24AGL(1,ℤ/8)12yes—
290C24#13714yes—
291C24#13712yes—
292C24C22×D411yescharacteristic
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—28no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—28no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—28no—
313C22×C4—28no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—28no—
317C22×C4—28no—
318C22×C4—216no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—48no—
322C22×C4—48no—
323C22×C4—48no—
324C22×C4—48no—
325C22×C4—48no—
326C22×C4—48no—
327C22×C4—48no—
328C22×C4—432no—
329C22×C4—432no—
330C22×C4—432no—
331C22×C4—432no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—48no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—48no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—48no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—88no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4—88no—
378C22×C4C2×C4⋊C414yes—
379C22×C4C2×C4⋊C412yes—
380C22×C4C4×D414yes—
381C22×C4C4×D414yes—
382C22×C4#12012yes—
383C22×C4C22⋊Q814yes—
384C22×C4C22⋊Q814yes—
385C22×C4#12312yes—
386C22×C4C23×C411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×Q811yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4H(5,𝔽2)14yes—
393C22×C4H(5,𝔽2)14yes—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—28no—
397C2×C8—28no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—44no—
403C2×C8—44no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C22⋊C4—432no—
419C22⋊C4—88no—
420C22⋊C4—88no—
421C22⋊C4—84no—
422C22⋊C4—84no—
423C23—24no—
424C23—24no—
425C23—22no—
426C23—22no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—22no—
438C23—22no—
439C23—232no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—432no—
459C23—432no—
460C23—416no—
461C23—416no—
462C23—416no—
463C23—416no—
464C23—416no—
465C23—432no—
466C23—432no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—416no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23#38514yes—
485C23#42714yes—
486C23#42818yes—
487C23#42814yes—
488C23#44914yes—
489C23#45214yes—
490C23#48214yes—
491C23#48514yes—
492C23C22×C4⋊C411yescharacteristic
493C23C2×#12011yescharacteristic
494C23C2×#12311yescharacteristic
495C23#53412yes—
496C23#53412yes—
497C23C2×#13511yescharacteristic
498C23C2×#13511yescharacteristic
499C23C2×AGL(1,ℤ/8)11yescharacteristic
500C23C2×AGL(1,ℤ/8)12yes—
501C23C2×#13711yescharacteristic
502C23C2×#13712yes—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—432no—
515C2×C4—416no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—48no—
521C2×C4—48no—
522C2×C4—48no—
523C2×C4—48no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—816no—
531C2×C4—816no—
532C2×C4—84no—
533C2×C4—84no—
534C2×C4—84no—
535C2×C4—84no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4#38514yes—
554C2×C4C2×C4×D411yescharacteristic
555C2×C4C2×C4×D411yescharacteristic
556C2×C4#51814yes—
557C2×C4C2×C22⋊Q811yescharacteristic
558C2×C4C2×C22⋊Q811yescharacteristic
559C2×C4#53412yes—
560C2×C4#53412yes—
561C2×C4#53512yes—
562C2×C4#53512yes—
563C2×C4#53814yes—
564C2×C4#53814yes—
565C2×C4#54314yes—
566C8—44no—
567C8—44no—
568C22—24no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—28no—
574C22—24no—
575C22—28no—
576C22—28no—
577C22—28no—
578C22—28no—
579C22—28no—
580C22—28no—
581C22—464no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22#185418yes—
595C22C2×#38511yescharacteristic
596C22#231512yes—
597C22#261311yescharacteristic
598C22#261311yescharacteristic
599C22#262212yes—
600C22C2×#42711yescharacteristic
601C22C2×#42811yescharacteristic
602C22C2×#42812yes—
603C22C2×#44911yescharacteristic
604C22C2×#45211yescharacteristic
605C22#300311yescharacteristic
606C22#300411yescharacteristic
607C22C2×#48211yescharacteristic
608C22C2×#48511yescharacteristic
609C22#309212yes—
610C22#318812yes—
611C22#318812yes—
612C22#319012yes—
613C22#319012yes—
614C22#319212yes—
615C22#319312yes—
616C22#319312yes—
617C22#319512yes—
618C4—24no—
619C4—44no—
620C4—84no—
621C4—84no—
622C4—161no—
623C4—161no—
624C4—162no—
625C4#232112yes—
626C4#232112yes—
627C4#262612yes—
628C4#262612yes—
629C2—28no—
630C2—48no—
631C2#1426211yescharacteristic
632C2C2×#185411yescharacteristic
633C2#2084411yescharacteristic
634C2#2194112yes—
635C2#2194212yes—
636C2#2959411yescharacteristic
637C2#2959511yescharacteristic
638C2#2959811yescharacteristic
639C2#2959911yescharacteristic
640C2#2989112yes—
641C2#2989412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1853C2semidirect
3. C2×#1852C2semidirect
6. #11096C2semidirect
7. #11095C2semidirect
8. #10730C2semidirect
9. #10730C2semidirect
14. C22×#107C22semidirect
15. C22×#106C22semidirect
31. #1779C4semidirect
32. C2×#340C22semidirect
33. C2×#340C22semidirect
34. C2×#340C22semidirect
35. C2×#340C22semidirect
87. C2×C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3802
C2×#18532
C2×#18522
C2×#17792
#146802
#110962
#110952
#107302