Finite Group Databasebeta

← #388068

Subgroups — #388068

Order 512.

1492 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1870C211yesmaximal, characteristic
2#13906C211yesmaximal, characteristic
3#13872C211yesmaximal, characteristic
4#11267C212yesmaximal, complemented
5#11095C212yesmaximal, complemented
6#11095C212yesmaximal, complemented
7#10908C212yesmaximal, complemented
8#10730C212yesmaximal, complemented
9#10722C212yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×#106C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2211yescharacteristic
16C2×#390—21no—
17C2×#386—21no—
18C2×#384C2211yescharacteristic
19C2×#380C2212yes—
20C2×#380C2212yes—
21C2×#380C2212yes—
22C2×#380C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1870C418yescomplemented
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33#1441—21no—
34#1439—21no—
35#1439—21no—
36#1439—21no—
37C4⋊#95—21no—
38C4⋊#95C2212yes—
39C22⋊#95—21no—
40C23×D4D411yescharacteristic
41C24×C4—21no—
42C24×C4C2311yescharacteristic
43C24×C4C2311yescharacteristic
44C22×M16—41no—
45C22×M16—41no—
46C22×M16—42no—
47C22×M16—42no—
48C22×M16C2×C414yescomplemented
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4×D4—41no—
52C2×C4×D4—41no—
53C2×C4×D4—41no—
54C2×C4×D4—41no—
55C2×C4×D4C2×C414yes—
56C2×C4×D4Q814yes—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—41no—
62C22×C4⋊C4—41no—
63C22×C4⋊C4—42no—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4D411yescharacteristic
74C22×C22⋊C4D411yescharacteristic
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77C2×#106—24no—
78C2×#106—42no—
79C2×#106—42no—
80C2×C22⋊C8—42no—
81C2×C22⋊C8—42no—
82C2×C22⋊C8—41no—
83C2×C22⋊C8—41no—
84C2×C22⋊C8C2×C418yes—
85C2×C22⋊C8C2×C414yescomplemented
86#390—44no—
87#386—44no—
88#384—41no—
89#384—41no—
90#380—24no—
91#380—24no—
92#380—24no—
93#380—42no—
94#380—42no—
95#380—42no—
96#380—41no—
97#380—41no—
98C4×C4⋊C4—41no—
99C4×C4⋊C4—41no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—22no—
107#340—24no—
108#340—42no—
109#340—42no—
110#340—42no—
111#340—42no—
112#340—42no—
113#340—42no—
114#336C2×C418yescomplemented
115C25C2×D411yescharacteristic
116C25C2×D411yescharacteristic
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—41no—
122C22×D4—41no—
123C22×D4C4⋊C414yes—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—44no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—82no—
149C23×C4C22×C412yes—
150C23×C4C22×C412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×Q812yes—
156C23×C4C2×Q812yes—
157C23×C4C2411yescharacteristic
158C2×M16—48no—
159C2×M16—44no—
160C2×M16—44no—
161C2×M16—48no—
162C2×M16—44no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8C22×C414yes—
176C22×C8C22×C414yes—
177C4×D4—44no—
178C4×D4—44no—
179C4×D4—44no—
180C4×D4—44no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—48no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4C4⋊C414yes—
216C2×C4⋊C4#4114yes—
217C2×C4⋊C4#4114yes—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—82no—
232C2×C22⋊C4—82no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4C4⋊C414yes—
235C2×C22⋊C4C4⋊C414yes—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42C22×C412yes—
244C2×C42C2×Q812yes—
245#106—48no—
246C22⋊C8—48no—
247C22⋊C8—44no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—42no—
252#95—42no—
253#95—44no—
254#95—82no—
255C24—21no—
256C24—21no—
257C24—22no—
258C24—22no—
259C24—21no—
260C24—21no—
261C24—22no—
262C24—22no—
263C24—21no—
264C24—21no—
265C24—22no—
266C24—22no—
267C24—21no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—22no—
272C24—22no—
273C24C2×C4⋊C412yes—
274C24C2×C4⋊C412yes—
275C24AGL(1,ℤ/8)12yes—
276C24#13712yes—
277C24C22×D411yescharacteristic
278C2×D4—28no—
279C2×D4—28no—
280C2×D4—44no—
281C2×D4—44no—
282C2×D4—44no—
283C2×D4—44no—
284C2×D4—44no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—48no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
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—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—88no—
367C22×C4—88no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4C2×C4⋊C412yes—
371C22×C4C2×C4⋊C412yes—
372C22×C4C2×C4⋊C412yes—
373C22×C4C2×C4⋊C412yes—
374C22×C4AGL(1,ℤ/8)12yes—
375C22×C4#13712yes—
376C22×C4C23×C411yescharacteristic
377C22×C4C22×Q811yescharacteristic
378C22×C4C2×#4112yes—
379C22×C4C2×#4112yes—
380C22×C4H(5,𝔽2)11yescharacteristic
381C22×C4H(5,𝔽2)11yescharacteristic
382C22×C4H(5,𝔽2)12yes—
383C22×C4H(5,𝔽2)12yes—
384C22×C4H(5,𝔽2)11yescharacteristic
385C22×C4H(5,𝔽2)11yescharacteristic
386M16—416no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—44no—
406C4⋊C4—48no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—84no—
420C4⋊C4—84no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—88no—
424C22⋊C4—88no—
425C22⋊C4—88no—
426C22⋊C4—84no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—84no—
432C23—22no—
433C23—24no—
434C23—22no—
435C23—24no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—22no—
445C23—24no—
446C23—22no—
447C23—22no—
448C23—24no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—41no—
453C23—42no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23—41no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23—42no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23#42814yes—
478C23C22×C4⋊C411yescharacteristic
479C23#53411yescharacteristic
480C23#53411yescharacteristic
481C23#54612yes—
482C23#54612yes—
483C23C2×AGL(1,ℤ/8)11yescharacteristic
484C23C2×AGL(1,ℤ/8)11yescharacteristic
485C23C2×#13711yescharacteristic
486C23C2×#13711yescharacteristic
487C23#57712yes—
488C23#57712yes—
489D4—416no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—816no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
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—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4#42814yes—
553C2×C4C4×#4112yes—
554C2×C4#51811yescharacteristic
555C2×C4#51812yes—
556C2×C4#51811yescharacteristic
557C2×C4#53812yes—
558C2×C4#53812yes—
559C2×C4#54012yes—
560C2×C4#54012yes—
561C2×C4#54311yescharacteristic
562C2×C4#54311yescharacteristic
563C2×C4#55111yescharacteristic
564C2×C4#55312yes—
565C2×C4#55412yes—
566C2×C4#55412yes—
567C2×C4#55911yescharacteristic
568C2×C4#56011yescharacteristic
569C2×C4#56012yes—
570C2×C4H(3,𝔽4)11yescharacteristic
571C8—44no—
572C8—44no—
573C22—24no—
574C22—28no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—44no—
580C22—44no—
581C22—42no—
582C22—42no—
583C22—44no—
584C22—42no—
585C22—42no—
586C22—44no—
587C22—42no—
588C22—44no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—82no—
594C22—82no—
595C22#231511yescharacteristic
596C22#234912yes—
597C22#268411yescharacteristic
598C22#268812yes—
599C22#270911yescharacteristic
600C22C2×#42811yescharacteristic
601C22C2×#42811yescharacteristic
602C22#291212yes—
603C22#319211yescharacteristic
604C22#319311yescharacteristic
605C22#319311yescharacteristic
606C22#319511yescharacteristic
607C22#320011yescharacteristic
608C22#320011yescharacteristic
609C22#320412yes—
610C22#330612yes—
611C22#331012yes—
612C22#331612yes—
613C22#332012yes—
614C22#332412yes—
615C22#332912yes—
616C22#333112yes—
617C22#333212yes—
618C4—24no—
619C4—42no—
620C4—42no—
621C4—42no—
622C4—84no—
623C4—84no—
624C4—162no—
625C4—162no—
626C4#233312yes—
627C4#237411yescharacteristic
628C4#237511yescharacteristic
629C4#274811yescharacteristic
630C4#278012yes—
631C4#280411yescharacteristic
632C2—44no—
633C2—82no—
634C2#1451311yescharacteristic
635C2#2194211yescharacteristic
636C2#2194611yescharacteristic
637C2#2271112yes—
638C2#2271512yes—
639C2#3127111yescharacteristic
640C2#3127211yescharacteristic
641C2#3140012yes—
642C2#3140212yes—
643C2#3203711yescharacteristic
644C2#3203811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11267C2semidirect
5. #11095C2semidirect
6. #11095C2semidirect
7. #10908C2semidirect
8. #10730C2semidirect
9. #10722C2semidirect
24. #1870C4semidirect
48. C22×M16C2×C4semidirect
85. C2×C22⋊C8C2×C4semidirect
114. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18702
#139062
#138722
#112672
#110952
#109082
#107302
#107222