Finite Group Databasebeta

← #387930

Subgroups — #387930

Order 512.

1608 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2C2×#1955C211yesmaximal, characteristic
3#13871C211yesmaximal, characteristic
4#11228C212yesmaximal, complemented
5#11200C212yesmaximal, complemented
6#11199C212yesmaximal, complemented
7#10881C212yesmaximal, complemented
8#10731C212yesmaximal, complemented
9#10730C212yesmaximal, complemented
10C22×#123—21no—
11C22×C4×D4C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×#97C2211yescharacteristic
18#2486—24no—
19#2486C418yescomplemented
20C2×#398—21no—
21C2×#397—21no—
22C2×#389C2212yes—
23C2×#381—21no—
24C2×#380C2212yes—
25C2×#380C2211yescharacteristic
26C2×#380C2212yes—
27C2×#380C2211yescharacteristic
28#1955—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37#1440—21no—
38#1438—21no—
39#1437—21no—
40#1436—21no—
41C4⋊#95C2212yes—
42C23×D4D411yescharacteristic
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×#123—28no—
49C2×#123—48no—
50C2×C4×D4—41no—
51C2×C4×D4—41no—
52C2×C4×D4—41no—
53C2×C4×D4—41no—
54C2×C4×D4C2×C414yes—
55C2×C4×D4D414yes—
56C2×#117—24no—
57C2×#117—42no—
58C2×#117—41no—
59C2×#117—41no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—41no—
64C22×C4⋊C4—41no—
65C22×C4⋊C4—41no—
66C22×C4⋊C4—41no—
67C22×C4⋊C4C2×C414yes—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4D411yescharacteristic
79C22×C22⋊C4D411yescharacteristic
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82C2×C4⋊C8—22no—
83C2×C4⋊C8—41no—
84C2×C4⋊C8—41no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—42no—
90C2×C22⋊C8—42no—
91C2×#97—22no—
92C2×#97—41no—
93C2×#97—41no—
94#398—28no—
95#397—28no—
96#389—24no—
97#389C2×C418yes—
98#381—28no—
99#380—24no—
100#380—22no—
101#380—24no—
102#380—42no—
103#380—41no—
104#380—41no—
105#380C2×C418yes—
106#380C2×C414yes—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114#340—24no—
115#340—24no—
116#340—42no—
117#340—42no—
118#340—42no—
119#340—42no—
120#340—42no—
121#340—42no—
122C25C2×D411yescharacteristic
123C25C2×D411yescharacteristic
124C22×D4—24no—
125C22×D4—41no—
126C22×D4—41no—
127C22×D4—42no—
128C22×D4C22⋊C414yes—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4C22×C412yes—
155C23×C4C22×C412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2411yescharacteristic
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×C8—22no—
176C22×C8#4114yes—
177#123—432no—
178C4×D4—44no—
179C4×D4—44no—
180C4×D4—44no—
181C4×D4—44no—
182#117—44no—
183#117—48no—
184#117—44no—
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—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4C22⋊C414yes—
219C2×C4⋊C4C22×C414yes—
220C2×C4⋊C4C22×C414yes—
221C2×C4⋊C4#4114yes—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—82no—
237C2×C22⋊C4—82no—
238C2×C22⋊C4C22⋊C414yes—
239C2×C22⋊C4C22⋊C414yes—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—44no—
245C2×C42—42no—
246C2×C42C22×C412yes—
247C2×C42C2×D412yes—
248C4⋊C8—44no—
249C22⋊C8—48no—
250C22⋊C8—48no—
251#97—44no—
252#95—28no—
253#95—24no—
254#95—28no—
255#95—44no—
256#95—44no—
257#95—42no—
258#95—82no—
259C24—21no—
260C24—21no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—21no—
267C24—21no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24C2×C22⋊C412yes—
275C24C2×C22⋊C412yes—
276C24AGL(1,ℤ/8)14yes—
277C24C22×D411yescharacteristic
278C2×D4—28no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—48no—
282C2×D4—48no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—48no—
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—48no—
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×C4C2×C22⋊C412yes—
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C412yes—
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4112yes—
375C22×C4C2×#4112yes—
376C22×C4H(5,𝔽2)11yescharacteristic
377C22×C4H(5,𝔽2)11yescharacteristic
378C22×C4H(5,𝔽2)11yescharacteristic
379C22×C4H(5,𝔽2)11yescharacteristic
380C22×C4H(5,𝔽2)12yes—
381C22×C42−1+412yes—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—44no—
403C4⋊C4—48no—
404C4⋊C4—416no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—416no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—88no—
421C22⋊C4—84no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—48no—
427C23—22no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—22no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—22no—
438C23—24no—
439C23—22no—
440C23—22no—
441C23—24no—
442C23—42no—
443C23—42no—
444C23—41no—
445C23—42no—
446C23—41no—
447C23—41no—
448C23—42no—
449C23—41no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—41no—
459C23—42no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—41no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23#41814yes—
469C23C22×C22⋊C411yescharacteristic
470C23#53411yescharacteristic
471C23#53411yescharacteristic
472C23#54612yes—
473C23#54712yes—
474C23C2×AGL(1,ℤ/8)12yes—
475C23#57512yes—
476C23#57712yes—
477C23#57812yes—
478D4—416no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—816no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
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×C4C4×#4112yes—
541C2×C4#51811yescharacteristic
542C2×C4#51811yescharacteristic
543C2×C4#52012yes—
544C2×C4#53411yescharacteristic
545C2×C4#53411yescharacteristic
546C2×C4#53812yes—
547C2×C4#53912yes—
548C2×C4#54012yes—
549C2×C4#54212yes—
550C2×C4#54612yes—
551C2×C4#54712yes—
552C2×C4#55111yescharacteristic
553C2×C4#55312yes—
554C2×C4#55911yescharacteristic
555C2×C4#56011yescharacteristic
556C2×C4#56212yes—
557C2×C4#56311yescharacteristic
558C8—44no—
559C8—82no—
560C22—24no—
561C22—28no—
562C22—42no—
563C22—44no—
564C22—44no—
565C22—42no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—44no—
573C22—44no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22#230811yescharacteristic
579C22#233912yes—
580C22#269711yescharacteristic
581C22#270112yes—
582C22#272411yescharacteristic
583C22C2×#41811yescharacteristic
584C22#289611yescharacteristic
585C22#289812yes—
586C22#319412yes—
587C22#319412yes—
588C22#320011yescharacteristic
589C22#320112yes—
590C22#320211yescharacteristic
591C22#330612yes—
592C22#330812yes—
593C22#331512yes—
594C22#331712yes—
595C22#334312yes—
596C22#334512yes—
597C22#334812yes—
598C22#334912yes—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—84no—
604C4—84no—
605C4—84no—
606C4—162no—
607C4#233512yes—
608C4#237411yescharacteristic
609C4#237611yescharacteristic
610C4#270812yes—
611C4#270911yescharacteristic
612C4#271711yescharacteristic
613C2—44no—
614C2—82no—
615C2#1450511yescharacteristic
616C2#2176711yescharacteristic
617C2#2177111yescharacteristic
618C2#2249512yes—
619C2#2249612yes—
620C2#3167912yes—
621C2#3179112yes—
622C2#3179312yes—
623C2#3246612yes—
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. #11228C2semidirect
5. #11200C2semidirect
6. #11199C2semidirect
7. #10881C2semidirect
8. #10731C2semidirect
9. #10730C2semidirect
19. #2486C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
C2×#19552
#138712
#112282
#112002
#111992
#108812
#107312
#107302