Finite Group Databasebeta

← #387949

Subgroups — #387949

Order 512.

1540 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2C2×#1870C211yesmaximal, characteristic
3#13905C211yesmaximal, characteristic
4#11208C212yesmaximal, complemented
5#11200C212yesmaximal, complemented
6#11199C212yesmaximal, complemented
7#10884C212yesmaximal, complemented
8#10730C212yesmaximal, complemented
9#10722C212yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2211yescharacteristic
15#2487—24no—
16#2487C418yescomplemented
17C2×#400—21no—
18C2×#399—21no—
19C2×#389C2212yes—
20C2×#386—21no—
21C2×#385—21no—
22C2×#382C2211yescharacteristic
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1870—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38#1445—21no—
39#1438—21no—
40#1437—21no—
41#1437—21no—
42C23×D4D411yescharacteristic
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C22×M16—24no—
48C22×M16—41no—
49C22×M16—41no—
50C22×M16—42no—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4×D4—41no—
54C2×C4×D4—41no—
55C2×C4×D4—41no—
56C2×C4×D4—41no—
57C2×C4×D4C2×C414yes—
58C2×C4×D4D414yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—41no—
63C22×C4⋊C4—41no—
64C22×C4⋊C4—41no—
65C22×C4⋊C4—41no—
66C22×C4⋊C4C2×C414yes—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4D411yescharacteristic
77C22×C22⋊C4D411yescharacteristic
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80C2×C22⋊C8—24no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—42no—
84C2×C22⋊C8—41no—
85C2×C22⋊C8—41no—
86#400—28no—
87#399—28no—
88#389—24no—
89#389C2×C418yes—
90#386—28no—
91#385—28no—
92#382—41no—
93#382—41no—
94#380—24no—
95#380—22no—
96#380—24no—
97#380—24no—
98#380—42no—
99#380—42no—
100#380C2×C418yes—
101#380C2×C414yes—
102C4×C4⋊C4—41no—
103C4×C4⋊C4—41no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—21no—
112#340—24no—
113#340—24no—
114#340—42no—
115#340—42no—
116#340—42no—
117#340—42no—
118#340—42no—
119#340—42no—
120#336—24no—
121C25C2×D411yescharacteristic
122C25C2×D411yescharacteristic
123C22×D4—24no—
124C22×D4—41no—
125C22×D4—41no—
126C22×D4—42no—
127C22×D4C22⋊C414yes—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—82no—
152C23×C4—82no—
153C23×C4C22×C412yes—
154C23×C4C22×C412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2411yescharacteristic
162C2×M16—44no—
163C2×M16—48no—
164C2×M16—48no—
165C2×M16—44no—
166C2×M16—44no—
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—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8#4114yes—
181C4×D4—44no—
182C4×D4—44no—
183C4×D4—44no—
184C4×D4—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—22no—
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—28no—
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—82no—
217C2×C4⋊C4—82no—
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—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4—82no—
235C2×C22⋊C4—82no—
236C2×C22⋊C4—82no—
237C2×C22⋊C4C22⋊C414yes—
238C2×C22⋊C4C22⋊C414yes—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—42no—
244C2×C42—82no—
245C2×C42C22×C412yes—
246C2×C42C2×D412yes—
247C22⋊C8—48no—
248C22⋊C8—44no—
249#95—28no—
250#95—24no—
251#95—28no—
252#95—44no—
253#95—44no—
254#95—42no—
255#95—42no—
256#95—42no—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—21no—
265C24—21no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24C2×C22⋊C412yes—
273C24C2×C22⋊C412yes—
274C24#13714yes—
275C24C22×D411yescharacteristic
276C2×D4—28no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—48no—
280C2×D4—48no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—48no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
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—88no—
363C22×C4—88no—
364C22×C4—84no—
365C22×C4—84no—
366C22×C4C2×C22⋊C412yes—
367C22×C4C2×C22⋊C412yes—
368C22×C4C2×C22⋊C412yes—
369C22×C4C2×C22⋊C412yes—
370C22×C4C23×C411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C2×#4112yes—
373C22×C4C2×#4112yes—
374C22×C4H(5,𝔽2)11yescharacteristic
375C22×C4H(5,𝔽2)11yescharacteristic
376C22×C4H(5,𝔽2)11yescharacteristic
377C22×C4H(5,𝔽2)11yescharacteristic
378C22×C4H(5,𝔽2)12yes—
379C22×C42−1+412yes—
380M16—416no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—42no—
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—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—44no—
402C4⋊C4—48no—
403C4⋊C4—42no—
404C4⋊C4—42no—
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—88no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—48no—
417C22⋊C4—88no—
418C22⋊C4—88no—
419C22⋊C4—88no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—84no—
425C23—22no—
426C23—24no—
427C23—24no—
428C23—24no—
429C23—22no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—22no—
436C23—24no—
437C23—22no—
438C23—22no—
439C23—24no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—41no—
444C23—41no—
445C23—42no—
446C23—41no—
447C23—41no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—41no—
458C23—41no—
459C23—42no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23#41914yes—
467C23C22×C22⋊C411yescharacteristic
468C23#53411yescharacteristic
469C23#53411yescharacteristic
470C23#54612yes—
471C23#54712yes—
472C23C2×#13712yes—
473C23#57512yes—
474C23#57612yes—
475C23#57712yes—
476D4—416no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
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—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—816no—
519C2×C4—82no—
520C2×C4—82no—
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×C4C4×#4112yes—
539C2×C4#51811yescharacteristic
540C2×C4#51811yescharacteristic
541C2×C4#52012yes—
542C2×C4#53411yescharacteristic
543C2×C4#53411yescharacteristic
544C2×C4#53812yes—
545C2×C4#53912yes—
546C2×C4#54012yes—
547C2×C4#54212yes—
548C2×C4#54612yes—
549C2×C4#54712yes—
550C2×C4#55111yescharacteristic
551C2×C4#55312yes—
552C2×C4#55911yescharacteristic
553C2×C4#56011yescharacteristic
554C2×C4#56212yes—
555C2×C4#56311yescharacteristic
556C8—44no—
557C8—82no—
558C22—24no—
559C22—28no—
560C22—42no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—42no—
566C22—44no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—44no—
571C22—44no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22#230811yescharacteristic
577C22#233912yes—
578C22#270812yes—
579C22#270911yescharacteristic
580C22#271711yescharacteristic
581C22C2×#41911yescharacteristic
582C22#289611yescharacteristic
583C22#289712yes—
584C22#319612yes—
585C22#319612yes—
586C22#319812yes—
587C22#319911yescharacteristic
588C22#320011yescharacteristic
589C22#331412yes—
590C22#331612yes—
591C22#332212yes—
592C22#332612yes—
593C22#334412yes—
594C22#334612yes—
595C22#335112yes—
596C22#335512yes—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—84no—
602C4—84no—
603C4—84no—
604C4—162no—
605C4#233512yes—
606C4#237411yescharacteristic
607C4#237611yescharacteristic
608C4#269711yescharacteristic
609C4#270112yes—
610C4#272411yescharacteristic
611C2—44no—
612C2—82no—
613C2#1450511yescharacteristic
614C2#2176811yescharacteristic
615C2#2177011yescharacteristic
616C2#2249712yes—
617C2#2250112yes—
618C2#3201812yes—
619C2#3202212yes—
620C2#3204212yes—
621C2#3227412yes—
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. #11208C2semidirect
5. #11200C2semidirect
6. #11199C2semidirect
7. #10884C2semidirect
8. #10730C2semidirect
9. #10722C2semidirect
16. #2487C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C2×#18702
#139052
#112082
#112002
#111992
#108842
#107302
#107222