Finite Group Databasebeta

← #357999

Subgroups — #357999

Order 512.

2118 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic
3C2×#1736C211yesmaximal, characteristic
4C2×#1736C211yesmaximal, characteristic
5C2×#1735C211yesmaximal, characteristic
6C22×#340C211yesmaximal, characteristic
7C2×#1729C211yesmaximal, characteristic
8#11208C214yesmaximal, complemented
9#11207C214yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#2317—24no—
18#2317—24no—
19#2316—24no—
20#2316C418yescomplemented
21C2×#389—22no—
22C2×#389—22no—
23C2×#389C2214yes—
24C2×#385—22no—
25C2×#384C2212yes—
26C2×#382C2212yes—
27C2×#380—22no—
28#1736—24no—
29#1736—24no—
30#1735—24no—
31C2×#340—24no—
32C2×#340C2214yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2214yes—
37C2×#340C2212yes—
38#1729—24no—
39C2×#336C2214yes—
40C24×C4—21no—
41C24×C4—21no—
42C24×C4C2311yescharacteristic
43C22×M16—22no—
44C22×M16—22no—
45C22×M16—22no—
46C22×M16—22no—
47C22×M16—22no—
48C22×M16—22no—
49C22×M16Q814yes—
50C23×C8—22no—
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—22no—
54C23×C8—22no—
55C23×C8—22no—
56C23×C8Q814yes—
57C23×C8C2312yes—
58C23×C8C2312yes—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117C2×C414yes—
74C2×#117C2×C414yes—
75C2×#117D414yes—
76C2×#117D414yes—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4C2×C414yes—
90C22×C4⋊C4D414yes—
91C22×C4⋊C4D412yes—
92C22×C4⋊C4D414yes—
93C22×C4⋊C4Q814yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4—21no—
100C22×C22⋊C4—21no—
101C22×C22⋊C4D412yes—
102C22×C42—21no—
103C22×C42—21no—
104C22×C42C2312yes—
105#389—216no—
106#389—216no—
107#389—28no—
108#389C2×C4116yescomplemented
109#385—216no—
110#384—24no—
111#384—24no—
112#382—24no—
113#382—24no—
114#380—216no—
115C2×#95—24no—
116C2×#95—24no—
117#340—28no—
118#340—24no—
119#340—24no—
120#340—28no—
121#340—24no—
122#340—24no—
123#336—28no—
124C25C2×D411yescharacteristic
125C23×C4—22no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—24no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—82no—
137C23×C4—82no—
138C23×C4—82no—
139C23×C4—82no—
140C23×C4C22×C412yes—
141C23×C4C2×D414yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×Q812yes—
148C23×C4C2×Q812yes—
149C23×C4C2×Q812yes—
150C23×C4#4112yes—
151C23×C4#4112yes—
152C23×C4#4112yes—
153C23×C4#4112yes—
154C23×C4#4112yes—
155C23×C4#4112yes—
156C23×C4C2411yescharacteristic
157C2×M16—28no—
158C2×M16—28no—
159C2×M16—28no—
160C2×M16—44no—
161C2×M16—44no—
162C2×M16—44no—
163C2×M16—44no—
164C22×C8—28no—
165C22×C8—28no—
166C22×C8—28no—
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—44no—
180C22×C8—44no—
181C22×C8—44no—
182C22×C8—44no—
183C22×C8C2×Q814yes—
184C22×C8C2×Q814yes—
185#117—28no—
186#117—28no—
187#117—28no—
188#117—28no—
189#117—28no—
190#117—28no—
191#117—28no—
192#117—28no—
193#117—44no—
194#117—44no—
195#117—44no—
196#117—44no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—216no—
232C2×C4⋊C4—216no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—84no—
238C2×C4⋊C4—84no—
239C2×C4⋊C4—84no—
240C2×C4⋊C4—84no—
241C2×C4⋊C4C22⋊C418yes—
242C2×C4⋊C4C4⋊C4116yescomplemented
243C2×C4⋊C4C22×C414yes—
244C2×C4⋊C4C22×C414yes—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D414yes—
249C2×C4⋊C4C2×Q814yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4C22⋊C418yes—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42C22×C414yes—
270C2×C42C2×D414yes—
271#95—216no—
272#95—216no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24C2×C22⋊C412yes—
278C24#12012yes—
279C24#12112yes—
280C24C22⋊Q812yes—
281C24#12312yes—
282C24#12412yes—
283C24C4⋊Q812yes—
284C24C22×D411yescharacteristic
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—216no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—44no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—44no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—48no—
320C22×C4—88no—
321C22×C4—88no—
322C22×C4—84no—
323C22×C4—84no—
324C22×C4—88no—
325C22×C4—88no—
326C22×C4—84no—
327C22×C4—84no—
328C22×C4C2×C22⋊C418yes—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C4⋊C414yes—
331C22×C4#11714yes—
332C22×C4C4×D414yes—
333C22×C4C4×D414yes—
334C22×C4C4×Q814yes—
335C22×C4C4×Q814yes—
336C22×C4#12012yes—
337C22×C4#12114yes—
338C22×C4#12114yes—
339C22×C4#12114yes—
340C22×C4#12114yes—
341C22×C4#12114yes—
342C22×C4#12112yes—
343C22×C4C22⋊Q814yes—
344C22×C4C22⋊Q814yes—
345C22×C4C22⋊Q814yes—
346C22×C4C22⋊Q814yes—
347C22×C4C22⋊Q814yes—
348C22×C4C22⋊Q814yes—
349C22×C4C22⋊Q814yes—
350C22×C4C22⋊Q812yes—
351C22×C4#12314yes—
352C22×C4#12314yes—
353C22×C4#12312yes—
354C22×C4#12412yes—
355C22×C4#12514yes—
356C22×C4#12514yes—
357C22×C4#12514yes—
358C22×C4#12514yes—
359C22×C4#12614yes—
360C22×C4#12614yes—
361C22×C4C4⋊Q814yes—
362C22×C4C4⋊Q814yes—
363C22×C4C4⋊Q812yes—
364C22×C4C23×C411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×Q811yescharacteristic
370C22×C4C22×Q811yescharacteristic
371C22×C4C22×Q811yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378M16—416no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
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—416no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—216no—
398C4⋊C4—216no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—42no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—48no—
416C4⋊C4—816no—
417C4⋊C4—816no—
418C4⋊C4C2×C4⋊C418yes—
419C22⋊C4—216no—
420C22⋊C4—48no—
421C22⋊C4—48no—
422C42—24no—
423C42—24no—
424C42—24no—
425C42—24no—
426C42—48no—
427C42—48no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23#38014yes—
445C23#38614yes—
446C23#39114yes—
447C23#39414yes—
448C23#39514yes—
449C23#39814yes—
450C23#40014yes—
451C23C22×C22⋊C411yescharacteristic
452C23C2×#12011yescharacteristic
453C23C2×#12111yescharacteristic
454C23C2×C22⋊Q811yescharacteristic
455C23C2×#12311yescharacteristic
456C23C2×#12411yescharacteristic
457C23C2×C4⋊Q811yescharacteristic
458C23#57612yes—
459C23#57712yes—
460C23#57712yes—
461C23#57812yes—
462C2×C4—28no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—48no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—48no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—48no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—816no—
484C2×C4—816no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—84no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—81no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—82no—
497C2×C4—81no—
498C2×C4#38014yes—
499C2×C4#38614yes—
500C2×C4#38718yes—
501C2×C4#38718yes—
502C2×C4#38918yes—
503C2×C4#38918yes—
504C2×C4#39114yes—
505C2×C4#39414yes—
506C2×C4#39514yes—
507C2×C4#39618yes—
508C2×C4#39618yes—
509C2×C4#39818yes—
510C2×C4#39818yes—
511C2×C4#39814yes—
512C2×C4#40014yes—
513C2×C4C22×C4⋊C411yescharacteristic
514C2×C4C2×#11711yescharacteristic
515C2×C4C2×C4×D411yescharacteristic
516C2×C4C2×C4×D411yescharacteristic
517C2×C4C2×C4×Q811yescharacteristic
518C2×C4C2×C4×Q811yescharacteristic
519C2×C4C2×#12111yescharacteristic
520C2×C4C2×#12111yescharacteristic
521C2×C4C2×#12111yescharacteristic
522C2×C4C2×#12111yescharacteristic
523C2×C4C2×#12111yescharacteristic
524C2×C4C2×C22⋊Q811yescharacteristic
525C2×C4C2×C22⋊Q811yescharacteristic
526C2×C4C2×C22⋊Q811yescharacteristic
527C2×C4C2×C22⋊Q811yescharacteristic
528C2×C4C2×C22⋊Q811yescharacteristic
529C2×C4C2×C22⋊Q811yescharacteristic
530C2×C4C2×C22⋊Q811yescharacteristic
531C2×C4C2×#12311yescharacteristic
532C2×C4C2×#12311yescharacteristic
533C2×C4C2×#12511yescharacteristic
534C2×C4C2×#12511yescharacteristic
535C2×C4C2×#12511yescharacteristic
536C2×C4C2×#12511yescharacteristic
537C2×C4C2×#12611yescharacteristic
538C2×C4C2×#12611yescharacteristic
539C2×C4C2×C4⋊Q811yescharacteristic
540C2×C4C2×C4⋊Q811yescharacteristic
541C8—44no—
542C8—44no—
543C22—28no—
544C22—28no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—24no—
549C22—28no—
550C22—28no—
551C22—24no—
552C22—24no—
553C22#144118yes—
554C22C2×#38011yescharacteristic
555C22C2×#38611yescharacteristic
556C22C2×#39111yescharacteristic
557C22C2×#39411yescharacteristic
558C22C2×#39511yescharacteristic
559C22C2×#39811yescharacteristic
560C22C2×#40011yescharacteristic
561C22#289712yes—
562C22#289812yes—
563C22#301212yes—
564C22#301312yes—
565C22#304111yescharacteristic
566C22#304211yescharacteristic
567C22#304311yescharacteristic
568C22#304411yescharacteristic
569C22#308112yes—
570C22#308212yes—
571C22#309812yes—
572C22#309912yes—
573C22#313612yes—
574C22#313712yes—
575C22#316312yes—
576C22#316412yes—
577C4—48no—
578C4—48no—
579C4—84no—
580C4—82no—
581C4—84no—
582C4—82no—
583C4#144118yes—
584C4C2×#38711yescharacteristic
585C4C2×#38711yescharacteristic
586C4C2×#38911yescharacteristic
587C4C2×#38911yescharacteristic
588C4C2×#39611yescharacteristic
589C4C2×#39611yescharacteristic
590C4C2×#39811yescharacteristic
591C4C2×#39811yescharacteristic
592C2—28no—
593C2C2×#144111yescharacteristic
594C2#2065011yescharacteristic
595C2#2065111yescharacteristic
596C2#2091512yes—
597C2#2132612yes—
598C2#2351111yescharacteristic
599C2#2351211yescharacteristic
600C2#2355712yes—
601C2#2366011yescharacteristic
602C2#2366111yescharacteristic
603C2#2374912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #11208C2semidirect
9. #11207C2semidirect
20. #2316C4semidirect
108. #389C2×C4semidirect
242. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#23162
C2×#17362
C2×#17352
C22×#3402
C2×#17292
#112082
#112072