Finite Group Databasebeta

← #358092

Subgroups — #358092

Order 512.

2090 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#427C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3C2×#1736C211yesmaximal, characteristic
4C2×#1736C211yesmaximal, characteristic
5C22×#340C211yesmaximal, characteristic
6C2×#1729C211yesmaximal, characteristic
7#13795C211yesmaximal, characteristic
8#11220C214yesmaximal, complemented
9#11219C214yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#427—24no—
18C2×#427—24no—
19C2×#427C418yescomplemented
20C22×#107C2212yes—
21C22×#106C2212yes—
22#2317—24no—
23#2317—24no—
24C2×#389C2214yes—
25C2×#385—22no—
26C2×#384—21no—
27C2×#382—21no—
28#1736—24no—
29#1736—24no—
30C2×#340—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2214yes—
34C2×#340C2214yes—
35#1729—24no—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C4⋊#95—21no—
39C4⋊#95—21no—
40C4⋊#95C2214yes—
41C22⋊#95—22no—
42C24×C4—21no—
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C22×M16—22no—
46C22×M16—22no—
47C22×M16—22no—
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—22no—
51C22×M16Q814yes—
52C23×C8—22no—
53C23×C8—22no—
54C23×C8—22no—
55C23×C8—22no—
56C23×C8C2×C414yes—
57C23×C8D414yes—
58C23×C8Q814yes—
59C23×C8C2312yes—
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
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×#117—22no—
74C2×#117—22no—
75C2×#117—22no—
76C2×#117C2×C414yes—
77C2×#117C2×C414yes—
78C2×#117D414yes—
79C2×#117D414yes—
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—22no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—22no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4D411yescharacteristic
94C22×C4⋊C4D411yescharacteristic
95C22×C4⋊C4D414yes—
96C22×C4⋊C4Q814yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C22⋊C4—21no—
102C22×C22⋊C4—21no—
103C22×C22⋊C4D412yes—
104C22×C42—21no—
105C22×C42—21no—
106C22×C42C2312yes—
107#427—232no—
108C2×#107—24no—
109C2×#107—24no—
110C2×#107C2×C418yescomplemented
111C2×#106—24no—
112C2×#106—24no—
113C2×#106C2×C418yescomplemented
114#389—28no—
115#389—28no—
116#385—216no—
117#384—28no—
118#382—28no—
119C2×#95—24no—
120C2×#95—24no—
121C2×#95—24no—
122#340—24no—
123#340—24no—
124#340—28no—
125#340—28no—
126#336—24no—
127#336—24no—
128C25C2×D411yescharacteristic
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—82no—
140C23×C4—82no—
141C23×C4—82no—
142C23×C4—82no—
143C23×C4C22×C412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4C2×Q812yes—
152C23×C4C2×Q812yes—
153C23×C4C2×Q812yes—
154C23×C4#4112yes—
155C23×C4#4112yes—
156C23×C4#4112yes—
157C23×C4#4112yes—
158C23×C4#4112yes—
159C23×C4#4112yes—
160C23×C4C2411yescharacteristic
161C2×M16—28no—
162C2×M16—28no—
163C2×M16—28no—
164C2×M16—44no—
165C2×M16—44no—
166C2×M16—44no—
167C2×M16—44no—
168C22×C8—28no—
169C22×C8—28no—
170C22×C8—28no—
171C22×C8—28no—
172C22×C8—28no—
173C22×C8—28no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8C4⋊C4116yescomplemented
189C22×C8C22×C412yes—
190C22×C8C22×C412yes—
191C22×C8C2×D412yes—
192C22×C8C2×D412yes—
193C22×C8C2×Q814yes—
194C22×C8C2×Q812yes—
195C22×C8C2×Q812yes—
196#117—28no—
197#117—28no—
198#117—28no—
199#117—28no—
200#117—28no—
201#117—44no—
202#117—44no—
203#117—44no—
204#117—44no—
205#117—44no—
206#117—44no—
207#117C22⋊C4116yes—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—216no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—84no—
248C2×C4⋊C4—84no—
249C2×C4⋊C4C22×C414yes—
250C2×C4⋊C4C22×C414yes—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×Q814yes—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—42no—
275C2×C42—42no—
276#107—216no—
277#106—216no—
278#95—216no—
279#95—84no—
280#95—84no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24C4×D412yes—
286C24#12012yes—
287C24#12112yes—
288C24#12112yes—
289C24C22⋊Q812yes—
290C24#12312yes—
291C24C4⋊Q812yes—
292C24C22×D411yescharacteristic
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—48no—
328C22×C4—48no—
329C22×C4—88no—
330C22×C4—88no—
331C22×C4—88no—
332C22×C4—88no—
333C22×C4—84no—
334C22×C4—84no—
335C22×C4—84no—
336C22×C4—84no—
337C22×C4C2×C22⋊C414yes—
338C22×C4C2×C4⋊C414yes—
339C22×C4#11714yes—
340C22×C4C4×D414yes—
341C22×C4C4×D412yes—
342C22×C4C4×Q814yes—
343C22×C4C4×Q814yes—
344C22×C4#12012yes—
345C22×C4#12114yes—
346C22×C4#12114yes—
347C22×C4#12114yes—
348C22×C4#12114yes—
349C22×C4#12112yes—
350C22×C4#12112yes—
351C22×C4C22⋊Q814yes—
352C22×C4C22⋊Q814yes—
353C22×C4C22⋊Q814yes—
354C22×C4C22⋊Q814yes—
355C22×C4C22⋊Q814yes—
356C22×C4C22⋊Q814yes—
357C22×C4C22⋊Q814yes—
358C22×C4C22⋊Q812yes—
359C22×C4#12314yes—
360C22×C4#12314yes—
361C22×C4#12312yes—
362C22×C4#12414yes—
363C22×C4#12514yes—
364C22×C4#12514yes—
365C22×C4#12514yes—
366C22×C4#12514yes—
367C22×C4#12614yes—
368C22×C4#12614yes—
369C22×C4C4⋊Q814yes—
370C22×C4C4⋊Q814yes—
371C22×C4C4⋊Q812yes—
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×Q811yescharacteristic
378C22×C4C22×Q811yescharacteristic
379C22×C4C22×Q811yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386M16—416no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—232no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8C2×C4⋊C414yes—
408C2×C8C2×C4⋊C414yes—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—48no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—48no—
428C4⋊C4—816no—
429C4⋊C4C2×C22⋊C418yes—
430C22⋊C4—216no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C42—216no—
434C42—42no—
435C42—42no—
436C42—42no—
437C42—42no—
438C42—48no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23#38614yes—
456C23#38714yes—
457C23#38914yes—
458C23#39414yes—
459C23#39514yes—
460C23#39614yes—
461C23#39814yes—
462C23C2×C4×D411yescharacteristic
463C23C2×#12011yescharacteristic
464C23C2×#12111yescharacteristic
465C23C2×#12111yescharacteristic
466C23C2×C22⋊Q811yescharacteristic
467C23C2×#12311yescharacteristic
468C23C2×C4⋊Q811yescharacteristic
469C23#57612yes—
470C23#57712yes—
471C23#57712yes—
472C23#57812yes—
473C2×C4—28no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—48no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—48no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—816no—
498C2×C4—816no—
499C2×C4—84no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—84no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4#38018yes—
510C2×C4#38614yes—
511C2×C4#38714yes—
512C2×C4#38718yes—
513C2×C4#38914yes—
514C2×C4#38918yes—
515C2×C4#39118yes—
516C2×C4#39414yes—
517C2×C4#39514yes—
518C2×C4#39618yes—
519C2×C4#39614yes—
520C2×C4#39818yes—
521C2×C4#39818yes—
522C2×C4#39814yes—
523C2×C4#40018yes—
524C2×C4C22×C22⋊C411yescharacteristic
525C2×C4C22×C4⋊C411yescharacteristic
526C2×C4C2×#11711yescharacteristic
527C2×C4C2×C4×D411yescharacteristic
528C2×C4C2×C4×Q811yescharacteristic
529C2×C4C2×C4×Q811yescharacteristic
530C2×C4C2×#12111yescharacteristic
531C2×C4C2×#12111yescharacteristic
532C2×C4C2×#12111yescharacteristic
533C2×C4C2×#12111yescharacteristic
534C2×C4C2×C22⋊Q811yescharacteristic
535C2×C4C2×C22⋊Q811yescharacteristic
536C2×C4C2×C22⋊Q811yescharacteristic
537C2×C4C2×C22⋊Q811yescharacteristic
538C2×C4C2×C22⋊Q811yescharacteristic
539C2×C4C2×C22⋊Q811yescharacteristic
540C2×C4C2×C22⋊Q811yescharacteristic
541C2×C4C2×#12311yescharacteristic
542C2×C4C2×#12311yescharacteristic
543C2×C4C2×#12411yescharacteristic
544C2×C4C2×#12511yescharacteristic
545C2×C4C2×#12511yescharacteristic
546C2×C4C2×#12511yescharacteristic
547C2×C4C2×#12511yescharacteristic
548C2×C4C2×#12611yescharacteristic
549C2×C4C2×#12611yescharacteristic
550C2×C4C2×C4⋊Q811yescharacteristic
551C2×C4C2×C4⋊Q811yescharacteristic
552C8—24no—
553C8—24no—
554C8—44no—
555C22—28no—
556C22—28no—
557C22—24no—
558C22—24no—
559C22—28no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—24no—
564C22—28no—
565C22#144118yes—
566C22C2×#38611yescharacteristic
567C22C2×#38711yescharacteristic
568C22C2×#38911yescharacteristic
569C22C2×#39411yescharacteristic
570C22C2×#39511yescharacteristic
571C22C2×#39611yescharacteristic
572C22C2×#39811yescharacteristic
573C22#295112yes—
574C22#295212yes—
575C22#301212yes—
576C22#301312yes—
577C22#304512yes—
578C22#304612yes—
579C22#305711yescharacteristic
580C22#305711yescharacteristic
581C22#305811yescharacteristic
582C22#305911yescharacteristic
583C22#308112yes—
584C22#308212yes—
585C22#309612yes—
586C22#309712yes—
587C22#316312yes—
588C22#316412yes—
589C4—48no—
590C4—84no—
591C4—82no—
592C4—84no—
593C4—82no—
594C4—84no—
595C4#144118yes—
596C4C2×#38011yescharacteristic
597C4C2×#38711yescharacteristic
598C4C2×#38911yescharacteristic
599C4C2×#39111yescharacteristic
600C4C2×#39611yescharacteristic
601C4C2×#39811yescharacteristic
602C4C2×#39811yescharacteristic
603C4C2×#40011yescharacteristic
604C2—28no—
605C2C2×#144111yescharacteristic
606C2#2093712yes—
607C2#2103912yes—
608C2#2115511yescharacteristic
609C2#2115611yescharacteristic
610C2#2348611yescharacteristic
611C2#2348711yescharacteristic
612C2#2355712yes—
613C2#2361211yescharacteristic
614C2#2361311yescharacteristic
615C2#2366212yes—
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. #11220C2semidirect
9. #11219C2semidirect
19. C2×#427C4semidirect
110. C2×#107C2×C4semidirect
113. C2×#106C2×C4semidirect
188. C22×C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4272
C2×#23172
C2×#17362
C22×#3402
C2×#17292
#137952
#112202
#112192