Finite Group Databasebeta

← #358091

Subgroups — #358091

Order 512.

2090 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic
3C2×#1736C211yesmaximal, characteristic
4C2×#1735C211yesmaximal, 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×#428—24no—
18C2×#428—24no—
19C2×#428C418yescomplemented
20C22×#107C2212yes—
21C22×#106C2212yes—
22#2316—24no—
23#2316—24no—
24C2×#389—22no—
25C2×#389C2214yes—
26C2×#380—22no—
27#1736—24no—
28#1735—24no—
29C2×#340—24no—
30C2×#340C2214yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2214yes—
34#1729—24no—
35C2×#336C2214yes—
36C4⋊#95—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C22⋊#95—22no—
40C24×C4—21no—
41C24×C4—21no—
42C24×C4C2311yescharacteristic
43C22×M16—22no—
44C22×M16—22no—
45C22×M16—22no—
46C22×M16—22no—
47C22×M16C2×C414yes—
48C22×M16D414yes—
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×#117—22no—
74C2×#117—22no—
75C2×#117C2×C414yes—
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—21no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4C2×C414yes—
89C22×C4⋊C4D412yes—
90C22×C4⋊C4D414yes—
91C22×C4⋊C4D414yes—
92C22×C4⋊C4Q814yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C22⋊C4—21no—
99C22×C22⋊C4—21no—
100C22×C22⋊C4D412yes—
101C22×C42—21no—
102C22×C42—21no—
103C22×C42C2312yes—
104#428—232no—
105C2×#107—24no—
106C2×#107—24no—
107C2×#107C2×C418yescomplemented
108C2×#106—24no—
109C2×#106—24no—
110C2×#106C2×C418yescomplemented
111#389—216no—
112#389—28no—
113#389—28no—
114#380—216no—
115C2×#95—24no—
116C2×#95—24no—
117C2×#95—24no—
118#340—28no—
119#340—28no—
120#340—24no—
121#340—24no—
122#336—28no—
123C25C2×D411yescharacteristic
124C23×C4—22no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—82no—
136C23×C4—82no—
137C23×C4—82no—
138C23×C4—82no—
139C23×C4C22×C412yes—
140C23×C4C2×D414yes—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×Q812yes—
147C23×C4C2×Q812yes—
148C23×C4C2×Q812yes—
149C23×C4#4112yes—
150C23×C4#4112yes—
151C23×C4#4112yes—
152C23×C4#4112yes—
153C23×C4#4112yes—
154C23×C4#4112yes—
155C23×C4C2411yescharacteristic
156C2×M16—28no—
157C2×M16—28no—
158C2×M16—28no—
159C2×M16—28no—
160C2×M16—28no—
161C2×M16—28no—
162C2×M16C4⋊C4116yescomplemented
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—28no—
168C22×C8—28no—
169C22×C8—28no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—44no—
177C22×C8—44no—
178C22×C8—44no—
179C22×C8—44no—
180C22×C8C22×C414yes—
181C22×C8C2×D414yes—
182C22×C8C2×Q814yes—
183C22×C8C2×Q814yes—
184#117—28no—
185#117—28no—
186#117—28no—
187#117—28no—
188#117—44no—
189#117—44no—
190#117—44no—
191#117—44no—
192#117—44no—
193#117—44no—
194#117—44no—
195#117—44no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—216no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—84no—
237C2×C4⋊C4—84no—
238C2×C4⋊C4C22⋊C4116yes—
239C2×C4⋊C4C22×C412yes—
240C2×C4⋊C4C22×C412yes—
241C2×C4⋊C4C22×C414yes—
242C2×C4⋊C4C2×D414yes—
243C2×C4⋊C4C2×D414yes—
244C2×C4⋊C4C2×D414yes—
245C2×C4⋊C4C2×Q814yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—42no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42—42no—
267#107—216no—
268#106—216no—
269#95—216no—
270#95—84no—
271#95—84no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24C4×D412yes—
277C24#12012yes—
278C24#12112yes—
279C24#12112yes—
280C24C22⋊Q812yes—
281C24#12312yes—
282C24C4⋊Q812yes—
283C24C22×D411yescharacteristic
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—216no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—42no—
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—84no—
326C22×C4—84no—
327C22×C4—88no—
328C22×C4C2×C22⋊C414yes—
329C22×C4C2×C4⋊C414yes—
330C22×C4#11714yes—
331C22×C4C4×D414yes—
332C22×C4C4×D412yes—
333C22×C4C4×Q814yes—
334C22×C4C4×Q814yes—
335C22×C4#12012yes—
336C22×C4#12114yes—
337C22×C4#12114yes—
338C22×C4#12114yes—
339C22×C4#12114yes—
340C22×C4#12112yes—
341C22×C4#12112yes—
342C22×C4C22⋊Q814yes—
343C22×C4C22⋊Q814yes—
344C22×C4C22⋊Q814yes—
345C22×C4C22⋊Q814yes—
346C22×C4C22⋊Q814yes—
347C22×C4C22⋊Q814yes—
348C22×C4C22⋊Q814yes—
349C22×C4C22⋊Q812yes—
350C22×C4#12314yes—
351C22×C4#12314yes—
352C22×C4#12312yes—
353C22×C4#12414yes—
354C22×C4#12514yes—
355C22×C4#12514yes—
356C22×C4#12514yes—
357C22×C4#12514yes—
358C22×C4#12614yes—
359C22×C4#12614yes—
360C22×C4C4⋊Q814yes—
361C22×C4C4⋊Q814yes—
362C22×C4C4⋊Q812yes—
363C22×C4C23×C411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×Q811yescharacteristic
369C22×C4C22×Q811yescharacteristic
370C22×C4C22×Q811yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377M16—232no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—416no—
392C2×C8C2×C4⋊C418yes—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—216no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—816no—
416C4⋊C4C2×C22⋊C418yes—
417C22⋊C4—48no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C42—42no—
421C42—42no—
422C42—42no—
423C42—42no—
424C42—48no—
425C42—48no—
426C23—24no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—24no—
441C23—24no—
442C23#38614yes—
443C23#38714yes—
444C23#38914yes—
445C23#39414yes—
446C23#39514yes—
447C23#39614yes—
448C23#39814yes—
449C23C2×C4×D411yescharacteristic
450C23C2×#12011yescharacteristic
451C23C2×#12111yescharacteristic
452C23C2×#12111yescharacteristic
453C23C2×C22⋊Q811yescharacteristic
454C23C2×#12311yescharacteristic
455C23C2×C4⋊Q811yescharacteristic
456C23#57612yes—
457C23#57712yes—
458C23#57712yes—
459C23#57812yes—
460C2×C4—28no—
461C2×C4—48no—
462C2×C4—48no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—48no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—816no—
482C2×C4—816no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—82no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—82no—
492C2×C4—81no—
493C2×C4—82no—
494C2×C4—84no—
495C2×C4—82no—
496C2×C4#38018yes—
497C2×C4#38614yes—
498C2×C4#38714yes—
499C2×C4#38718yes—
500C2×C4#38914yes—
501C2×C4#38918yes—
502C2×C4#39118yes—
503C2×C4#39414yes—
504C2×C4#39514yes—
505C2×C4#39618yes—
506C2×C4#39614yes—
507C2×C4#39818yes—
508C2×C4#39818yes—
509C2×C4#39814yes—
510C2×C4#40018yes—
511C2×C4C22×C22⋊C411yescharacteristic
512C2×C4C22×C4⋊C411yescharacteristic
513C2×C4C2×#11711yescharacteristic
514C2×C4C2×C4×D411yescharacteristic
515C2×C4C2×C4×Q811yescharacteristic
516C2×C4C2×C4×Q811yescharacteristic
517C2×C4C2×#12111yescharacteristic
518C2×C4C2×#12111yescharacteristic
519C2×C4C2×#12111yescharacteristic
520C2×C4C2×#12111yescharacteristic
521C2×C4C2×C22⋊Q811yescharacteristic
522C2×C4C2×C22⋊Q811yescharacteristic
523C2×C4C2×C22⋊Q811yescharacteristic
524C2×C4C2×C22⋊Q811yescharacteristic
525C2×C4C2×C22⋊Q811yescharacteristic
526C2×C4C2×C22⋊Q811yescharacteristic
527C2×C4C2×C22⋊Q811yescharacteristic
528C2×C4C2×#12311yescharacteristic
529C2×C4C2×#12311yescharacteristic
530C2×C4C2×#12411yescharacteristic
531C2×C4C2×#12511yescharacteristic
532C2×C4C2×#12511yescharacteristic
533C2×C4C2×#12511yescharacteristic
534C2×C4C2×#12511yescharacteristic
535C2×C4C2×#12611yescharacteristic
536C2×C4C2×#12611yescharacteristic
537C2×C4C2×C4⋊Q811yescharacteristic
538C2×C4C2×C4⋊Q811yescharacteristic
539C8—28no—
540C8—44no—
541C22—28no—
542C22—28no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—28no—
548C22—28no—
549C22—24no—
550C22—24no—
551C22#144118yes—
552C22C2×#38611yescharacteristic
553C22C2×#38711yescharacteristic
554C22C2×#38911yescharacteristic
555C22C2×#39411yescharacteristic
556C22C2×#39511yescharacteristic
557C22C2×#39611yescharacteristic
558C22C2×#39811yescharacteristic
559C22#294612yes—
560C22#294712yes—
561C22#301212yes—
562C22#301312yes—
563C22#304111yescharacteristic
564C22#304211yescharacteristic
565C22#304311yescharacteristic
566C22#304411yescharacteristic
567C22#306212yes—
568C22#306312yes—
569C22#308112yes—
570C22#308212yes—
571C22#309812yes—
572C22#309912yes—
573C22#316312yes—
574C22#316412yes—
575C4—48no—
576C4—84no—
577C4—84no—
578C4—84no—
579C4—82no—
580C4—82no—
581C4#144118yes—
582C4C2×#38011yescharacteristic
583C4C2×#38711yescharacteristic
584C4C2×#38911yescharacteristic
585C4C2×#39111yescharacteristic
586C4C2×#39611yescharacteristic
587C4C2×#39811yescharacteristic
588C4C2×#39811yescharacteristic
589C4C2×#40011yescharacteristic
590C2—28no—
591C2C2×#144111yescharacteristic
592C2#2092911yescharacteristic
593C2#2093011yescharacteristic
594C2#2103612yes—
595C2#2116012yes—
596C2#2348512yes—
597C2#2355712yes—
598C2#2361411yescharacteristic
599C2#2361511yescharacteristic
600C2#2366011yescharacteristic
601C2#2366111yescharacteristic
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×#428C4semidirect
107. C2×#107C2×C4semidirect
110. C2×#106C2×C4semidirect
162. C2×M16C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C2×#23162
C2×#17362
C2×#17352
C22×#3402
C2×#17292
#137952
#112202
#112192