Finite Group Databasebeta

← #358233

Subgroups — #358233

Order 512.

1938 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2C2×#1736C211yesmaximal, characteristic
3C2×#1736C211yesmaximal, characteristic
4C2×#1735C211yesmaximal, characteristic
5C22×#340C211yesmaximal, characteristic
6C2×#1729C211yesmaximal, characteristic
7#13795C211yesmaximal, characteristic
8#11268C214yesmaximal, complemented
9#11267C214yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#2317—24no—
18#2317—24no—
19C2×#389—22no—
20C2×#385—22no—
21C2×#384C2212yes—
22C2×#382C2212yes—
23#1736—24no—
24#1736—24no—
25#1735—24no—
26C2×#340—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2214yes—
30C2×#340C2214yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33#1729C418yescomplemented
34C2×#336C2214yes—
35C4⋊#95—22no—
36C4⋊#95C2214yes—
37C22⋊#95—22no—
38C24×C4—21no—
39C24×C4—21no—
40C24×C4C2311yescharacteristic
41C22×M16—22no—
42C22×M16—22no—
43C22×M16—22no—
44C22×M16—22no—
45C22×M16—22no—
46C22×M16—22no—
47C22×M16C2×C414yes—
48C23×C8—22no—
49C23×C8—22no—
50C23×C8—22no—
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—22no—
54C23×C8C2×C414yes—
55C23×C8C2312yes—
56C23×C8C2312yes—
57C2×#117—22no—
58C2×#117—22no—
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×#117C2×C414yes—
72C2×#117D414yes—
73C2×#117Q814yes—
74C2×#117Q814yes—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4D414yes—
89C22×C4⋊C4D414yes—
90C22×C4⋊C4D412yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4D412yes—
99C22×C42—21no—
100C22×C42—21no—
101C22×C42C2312yes—
102#389—216no—
103#385—216no—
104#384—24no—
105#384—24no—
106#382—24no—
107#382—24no—
108C2×#95—24no—
109C2×#95—24no—
110C2×#95—24no—
111#340—28no—
112#340—24no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—28no—
117#336C2×C4116yescomplemented
118C25C2×D411yescharacteristic
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—42no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4—42no—
130C23×C4—82no—
131C23×C4—82no—
132C23×C4—82no—
133C23×C4—82no—
134C23×C4C22×C412yes—
135C23×C4C2×D414yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×Q812yes—
141C23×C4C2×Q812yes—
142C23×C4#4112yes—
143C23×C4#4112yes—
144C23×C4#4112yes—
145C23×C4#4112yes—
146C23×C4#4112yes—
147C23×C4#4112yes—
148C23×C4#4112yes—
149C23×C4#4112yes—
150C23×C4C2411yescharacteristic
151C2×M16—28no—
152C2×M16—28no—
153C2×M16—28no—
154C2×M16—44no—
155C2×M16—44no—
156C2×M16—44no—
157C2×M16—44no—
158C22×C8—28no—
159C22×C8—28no—
160C22×C8—28no—
161C22×C8—22no—
162C22×C8—22no—
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—44no—
174C22×C8—44no—
175C22×C8—44no—
176C22×C8—44no—
177C22×C8C22×C414yes—
178C22×C8C22×C414yes—
179#117—28no—
180#117—28no—
181#117—28no—
182#117—28no—
183#117—28no—
184#117—28no—
185#117—28no—
186#117—28no—
187#117—44no—
188#117—44no—
189#117—44no—
190#117—44no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
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—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—216no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—84no—
234C2×C4⋊C4—84no—
235C2×C4⋊C4C4⋊C418yes—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×Q814yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4C4⋊C418yes—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42C22×C414yes—
261C2×C42C2×Q814yes—
262#95—216no—
263#95—84no—
264#95—84no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24C2×C4⋊C412yes—
270C24#12012yes—
271C24#12112yes—
272C24C22⋊Q812yes—
273C24#12312yes—
274C24#12412yes—
275C24#12412yes—
276C24C22×D411yescharacteristic
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—48no—
311C22×C4—48no—
312C22×C4—88no—
313C22×C4—84no—
314C22×C4—84no—
315C22×C4—88no—
316C22×C4—88no—
317C22×C4—88no—
318C22×C4—84no—
319C22×C4—84no—
320C22×C4C2×C4⋊C418yes—
321C22×C4C2×C4⋊C412yes—
322C22×C4#11714yes—
323C22×C4#11714yes—
324C22×C4C4×D414yes—
325C22×C4C4×D414yes—
326C22×C4C4×D414yes—
327C22×C4C4×Q814yes—
328C22×C4#12012yes—
329C22×C4#12114yes—
330C22×C4#12114yes—
331C22×C4#12112yes—
332C22×C4C22⋊Q814yes—
333C22×C4C22⋊Q814yes—
334C22×C4C22⋊Q814yes—
335C22×C4C22⋊Q814yes—
336C22×C4C22⋊Q812yes—
337C22×C4#12314yes—
338C22×C4#12314yes—
339C22×C4#12314yes—
340C22×C4#12314yes—
341C22×C4#12312yes—
342C22×C4#12414yes—
343C22×C4#12414yes—
344C22×C4#12412yes—
345C22×C4#12412yes—
346C22×C4#12514yes—
347C22×C4#12514yes—
348C22×C4#12514yes—
349C22×C4#12514yes—
350C22×C4#12514yes—
351C22×C4#12614yes—
352C22×C4#12614yes—
353C22×C4#12614yes—
354C22×C4#12614yes—
355C22×C4C4⋊Q814yes—
356C22×C4C23×C411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×Q811yescharacteristic
361C22×C4C22×Q811yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370M16—416no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—416no—
386C4⋊C4—216no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—42no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—48no—
408C4⋊C4—48no—
409C4⋊C4—816no—
410C22⋊C4—216no—
411C22⋊C4—48no—
412C22⋊C4—48no—
413C42—24no—
414C42—24no—
415C42—24no—
416C42—24no—
417C42—48no—
418C42—48no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23#38214yes—
436C23#38514yes—
437C23#38914yes—
438C23#39314yes—
439C23#39414yes—
440C23#39714yes—
441C23#40014yes—
442C23C22×C4⋊C411yescharacteristic
443C23C2×#12011yescharacteristic
444C23C2×#12111yescharacteristic
445C23C2×C22⋊Q811yescharacteristic
446C23C2×#12311yescharacteristic
447C23C2×#12411yescharacteristic
448C23C2×#12411yescharacteristic
449C23#57612yes—
450C23#57712yes—
451C23#57712yes—
452C23#57812yes—
453C2×C4—28no—
454C2×C4—28no—
455C2×C4—28no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—48no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—816no—
473C2×C4—816no—
474C2×C4—84no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—84no—
478C2×C4—84no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—81no—
484C2×C4—82no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—82no—
488C2×C4—81no—
489C2×C4#38214yes—
490C2×C4#38514yes—
491C2×C4#38718yes—
492C2×C4#38718yes—
493C2×C4#38818yes—
494C2×C4#38818yes—
495C2×C4#38914yes—
496C2×C4#39314yes—
497C2×C4#39414yes—
498C2×C4#39714yes—
499C2×C4#39818yes—
500C2×C4#39818yes—
501C2×C4#40018yes—
502C2×C4#40018yes—
503C2×C4#40014yes—
504C2×C4C2×#11711yescharacteristic
505C2×C4C2×#11711yescharacteristic
506C2×C4C2×C4×D411yescharacteristic
507C2×C4C2×C4×D411yescharacteristic
508C2×C4C2×C4×D411yescharacteristic
509C2×C4C2×C4×Q811yescharacteristic
510C2×C4C2×#12111yescharacteristic
511C2×C4C2×#12111yescharacteristic
512C2×C4C2×C22⋊Q811yescharacteristic
513C2×C4C2×C22⋊Q811yescharacteristic
514C2×C4C2×C22⋊Q811yescharacteristic
515C2×C4C2×C22⋊Q811yescharacteristic
516C2×C4C2×#12311yescharacteristic
517C2×C4C2×#12311yescharacteristic
518C2×C4C2×#12311yescharacteristic
519C2×C4C2×#12311yescharacteristic
520C2×C4C2×#12411yescharacteristic
521C2×C4C2×#12411yescharacteristic
522C2×C4C2×#12511yescharacteristic
523C2×C4C2×#12511yescharacteristic
524C2×C4C2×#12511yescharacteristic
525C2×C4C2×#12511yescharacteristic
526C2×C4C2×#12511yescharacteristic
527C2×C4C2×#12611yescharacteristic
528C2×C4C2×#12611yescharacteristic
529C2×C4C2×#12611yescharacteristic
530C2×C4C2×#12611yescharacteristic
531C2×C4C2×C4⋊Q811yescharacteristic
532C8—44no—
533C8—44no—
534C22—28no—
535C22—24no—
536C22—24no—
537C22—28no—
538C22—28no—
539C22—24no—
540C22—24no—
541C22—28no—
542C22—24no—
543C22—24no—
544C22#144218yes—
545C22C2×#38211yescharacteristic
546C22C2×#38511yescharacteristic
547C22C2×#38911yescharacteristic
548C22C2×#39311yescharacteristic
549C22C2×#39411yescharacteristic
550C22C2×#39711yescharacteristic
551C22C2×#40011yescharacteristic
552C22#291212yes—
553C22#291312yes—
554C22#301212yes—
555C22#301312yes—
556C22#304111yescharacteristic
557C22#304211yescharacteristic
558C22#304311yescharacteristic
559C22#304411yescharacteristic
560C22#308312yes—
561C22#308412yes—
562C22#309612yes—
563C22#309712yes—
564C22#313612yes—
565C22#313612yes—
566C22#313712yes—
567C22#313712yes—
568C4—48no—
569C4—84no—
570C4—84no—
571C4—82no—
572C4—84no—
573C4—82no—
574C4#144218yes—
575C4C2×#38711yescharacteristic
576C4C2×#38711yescharacteristic
577C4C2×#38811yescharacteristic
578C4C2×#38811yescharacteristic
579C4C2×#39811yescharacteristic
580C4C2×#39811yescharacteristic
581C4C2×#40011yescharacteristic
582C4C2×#40011yescharacteristic
583C2—28no—
584C2C2×#144211yescharacteristic
585C2#2076312yes—
586C2#2084912yes—
587C2#2117311yescharacteristic
588C2#2117411yescharacteristic
589C2#2345812yes—
590C2#2351111yescharacteristic
591C2#2351211yescharacteristic
592C2#2363211yescharacteristic
593C2#2363311yescharacteristic
594C2#2374812yes—
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. #11268C2semidirect
9. #11267C2semidirect
33. #1729C4semidirect
117. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#17362
C2×#17352
C22×#3402
C2×#17292
#137952
#112682
#112672