Finite Group Databasebeta

← #416586

Subgroups — #416586

Order 512.

1996 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#491C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic
3C2×#2562C211yesmaximal, characteristic
4C2×#1989C211yesmaximal, characteristic, complemented
5C2×#1981C211yesmaximal, characteristic, complemented
6C2×#1955C211yesmaximal, characteristic
7#10970C214yesmaximal, complemented
8#10970C214yesmaximal, complemented
9#10703C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×#125C2211yescharacteristic
12C22×C22⋊Q8C2214yes—
13C22×C4×D4C2211yescharacteristic
14C2×#491—24no—
15C2×#491—24no—
16C2×#491—24no—
17C22×C4⋊C8C2211yescharacteristic
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2214yes—
21C22×C22⋊C8C2214yes—
22C22×#97C2211yescharacteristic
23#2562—24no—
24#2562—24no—
25#2542—24no—
26#2542—24no—
27C2×#400—21no—
28C2×#399—22no—
29C2×#394—21no—
30C2×#390—21no—
31C2×#388—22no—
32C2×#384—21no—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37#1989—24no—
38#1981—24no—
39#1955—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2214yes—
43C23×Q8C2312yes—
44C23×D4D411yescharacteristic
45C24×C4C2312yes—
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8D414yes—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125D414yes—
60C2×C22⋊Q8—28no—
61C2×C22⋊Q8—28no—
62C2×C22⋊Q8—28no—
63C2×C22⋊Q8—28no—
64C2×C22⋊Q8—44no—
65C2×C22⋊Q8—44no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—24no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4D412yes—
86C22×C42—21no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89#491—232no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—28no—
100C2×#103—44no—
101C2×#103—44no—
102C2×C22⋊C8—28no—
103C2×C22⋊C8—44no—
104C2×C22⋊C8—44no—
105C2×#97—22no—
106C2×#97—22no—
107C2×#97D414yes—
108#400—28no—
109#399—216no—
110#394—28no—
111#390—28no—
112#388—216no—
113#384—28no—
114#380—24no—
115#380—24no—
116#380—24no—
117#380—24no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4D414yes—
121C4×C4⋊C4D414yes—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126#340—24no—
127#340—24no—
128#340—28no—
129C25C2×D412yes—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—24no—
135C22×Q8—24no—
136C22×Q8—22no—
137C22×Q8—24no—
138C22×Q8—24no—
139C22×Q8C2×D414yes—
140C22×D4—24no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—42no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2411yescharacteristic
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—24no—
167C22×C8—24no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—24no—
172C22×C8—24no—
173C22×C8C2×D414yes—
174C4⋊Q8—28no—
175C4⋊Q8—28no—
176C4⋊Q8—44no—
177C4⋊Q8—44no—
178#125—28no—
179#125—28no—
180#125—44no—
181#125—44no—
182C22⋊Q8—416no—
183C22⋊Q8—416no—
184C22⋊Q8—416no—
185C22⋊Q8—416no—
186C4×D4—44no—
187C4×D4—44no—
188C4×D4—44no—
189C4×D4—44no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—216no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D414yes—
226C2×C4⋊C4C2×D414yes—
227C2×C4⋊C4#4118yes—
228C2×C22⋊C4—216no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—48no—
235C2×C22⋊C4—48no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—84no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—44no—
242C2×C42—82no—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42#4112yes—
249C4⋊C8—44no—
250#103—216no—
251#103—216no—
252#103—416no—
253C22⋊C8—416no—
254#97—28no—
255#95—28no—
256#95—28no—
257#95—24no—
258#95—24no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—24no—
263C24—24no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—24no—
268C24—24no—
269C24#13714yes—
270C24C22×D411yescharacteristic
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—28no—
274C2×Q8—28no—
275C2×Q8—42no—
276C2×Q8—44no—
277C2×Q8—42no—
278C2×Q8—44no—
279C2×Q8—44no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—44no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—44no—
286C2×Q8—44no—
287C2×Q8—44no—
288C2×Q8—44no—
289C2×D4—28no—
290C2×D4—48no—
291C2×D4—48no—
292C2×D4—44no—
293C2×D4—44no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—28no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—44no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4#12012yes—
347C22×C4#12012yes—
348C22×C4#12012yes—
349C22×C4#12012yes—
350C22×C4#12112yes—
351C22×C4#12112yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4114yes—
359C22×C4H(5,𝔽2)14yes—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—44no—
364C2×C8—44no—
365C2×C8—42no—
366C2×C8—44no—
367C2×C8—44no—
368C2×C8—42no—
369C2×C8—42no—
370C4⋊C4—24no—
371C4⋊C4—28no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—432no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—48no—
386C4⋊C4—48no—
387C4⋊C4—48no—
388C4⋊C4—48no—
389C4⋊C4#12418yes—
390C22⋊C4—432no—
391C22⋊C4—416no—
392C22⋊C4—48no—
393C22⋊C4—88no—
394C42—48no—
395C42—84no—
396C42#12014yes—
397C42#12114yes—
398C42#12114yes—
399C42#12714yes—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—28no—
404C23—28no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—44no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—42no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23C2×#12011yescharacteristic
424C23C2×#12011yescharacteristic
425C23C2×#12111yescharacteristic
426C23#54614yes—
427C23C2×#13712yes—
428C23#57512yes—
429C23#57712yes—
430C23#57812yes—
431Q8—44no—
432Q8—44no—
433Q8—48no—
434Q8—82no—
435Q8—82no—
436Q8—84no—
437D4—416no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—28no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—48no—
450C2×C4—48no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—48no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—84no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—82no—
470C2×C4—84no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—84no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—84no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—84no—
482C2×C4#39214yes—
483C2×C4#39214yes—
484C2×C4C2×#12011yescharacteristic
485C2×C4C2×#12111yescharacteristic
486C2×C4C2×#12111yescharacteristic
487C2×C4C2×#12412yes—
488C2×C4C2×#12711yescharacteristic
489C2×C4#53212yes—
490C2×C4#53212yes—
491C2×C4#53512yes—
492C2×C4#53512yes—
493C2×C4#53512yes—
494C2×C4#54614yes—
495C2×C4#55014yes—
496C2×C4#55914yes—
497C8—44no—
498C8—82no—
499C22—48no—
500C22—44no—
501C22—48no—
502C22—44no—
503C22—44no—
504C22—84no—
505C22—82no—
506C22—82no—
507C22—84no—
508C22—82no—
509C22—82no—
510C22C2×#39211yescharacteristic
511C22#243112yes—
512C22#243512yes—
513C22#254412yes—
514C22#300411yescharacteristic
515C22#300511yescharacteristic
516C22#300712yes—
517C22#300911yescharacteristic
518C22#301111yescharacteristic
519C22#301312yes—
520C22#305612yes—
521C22#306312yes—
522C22#330714yes—
523C22#331214yes—
524C22#331514yes—
525C22#332014yes—
526C4—24no—
527C4—24no—
528C4—48no—
529C4—82no—
530C4—82no—
531C4—82no—
532C4—84no—
533C4—84no—
534C4#243112yes—
535C4#254412yes—
536C4#254812yes—
537C4#265711yescharacteristic
538C4#285811yescharacteristic
539C2—84no—
540C2#1522711yescharacteristic
541C2#2339211yescharacteristic
542C2#2339411yescharacteristic
543C2#2558212yes—
544C2#2558412yes—
545C2#2589212yes—
546C2#2589412yes—
547C2#2861114yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×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
1. C22×#491C2semidirect
4. C2×#1989C2semidirect
5. C2×#1981C2semidirect
7. #10970C2semidirect
8. #10970C2semidirect
9. #10703C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4912
C2×#25422
C2×#25622
C2×#19892
C2×#19812
C2×#19552
#109702
#107032