Finite Group Databasebeta

← #409314

Subgroups — #409314

Order 512.

1138 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
1C2×#1834C211yesmaximal, characteristic, complemented
2C2×#1753C211yesmaximal, characteristic
3#15195C211yesmaximal, characteristic
4#15135C211yesmaximal, characteristic
5#11371C211yesmaximal, characteristic, complemented
6#11357C211yesmaximal, characteristic, complemented
7#11292C212yesmaximal, complemented
8#11199C212yesmaximal
9#10881C212yesmaximal, complemented
10#10820C211yesmaximal, characteristic, complemented
11#10722C212yesmaximal
12C23×M16C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C22×#97C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#399—21no—
17C2×#389C2212yes—
18C2×#384C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#380C2212yes—
21C2×#380C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23#1834—24no—
24#1753—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35#1445—21no—
36#1444—21no—
37#1442—21no—
38#1442—21no—
39#1439—21no—
40#1439—21no—
41#1439—21no—
42C4⋊#95C2212yes—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C22×M16—22no—
48C22×M16—22no—
49C22×M16—24no—
50C22×M16—42no—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4D411yescharacteristic
67C22×C22⋊C4D411yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×#97—22no—
74C2×#97—42no—
75C2×C4×C8—24no—
76C2×C4×C8D414yes—
77#399—44no—
78#389—24no—
79#389—42no—
80#384—22no—
81#384—22no—
82#384—22no—
83#384—22no—
84#380—24no—
85#380—24no—
86#380—24no—
87#380—42no—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—22no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95D411yescharacteristic
100C2×#95D411yescharacteristic
101#340—24no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—42no—
108#340—42no—
109#336—22no—
110#336—22no—
111C25C2×D411yescharacteristic
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4#4112yes—
130C23×C4#4112yes—
131C23×C4C2411yescharacteristic
132C2×M16—28no—
133C2×M16—48no—
134C2×M16—48no—
135C2×M16—44no—
136C2×M16—44no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8C2×D412yes—
153C22×C8C2×D412yes—
154C22×C8#4112yes—
155C22×C8#4112yes—
156C22×C8#4114yes—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4#4114yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—82no—
203C2×C22⋊C4—82no—
204C2×C22⋊C4—82no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42C2×D412yes—
212C2×C42#4112yes—
213C22⋊C8—28no—
214#97—44no—
215C4×C8—28no—
216#95—22no—
217#95—22no—
218#95—44no—
219#95—44no—
220#95—44no—
221#95—42no—
222#95—42no—
223#95—42no—
224#95—42no—
225#95—82no—
226#95—82no—
227C24—21no—
228C24—21no—
229C24—22no—
230C24—22no—
231C24—21no—
232C24—21no—
233C24—21no—
234C24—21no—
235C24—22no—
236C24—22no—
237C24#12412yes—
238C24C22×D411yescharacteristic
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4#12112yes—
304C22×C4#12112yes—
305C22×C4#12412yes—
306C22×C4AGL(1,ℤ/8)12yes—
307C22×C4#13712yes—
308C22×C4#13712yes—
309C22×C4#13712yes—
310C22×C4C22×D411yescharacteristic
311C22×C4C2×#4112yes—
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4112yes—
317C22×C4C2×#4111yescharacteristic
318C22×C4H(5,𝔽2)12yes—
319C22×C42−1+412yes—
320M16—416no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8#12314yes—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—84no—
358C4⋊C4—84no—
359C4⋊C4—84no—
360C4⋊C4—88no—
361C4⋊C4—84no—
362C22⋊C4—44no—
363C22⋊C4—44no—
364C22⋊C4—88no—
365C22⋊C4—88no—
366C42—22no—
367C42—22no—
368C42—24no—
369C42—41no—
370C42—41no—
371C42—42no—
372C23—22no—
373C23—24no—
374C23—22no—
375C23—22no—
376C23—24no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—42no—
394C23—42no—
395C23C2×#12111yescharacteristic
396C23C2×#12411yescharacteristic
397C23#53211yescharacteristic
398C23#54612yes—
399C23#54712yes—
400C23C2×AGL(1,ℤ/8)11yescharacteristic
401C23C2×#13711yescharacteristic
402C23C2×#13711yescharacteristic
403C23C2×#13711yescharacteristic
404C23#57612yes—
405C23#57712yes—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4#46814yes—
464C2×C4#47014yes—
465C2×C4C2×#12311yescharacteristic
466C2×C4#52511yescharacteristic
467C2×C4#52511yescharacteristic
468C2×C4#52911yescharacteristic
469C2×C4#52912yes—
470C2×C4#52912yes—
471C2×C4#52911yescharacteristic
472C2×C4#53211yescharacteristic
473C2×C4#53511yescharacteristic
474C2×C4#53611yescharacteristic
475C2×C4#53912yes—
476C2×C4#53912yes—
477C2×C4#55112yes—
478C2×C4#55112yes—
479C2×C4#55212yes—
480C2×C4#55212yes—
481C2×C4#55212yes—
482C2×C4#55912yes—
483C8—24no—
484C8—42no—
485C8—42no—
486C8—42no—
487C22—42no—
488C22—42no—
489C22—44no—
490C22—44no—
491C22—42no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—82no—
497C22—82no—
498C22#248611yescharacteristic
499C22#248812yes—
500C22#254411yescharacteristic
501C22#255312yes—
502C22#285011yescharacteristic
503C22C2×#46811yescharacteristic
504C22C2×#47011yescharacteristic
505C22#305711yescharacteristic
506C22#305811yescharacteristic
507C22#313612yes—
508C22#313911yescharacteristic
509C22#314111yescharacteristic
510C22#317912yes—
511C22#318211yescharacteristic
512C22#318511yescharacteristic
513C22#332512yes—
514C22#332912yes—
515C22#333112yes—
516C22#333312yes—
517C22#334412yes—
518C22#334512yes—
519C22#334612yes—
520C22#334612yes—
521C4—24no—
522C4—24no—
523C4—42no—
524C4—81no—
525C4—81no—
526C4—84no—
527C4—162no—
528C4—162no—
529C4—162no—
530C4#239511yescharacteristic
531C4#248211yescharacteristic
532C4#251712yes—
533C4#257512yes—
534C4#261111yescharacteristic
535C4#263311yescharacteristic
536C2—82no—
537C2#1501611yescharacteristic
538C2#2766311yescharacteristic
539C2#2766611yescharacteristic
540C2#2770412yes—
541C2#2770712yes—
542C2#2860511yescharacteristic
543C2#2860811yescharacteristic
544C2#2886512yes—
545C2#2887112yes—
546C2#3304511yescharacteristic
547C2#3304911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1834C2semidirect
5. #11371C2semidirect
6. #11357C2semidirect
7. #11292C2semidirect
9. #10881C2semidirect
10. #10820C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18342
C2×#17532
#151952
#151352
#113712
#113572
#112922
#111992
#108812
#108202
#107222