Finite Group Databasebeta

← #416681

Subgroups — #416681

Order 512.

1022 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic
2#20801C211yesmaximal, characteristic
3#20765C211yesmaximal, characteristic
4#13909C211yesmaximal, characteristic
5#11340C212yesmaximal
6#11339C212yesmaximal
7#11217C212yesmaximal
8#11217C212yesmaximal
9#10851C211yesmaximal, characteristic
10#10851C211yesmaximal, characteristic
11C4×#343C211yesmaximal, characteristic
12C2×#428C2212yes—
13C2×#428C2212yes—
14C2×C4×M16C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2851—22no—
17#2646—22no—
18#2571—21no—
19#2571—21no—
20#2570—22no—
21#2570—22no—
22#2552—21no—
23#2552—21no—
24#2317C2212yes—
25#2317C2212yes—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1847—21no—
34#1847—21no—
35#1846—22no—
36#1836—21no—
37#1836—21no—
38#1835—22no—
39C2×#343C2212yes—
40C2×#343C2212yes—
41#1736C2212yes—
42#1736C2212yes—
43C2×#339C2212yes—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2212yes—
47#1441—22no—
48#1436—22no—
49C4⋊#95C2211yescharacteristic
50C4⋊#95C2211yescharacteristic
51C22×M16C2312yes—
52C22×M16C2312yes—
53C2×#125—22no—
54C2×#117C2312yes—
55C2×#117C2312yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C42D412yes—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66#428—24no—
67#428—24no—
68#428—42no—
69#428—42no—
70C2×#107—22no—
71C2×#107—22no—
72C2×#106—22no—
73C2×#106—22no—
74C4×M16—22no—
75C4×M16—22no—
76C4×M16Q814yes—
77C4×M16Q814yes—
78C2×#97—21no—
79C2×#97—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82#400—24no—
83#400—24no—
84#398—24no—
85#398—24no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#387—24no—
93#387—24no—
94#385—24no—
95#385—24no—
96#384—22no—
97#384—22no—
98#384—42no—
99#384D411yescharacteristic
100#384D411yescharacteristic
101#384D411yescharacteristic
102#384D411yescharacteristic
103#382—22no—
104#382—22no—
105#382—22no—
106#382—21no—
107#382—21no—
108#382—21no—
109#382—21no—
110#382—42no—
111#382D412yes—
112#382D412yes—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4Q814yes—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95D412yes—
134#343—24no—
135#343—24no—
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140#339—24no—
141#339—24no—
142#339—42no—
143#339—42no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2411yescharacteristic
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16C2×Q814yes—
162C2×M16C2×Q814yes—
163C2×M16#4114yes—
164C2×M16#4114yes—
165C22×C8—22no—
166C22×C8—22no—
167#125—84no—
168#117—22no—
169#117—22no—
170#117—22no—
171#117—22no—
172#117—22no—
173#117—22no—
174#117—22no—
175#117—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—82no—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×Q812yes—
232C2×C42C2×Q812yes—
233C2×C42C2×Q812yes—
234C2×C42C2×Q812yes—
235C2×C42C2×Q812yes—
236C2×C42#4112yes—
237#107—44no—
238#107—44no—
239#106—44no—
240#106—44no—
241#97—24no—
242#97—24no—
243C4×C8—24no—
244C4×C8—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—24no—
256#95—44no—
257#95—44no—
258#95—41no—
259#95—41no—
260#95—41no—
261#95—41no—
262#95—82no—
263C24C22×D411yescharacteristic
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4#12011yescharacteristic
307C22×C4#12011yescharacteristic
308C22×C4#12011yescharacteristic
309C22×C4#12011yescharacteristic
310C22×C4C22⋊Q812yes—
311C22×C4C22⋊Q812yes—
312C22×C4C4⋊Q812yes—
313C22×C4C4⋊Q812yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×Q811yescharacteristic
317C22×C4C22×Q811yescharacteristic
318C22×C4C22×Q811yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4112yes—
321C22×C4C2×#4112yes—
322C22×C4H(5,𝔽2)12yes—
323C22×C42−1+412yes—
324M16—24no—
325M16—24no—
326M16—24no—
327M16—24no—
328M16—24no—
329M16—42no—
330M16—42no—
331M16#12518yescomplemented
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—42no—
337C2×C8—42no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—48no—
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—42no—
358C4⋊C4—42no—
359C4⋊C4—81no—
360C4⋊C4—81no—
361C4⋊C4—81no—
362C4⋊C4—81no—
363C4⋊C4—84no—
364C4⋊C4—84no—
365C4⋊C4—84no—
366C4⋊C4—84no—
367C4⋊C4—88no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C42—24no—
373C42—24no—
374C42—44no—
375C42—42no—
376C42—42no—
377C42—42no—
378C42—42no—
379C42—42no—
380C42—42no—
381C42—82no—
382C42C22⋊Q812yes—
383C42C22⋊Q812yes—
384C42C22⋊Q812yes—
385C42C22⋊Q812yes—
386C42C4⋊Q812yes—
387C42C4⋊Q812yes—
388C42C4⋊Q812yes—
389C42C4⋊Q812yes—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23C2×#12011yescharacteristic
395C23C2×C22⋊Q811yescharacteristic
396C23C2×C4⋊Q811yescharacteristic
397C23#53211yescharacteristic
398C23#53211yescharacteristic
399C23#53511yescharacteristic
400C23#53611yescharacteristic
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—24no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4#39512yes—
460C2×C4#39512yes—
461C2×C4#39512yes—
462C2×C4#39512yes—
463C2×C4C2×C22⋊Q811yescharacteristic
464C2×C4C2×C22⋊Q811yescharacteristic
465C2×C4C2×#12512yes—
466C2×C4C2×C4⋊Q811yescharacteristic
467C2×C4C2×C4⋊Q811yescharacteristic
468C2×C4#53312yes—
469C2×C4#54412yes—
470C2×C4#54412yes—
471C2×C4#54612yes—
472C2×C4#54712yes—
473C2×C4#54812yes—
474C2×C4#55012yes—
475C2×C4#55212yes—
476C2×C4#55312yes—
477C2×C4#55412yes—
478C2×C4#55712yes—
479C8—28no—
480C8—44no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22C2×#39511yescharacteristic
494C22#243111yescharacteristic
495C22#243211yescharacteristic
496C22#256511yescharacteristic
497C22#256611yescharacteristic
498C22#266011yescharacteristic
499C22#285911yescharacteristic
500C22#301811yescharacteristic
501C22#301811yescharacteristic
502C22#301911yescharacteristic
503C22#302011yescharacteristic
504C4—22no—
505C4—22no—
506C4—22no—
507C4—22no—
508C4—44no—
509C4—82no—
510C4—82no—
511C4—82no—
512C4—82no—
513C4—82no—
514C4—82no—
515C4—82no—
516C4—161no—
517C4—161no—
518C4#244112yes—
519C4#244212yes—
520C4#255012yes—
521C4#255212yes—
522C2—21no—
523C2—21no—
524C2—21no—
525C2—21no—
526C2#1522811yescharacteristic
527C2#2356011yescharacteristic
528C2#2356111yescharacteristic
529C2#2562511yescharacteristic
530C2#2562611yescharacteristic
531C2#2569711yescharacteristic
532C2#2569811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
331. M16#125semidirect

Maximal subgroups

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

subgroupindex
#351812
#208012
#207652
#139092
#113402
#113392
#112172
#108512
C4×#3432