Finite Group Databasebeta

← #413586

Subgroups — #413586

Order 512.

1348 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2076C212yesmaximal
2C2×#2077C212yesmaximal
3C2×#1847C211yesmaximal, characteristic, complemented
4C2×#1749C211yesmaximal, characteristic
5#14686C211yesmaximal, characteristic
6#14682C211yesmaximal, characteristic
7#11379C212yesmaximal, complemented
8#11378C212yesmaximal, complemented
9#10843C211yesmaximal, characteristic, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10843C211yesmaximal, characteristic, complemented
12C23×C42C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C22⋊C8C2212yes—
16C22×#97C2211yescharacteristic
17C22×#97C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#380—22no—
23C2×#380C2211yescharacteristic
24C2×#380C2212yes—
25C2×#380C2211yescharacteristic
26#2077—28no—
27#2076—28no—
28#1847—24no—
29#1749—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C4⋊#95—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41#1434—22no—
42C22⋊#95—22no—
43C24×C4D412yes—
44C24×C4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C42—22no—
59C22×C42—22no—
60C22×C42—22no—
61C22×C42—22no—
62C22×C42—22no—
63C22×C42—22no—
64C22×C42D412yes—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×#107—24no—
68C2×#107—24no—
69C2×#107—24no—
70C2×#106—24no—
71C2×#106—24no—
72C2×#106—24no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8D418yes—
76C2×#97—22no—
77C2×#97—22no—
78C2×#97—22no—
79C2×#97—22no—
80C2×#97—22no—
81C2×#97D414yes—
82#384—22no—
83#384—22no—
84#384—22no—
85#384—22no—
86#382—22no—
87#382—22no—
88#382—22no—
89#382—22no—
90#380—24no—
91#380—24no—
92#380—24no—
93#380—24no—
94#380—48no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—24no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—24no—
109#336—24no—
110C25C2×D411yescharacteristic
111C23×C4—24no—
112C23×C4—24no—
113C23×C4—24no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D414yes—
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4C2411yescharacteristic
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8C2×D414yes—
149C22×C8C2×D414yes—
150C22×C8C2×D414yes—
151C22×C8#4114yes—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
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—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—48no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—44no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—44no—
202C2×C42—44no—
203C2×C42—44no—
204C2×C42—44no—
205C2×C42C2×D412yes—
206C2×C42#4112yes—
207C2×C42#4112yes—
208C2×C42#4112yes—
209C2×C42#4112yes—
210C2×C42#4112yes—
211#107—48no—
212#106—48no—
213C22⋊C8—216no—
214#97—28no—
215#97—44no—
216#95—44no—
217#95—44no—
218#95—44no—
219#95—42no—
220#95—42no—
221#95—44no—
222#95—44no—
223#95—48no—
224C24—21no—
225C24—21no—
226C24—21no—
227C24—21no—
228C24—21no—
229C24—21no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24#12712yes—
234C24#13714yes—
235C24C22×D411yescharacteristic
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—216no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—48no—
260C22×C4—48no—
261C22×C4—48no—
262C22×C4—48no—
263C22×C4—48no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—48no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—44no—
294C22×C4—44no—
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—44no—
304C22×C4—44no—
305C22×C4#12114yes—
306C22×C4#12314yes—
307C22×C4#12314yes—
308C22×C4#12712yes—
309C22×C4AGL(1,ℤ/8)14yes—
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D412yes—
312C22×C4C2×#4112yes—
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C22×C42−1+412yes—
319C22×C42−1+412yes—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8#12118yes—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—88no—
351C4⋊C4—88no—
352C22⋊C4—88no—
353C22⋊C4—84no—
354C22⋊C4—84no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—48no—
360C42#12314yes—
361C42#12414yes—
362C42#12614yes—
363C42#12614yes—
364C23—24no—
365C23—24no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—24no—
371C23—24no—
372C23—24no—
373C23—22no—
374C23—22no—
375C23—24no—
376C23—42no—
377C23—42no—
378C23—41no—
379C23—41no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—41no—
384C23—41no—
385C23—42no—
386C23#49714yes—
387C23C2×#12111yescharacteristic
388C23C2×#12311yescharacteristic
389C23C2×#12311yescharacteristic
390C23C2×#12711yescharacteristic
391C23#53211yescharacteristic
392C23#53611yescharacteristic
393C23#53611yescharacteristic
394C23C2×AGL(1,ℤ/8)12yes—
395C23C2×#13712yes—
396C23#57512yes—
397C23#57512yes—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—432no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—84no—
426C2×C4—81no—
427C2×C4—82no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—82no—
433C2×C4—81no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—82no—
441C2×C4—84no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—84no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—84no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4#39718yes—
454C2×C4C2×#12112yes—
455C2×C4C2×#12311yescharacteristic
456C2×C4C2×#12411yescharacteristic
457C2×C4C2×#12611yescharacteristic
458C2×C4C2×#12611yescharacteristic
459C2×C4#53212yes—
460C2×C4#53612yes—
461C2×C4#54112yes—
462C2×C4#54712yes—
463C2×C4#54712yes—
464C2×C4#54712yes—
465C2×C4#54712yes—
466C2×C4#55212yes—
467C2×C4#55212yes—
468C2×C4#55612yes—
469C2×C4#55612yes—
470C8—28no—
471C8—44no—
472C22—24no—
473C22—24no—
474C22—24no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—44no—
479C22—42no—
480C22—44no—
481C22—44no—
482C22—44no—
483C22C2×#39711yescharacteristic
484C22#249911yescharacteristic
485C22#249911yescharacteristic
486C22#250111yescharacteristic
487C22#250111yescharacteristic
488C22#262111yescharacteristic
489C22#285611yescharacteristic
490C22#305512yes—
491C22#305612yes—
492C22#309212yes—
493C22#309312yes—
494C22#309412yes—
495C22#309512yes—
496C22C2×#49711yescharacteristic
497C22#315011yescharacteristic
498C22#315111yescharacteristic
499C22#315311yescharacteristic
500C22#317112yes—
501C22#318112yes—
502C22#323512yes—
503C22#323612yes—
504C22#323912yes—
505C22#324012yes—
506C4—28no—
507C4—42no—
508C4—44no—
509C4—161no—
510C4—161no—
511C4—162no—
512C4—161no—
513C4—162no—
514C4—161no—
515C4#248612yes—
516C4#249612yes—
517C4#249612yes—
518C4#254712yes—
519C2—44no—
520C2#1515011yescharacteristic
521C2#2364212yes—
522C2#2788112yes—
523C2#2788212yes—
524C2#2791311yescharacteristic
525C2#2791511yescharacteristic
526C2#2791711yescharacteristic
527C2#2791911yescharacteristic
528C2#2988412yes—
529C2#3318411yescharacteristic
530C2#3318511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1847C2semidirect
7. #11379C2semidirect
8. #11378C2semidirect
9. #10843C2semidirect
10. #10843C2semidirect
11. #10843C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20762
C2×#20772
C2×#18472
C2×#17492
#146862
#146822
#113792
#113782
#108432