Finite Group Databasebeta

← #415467

Subgroups — #415467

Order 512.

1134 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1847C211yesmaximal, characteristic, complemented
2C2×#1750C211yesmaximal, characteristic
3#14340C211yesmaximal, characteristic
4#14340C211yesmaximal, characteristic
5#11257C212yesmaximal, complemented
6#11256C212yesmaximal, complemented
7#11096C212yesmaximal
8#11095C212yesmaximal
9#10843C211yesmaximal, characteristic, complemented
10#10842C211yesmaximal, characteristic, complemented
11#10841C211yesmaximal, characteristic, complemented
12C22×#117C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C22⋊C8C2212yes—
16C22×#97C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×#380C2212yes—
24C2×#380C2211yescharacteristic
25#1847—24no—
26#1750—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34#1441—22no—
35#1441—22no—
36#1439—22no—
37#1439—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C24×C4C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×#117—22no—
45C2×#117—22no—
46C2×#117—22no—
47C2×#117—22no—
48C2×#117—22no—
49C2×#117—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C22⋊C4—21no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4D412yes—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×#107—24no—
66C2×#107—42no—
67C2×#107—42no—
68C2×#106—24no—
69C2×#106—42no—
70C2×#106—42no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8Q818yes—
74C2×#97—22no—
75C2×#97—22no—
76C2×#97D414yes—
77C2×C4×C8—22no—
78C2×C4×C8—22no—
79C2×C4×C8—22no—
80#384—22no—
81#384—22no—
82#384—22no—
83#384—22no—
84#382—22no—
85#382—22no—
86#382—22no—
87#382—22no—
88#380—42no—
89#380—42no—
90#380—42no—
91#380—42no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101#340—24no—
102#340—24no—
103#340—42no—
104#340—42no—
105#340—42no—
106#340—42no—
107#336—24no—
108C25C2×D411yescharacteristic
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2×Q814yes—
128C23×C4C2411yescharacteristic
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8C2×D414yes—
140C22×C8C2×Q814yes—
141C22×C8C2×Q814yes—
142C22×C8#4114yes—
143#117—44no—
144#117—44no—
145#117—44no—
146#117—44no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
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—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—82no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—82no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4—82no—
193C2×C22⋊C4—82no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42C2×D412yes—
201C2×C42#4112yes—
202C2×C42#4112yes—
203C2×C42#4112yes—
204C2×C42#4112yes—
205C2×C42#4112yes—
206#107—48no—
207#106—48no—
208C22⋊C8—216no—
209#97—28no—
210C4×C8—44no—
211#95—42no—
212#95—42no—
213#95—44no—
214#95—44no—
215#95—44no—
216#95—82no—
217#95—82no—
218#95—82no—
219#95—82no—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—21no—
224C24—21no—
225C24—21no—
226C24—22no—
227C24—22no—
228C24—22no—
229C24—22no—
230C24C4⋊Q812yes—
231C24C22×D411yescharacteristic
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
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#12114yes—
300C22×C4#12314yes—
301C22×C4#12314yes—
302C22×C4C4⋊Q812yes—
303C22×C4C22×D411yescharacteristic
304C22×C4C22×Q812yes—
305C22×C4C2×#4112yes—
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4H(5,𝔽2)12yes—
312C22×C4H(5,𝔽2)12yes—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8C22⋊Q818yes—
327C4⋊C4—48no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
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—88no—
345C4⋊C4—88no—
346C22⋊C4—48no—
347C22⋊C4—84no—
348C22⋊C4—84no—
349C22⋊C4—88no—
350C42—41no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42#12314yes—
355C42#12414yes—
356C42#12614yes—
357C42#12614yes—
358C23—22no—
359C23—24no—
360C23—24no—
361C23—22no—
362C23—22no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—41no—
368C23—41no—
369C23—42no—
370C23—42no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—42no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—42no—
380C23—42no—
381C23C2×#12111yescharacteristic
382C23C2×#12311yescharacteristic
383C23C2×#12311yescharacteristic
384C23C2×C4⋊Q811yescharacteristic
385C23#53211yescharacteristic
386C23#53511yescharacteristic
387C23#53511yescharacteristic
388C23#57612yes—
389C23#57712yes—
390C23#57712yes—
391C23#57812yes—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—82no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—82no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4#39718yes—
451C2×C4C2×C22⋊Q812yes—
452C2×C4C2×#12311yescharacteristic
453C2×C4C2×#12411yescharacteristic
454C2×C4C2×#12611yescharacteristic
455C2×C4C2×#12611yescharacteristic
456C2×C4#53312yes—
457C2×C4#53512yes—
458C2×C4#54012yes—
459C2×C4#55312yes—
460C2×C4#55312yes—
461C2×C4#55412yes—
462C2×C4#55412yes—
463C2×C4#55412yes—
464C2×C4#55412yes—
465C2×C4#55512yes—
466C2×C4#55512yes—
467C8—28no—
468C8—44no—
469C22—42no—
470C22—42no—
471C22—44no—
472C22—42no—
473C22—44no—
474C22—82no—
475C22—82no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22C2×#39711yescharacteristic
481C22#250011yescharacteristic
482C22#250011yescharacteristic
483C22#250211yescharacteristic
484C22#250211yescharacteristic
485C22#261911yescharacteristic
486C22#285711yescharacteristic
487C22#306212yes—
488C22#306312yes—
489C22#309612yes—
490C22#309712yes—
491C22#309812yes—
492C22#309912yes—
493C22#316312yes—
494C22#316412yes—
495C22#317812yes—
496C22#317912yes—
497C22#322011yescharacteristic
498C22#322111yescharacteristic
499C22#322211yescharacteristic
500C22#322311yescharacteristic
501C22#322412yes—
502C22#322512yes—
503C4—28no—
504C4—42no—
505C4—82no—
506C4—162no—
507C4—161no—
508C4—162no—
509C4—161no—
510C4—161no—
511C4—161no—
512C4#248712yes—
513C4#249712yes—
514C4#249712yes—
515C4#256312yes—
516C2—82no—
517C2#1519511yescharacteristic
518C2#2364312yes—
519C2#2790211yescharacteristic
520C2#2790311yescharacteristic
521C2#2790412yes—
522C2#2795711yescharacteristic
523C2#2795811yescharacteristic
524C2#2795912yes—
525C2#2981611yescharacteristic
526C2#2981711yescharacteristic
527C2#3321912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
1. C2×#1847C2semidirect
5. #11257C2semidirect
6. #11256C2semidirect
9. #10843C2semidirect
10. #10842C2semidirect
11. #10841C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18472
C2×#17502
#143402
#112572
#112562
#110962
#110952
#108432
#108422
#108412