Finite Group Databasebeta

← #324016

Subgroups — #324016

Order 512.

616 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33212C211yesmaximal, characteristic
2#24837C211yesmaximal, characteristic
3#24702C211yesmaximal, characteristic
4#21596C211yesmaximal, characteristic
5#10653C211yesmaximal, characteristic, complemented
6#10620C211yesmaximal, characteristic, complemented
7#10543C211yesmaximal, characteristic, complemented
8#10279C211yesmaximal, characteristic, complemented
9#9576C211yesmaximal, characteristic, complemented
10#9576C211yesmaximal, characteristic, complemented
11#9216C211yesmaximal, characteristic, complemented
12#9216C211yesmaximal, characteristic, complemented
13#9128C211yesmaximal, characteristic
14#9127C211yesmaximal, characteristic
15#7873C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17C2×#477C2211yescharacteristic
18C2×#477C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25C4×C22⋊Q8C2211yescharacteristic
26#2087—21no—
27#2076—21no—
28#2068—21no—
29#2036—21no—
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1987C2211yescharacteristic
33#1987C2211yescharacteristic
34#1984C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1944—21no—
38#1944—21no—
39#1942—21no—
40#1939C2211yescharacteristic
41#1933—21no—
42#1917—21no—
43#1892—21no—
44#1851—21no—
45#1844C2211yescharacteristic
46#1809—21no—
47#1809—21no—
48#1808—21no—
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1797C2211yescharacteristic
52#1771—21no—
53#1770C2211yescharacteristic
54#1753—21no—
55C2×#333C2211yescharacteristic
56C2×#333C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16D411yescharacteristic
68C22×M16D411yescharacteristic
69C23×C8—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4⋊Q8C2311yescharacteristic
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8D411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C22×C42C2311yescharacteristic
77#500—41no—
78#500—41no—
79#477—22no—
80#477—22no—
81#477—41no—
82#477—41no—
83C8×Q8—41no—
84C8×Q8—41no—
85#431—22no—
86#431—41no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#107—21no—
92C2×#107—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—41no—
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C4×M16—41no—
113C4×M16—41no—
114C2×#97—21no—
115C2×#97D411yescharacteristic
116C2×#97D411yescharacteristic
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#391—21no—
122#391—21no—
123#391D411yescharacteristic
124#389—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#389D411yescharacteristic
129#387—21no—
130#385—21no—
131#384C2311yescharacteristic
132#384C2311yescharacteristic
133#380—21no—
134#380—21no—
135#380—21no—
136#380—21no—
137#380D411yescharacteristic
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2311yescharacteristic
140C4×C22⋊C4—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#333—22no—
159#333—22no—
160#333—41no—
161#333—41no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4#4111yescharacteristic
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—41no—
171C2×M16—41no—
172C2×M16—41no—
173C2×M16Q1614yes—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—42no—
181C22×C8—42no—
182C22×C8—81no—
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C22⋊Q8—22no—
196C22⋊Q8—41no—
197C22⋊Q8—82no—
198C22⋊Q8—82no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42#4111yescharacteristic
234C2×C42C2411yescharacteristic
235#107—42no—
236#107—42no—
237#107—42no—
238#107—42no—
239#107—42no—
240#107—81no—
241#106—42no—
242#106—42no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—42no—
254C4⋊C8—42no—
255C4⋊C8—42no—
256C4⋊C8#4112yes—
257C4⋊C8#4112yes—
258#103—42no—
259#103—42no—
260#103—42no—
261#103—81no—
262C22⋊C8—42no—
263C22⋊C8—42no—
264#97—41no—
265#97—41no—
266#97—42no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—42no—
272C4×C8—42no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—41no—
279#95—41no—
280#95—41no—
281#95—41no—
282C24#12111yescharacteristic
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—81no—
293C2×Q8—81no—
294C2×Q8—81no—
295C2×Q8—81no—
296C2×Q8#13712yes—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4C2×Q1612yes—
323C22×C4#13712yes—
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C2×#4111yescharacteristic
327M16—44no—
328M16—44no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—84no—
357C2×C8C2×Q1612yes—
358C2×C8C2×Q1612yes—
359C2×C8#13512yes—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—81no—
386C4⋊C4—81no—
387C4⋊C4—81no—
388C4⋊C4—82no—
389C4⋊C4—82no—
390C22⋊C4—42no—
391C22⋊C4—81no—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42C2×#4111yescharacteristic
405C42C2×#4111yescharacteristic
406C42H(5,𝔽2)11yescharacteristic
407C42H(5,𝔽2)11yescharacteristic
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23#46212yes—
415C23C2×#12111yescharacteristic
416Q8—22no—
417Q8—22no—
418Q8—41no—
419Q8—41no—
420Q8—161no—
421Q8—161no—
422Q8—161no—
423Q8—161no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4#46212yes—
469C2×C4#53211yescharacteristic
470C2×C4#53411yescharacteristic
471C2×C4#54611yescharacteristic
472C2×C4#54611yescharacteristic
473C2×C4#55311yescharacteristic
474C2×C4#55311yescharacteristic
475C2×C4C22×Q1611yescharacteristic
476C2×C4C2×#13511yescharacteristic
477C2×C4C2×#13711yescharacteristic
478C2×C4C2×#13711yescharacteristic
479C2×C4#57511yescharacteristic
480C2×C4#57511yescharacteristic
481C2×C4#57711yescharacteristic
482C2×C4#57811yescharacteristic
483C8—22no—
484C8—22no—
485C8—22no—
486C8—41no—
487C8—42no—
488C8—42no—
489C8—81no—
490C8—81no—
491C8—81no—
492C8—81no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22#254311yescharacteristic
501C22C2×#46211yescharacteristic
502C22#303511yescharacteristic
503C22#305611yescharacteristic
504C22#306311yescharacteristic
505C22#306811yescharacteristic
506C22#306811yescharacteristic
507C4—22no—
508C4—21no—
509C4—21no—
510C4—41no—
511C4—42no—
512C4—41no—
513C4—81no—
514C4—81no—
515C4—81no—
516C4—161no—
517C4—161no—
518C4—161no—
519C4—161no—
520C4#317011yescharacteristic
521C4#318111yescharacteristic
522C4#318911yescharacteristic
523C4#319011yescharacteristic
524C4#331111yescharacteristic
525C4#331211yescharacteristic
526C4#331511yescharacteristic
527C4#331611yescharacteristic
528C2—42no—
529C2#1917311yescharacteristic
530C2#1918311yescharacteristic
531C2#1924111yescharacteristic
532C2#1924711yescharacteristic
533C2#2855811yescharacteristic
534C2#2857511yescharacteristic
535C2#2859511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10653C2semidirect
6. #10620C2semidirect
7. #10543C2semidirect
8. #10279C2semidirect
9. #9576C2semidirect
10. #9576C2semidirect
11. #9216C2semidirect
12. #9216C2semidirect

Maximal subgroups

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

subgroupindex
#332122
#248372
#247022
#215962
#106532
#106202
#105432
#102792
#95762
#92162
#91282
#91272
#78732