Finite Group Databasebeta

← #203579

Subgroups — #203579

Order 512.

724 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21974C211yesmaximal, characteristic
2#21953C211yesmaximal, characteristic
3C4×#428C211yesmaximal, characteristic
4C2×#1569C211yesmaximal, characteristic
5C2×#1453C211yesmaximal, characteristic
6#9600C211yesmaximal, characteristic, complemented
7#9599C211yesmaximal, characteristic, complemented
8#9598C211yesmaximal, characteristic, complemented
9#9597C211yesmaximal, characteristic, complemented
10#9596C211yesmaximal, characteristic, complemented
11#9595C211yesmaximal, characteristic, complemented
12#9594C211yesmaximal, characteristic, complemented
13#9593C211yesmaximal, characteristic, complemented
14#7902C211yesmaximal, characteristic
15#7901C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#428C411yescharacteristic
20C2×#428C411yescharacteristic
21C2×#423C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23C4×#117C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1940—21no—
26#1940—21no—
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1939C2211yescharacteristic
35#1851—21no—
36#1851—21no—
37#1842—21no—
38#1831—21no—
39#1797—21no—
40#1797—21no—
41#1776C411yescharacteristic
42#1776C411yescharacteristic
43#1776C411yescharacteristic
44#1776C411yescharacteristic
45C4×#107C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C4×#106C2211yescharacteristic
48C4×#106C2211yescharacteristic
49#1736C411yescharacteristic
50#1736C411yescharacteristic
51#1569—24no—
52C2×#335C2211yescharacteristic
53C2×#335C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×#334C2211yescharacteristic
56#1453—24no—
57C2×#322C2211yescharacteristic
58C2×#322C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68#1279C2211yescharacteristic
69C22×M16—21no—
70C22×M16C2×C411yescharacteristic
71C22×M16C2×C411yescharacteristic
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78#432—22no—
79#432—22no—
80#431—22no—
81#431—22no—
82#431—22no—
83#431Q814yes—
84#428—24no—
85#428—24no—
86C2×#107C2×C411yescharacteristic
87C2×#107C2×C411yescharacteristic
88C2×#107C2×C411yescharacteristic
89C2×#107C2×C411yescharacteristic
90C2×#106C2×C411yescharacteristic
91C2×#106C2×C411yescharacteristic
92C2×#106C2×C411yescharacteristic
93C2×#106C2×C411yescharacteristic
94#423—22no—
95#423—22no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8D411yescharacteristic
107C2×C4⋊C8D411yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8D411yescharacteristic
115C2×C22⋊C8D411yescharacteristic
116C4×M16D414yes—
117C4×M16Q814yes—
118C2×#97—21no—
119C2×#97C2×C411yescharacteristic
120C2×#97C2×C411yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139#376—21no—
140C2×#95—21no—
141C43—21no—
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#335—22no—
159#335—22no—
160#334—22no—
161#334—22no—
162#322—22no—
163#322—22no—
164#322—22no—
165C82—22no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4C22×C411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4#4111yescharacteristic
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—41no—
174C2×M16—41no—
175C2×M16—42no—
176C2×M16—81no—
177C2×M16—81no—
178C2×M16C4⋊C414yes—
179C2×M16C4⋊C414yes—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8C22×C411yescharacteristic
187C22×C8C22×C411yescharacteristic
188C22×C8C22×C411yescharacteristic
189C22×C8C22×C411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194#117—22no—
195#117—22no—
196#117—41no—
197#117—41no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—81no—
225C2×C42—81no—
226C2×C42C22×C411yescharacteristic
227C2×C42C2×D412yes—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×Q812yes—
230C2×C42C2×Q812yes—
231C2×C42#4112yes—
232C2×C42#4111yescharacteristic
233C2×C42C2411yescharacteristic
234#107—22no—
235#107—22no—
236#107—22no—
237#107M1614yescomplemented
238#106—22no—
239#106—22no—
240#106—22no—
241#106M1614yescomplemented
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—24no—
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—81no—
255C4⋊C8—81no—
256C4⋊C8—81no—
257C4⋊C8—81no—
258C4⋊C8—81no—
259C4⋊C8—81no—
260C4⋊C8—81no—
261C4⋊C8—81no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—24no—
265C22⋊C8—42no—
266#97—42no—
267#97C4⋊C414yes—
268#97C4⋊C414yes—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8C2×D412yes—
276C4×C8C2×D412yes—
277C4×C8C2×Q812yes—
278C4×C8C2×Q812yes—
279C4×C8C2×Q812yes—
280C4×C8C2×Q812yes—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289C24C4×D411yescharacteristic
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
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—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—81no—
313C22×C4—81no—
314C22×C4C2×C4⋊C412yes—
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4C4×D411yescharacteristic
318C22×C4C4×Q812yes—
319C22×C4#12112yes—
320C22×C4C22⋊Q812yes—
321C22×C4#12512yes—
322C22×C4C4⋊Q812yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C2×#4111yescharacteristic
326M16—28no—
327M16—44no—
328M16—84no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8C2×C4⋊C412yes—
366C2×C8C2×C4⋊C412yes—
367C2×C8C2×C4⋊C412yes—
368C2×C8C2×C4⋊C412yes—
369C2×C8C2×M1612yes—
370C2×C8C2×M1612yes—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4C2×M1612yes—
386C4⋊C4C2×M1612yes—
387C4⋊C4C2×M1612yes—
388C4⋊C4C2×M1612yes—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C42—42no—
392C42—42no—
393C42—42no—
394C42—42no—
395C42—41no—
396C42—41no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42—81no—
404C42—81no—
405C42C2×C4⋊C412yes—
406C42C4×Q812yes—
407C42#12112yes—
408C42C22⋊Q812yes—
409C42#12512yes—
410C42C4⋊Q812yes—
411C42C22×D411yescharacteristic
412C42C22×Q811yescharacteristic
413C42C22×Q811yescharacteristic
414C42C2×#4111yescharacteristic
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23#38912yes—
422C23C2×C4×D411yescharacteristic
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—42no—
444C2×C4—42no—
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—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—84no—
466C2×C4—84no—
467C2×C4#38912yes—
468C2×C4#38912yes—
469C2×C4#38912yes—
470C2×C4C22×C4⋊C411yescharacteristic
471C2×C4C2×C4×Q811yescharacteristic
472C2×C4C2×#12111yescharacteristic
473C2×C4C2×C22⋊Q811yescharacteristic
474C2×C4C2×#12511yescharacteristic
475C2×C4C2×C4⋊Q811yescharacteristic
476C2×C4C22×M1611yescharacteristic
477C2×C4C22×M1611yescharacteristic
478C2×C4#56811yescharacteristic
479C2×C4#56811yescharacteristic
480C2×C4#57611yescharacteristic
481C2×C4#57711yescharacteristic
482C2×C4#57711yescharacteristic
483C2×C4#57811yescharacteristic
484C8—22no—
485C8—22no—
486C8—22no—
487C8—42no—
488C8—42no—
489C8—41no—
490C8—41no—
491C8—41no—
492C8—41no—
493C8—81no—
494C8—81no—
495C8—81no—
496C8—81no—
497C8#42314yes—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22C2×#38911yescharacteristic
506C22D4×M1611yescharacteristic
507C22D4×M1611yescharacteristic
508C22#294611yescharacteristic
509C22#294711yescharacteristic
510C22#295511yescharacteristic
511C22#295511yescharacteristic
512C4—22no—
513C4—22no—
514C4—21no—
515C4—21no—
516C4—22no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—81no—
522C4—81no—
523C4—81no—
524C4—81no—
525C4C2×#42311yescharacteristic
526C4#290511yescharacteristic
527C4Q8×M1611yescharacteristic
528C4Q8×M1611yescharacteristic
529C4#306211yescharacteristic
530C4#306311yescharacteristic
531C4#316311yescharacteristic
532C4#316411yescharacteristic
533C2—42no—
534C2#1805011yescharacteristic
535C2#1805111yescharacteristic
536C2#1805211yescharacteristic
537C2#1805311yescharacteristic
538C2#2114611yescharacteristic
539C2#2116011yescharacteristic
540C2#2116411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9600C2semidirect
7. #9599C2semidirect
8. #9598C2semidirect
9. #9597C2semidirect
10. #9596C2semidirect
11. #9595C2semidirect
12. #9594C2semidirect
13. #9593C2semidirect
237. #107M16semidirect
241. #106M16semidirect

Maximal subgroups

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

subgroupindex
#219742
#219532
C4×#4282
C2×#15692
C2×#14532
#96002
#95992
#95982
#95972
#95962
#95952
#95942
#95932
#79022
#79012