Finite Group Databasebeta

← #314859

Subgroups — #314859

Order 512.

591 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27649C211yesmaximal, characteristic
2#27647C211yesmaximal, characteristic
3#22098C211yesmaximal, characteristic
4#21982C211yesmaximal, characteristic
5#10655C211yesmaximal, characteristic, complemented
6#10651C211yesmaximal, characteristic, complemented
7#10623C211yesmaximal, characteristic, complemented
8#10617C211yesmaximal, characteristic, complemented
9#9616C211yesmaximal, characteristic, complemented
10#9615C211yesmaximal, characteristic, complemented
11#8726C211yesmaximal, characteristic
12#8724C211yesmaximal, characteristic
13#8649C211yesmaximal, characteristic, complemented
14#8645C211yesmaximal, characteristic, complemented
15#8599C211yesmaximal, characteristic
16C2×#485—21no—
17C2×#483—21no—
18C2×C8×Q8C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C4×C22⋊Q8C2211yescharacteristic
21#2285C2211yescharacteristic
22C2×#384C2211yescharacteristic
23#2085—21no—
24#2084—21no—
25#2074—21no—
26#2072—21no—
27#1984C2211yescharacteristic
28#1948C2211yescharacteristic
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1947—21no—
32#1947—21no—
33#1947C2211yescharacteristic
34#1939—21no—
35#1939C2211yescharacteristic
36#1923—21no—
37#1923—21no—
38#1916—21no—
39#1852—21no—
40#1852—21no—
41#1845C2211yescharacteristic
42#1845C2211yescharacteristic
43#1834C2211yescharacteristic
44#1834C2211yescharacteristic
45#1825—21no—
46#1823—21no—
47#1803C2211yescharacteristic
48#1803C2211yescharacteristic
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53#1797C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57#1296C2211yescharacteristic
58#1296C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C22⋊Q8D411yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C42C2311yescharacteristic
74#485—44no—
75#483—44no—
76C8×Q8—22no—
77C8×Q8—22no—
78#431—22no—
79#431—22no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8D411yescharacteristic
91C2×C4⋊C8D411yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#103—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8D411yescharacteristic
106C2×C22⋊C8D411yescharacteristic
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#391D411yescharacteristic
112#389D411yescharacteristic
113#387—21no—
114#385—21no—
115#384—21no—
116#384—21no—
117#384—21no—
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#382—21no—
123#380—21no—
124#380—21no—
125#380D411yescharacteristic
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2311yescharacteristic
130C4×C22⋊C4—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#335—22no—
151#335—22no—
152C22×Q8C2×D411yescharacteristic
153C23×C4—41no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4#4111yescharacteristic
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22⋊Q8—22no—
168C22⋊Q8D814yes—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173#117—42no—
174#117—42no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—81no—
196C2×C4⋊C4—81no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—41no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—41no—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42#4111yescharacteristic
214C2×C42C2411yescharacteristic
215#107—42no—
216#107—81no—
217#107—81no—
218#107—81no—
219#107—81no—
220#106—42no—
221#106—42no—
222#106—42no—
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—42no—
238C4⋊C8—81no—
239#103—42no—
240#103—42no—
241#103—42no—
242#103—42no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C22⋊C8—42no—
246C22⋊C8—42no—
247C22⋊C8—81no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—42no—
251C4×C8—81no—
252C4×C8#4112yes—
253C4×C8#4112yes—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—41no—
260#95—41no—
261C24#12111yescharacteristic
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8C2×D812yes—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4#12111yescharacteristic
294C22×C4#12111yescharacteristic
295C22×C4#12111yescharacteristic
296C22×C4C2×D812yes—
297C22×C4#13512yes—
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C2×#4111yescharacteristic
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8#12312yes—
330C2×C8#12312yes—
331C4⋊C4—21no—
332C4⋊C4—24no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—81no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C4⋊C4—81no—
359C4⋊C4—81no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4C2×D812yes—
364C22⋊C4—24no—
365C22⋊C4—81no—
366C22⋊C4—82no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—82no—
378C42C2×#4111yescharacteristic
379C42C2×#4111yescharacteristic
380C422−1+411yescharacteristic
381C422−1+411yescharacteristic
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23#46612yes—
389C23C2×#12111yescharacteristic
390Q8—22no—
391Q8—22no—
392Q8—41no—
393Q8—41no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—164no—
439C2×C4#46612yes—
440C2×C4C2×#12311yescharacteristic
441C2×C4#53611yescharacteristic
442C2×C4#54711yescharacteristic
443C2×C4#54711yescharacteristic
444C2×C4#55211yescharacteristic
445C2×C4#55211yescharacteristic
446C2×C4C22×D811yescharacteristic
447C2×C4C2×#13511yescharacteristic
448C2×C4#57711yescharacteristic
449C2×C4#57711yescharacteristic
450C2×C4#57711yescharacteristic
451C2×C4#57811yescharacteristic
452C2×C4#57811yescharacteristic
453C2×C4#57811yescharacteristic
454C8—22no—
455C8—22no—
456C8—41no—
457C8—41no—
458C8—81no—
459C8—81no—
460C8—81no—
461C8—81no—
462C8—81no—
463C8—81no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22#248611yescharacteristic
472C22#304311yescharacteristic
473C22#304611yescharacteristic
474C22C2×#46611yescharacteristic
475C22#305911yescharacteristic
476C22#306811yescharacteristic
477C22#306911yescharacteristic
478C4—22no—
479C4—21no—
480C4—21no—
481C4—41no—
482C4—42no—
483C4—41no—
484C4—42no—
485C4—81no—
486C4—81no—
487C4—161no—
488C4—161no—
489C4—161no—
490C4—161no—
491C4#309711yescharacteristic
492C4#309911yescharacteristic
493C4#324211yescharacteristic
494C4#324411yescharacteristic
495C4#333411yescharacteristic
496C4#333411yescharacteristic
497C4#333611yescharacteristic
498C4#333611yescharacteristic
499C2—42no—
500C2#1939811yescharacteristic
501C2#1941611yescharacteristic
502C2#1952311yescharacteristic
503C2#1954711yescharacteristic
504C2#2765711yescharacteristic
505C2#2766111yescharacteristic
506C2#2767611yescharacteristic
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. #10655C2semidirect
6. #10651C2semidirect
7. #10623C2semidirect
8. #10617C2semidirect
9. #9616C2semidirect
10. #9615C2semidirect
13. #8649C2semidirect
14. #8645C2semidirect

Maximal subgroups

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

subgroupindex
#276492
#276472
#220982
#219822
#106552
#106512
#106232
#106172
#96162
#96152
#87262
#87242
#86492
#86452
#85992