Finite Group Databasebeta

← #314828

Subgroups — #314828

Order 512.

587 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27647C211yesmaximal, characteristic
2#27629C211yesmaximal, characteristic
3#21598C211yesmaximal, characteristic
4#20764C211yesmaximal, characteristic
5#10658C211yesmaximal, characteristic, complemented
6#10655C211yesmaximal, characteristic, complemented
7#10626C211yesmaximal, characteristic, complemented
8#10624C211yesmaximal, characteristic, complemented
9#9616C211yesmaximal, characteristic, complemented
10#9613C211yesmaximal, characteristic, complemented
11#8727C211yesmaximal, characteristic
12#8724C211yesmaximal, characteristic
13#8657C211yesmaximal, characteristic, complemented
14#8655C211yesmaximal, characteristic, complemented
15#8597C211yesmaximal, characteristic
16C2×#483—21no—
17C2×#446C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19C4×C22⋊Q8C2211yescharacteristic
20#2285C2211yescharacteristic
21C2×#384C2211yescharacteristic
22#2084—21no—
23#2076—21no—
24#2072—21no—
25#2036—21no—
26#1984C2211yescharacteristic
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1947C2211yescharacteristic
30#1947C2211yescharacteristic
31#1939C2211yescharacteristic
32#1928—21no—
33#1921—21no—
34#1917—21no—
35#1851—21no—
36#1836—21no—
37#1836—21no—
38#1834C2211yescharacteristic
39#1834C2211yescharacteristic
40#1834C2211yescharacteristic
41#1834C2211yescharacteristic
42#1823—21no—
43#1803C2211yescharacteristic
44#1803C2211yescharacteristic
45#1803C2211yescharacteristic
46#1803C2211yescharacteristic
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50#1770—21no—
51C8×C4⋊C4C2211yescharacteristic
52#1753—21no—
53#1736—21no—
54#1736—21no—
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57#1296C2211yescharacteristic
58#1296C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16D411yescharacteristic
68C22×M16D411yescharacteristic
69C2×C22⋊Q8D411yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×#117—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C42C2311yescharacteristic
76#483—44no—
77#446—22no—
78#446—22no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#106—21no—
82C2×#106—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×#103—21no—
92C2×#103—21no—
93C2×#103—21no—
94C2×#103—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C4×M16—22no—
99C4×M16—22no—
100C2×#97—21no—
101C2×#97D411yescharacteristic
102C2×#97D411yescharacteristic
103C2×C4×C8—21no—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106#391D411yescharacteristic
107#389D411yescharacteristic
108#387—21no—
109#385—21no—
110#384—21no—
111#384—21no—
112#384—21no—
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#382—21no—
118#380—21no—
119#380—21no—
120#380D411yescharacteristic
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2311yescharacteristic
125C4×C22⋊C4—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—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#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#335—22no—
150#334—22no—
151C22×Q8C2×D411yescharacteristic
152C23×C4—41no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4#4111yescharacteristic
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—41no—
161C2×M16—41no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22⋊Q8—22no—
173C22⋊Q8—41no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178#117—42no—
179#117—42no—
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—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—41no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—41no—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42#4111yescharacteristic
219C2×C42C2411yescharacteristic
220#107—42no—
221#107—81no—
222#106—42no—
223#106—42no—
224C4⋊C8—21no—
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—42no—
237C4⋊C8—42no—
238#103—42no—
239#103—42no—
240#103—42no—
241#103—81no—
242C22⋊C8—42no—
243C22⋊C8—42no—
244C22⋊C8—81no—
245#97—22no—
246#97—22no—
247#97—42no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—42no—
251C4×C8#4112yes—
252C4×C8#4112yes—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—41no—
259#95—41no—
260C24#12111yescharacteristic
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—21no—
267C22×C4—21no—
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—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4#12111yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4#12111yescharacteristic
295C22×C4#13512yes—
296C22×C4#13712yes—
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C2×#4111yescharacteristic
300M16—44no—
301M16—44no—
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—21no—
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—41no—
330C2×C8#12312yes—
331C2×C8#12312yes—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—42no—
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—41no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C4⋊C4—81no—
359C4⋊C4—81no—
360C4⋊C4—81no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4AGL(1,ℤ/8)12yes—
365C22⋊C4—42no—
366C22⋊C4—81no—
367C22⋊C4—82no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—82no—
379C42C2×#4111yescharacteristic
380C42C2×#4111yescharacteristic
381C422−1+411yescharacteristic
382C422−1+411yescharacteristic
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23#46412yes—
390C23C2×#12111yescharacteristic
391Q8—41no—
392Q8—41no—
393Q8—41no—
394Q8—41no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
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—41no—
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—81no—
439C2×C4—164no—
440C2×C4#46412yes—
441C2×C4C2×#12311yescharacteristic
442C2×C4#53611yescharacteristic
443C2×C4#54711yescharacteristic
444C2×C4#54711yescharacteristic
445C2×C4#55211yescharacteristic
446C2×C4#55211yescharacteristic
447C2×C4C2×#13511yescharacteristic
448C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
449C2×C4C2×#13711yescharacteristic
450C2×C4#57511yescharacteristic
451C2×C4#57511yescharacteristic
452C2×C4#57511yescharacteristic
453C2×C4#57711yescharacteristic
454C2×C4#57811yescharacteristic
455C8—22no—
456C8—22no—
457C8—41no—
458C8—41no—
459C8—81no—
460C8—81no—
461C8—81no—
462C8—81no—
463C8—81no—
464C8—81no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22#248611yescharacteristic
473C22C2×#46411yescharacteristic
474C22#304011yescharacteristic
475C22#305511yescharacteristic
476C22#306311yescharacteristic
477C22#306811yescharacteristic
478C22#306911yescharacteristic
479C4—22no—
480C4—21no—
481C4—21no—
482C4—41no—
483C4—41no—
484C4—42no—
485C4—81no—
486C4—81no—
487C4—81no—
488C4—161no—
489C4—161no—
490C4—161no—
491C4—161no—
492C4#309111yescharacteristic
493C4#309511yescharacteristic
494C4#323211yescharacteristic
495C4#323611yescharacteristic
496C4#335111yescharacteristic
497C4#335211yescharacteristic
498C4#335311yescharacteristic
499C4#335411yescharacteristic
500C2—42no—
501C2#1917411yescharacteristic
502C2#1924311yescharacteristic
503C2#1931911yescharacteristic
504C2#1938211yescharacteristic
505C2#2765311yescharacteristic
506C2#2767011yescharacteristic
507C2#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. #10658C2semidirect
6. #10655C2semidirect
7. #10626C2semidirect
8. #10624C2semidirect
9. #9616C2semidirect
10. #9613C2semidirect
13. #8657C2semidirect
14. #8655C2semidirect

Maximal subgroups

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

subgroupindex
#276472
#276292
#215982
#207642
#106582
#106552
#106262
#106242
#96162
#96132
#87272
#87242
#86572
#86552
#85972