Finite Group Databasebeta

← #314778

Subgroups — #314778

Order 512.

568 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27649C211yesmaximal, characteristic
2#27631C211yesmaximal, characteristic
3#21596C211yesmaximal, characteristic
4#20824C211yesmaximal, characteristic
5#10657C211yesmaximal, characteristic, complemented
6#10656C211yesmaximal, characteristic, complemented
7#10626C211yesmaximal, characteristic, complemented
8#10623C211yesmaximal, characteristic, complemented
9#9621C211yesmaximal, characteristic, complemented
10#9617C211yesmaximal, characteristic, complemented
11#8728C211yesmaximal, characteristic
12#8727C211yesmaximal, characteristic
13#8645C211yesmaximal, characteristic, complemented
14#8643C211yesmaximal, characteristic, complemented
15#8597C211yesmaximal, characteristic
16C2×#485—21no—
17C2×C8×Q8C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19C4×C22⋊Q8C2211yescharacteristic
20#2285C2211yescharacteristic
21#2285C2211yescharacteristic
22#2085—21no—
23#2075—21no—
24#2074—21no—
25#2035—21no—
26#1984C2211yescharacteristic
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1947C2211yescharacteristic
30#1947C2211yescharacteristic
31#1939C2211yescharacteristic
32#1928—21no—
33#1919—21no—
34#1917—21no—
35#1851—21no—
36#1845C2211yescharacteristic
37#1844C2211yescharacteristic
38#1834C2211yescharacteristic
39#1834C2211yescharacteristic
40#1825—21no—
41#1809—21no—
42#1809—21no—
43#1803C2211yescharacteristic
44#1803C2211yescharacteristic
45#1803C2211yescharacteristic
46#1803C2211yescharacteristic
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50#1776—21no—
51#1776—21no—
52#1771—21no—
53#1770C2211yescharacteristic
54#1753—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—
72C2×#117—21no—
73C22×C42C2311yescharacteristic
74#485—44no—
75C8×Q8—22no—
76C8×Q8—22no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#106—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×#103—21no—
94C2×#103—21no—
95C2×#103—21no—
96C2×#103—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C4×M16—22no—
101C4×M16—22no—
102C2×#97—21no—
103C2×#97D411yescharacteristic
104C2×#97D411yescharacteristic
105C2×C4×C8—21no—
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#391D411yescharacteristic
109#389D411yescharacteristic
110#387—21no—
111#385—21no—
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116#382—21no—
117#382—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380—21no—
122#380D411yescharacteristic
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4C2311yescharacteristic
129C4×C22⋊C4—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×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4#4111yescharacteristic
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—41no—
160C2×M16—41no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22⋊Q8—22no—
172C22⋊Q8—41no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—21no—
177#117—42no—
178#117—42no—
179#117—82no—
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⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—41no—
209C2×C42—41no—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42#4111yescharacteristic
213C2×C42C2411yescharacteristic
214#107—42no—
215#107—42no—
216#107—42no—
217#107—81no—
218#106—42no—
219#106—42no—
220#106—42no—
221#106—42no—
222C4⋊C8—21no—
223C4⋊C8—21no—
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—42no—
235C4⋊C8—42no—
236#103—42no—
237#103—42no—
238#103—42no—
239#103—81no—
240C22⋊C8—42no—
241C22⋊C8—42no—
242C22⋊C8—81no—
243#97—22no—
244#97—22no—
245#97—42no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—42no—
249C4×C8#4112yes—
250C4×C8#4112yes—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—41no—
257#95—41no—
258#95—41no—
259#95—41no—
260C24#12111yescharacteristic
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8AGL(1,ℤ/8)12yes—
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—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C2×#4111yescharacteristic
297M16—44no—
298M16—44no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
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#12312yes—
326C2×C8#12312yes—
327C2×C8#13512yes—
328C2×C8#13712yes—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—42no—
336C4⋊C4—41no—
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—81no—
355C4⋊C4—81no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C4⋊C4—81no—
359C22⋊C4—42no—
360C22⋊C4—81no—
361C22⋊C4—82no—
362C22⋊C4—82no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—82no—
374C42—82no—
375C42C2×#4111yescharacteristic
376C42C2×#4111yescharacteristic
377C422−1+411yescharacteristic
378C422−1+411yescharacteristic
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23C2×#12111yescharacteristic
387Q8—22no—
388Q8—22no—
389Q8—41no—
390Q8—41no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
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—81no—
425C2×C4—81no—
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×C4C2×#12311yescharacteristic
437C2×C4#53611yescharacteristic
438C2×C4#54711yescharacteristic
439C2×C4#54711yescharacteristic
440C2×C4#55211yescharacteristic
441C2×C4#55211yescharacteristic
442C2×C4C2×#13511yescharacteristic
443C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
444C2×C4C2×#13711yescharacteristic
445C2×C4#57511yescharacteristic
446C2×C4#57511yescharacteristic
447C2×C4#57511yescharacteristic
448C2×C4#57711yescharacteristic
449C2×C4#57811yescharacteristic
450C8—22no—
451C8—41no—
452C8—41no—
453C8—81no—
454C8—81no—
455C8—81no—
456C8—81no—
457C8—81no—
458C8—81no—
459C8#48514yescomplemented
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22#248611yescharacteristic
468C22#303511yescharacteristic
469C22#303711yescharacteristic
470C22#305511yescharacteristic
471C22#306311yescharacteristic
472C22#306811yescharacteristic
473C22#306911yescharacteristic
474C4—21no—
475C4—21no—
476C4—41no—
477C4—41no—
478C4—42no—
479C4—41no—
480C4—81no—
481C4—81no—
482C4—81no—
483C4—161no—
484C4—161no—
485C4—161no—
486C4—161no—
487C4C2×#48511yescharacteristic
488C4#309311yescharacteristic
489C4#323311yescharacteristic
490C4#323711yescharacteristic
491C4#334311yescharacteristic
492C4#334511yescharacteristic
493C4#334811yescharacteristic
494C4#334911yescharacteristic
495C2—81no—
496C2#1918211yescharacteristic
497C2#1925311yescharacteristic
498C2#1931011yescharacteristic
499C2#1937511yescharacteristic
500C2#2764911yescharacteristic
501C2#2766911yescharacteristic
502C2#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. #10657C2semidirect
6. #10656C2semidirect
7. #10626C2semidirect
8. #10623C2semidirect
9. #9621C2semidirect
10. #9617C2semidirect
13. #8645C2semidirect
14. #8643C2semidirect
459. C8#485semidirect

Maximal subgroups

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

subgroupindex
#276492
#276312
#215962
#208242
#106572
#106562
#106262
#106232
#96212
#96172
#87282
#87272
#86452
#86432
#85972