Finite Group Databasebeta

← #275294

Subgroups — #275294

Order 512.

600 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33114C211yesmaximal, characteristic
2#27876C211yesmaximal, characteristic
3#22632C211yesmaximal, characteristic
4#21169C211yesmaximal, characteristic
5#10658C211yesmaximal, characteristic, complemented
6#10654C211yesmaximal, characteristic, complemented
7#10598C211yesmaximal, characteristic, complemented
8#10590C211yesmaximal, characteristic, complemented
9#9613C211yesmaximal, characteristic, complemented
10#9611C211yesmaximal, characteristic, complemented
11#9566C211yesmaximal, characteristic, complemented
12#9565C211yesmaximal, characteristic, complemented
13#9131C211yesmaximal, characteristic
14#8830C211yesmaximal, characteristic
15#8727C211yesmaximal, characteristic
16C2×#485—21no—
17C2×#427—21no—
18C22×C4⋊C8C2211yescharacteristic
19#2857C2211yescharacteristic
20#2285C2211yescharacteristic
21C4×#117C2211yescharacteristic
22#2077—21no—
23#2077—21no—
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1981C2211yescharacteristic
27#1981C2211yescharacteristic
28#1949—21no—
29#1949—21no—
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1928—21no—
39#1927—21no—
40#1923—21no—
41#1923—21no—
42#1919—21no—
43#1919—21no—
44#1873—21no—
45#1844C2211yescharacteristic
46#1834C2211yescharacteristic
47#1831—21no—
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53#1744—21no—
54#1735—21no—
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×C22⋊Q8—21no—
70C2×#117—21no—
71C2×#117D411yescharacteristic
72C2×#117D411yescharacteristic
73C22×C42C2311yescharacteristic
74#485—44no—
75#427—44no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#106—21no—
80C2×#106—21no—
81C2×#106—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8D411yescharacteristic
102C2×C22⋊C8D411yescharacteristic
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#391—21no—
108#389—21no—
109#389—21no—
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#382—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4C2311yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126#376—21no—
127C43—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#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336D411yescharacteristic
147#336D411yescharacteristic
148#334—22no—
149#334—22no—
150#332—22no—
151#332—22no—
152C22×Q8—21no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4#4111yescharacteristic
156C22×C8—24no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—42no—
164C22×C8—42no—
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C22⋊Q8—42no—
174C22⋊Q8—42no—
175#117—22no—
176#117—22no—
177#117—41no—
178#117—42no—
179#117—42no—
180#117SD1614yes—
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—41no—
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—41no—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42#4111yescharacteristic
215C2×C42C2411yescharacteristic
216#107—42no—
217#107—42no—
218#107—42no—
219#106—42no—
220#106—42no—
221#106—42no—
222#106—42no—
223#106—42no—
224C4⋊C8—24no—
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—81no—
236C4⋊C8—81no—
237C4⋊C8#4112yes—
238C4⋊C8#4112yes—
239#103—42no—
240#103—42no—
241#103—42no—
242#103—42no—
243C22⋊C8—22no—
244C22⋊C8—41no—
245C22⋊C8—42no—
246C22⋊C8—81no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—42no—
252C4×C8—42no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—41no—
258#95—41no—
259#95—41no—
260#95—41no—
261C24#12111yescharacteristic
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
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#12111yescharacteristic
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4C2×SD1612yes—
294C22×C4#13512yes—
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—22no—
301C2×C8—22no—
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—48no—
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#13512yes—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—41no—
335C4⋊C4—41no—
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—81no—
351C4⋊C4—81no—
352C4⋊C4—81no—
353C4⋊C4—82no—
354C4⋊C4C2×SD1612yes—
355C4⋊C4C2×SD1612yes—
356C4⋊C4AGL(1,ℤ/8)12yes—
357C22⋊C4—24no—
358C22⋊C4—42no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—24no—
368C42—42no—
369C42—42no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—81no—
375C42—82no—
376C42C2×#4111yescharacteristic
377C42C2×#4111yescharacteristic
378C422−1+411yescharacteristic
379C422−1+411yescharacteristic
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23#46512yes—
387C23C2×#12111yescharacteristic
388Q8—81no—
389Q8—81no—
390Q8—81no—
391Q8—81no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
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—81no—
423C2×C4—81no—
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×C4#46512yes—
437C2×C4#52511yescharacteristic
438C2×C4#53611yescharacteristic
439C2×C4#54711yescharacteristic
440C2×C4#54711yescharacteristic
441C2×C4#55211yescharacteristic
442C2×C4#55211yescharacteristic
443C2×C4C22×SD1611yescharacteristic
444C2×C4C2×#13511yescharacteristic
445C2×C4C2×#13511yescharacteristic
446C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
447C2×C4#57511yescharacteristic
448C2×C4#57511yescharacteristic
449C2×C4#57711yescharacteristic
450C2×C4#57811yescharacteristic
451C8—22no—
452C8—41no—
453C8—41no—
454C8—41no—
455C8—42no—
456C8—42no—
457C8—81no—
458C8—81no—
459C8—81no—
460C8—81no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22#241311yescharacteristic
469C22#303511yescharacteristic
470C22#304611yescharacteristic
471C22C2×#46511yescharacteristic
472C22#305511yescharacteristic
473C22#306711yescharacteristic
474C22#306911yescharacteristic
475C4—22no—
476C4—21no—
477C4—21no—
478C4—41no—
479C4—41no—
480C4—42no—
481C4—42no—
482C4—42no—
483C4—81no—
484C4—161no—
485C4—161no—
486C4—161no—
487C4—161no—
488C4#310311yescharacteristic
489C4#310511yescharacteristic
490C4#322711yescharacteristic
491C4#323211yescharacteristic
492C4#333611yescharacteristic
493C4#334011yescharacteristic
494C4#334811yescharacteristic
495C4#334911yescharacteristic
496C2—42no—
497C2#1912411yescharacteristic
498C2#1917811yescharacteristic
499C2#1941511yescharacteristic
500C2#1949811yescharacteristic
501C2#2458611yescharacteristic
502C2#2460311yescharacteristic
503C2#2462411yescharacteristic
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. #10654C2semidirect
7. #10598C2semidirect
8. #10590C2semidirect
9. #9613C2semidirect
10. #9611C2semidirect
11. #9566C2semidirect
12. #9565C2semidirect

Maximal subgroups

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

subgroupindex
#331142
#278762
#226322
#211692
#106582
#106542
#105982
#105902
#96132
#96112
#95662
#95652
#91312
#88302
#87272