Finite Group Databasebeta

← #275362

Subgroups — #275362

Order 512.

562 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33114C211yesmaximal, characteristic
2#27962C211yesmaximal, characteristic
3#22632C211yesmaximal, characteristic
4#21364C211yesmaximal, characteristic
5#10658C211yesmaximal, characteristic, complemented
6#10656C211yesmaximal, characteristic, complemented
7#10602C211yesmaximal, characteristic, complemented
8#10598C211yesmaximal, characteristic, complemented
9#9616C211yesmaximal, characteristic, complemented
10#9613C211yesmaximal, characteristic, complemented
11#9566C211yesmaximal, characteristic, complemented
12#9566C211yesmaximal, characteristic, complemented
13#9140C211yesmaximal, characteristic
14#8834C211yesmaximal, characteristic
15#8727C211yesmaximal, characteristic
16C2×#485—21no—
17C2×#432C2211yescharacteristic
18#2857C2211yescharacteristic
19#2285C2211yescharacteristic
20C4×#117C2211yescharacteristic
21#2101—21no—
22#2101—21no—
23#1990C2211yescharacteristic
24#1990C2211yescharacteristic
25#1981C2211yescharacteristic
26#1981C2211yescharacteristic
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1948C2211yescharacteristic
30#1947C2211yescharacteristic
31#1947C2211yescharacteristic
32#1947C2211yescharacteristic
33#1939C2211yescharacteristic
34#1939C2211yescharacteristic
35#1928—21no—
36#1927—21no—
37#1923—21no—
38#1923—21no—
39#1873—21no—
40#1854—21no—
41#1854—21no—
42#1834C2211yescharacteristic
43#1834C2211yescharacteristic
44#1831—21no—
45#1825—21no—
46#1825—21no—
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1776—21no—
50#1776—21no—
51C8×C4⋊C4C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C4×C22⋊C8C2211yescharacteristic
54#1744—21no—
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×C22⋊Q8—21no—
69C2×#117—21no—
70C2×#117D411yescharacteristic
71C2×#117D411yescharacteristic
72C22×C42C2311yescharacteristic
73#485—44no—
74#432—22no—
75#432—22no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#106—21no—
80C2×#106—21no—
81C2×#106—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#103—21no—
91C2×#103—21no—
92C2×#103—21no—
93C2×#103—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8D411yescharacteristic
98C2×C22⋊C8D411yescharacteristic
99C2×#97—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#391—21no—
105#389—21no—
106#389—21no—
107#384C2311yescharacteristic
108#384C2311yescharacteristic
109#384C2311yescharacteristic
110#382—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4D411yescharacteristic
118C4×C4⋊C4D411yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2311yescharacteristic
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123#376—21no—
124C43—21no—
125#340—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#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336D411yescharacteristic
146#336D411yescharacteristic
147#335—22no—
148#334—22no—
149#332—22no—
150#332—22no—
151C22×Q8—21no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4#4111yescharacteristic
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C22⋊Q8—42no—
170C22⋊Q8—42no—
171#117—22no—
172#117—22no—
173#117—22no—
174#117—41no—
175#117—42no—
176#117—42no—
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—41no—
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—41no—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42#4111yescharacteristic
211C2×C42C2411yescharacteristic
212#107—42no—
213#107—42no—
214#107—42no—
215#106—42no—
216#106—42no—
217#106—42no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—81no—
229C4⋊C8—81no—
230C4⋊C8#4112yes—
231C4⋊C8#4112yes—
232#103—42no—
233#103—42no—
234#103—42no—
235#103—42no—
236C22⋊C8—24no—
237C22⋊C8—41no—
238C22⋊C8—41no—
239C22⋊C8—42no—
240C22⋊C8—81no—
241#97—24no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—42no—
247C4×C8—42no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256C24#12111yescharacteristic
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—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—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4#12111yescharacteristic
288C22×C4#12111yescharacteristic
289C22×C4#12111yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C2×#4111yescharacteristic
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
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—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
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—81no—
345C4⋊C4—81no—
346C4⋊C4—81no—
347C4⋊C4—82no—
348C4⋊C4#13512yes—
349C4⋊C4#13512yes—
350C4⋊C4AGL(1,ℤ/8)12yes—
351C22⋊C4—24no—
352C22⋊C4—42no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C42—24no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—42no—
363C42—42no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—81no—
369C42—82no—
370C42C2×#4111yescharacteristic
371C42C2×#4111yescharacteristic
372C422−1+411yescharacteristic
373C422−1+411yescharacteristic
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23C2×#12111yescharacteristic
382Q8—81no—
383Q8—81no—
384Q8—81no—
385Q8—81no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
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—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
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#52511yescharacteristic
432C2×C4#53611yescharacteristic
433C2×C4#54711yescharacteristic
434C2×C4#54711yescharacteristic
435C2×C4#55211yescharacteristic
436C2×C4#55211yescharacteristic
437C2×C4C2×#13511yescharacteristic
438C2×C4C2×#13511yescharacteristic
439C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
440C2×C4#57511yescharacteristic
441C2×C4#57511yescharacteristic
442C2×C4#57511yescharacteristic
443C2×C4#57711yescharacteristic
444C2×C4#57811yescharacteristic
445C8—41no—
446C8—41no—
447C8—41no—
448C8—41no—
449C8—42no—
450C8—42no—
451C8—81no—
452C8—81no—
453C8—81no—
454C8—81no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22#241311yescharacteristic
463C22#304011yescharacteristic
464C22#304611yescharacteristic
465C22#305011yescharacteristic
466C22#305511yescharacteristic
467C22#306711yescharacteristic
468C22#306911yescharacteristic
469C4—21no—
470C4—21no—
471C4—41no—
472C4—41no—
473C4—41no—
474C4—42no—
475C4—42no—
476C4—42no—
477C4—81no—
478C4—161no—
479C4—161no—
480C4—161no—
481C4—161no—
482C4#310211yescharacteristic
483C4#310611yescharacteristic
484C4#322811yescharacteristic
485C4#323311yescharacteristic
486C4#334811yescharacteristic
487C4#334811yescharacteristic
488C4#335211yescharacteristic
489C4#335211yescharacteristic
490C2—81no—
491C2#1927811yescharacteristic
492C2#1931911yescharacteristic
493C2#1941811yescharacteristic
494C2#1949811yescharacteristic
495C2#2459511yescharacteristic
496C2#2460511yescharacteristic
497C2#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. #10656C2semidirect
7. #10602C2semidirect
8. #10598C2semidirect
9. #9616C2semidirect
10. #9613C2semidirect
11. #9566C2semidirect
12. #9566C2semidirect

Maximal subgroups

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

subgroupindex
#331142
#279622
#226322
#213642
#106582
#106562
#106022
#105982
#96162
#96132
#95662
#91402
#88342
#87272