Finite Group Databasebeta

← #275427

Subgroups — #275427

Order 512.

586 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33109C211yesmaximal, characteristic
2#27949C211yesmaximal, characteristic
3#22629C211yesmaximal, characteristic
4#21358C211yesmaximal, characteristic
5#10653C211yesmaximal, characteristic, complemented
6#10649C211yesmaximal, characteristic, complemented
7#10594C211yesmaximal, characteristic, complemented
8#10590C211yesmaximal, characteristic, complemented
9#9620C211yesmaximal, characteristic, complemented
10#9617C211yesmaximal, characteristic, complemented
11#9574C211yesmaximal, characteristic, complemented
12#9574C211yesmaximal, characteristic, complemented
13#9132C211yesmaximal, characteristic
14#8831C211yesmaximal, characteristic
15#8727C211yesmaximal, characteristic
16C2×#484—21no—
17C2×#431C2211yescharacteristic
18#2857C2211yescharacteristic
19#2285C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#2085—21no—
22#2085—21no—
23#1990C2211yescharacteristic
24#1990C2211yescharacteristic
25#1981C2211yescharacteristic
26#1981C2211yescharacteristic
27#1947C2211yescharacteristic
28#1947C2211yescharacteristic
29#1947C2211yescharacteristic
30#1947C2211yescharacteristic
31#1947C2211yescharacteristic
32#1947C2211yescharacteristic
33#1939C2211yescharacteristic
34#1939C2211yescharacteristic
35#1928—21no—
36#1928—21no—
37#1917—21no—
38#1917—21no—
39#1872—21no—
40#1854—21no—
41#1854—21no—
42#1852—21no—
43#1852—21no—
44#1845C2211yescharacteristic
45#1844C2211yescharacteristic
46#1797—21no—
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49C4×#107—21no—
50C4×#107—21no—
51#1770C2211yescharacteristic
52#1770C2211yescharacteristic
53#1749—21no—
54C4×C22⋊C8C2211yescharacteristic
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—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74#484—44no—
75#431—22no—
76#431—22no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#106—21no—
86C2×#106—21no—
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⋊C8—21no—
102C2×C22⋊C8D411yescharacteristic
103C2×C22⋊C8D411yescharacteristic
104C2×#97—21no—
105C2×#97—21no—
106C2×C4×C8—21no—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109#391—21no—
110#389—21no—
111#389—21no—
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#382—21no—
116#380—21no—
117#380—21no—
118#380—21no—
119#380—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C2×#95—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×C4—21no—
153C23×C4—21no—
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
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C22⋊Q8—42no—
172C22⋊Q8—42no—
173#117—81no—
174#117—81no—
175C2×C4⋊C4—22no—
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—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
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×C42—81no—
209C2×C42—81no—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42#4111yescharacteristic
213C2×C42C2411yescharacteristic
214#107—42no—
215#107—42no—
216#107—42no—
217#107—42no—
218#107—42no—
219#107—42no—
220#107—81no—
221#107—81no—
222#106—81no—
223#106—81no—
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—81no—
235C4⋊C8—81no—
236C4⋊C8#4112yes—
237C4⋊C8#4112yes—
238#103—42no—
239#103—42no—
240#103—42no—
241#103—42no—
242C22⋊C8—24no—
243C22⋊C8—22no—
244C22⋊C8—42no—
245C22⋊C8—42no—
246C22⋊C8Q1614yes—
247#97—81no—
248#97—81no—
249C4×C8—24no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—41no—
259#95—41no—
260#95—41no—
261#95—41no—
262C24#12111yescharacteristic
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
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—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4#12111yescharacteristic
298C22×C4#12111yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4C2×Q1612yes—
301C22×C4#13712yes—
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C2×#4111yescharacteristic
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
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—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8C2×Q1612yes—
334C2×C8C2×Q1612yes—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—42no—
343C4⋊C4—42no—
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—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—81no—
361C4⋊C4—81no—
362C4⋊C4—81no—
363C4⋊C4—82no—
364C4⋊C4AGL(1,ℤ/8)12yes—
365C22⋊C4—82no—
366C22⋊C4—82no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—81no—
378C42—81no—
379C42—82no—
380C42C2×#4111yescharacteristic
381C42C2×#4111yescharacteristic
382C422−1+411yescharacteristic
383C422−1+411yescharacteristic
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23#46212yes—
391C23C2×#12111yescharacteristic
392Q8—81no—
393Q8—81no—
394Q8—81no—
395Q8—81no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
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—41no—
425C2×C4—41no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—84no—
429C2×C4—84no—
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—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4#46212yes—
443C2×C4#52511yescharacteristic
444C2×C4#53611yescharacteristic
445C2×C4#54711yescharacteristic
446C2×C4#54711yescharacteristic
447C2×C4#55211yescharacteristic
448C2×C4#55211yescharacteristic
449C2×C4C22×Q1611yescharacteristic
450C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
451C2×C4C2×#13711yescharacteristic
452C2×C4#57511yescharacteristic
453C2×C4#57511yescharacteristic
454C2×C4#57511yescharacteristic
455C2×C4#57611yescharacteristic
456C2×C4#57611yescharacteristic
457C8—22no—
458C8—22no—
459C8—41no—
460C8—41no—
461C8—42no—
462C8—42no—
463C8—81no—
464C8—81no—
465C8—81no—
466C8—81no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22#241311yescharacteristic
475C22C2×#46211yescharacteristic
476C22#304111yescharacteristic
477C22#305411yescharacteristic
478C22#305511yescharacteristic
479C22#306711yescharacteristic
480C22#306911yescharacteristic
481C4—22no—
482C4—21no—
483C4—21no—
484C4—42no—
485C4—41no—
486C4—41no—
487C4—81no—
488C4—81no—
489C4—81no—
490C4—161no—
491C4—161no—
492C4—161no—
493C4—161no—
494C4#311111yescharacteristic
495C4#311511yescharacteristic
496C4#323511yescharacteristic
497C4#324011yescharacteristic
498C4#333811yescharacteristic
499C4#333811yescharacteristic
500C4#334611yescharacteristic
501C4#334611yescharacteristic
502C2—42no—
503C2#1915011yescharacteristic
504C2#1917211yescharacteristic
505C2#1942411yescharacteristic
506C2#1946111yescharacteristic
507C2#2458211yescharacteristic
508C2#2461011yescharacteristic
509C2#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. #10653C2semidirect
6. #10649C2semidirect
7. #10594C2semidirect
8. #10590C2semidirect
9. #9620C2semidirect
10. #9617C2semidirect
11. #9574C2semidirect
12. #9574C2semidirect

Maximal subgroups

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

subgroupindex
#331092
#279492
#226292
#213582
#106532
#106492
#105942
#105902
#96202
#96172
#95742
#91322
#88312
#87272