Finite Group Databasebeta

← #275397

Subgroups — #275397

Order 512.

592 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33108C211yesmaximal, characteristic
2#27962C211yesmaximal, characteristic
3#22603C211yesmaximal, characteristic
4#21358C211yesmaximal, characteristic
5#10658C211yesmaximal, characteristic, complemented
6#10655C211yesmaximal, characteristic, complemented
7#10594C211yesmaximal, characteristic, complemented
8#10590C211yesmaximal, characteristic, complemented
9#9622C211yesmaximal, characteristic, complemented
10#9619C211yesmaximal, characteristic, complemented
11#9574C211yesmaximal, characteristic, complemented
12#9574C211yesmaximal, characteristic, complemented
13#9132C211yesmaximal, characteristic
14#8831C211yesmaximal, characteristic
15#8726C211yesmaximal, characteristic
16C2×#483—21no—
17C2×#432—21no—
18C2×#431C2211yescharacteristic
19#2857C2211yescharacteristic
20#2285C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#2101—21no—
23#2101—21no—
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1981C2211yescharacteristic
27#1981C2211yescharacteristic
28#1948C2211yescharacteristic
29#1948C2211yescharacteristic
30#1947C2211yescharacteristic
31#1947C2211yescharacteristic
32#1947C2211yescharacteristic
33#1947C2211yescharacteristic
34#1939C2211yescharacteristic
35#1939C2211yescharacteristic
36#1927—21no—
37#1927—21no—
38#1917—21no—
39#1917—21no—
40#1872—21no—
41#1852—21no—
42#1852—21no—
43#1834C2211yescharacteristic
44#1834C2211yescharacteristic
45#1825—21no—
46#1825—21no—
47#1817—21no—
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50C4×#107—21no—
51C4×#107—21no—
52#1770C2211yescharacteristic
53#1770C2211yescharacteristic
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#483—44no—
75#432—44no—
76#431—22no—
77#431—22no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×#103—21no—
93C2×#103—21no—
94C2×#103—21no—
95C2×#103—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8D411yescharacteristic
100C2×C22⋊C8D411yescharacteristic
101C2×#97—21no—
102C2×#97—21no—
103C2×C4×C8—21no—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106#391—21no—
107#389—21no—
108#389—21no—
109#384C2311yescharacteristic
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#382—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C2×#95—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#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336D411yescharacteristic
147#336D411yescharacteristic
148#335—22no—
149#334—22no—
150#332—22no—
151#332—22no—
152C22×Q8—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4#4111yescharacteristic
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C22⋊Q8—42no—
173C22⋊Q8—42no—
174#117—42no—
175#117—42no—
176C2×C4⋊C4—22no—
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—21no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42—41no—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42#4111yescharacteristic
214C2×C42C2411yescharacteristic
215#107—42no—
216#107—42no—
217#107—42no—
218#107—42no—
219#106—42no—
220#106—42no—
221C4⋊C8—21no—
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—42no—
232C4⋊C8—42no—
233C4⋊C8#4112yes—
234C4⋊C8#4112yes—
235#103—42no—
236#103—42no—
237#103—42no—
238#103—42no—
239C22⋊C8—24no—
240C22⋊C8—22no—
241C22⋊C8—22no—
242C22⋊C8—42no—
243C22⋊C8—42no—
244#97—42no—
245#97—42no—
246C4×C8—24no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—41no—
256#95—41no—
257#95—41no—
258#95—41no—
259C24#12111yescharacteristic
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
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—21no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—81no—
294C22×C4#12111yescharacteristic
295C22×C4#12111yescharacteristic
296C22×C4#12111yescharacteristic
297C22×C4#13512yes—
298C22×C4AGL(1,ℤ/8)12yes—
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C2×C8—22no—
303C2×C8—22no—
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—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
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—41no—
331C2×C8—41no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—42no—
340C4⋊C4—42no—
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—41no—
357C4⋊C4—81no—
358C4⋊C4—81no—
359C4⋊C4—81no—
360C4⋊C4—82no—
361C4⋊C4AGL(1,ℤ/8)12yes—
362C22⋊C4—82no—
363C22⋊C4—82no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—42no—
375C42—42no—
376C42—82no—
377C42C2×#4111yescharacteristic
378C42C2×#4111yescharacteristic
379C422−1+411yescharacteristic
380C422−1+411yescharacteristic
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23#46312yes—
388C23C2×#12111yescharacteristic
389Q8—81no—
390Q8—81no—
391Q8—81no—
392Q8—81no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
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—81no—
424C2×C4—81no—
425C2×C4—84no—
426C2×C4—84no—
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#46312yes—
440C2×C4#52511yescharacteristic
441C2×C4#53611yescharacteristic
442C2×C4#54711yescharacteristic
443C2×C4#54711yescharacteristic
444C2×C4#55211yescharacteristic
445C2×C4#55211yescharacteristic
446C2×C4C2×#13511yescharacteristic
447C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
448C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
449C2×C4#57511yescharacteristic
450C2×C4#57511yescharacteristic
451C2×C4#57511yescharacteristic
452C2×C4#57811yescharacteristic
453C2×C4#57811yescharacteristic
454C8—41no—
455C8—41no—
456C8—41no—
457C8—41no—
458C8—42no—
459C8—42no—
460C8—81no—
461C8—81no—
462C8—81no—
463C8—81no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22#241311yescharacteristic
472C22C2×#46311yescharacteristic
473C22#304411yescharacteristic
474C22#305411yescharacteristic
475C22#305511yescharacteristic
476C22#306711yescharacteristic
477C22#306911yescharacteristic
478C4—22no—
479C4—21no—
480C4—21no—
481C4—41no—
482C4—41no—
483C4—42no—
484C4—81no—
485C4—81no—
486C4—81no—
487C4—161no—
488C4—161no—
489C4—161no—
490C4—161no—
491C4#311111yescharacteristic
492C4#311511yescharacteristic
493C4#323511yescharacteristic
494C4#324011yescharacteristic
495C4#333611yescharacteristic
496C4#333611yescharacteristic
497C4#334311yescharacteristic
498C4#334311yescharacteristic
499C2—42no—
500C2#1914211yescharacteristic
501C2#1916911yescharacteristic
502C2#1944311yescharacteristic
503C2#1947911yescharacteristic
504C2#2457711yescharacteristic
505C2#2461011yescharacteristic
506C2#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. #10655C2semidirect
7. #10594C2semidirect
8. #10590C2semidirect
9. #9622C2semidirect
10. #9619C2semidirect
11. #9574C2semidirect
12. #9574C2semidirect

Maximal subgroups

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

subgroupindex
#331082
#279622
#226032
#213582
#106582
#106552
#105942
#105902
#96222
#96192
#95742
#91322
#88312
#87262