Finite Group Databasebeta

← #318878

Subgroups — #318878

Order 512.

586 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29275C211yesmaximal, characteristic, complemented
2#28286C211yesmaximal, characteristic
3#21675C211yesmaximal, characteristic, complemented
4#21646C211yesmaximal, characteristic
5#10597C211yesmaximal, characteristic, complemented
6#10595C211yesmaximal, characteristic
7#10593C211yesmaximal, characteristic, complemented
8#10591C211yesmaximal, characteristic
9#9747C211yesmaximal, characteristic
10#9735C211yesmaximal, characteristic
11#9622C211yesmaximal, characteristic, complemented
12#9616C211yesmaximal, characteristic, complemented
13#8808C211yesmaximal, characteristic, complemented
14#8592C211yesmaximal, characteristic
15#8590C211yesmaximal, characteristic, complemented
16C2×#486C2211yescharacteristic
17C2×#478—21no—
18C4×C4⋊Q8C2211yescharacteristic
19C4×#127C2211yescharacteristic
20C4×#125C2211yescharacteristic
21#2101—21no—
22#2067—21no—
23#2065—21no—
24#2059—21no—
25#1990C2211yescharacteristic
26#1987C2211yescharacteristic
27#1987C2211yescharacteristic
28#1982C2211yescharacteristic
29#1982C2211yescharacteristic
30#1957C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1939C2211yescharacteristic
36#1939C2211yescharacteristic
37#1936—21no—
38#1936—21no—
39#1920—21no—
40#1918—21no—
41#1917—21no—
42#1917—21no—
43#1842C2211yescharacteristic
44#1834C2211yescharacteristic
45#1831C2211yescharacteristic
46#1831C2211yescharacteristic
47#1804—21no—
48#1771—21no—
49#1770—21no—
50#1770C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1762—21no—
53#1762—21no—
54C4×#97C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#333C2211yescharacteristic
61C2×#331C2211yescharacteristic
62C2×#331C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×#127C2311yescharacteristic
69C2×#125D411yescharacteristic
70C2×C4×Q8—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74#486—22no—
75#486Q814yes—
76#478—44no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#106—21no—
80C2×#106—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#97—21no—
100C2×#97D411yescharacteristic
101C2×#97D411yescharacteristic
102C2×#97D411yescharacteristic
103C2×#97D411yescharacteristic
104C2×C4×C8—21no—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#391—21no—
108#390—21no—
109#390—21no—
110#390—21no—
111#389—21no—
112#389—21no—
113#389D411yescharacteristic
114#389D411yescharacteristic
115#387—21no—
116#387—21no—
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#382D411yescharacteristic
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C43C2311yescharacteristic
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#336—21no—
146#336—21no—
147#335—22no—
148#335—41no—
149#334—22no—
150#334—41no—
151#333—41no—
152#333—41no—
153#331—41no—
154#331—41no—
155C22×Q8—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C23×C4—41no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8C2×Q812yes—
174#127—21no—
175#127—21no—
176#127—21no—
177#127C2×Q812yes—
178#125—22no—
179#125—41no—
180C4×Q8—82no—
181C4×D4—42no—
182C4×D4—42no—
183C4×D4—42no—
184C4×D4—42no—
185C4×D4—82no—
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⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—41no—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42#4111yescharacteristic
214C2×C42#4111yescharacteristic
215C2×C42C2411yescharacteristic
216#107—42no—
217#107—42no—
218#106—42no—
219#106—42no—
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—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—42no—
233C4⋊C8—42no—
234C4⋊C8—42no—
235C4⋊C8—81no—
236#102—24no—
237#102—24no—
238#102—42no—
239#102—42no—
240#102—42no—
241#102—42no—
242#102—81no—
243#102—81no—
244#97—22no—
245#97—41no—
246#97—41no—
247#97—41no—
248#97—42no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—42no—
252C4×C8C2×Q812yes—
253C4×C8#4112yes—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—41no—
261#95—41no—
262C24—41no—
263C2×Q8—22no—
264C2×Q8—41no—
265C2×Q8—81no—
266C2×Q8—81no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—81no—
282C2×D4—81no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4#12111yescharacteristic
294C22×C4#12111yescharacteristic
295C22×C4#12111yescharacteristic
296C22×C4#12111yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
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—41no—
326C2×C8—41no—
327C2×C8C22⋊Q812yes—
328C2×C8C22⋊Q812yes—
329C2×C8#13512yes—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—22no—
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—81no—
352C4⋊C4—81no—
353C4⋊C4—81no—
354C4⋊C4—81no—
355C4⋊C4—82no—
356C4⋊C4AGL(1,ℤ/8)12yes—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C22⋊C4—82no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—82no—
375C42C22×Q811yescharacteristic
376C42C2×#4111yescharacteristic
377C42H(5,𝔽2)11yescharacteristic
378C422−1+411yescharacteristic
379C23—42no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23C2×#12111yescharacteristic
387Q8—82no—
388Q8—82no—
389D4—42no—
390D4—42no—
391D4—42no—
392D4—42no—
393D4—82no—
394D4—82no—
395D4—81no—
396D4—81no—
397D4—81no—
398D4—81no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
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—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—164no—
435C2×C4C2×C22⋊Q811yescharacteristic
436C2×C4#53711yescharacteristic
437C2×C4D4×Q811yescharacteristic
438C2×C4#55011yescharacteristic
439C2×C4#55411yescharacteristic
440C2×C4#55611yescharacteristic
441C2×C4C2×#13511yescharacteristic
442C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
443C2×C4#57511yescharacteristic
444C2×C4#57511yescharacteristic
445C2×C4#57511yescharacteristic
446C2×C4#57511yescharacteristic
447C2×C4#57711yescharacteristic
448C2×C4#57811yescharacteristic
449C8—22no—
450C8—22no—
451C8—22no—
452C8—41no—
453C8—81no—
454C8—81no—
455C8—81no—
456C8—81no—
457C8—81no—
458C8—81no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22—161no—
463C22—161no—
464C22—161no—
465C22—161no—
466C22#253811yescharacteristic
467C22#303511yescharacteristic
468C22#304011yescharacteristic
469C22#305511yescharacteristic
470C22#306311yescharacteristic
471C22#306811yescharacteristic
472C22#306811yescharacteristic
473C4—21no—
474C4—21no—
475C4—41no—
476C4—41no—
477C4—41no—
478C4—41no—
479C4—42no—
480C4—81no—
481C4—81no—
482C4—82no—
483C4—81no—
484C4—161no—
485C4—161no—
486C4#307611yescharacteristic
487C4#308011yescharacteristic
488C4#325011yescharacteristic
489C4#325611yescharacteristic
490C4#338611yescharacteristic
491C4#338811yescharacteristic
492C4#339711yescharacteristic
493C4#340211yescharacteristic
494C2—162no—
495C2#1918011yescharacteristic
496C2#1925311yescharacteristic
497C2#1932411yescharacteristic
498C2#1938211yescharacteristic
499C2#2827511yescharacteristic
500C2#2828511yescharacteristic
501C2#2829911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #29275C2semidirect
3. #21675C2semidirect
5. #10597C2semidirect
7. #10593C2semidirect
11. #9622C2semidirect
12. #9616C2semidirect
13. #8808C2semidirect
15. #8590C2semidirect

Maximal subgroups

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

subgroupindex
#292752
#282862
#216752
#216462
#105972
#105952
#105932
#105912
#97472
#97352
#96222
#96162
#88082
#85922
#85902