Finite Group Databasebeta

← #286673

Subgroups — #286673

Order 512.

610 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29347C211yesmaximal, characteristic, complemented
2#29254C211yesmaximal, characteristic
3#28718C211yesmaximal, characteristic
4#23149C211yesmaximal, characteristic, complemented
5#10593C211yesmaximal, characteristic, complemented
6#10590C211yesmaximal, characteristic, complemented
7#10540C211yesmaximal, characteristic
8#10539C211yesmaximal, characteristic
9#9738C211yesmaximal, characteristic
10#9735C211yesmaximal, characteristic
11#9693C211yesmaximal, characteristic, complemented
12#9603C211yesmaximal, characteristic, complemented
13#9568C211yesmaximal, characteristic, complemented
14#9566C211yesmaximal, characteristic, complemented
15C4×#331C211yesmaximal, characteristic
16C2×#486C2211yescharacteristic
17C2×#486C2211yescharacteristic
18C2×C8×Q8—21no—
19#2862C2211yescharacteristic
20C4×C4⋊Q8C2211yescharacteristic
21C4×#127C2211yescharacteristic
22#2093—21no—
23#2073—21no—
24#2073—21no—
25#2072—21no—
26#2024—21no—
27#2023—21no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1987C2211yescharacteristic
31#1987C2211yescharacteristic
32#1984—21no—
33#1982C2211yescharacteristic
34#1982C2211yescharacteristic
35#1981C2211yescharacteristic
36#1981C2211yescharacteristic
37#1947C2211yescharacteristic
38#1947C2211yescharacteristic
39#1939C2211yescharacteristic
40#1939C2211yescharacteristic
41#1933—21no—
42#1933—21no—
43#1916—21no—
44#1916—21no—
45#1862—21no—
46#1862—21no—
47#1831C2211yescharacteristic
48#1804—21no—
49#1804—21no—
50C8×C4⋊C4C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#331C2211yescharacteristic
62C2×#331C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C4⋊Q8D411yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×#127C2311yescharacteristic
71C2×C4×Q8—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75#486—22no—
76#486—22no—
77#486—41no—
78#486—41no—
79C8×Q8—44no—
80C2×#107—21no—
81C2×#107—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8D411yescharacteristic
85C2×C4⋊C8D411yescharacteristic
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×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#391—21no—
111#391—21no—
112#391—21no—
113#391—21no—
114#391—21no—
115#390—21no—
116#390—21no—
117#390—21no—
118#389—21no—
119#389D411yescharacteristic
120#389D411yescharacteristic
121#384D411yescharacteristic
122#384C2311yescharacteristic
123#382—21no—
124#382—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C43C2311yescharacteristic
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#336D411yescharacteristic
148#336D411yescharacteristic
149#335—41no—
150#334—41no—
151#333—22no—
152#333—41no—
153#332—22no—
154#332—41no—
155#331—22no—
156#331—22no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C23×C4—41no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184#127—21no—
185#127—21no—
186#127—21no—
187#127#4112yes—
188C4×Q8—42no—
189C4×Q8—42no—
190C4×D4—42no—
191C4×D4—42no—
192C4×D4—81no—
193C4×D4—81no—
194C4×D4—82no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42#4111yescharacteristic
221C2×C42#4111yescharacteristic
222C2×C42C2411yescharacteristic
223#107—81no—
224#107—81no—
225C4⋊C8—22no—
226C4⋊C8—22no—
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—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—42no—
239C4⋊C8—42no—
240C4⋊C8#4112yes—
241#103—42no—
242#103—42no—
243#103—81no—
244#103—81no—
245#102—42no—
246#102—42no—
247#102—42no—
248#102—42no—
249#102—42no—
250#102—42no—
251#102—81no—
252#102—81no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—42no—
258C4×C8—42no—
259C4×C8—42no—
260C4×C8—81no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—41no—
266#95—41no—
267#95—41no—
268#95—41no—
269C24—41no—
270C2×Q8—24no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—81no—
282C2×Q8—81no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—81no—
298C2×D4—81no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4#12111yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—22no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C4⋊C4—24no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—81no—
369C4⋊C4—81no—
370C4⋊C4#13512yes—
371C4⋊C4#13512yes—
372C22⋊C4—82no—
373C22⋊C4—82no—
374C22⋊C4—82no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42#13512yes—
388C42AGL(1,ℤ/8)12yes—
389C42C2×#4111yescharacteristic
390C42C2×#4111yescharacteristic
391C42H(5,𝔽2)11yescharacteristic
392C422−1+411yescharacteristic
393C23—42no—
394C23—81no—
395C23—81no—
396C23—81no—
397C23—81no—
398C23—81no—
399C23—81no—
400C23C2×#12111yescharacteristic
401Q8—48no—
402Q8—81no—
403Q8—81no—
404Q8—81no—
405Q8—81no—
406Q8—161no—
407Q8—161no—
408D4—42no—
409D4—42no—
410D4—42no—
411D4—42no—
412D4—82no—
413D4—82no—
414D4—81no—
415D4—81no—
416D4—81no—
417D4—81no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—164no—
452C2×C4#46312yes—
453C2×C4#46312yes—
454C2×C4#52911yescharacteristic
455C2×C4#54211yescharacteristic
456C2×C4#54611yescharacteristic
457C2×C4#54811yescharacteristic
458C2×C4#54811yescharacteristic
459C2×C4#55011yescharacteristic
460C2×C4C2×#13511yescharacteristic
461C2×C4C2×#13511yescharacteristic
462C2×C4C2×#13511yescharacteristic
463C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
464C2×C4#57611yescharacteristic
465C2×C4#57711yescharacteristic
466C2×C4#57811yescharacteristic
467C2×C4#57811yescharacteristic
468C8—41no—
469C8—41no—
470C8—42no—
471C8—41no—
472C8—41no—
473C8—81no—
474C8—81no—
475C8—81no—
476C8—81no—
477C8—81no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22—161no—
482C22—161no—
483C22—161no—
484C22—161no—
485C22#242611yescharacteristic
486C22C2×#46311yescharacteristic
487C22#304411yescharacteristic
488C22#305011yescharacteristic
489C22#306211yescharacteristic
490C22#306711yescharacteristic
491C22#306811yescharacteristic
492C4—22no—
493C4—22no—
494C4—21no—
495C4—21no—
496C4—41no—
497C4—42no—
498C4—42no—
499C4—41no—
500C4—81no—
501C4—81no—
502C4—161no—
503C4—161no—
504C4—161no—
505C4#330211yescharacteristic
506C4#333011yescharacteristic
507C4#336011yescharacteristic
508C4#336111yescharacteristic
509C4#336411yescharacteristic
510C4#336711yescharacteristic
511C4#339411yescharacteristic
512C4#339611yescharacteristic
513C2—162no—
514C2#1911011yescharacteristic
515C2#1923411yescharacteristic
516C2#1940611yescharacteristic
517C2#1957511yescharacteristic
518C2#2527511yescharacteristic
519C2#2529611yescharacteristic
520C2#2531611yescharacteristic
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. #29347C2semidirect
4. #23149C2semidirect
5. #10593C2semidirect
6. #10590C2semidirect
11. #9693C2semidirect
12. #9603C2semidirect
13. #9568C2semidirect
14. #9566C2semidirect

Maximal subgroups

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

subgroupindex
#293472
#292542
#287182
#231492
#105932
#105902
#105402
#105392
#97382
#97352
#96932
#96032
#95682
#95662
C4×#3312