Finite Group Databasebeta

← #211150

Subgroups — #211150

Order 512.

688 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22147C211yesmaximal, characteristic
2#22113C211yesmaximal, characteristic, complemented
3C4×#486C211yesmaximal, characteristic
4#20516C211yesmaximal, characteristic, complemented
5C2×#1678C211yesmaximal, characteristic
6#9735C211yesmaximal, characteristic
7#9613C211yesmaximal, characteristic, complemented
8#9575C211yesmaximal, characteristic, complemented
9#9571C211yesmaximal, characteristic
10#9230C211yesmaximal, characteristic
11#9224C211yesmaximal, characteristic, complemented
12#8647C211yesmaximal, characteristic, complemented
13C4×#334C211yesmaximal, characteristic, complemented
14#8235C211yesmaximal, characteristic, complemented
15#8234C211yesmaximal, characteristic
16C2×#486C2211yescharacteristic
17C2×#477C2211yescharacteristic
18C2×#460C2211yescharacteristic
19C2×#446C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C4×C4⋊Q8C2211yescharacteristic
22C4×#127C2211yescharacteristic
23#2295C2211yescharacteristic
24#1990C2211yescharacteristic
25#1987C2211yescharacteristic
26#1982C2211yescharacteristic
27#1968—21no—
28#1957C2211yescharacteristic
29#1948C2211yescharacteristic
30#1947C2211yescharacteristic
31#1942—21no—
32#1934—21no—
33#1934—21no—
34#1934—21no—
35#1933—21no—
36#1928—21no—
37#1921—21no—
38#1877—21no—
39#1864—21no—
40#1834C2211yescharacteristic
41#1803C2211yescharacteristic
42#1803C2211yescharacteristic
43#1771—21no—
44#1770C2211yescharacteristic
45C8×C4⋊C4—21no—
46C8×C4⋊C4C2211yescharacteristic
47#1768—21no—
48#1767C2211yescharacteristic
49C4×C4⋊C8C2211yescharacteristic
50C4×C4⋊C8C2211yescharacteristic
51C4×#102—21no—
52C4×#102—21no—
53#1726—21no—
54C42×C8C2211yescharacteristic
55#1678C418yescomplemented
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#331C2211yescharacteristic
61C2×#331C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×#127C2311yescharacteristic
71C2×C4×Q8—21no—
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76#486—22no—
77#486C2×C414yes—
78#477—22no—
79#477C2×C414yescomplemented
80#460—22no—
81#460C2×C414yes—
82#446—22no—
83#446—22no—
84C8×Q8—22no—
85C8×Q8C2×C414yescomplemented
86C2×#107—21no—
87C2×#107—21no—
88C2×#106—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#97—21no—
108C2×#97—21no—
109C2×#97—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8D411yescharacteristic
115C2×C4×C8D411yescharacteristic
116C2×C4×C8D411yescharacteristic
117C2×C4×C8D411yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120#391—21no—
121#391D411yescharacteristic
122#391D411yescharacteristic
123#390—21no—
124#390—21no—
125#390—21no—
126#389—21no—
127#389—21no—
128#389D411yescharacteristic
129#389D411yescharacteristic
130#387—21no—
131#387—21no—
132#384C2311yescharacteristic
133#384C2311yescharacteristic
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2311yescharacteristic
138#376—21no—
139C43C2311yescharacteristic
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#334—22no—
156#334C2×C414yescomplemented
157#333—22no—
158#333C2×C414yescomplemented
159#331—22no—
160#331C2×C414yes—
161C22×Q8—21no—
162C22×Q8C2×D411yescharacteristic
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C23×C4—41no—
167C2×SD16—24no—
168C2×SD16—81no—
169C2×SD16—81no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8C22×C412yes—
184#127—21no—
185#127—21no—
186#127—21no—
187#127C22×C412yes—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—82no—
191C4×Q8C22×C412yes—
192C4×Q8#4112yes—
193C4×D4—42no—
194C4×D4—42no—
195C4×D4—81no—
196C4×D4—81no—
197C4×D4—82no—
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⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—41no—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42#4111yescharacteristic
224C2×C42#4111yescharacteristic
225C2×C42C2411yescharacteristic
226#107—24no—
227#107—81no—
228#106—81no—
229C4⋊C8—24no—
230C4⋊C8—24no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8C22×C412yes—
240C4⋊C8C22×C412yes—
241C4⋊C8C22×C412yes—
242C4⋊C8#4112yes—
243#103—24no—
244#103—81no—
245#103—81no—
246#102—24no—
247#102—24no—
248#102—24no—
249#102—42no—
250#102—42no—
251#102—81no—
252#102—81no—
253#97—24no—
254#97—42no—
255#97—42no—
256C4×C8—24no—
257C4×C8—22no—
258C4×C8—22no—
259C4×C8—22no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—41no—
263C4×C8—42no—
264C4×C8—81no—
265C4×C8—81no—
266C4×C8C22×C412yes—
267C4×C8#4112yes—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—41no—
276#95—41no—
277C24—41no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—41no—
285C2×Q8—81no—
286C2×Q8—81no—
287C2×Q8C4×D412yes—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—81no—
303C2×D4—81no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4#12111yescharacteristic
315C22×C4#12111yescharacteristic
316C22×C4#12111yescharacteristic
317C22×C4#12111yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C2×#4111yescharacteristic
321SD16—84no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—21no—
332C2×C8—22no—
333C2×C8—21no—
334C2×C8—21no—
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—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8C4×D412yes—
350C2×C8C4×D412yes—
351C2×C8#13512yes—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—21no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—42no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—81no—
374C4⋊C4—81no—
375C4⋊C4—82no—
376C4⋊C4C4×D412yes—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C22⋊C4—82no—
380C42—22no—
381C42—22no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—22no—
389C42—41no—
390C42—41no—
391C42—41no—
392C42—82no—
393C42#13512yes—
394C42#13712yes—
395C42C23×C411yescharacteristic
396C42C2×#4111yescharacteristic
397C42C2×#4111yescharacteristic
398C42C2×#4111yescharacteristic
399C23—42no—
400C23—81no—
401C23—81no—
402C23—81no—
403C23—81no—
404C23—81no—
405C23—81no—
406C23C2×#12111yescharacteristic
407Q8—22no—
408Q8—22no—
409Q8—41no—
410Q8—41no—
411Q8—82no—
412Q8—82no—
413D4—42no—
414D4—42no—
415D4—42no—
416D4—42no—
417D4—82no—
418D4—82no—
419D4—81no—
420D4—81no—
421D4—81no—
422D4—81no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—164no—
456C2×C4#46412yes—
457C2×C4#46412yes—
458C2×C4#47212yes—
459C2×C4C2×C4×D411yescharacteristic
460C2×C4C2×C4×D411yescharacteristic
461C2×C4C4×#4111yescharacteristic
462C2×C4#52511yescharacteristic
463C2×C4#52911yescharacteristic
464C2×C4#53211yescharacteristic
465C2×C4C2×#13511yescharacteristic
466C2×C4C2×#13511yescharacteristic
467C2×C4C2×#13711yescharacteristic
468C2×C4#57511yescharacteristic
469C2×C4#57711yescharacteristic
470C2×C4#57711yescharacteristic
471C2×C4#57711yescharacteristic
472C2×C4#57711yescharacteristic
473C8—22no—
474C8—22no—
475C8—22no—
476C8—41no—
477C8—42no—
478C8—42no—
479C8—42no—
480C8—42no—
481C8—81no—
482C8—81no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—161no—
487C22—161no—
488C22—161no—
489C22—161no—
490C22C4×#12111yescharacteristic
491C22C2×#46411yescharacteristic
492C22#303511yescharacteristic
493C22#305711yescharacteristic
494C22#306011yescharacteristic
495C22C2×#47211yescharacteristic
496C22#306711yescharacteristic
497C4—22no—
498C4—22no—
499C4—22no—
500C4—21no—
501C4—21no—
502C4—42no—
503C4—41no—
504C4—42no—
505C4—81no—
506C4—81no—
507C4—82no—
508C4—161no—
509C4—161no—
510C4C4×#13511yescharacteristic
511C4C4×#13711yescharacteristic
512C4#294911yescharacteristic
513C4#295211yescharacteristic
514C4#310211yescharacteristic
515C4#310811yescharacteristic
516C4#317311yescharacteristic
517C4#317411yescharacteristic
518C2—162no—
519C2#1921211yescharacteristic
520C2#1921611yescharacteristic
521C2#1922411yescharacteristic
522C2#1922711yescharacteristic
523C2C4×#46411yescharacteristic
524C2#2152311yescharacteristic
525C2C4×#47211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ 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
2. #22113C2semidirect
4. #20516C2semidirect
7. #9613C2semidirect
8. #9575C2semidirect
11. #9224C2semidirect
12. #8647C2semidirect
13. C4×#334C2semidirect
14. #8235C2semidirect
55. #1678C4semidirect
79. #477C2×C4semidirect
85. C8×Q8C2×C4semidirect
156. #334C2×C4semidirect
158. #333C2×C4semidirect

Maximal subgroups

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

subgroupindex
#221472
#221132
C4×#4862
#205162
C2×#16782
#97352
#96132
#95752
#95712
#92302
#92242
#86472
C4×#3342
#82352
#82342