Finite Group Databasebeta

← #211153

Subgroups — #211153

Order 512.

700 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22111C211yesmaximal, characteristic
2#22110C211yesmaximal, characteristic
3C4×#487C211yesmaximal, characteristic, complemented
4#20485C211yesmaximal, characteristic
5C2×#1683C211yesmaximal, characteristic
6#9737C211yesmaximal, characteristic, complemented
7#9616C211yesmaximal, characteristic, complemented
8#9576C211yesmaximal, characteristic, complemented
9#9570C211yesmaximal, characteristic, complemented
10#9227C211yesmaximal, characteristic
11#9223C211yesmaximal, characteristic
12#8648C211yesmaximal, characteristic
13C4×#335C211yesmaximal, characteristic, complemented
14#8236C211yesmaximal, characteristic, complemented
15#8235C211yesmaximal, characteristic, complemented
16C2×#487C2211yescharacteristic
17C2×#476C2211yescharacteristic
18C2×#474C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C2×#405—21no—
22C4×C4⋊Q8C2211yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24#2293C2211yescharacteristic
25#1990C2211yescharacteristic
26#1988C2211yescharacteristic
27#1987C2211yescharacteristic
28#1956C2211yescharacteristic
29#1948C2211yescharacteristic
30#1947C2211yescharacteristic
31#1942—21no—
32#1941—21no—
33#1934—21no—
34#1934—21no—
35#1933—21no—
36#1933—21no—
37#1926—21no—
38#1925—21no—
39#1919—21no—
40#1918—21no—
41#1834C2211yescharacteristic
42#1802C2211yescharacteristic
43#1802C2211yescharacteristic
44#1770C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46#1767C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×C4⋊C8C2211yescharacteristic
49C4×#103—21no—
50C4×#103—21no—
51#1753—21no—
52C8×C22⋊C4—21no—
53#1750—21no—
54C42×C8C2211yescharacteristic
55#1683C418yescomplemented
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×M16—21no—
69C23×C8—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×D4C2311yescharacteristic
75C2×#117—21no—
76#487—22no—
77#487C2×C414yes—
78#476—22no—
79#476C2×C414yes—
80#474—22no—
81#474C2×C414yes—
82#436—22no—
83#436—22no—
84C8×D4—22no—
85C8×D4C2×C414yes—
86C2×#107—21no—
87C2×#107—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106#405—44no—
107C2×#97—21no—
108C2×#97—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8D411yescharacteristic
113C2×C4×C8D411yescharacteristic
114C2×C4×C8D411yescharacteristic
115C2×C4×C8D411yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#391—21no—
119#391—21no—
120#390D411yescharacteristic
121#390D411yescharacteristic
122#389—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#389D411yescharacteristic
127#389D411yescharacteristic
128#388—21no—
129#388—21no—
130#384C2311yescharacteristic
131#384C2311yescharacteristic
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4C2311yescharacteristic
137C4×C22⋊C4—21no—
138C43C2311yescharacteristic
139#340—21no—
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#340—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#335—22no—
157#335C2×C414yes—
158#333—22no—
159#333C2×C414yes—
160#332—22no—
161#332C2×C414yes—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×D4C2×D411yescharacteristic
165C23×C4—21no—
166C2×M16—24no—
167C2×M16—42no—
168C2×M16—42no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—24no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—42no—
177C22×C8—42no—
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8C22×C412yes—
189C4⋊Q8C22×C412yes—
190C4×Q8—42no—
191C4×Q8—42no—
192C4×Q8—82no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4C22×C412yes—
196C4×D4#4112yes—
197#117—42no—
198#117—42no—
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—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C42—21no—
222C2×C42—41no—
223C2×C42—41no—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42#4111yescharacteristic
230C2×C42C2411yescharacteristic
231#107—24no—
232#107—81no—
233#106—24no—
234#106—81no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8C22×C412yes—
244C4⋊C8C22×C412yes—
245C4⋊C8C22×C412yes—
246C4⋊C8#4112yes—
247#103—24no—
248#103—24no—
249#103—42no—
250#103—42no—
251#102—24no—
252#102—24no—
253#102—81no—
254#102—81no—
255C22⋊C8—24no—
256C22⋊C8—24no—
257#97—42no—
258#97—42no—
259C4×C8—22no—
260C4×C8—22no—
261C4×C8—22no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—41no—
265C4×C8—42no—
266C4×C8—81no—
267C4×C8—81no—
268C4×C8C22×C412yes—
269C4×C8#4112yes—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—41no—
274#95—41no—
275#95—41no—
276#95—41no—
277C24—21no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—81no—
287C2×Q8—81no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4C4×D412yes—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C2×#4111yescharacteristic
314M16—48no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—21no—
325C2×C8—22no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—48no—
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×C8C4×D412yes—
344C2×C8C4×D412yes—
345C2×C8#13512yes—
346C4⋊C4—21no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—81no—
370C4⋊C4—81no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C4⋊C4C4×D412yes—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—22no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—82no—
391C42—82no—
392C42#13512yes—
393C42AGL(1,ℤ/8)12yes—
394C42C23×C411yescharacteristic
395C42C2×#4111yescharacteristic
396C42C2×#4111yescharacteristic
397C42C2×#4111yescharacteristic
398C23—22no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23C2×#12111yescharacteristic
406Q8—42no—
407Q8—42no—
408Q8—82no—
409Q8—82no—
410Q8—81no—
411Q8—81no—
412D4—22no—
413D4—22no—
414D4—41no—
415D4—41no—
416C2×C4—21no—
417C2×C4—21no—
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—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
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—81no—
452C2×C4#46312yes—
453C2×C4#46312yes—
454C2×C4#47112yes—
455C2×C4C2×C4×D411yescharacteristic
456C2×C4C2×C4×D411yescharacteristic
457C2×C4C4×#4111yescharacteristic
458C2×C4#52511yescharacteristic
459C2×C4#52911yescharacteristic
460C2×C4#53211yescharacteristic
461C2×C4C2×#13511yescharacteristic
462C2×C4C2×#13511yescharacteristic
463C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
464C2×C4#57511yescharacteristic
465C2×C4#57611yescharacteristic
466C2×C4#57811yescharacteristic
467C2×C4#57811yescharacteristic
468C2×C4#57811yescharacteristic
469C8—22no—
470C8—22no—
471C8—22no—
472C8—41no—
473C8—42no—
474C8—42no—
475C8—42no—
476C8—42no—
477C8—81no—
478C8—81no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22C4×#12111yescharacteristic
487C22C2×#46311yescharacteristic
488C22#303511yescharacteristic
489C22#305911yescharacteristic
490C22#306111yescharacteristic
491C22C2×#47111yescharacteristic
492C22#306711yescharacteristic
493C4—22no—
494C4—22no—
495C4—21no—
496C4—21no—
497C4—22no—
498C4—41no—
499C4—42no—
500C4—81no—
501C4—82no—
502C4—82no—
503C4—81no—
504C4—161no—
505C4—161no—
506C4C4×#13511yescharacteristic
507C4C4×AGL(1,ℤ/8)11yescharacteristic
508C4#295011yescharacteristic
509C4#295111yescharacteristic
510C4#310211yescharacteristic
511C4#310511yescharacteristic
512C4#317511yescharacteristic
513C4#317711yescharacteristic
514C2—82no—
515C2#1919911yescharacteristic
516C2#1920111yescharacteristic
517C2#1922611yescharacteristic
518C2#1922911yescharacteristic
519C2C4×#46311yescharacteristic
520C2#2152511yescharacteristic
521C2C4×#47111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
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
3. C4×#487C2semidirect
6. #9737C2semidirect
7. #9616C2semidirect
8. #9576C2semidirect
9. #9570C2semidirect
13. C4×#335C2semidirect
14. #8236C2semidirect
15. #8235C2semidirect
55. #1683C4semidirect

Maximal subgroups

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

subgroupindex
#221112
#221102
C4×#4872
#204852
C2×#16832
#97372
#96162
#95762
#95702
#92272
#92232
#86482
C4×#3352
#82362
#82352