Finite Group Databasebeta

← #211458

Subgroups — #211458

Order 512.

720 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#486C211yesmaximal, characteristic
2#21616C211yesmaximal, characteristic, complemented
3#21505C211yesmaximal, characteristic
4C4×#446C211yesmaximal, characteristic, complemented
5C2×#1698C211yesmaximal, characteristic
6#9747C211yesmaximal, characteristic
7#9619C211yesmaximal, characteristic, complemented
8#9576C211yesmaximal, characteristic, complemented
9#9572C211yesmaximal, characteristic
10#9230C211yesmaximal, characteristic
11#9224C211yesmaximal, characteristic, complemented
12#8655C211yesmaximal, characteristic, complemented
13#8274C211yesmaximal, characteristic, complemented
14#8235C211yesmaximal, characteristic, complemented
15#8234C211yesmaximal, characteristic
16C2×#486C2211yescharacteristic
17C2×#477C2211yescharacteristic
18C2×#460C2211yescharacteristic
19C2×#446C2211yescharacteristic
20C2×#446C2211yescharacteristic
21C2×#440—21no—
22C4×C4⋊Q8C2211yescharacteristic
23C4×#127C2211yescharacteristic
24C42×Q8C2211yescharacteristic
25#1990C2211yescharacteristic
26#1987C2211yescharacteristic
27#1982C2211yescharacteristic
28#1957C2211yescharacteristic
29#1948C2211yescharacteristic
30#1947C2211yescharacteristic
31#1934—21no—
32#1934—21no—
33#1925—21no—
34#1918—21no—
35#1880—21no—
36#1865—21no—
37#1834C2211yescharacteristic
38#1803C2211yescharacteristic
39#1803C2211yescharacteristic
40#1776—21no—
41#1771—21no—
42#1771—21no—
43#1770C2211yescharacteristic
44#1770C2211yescharacteristic
45#1767C2211yescharacteristic
46#1767C2211yescharacteristic
47C4×C4⋊C8—21no—
48C4×C4⋊C8C2211yescharacteristic
49#1763—21no—
50#1762—21no—
51C4×#102—21no—
52C4×#102—21no—
53C4×#97—21no—
54C42×C8C2211yescharacteristic
55#1698C418yescomplemented
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
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×Q8D411yescharacteristic
73C2×C4×Q8D411yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78#486—22no—
79#486C2×C414yes—
80#477—22no—
81#477C2×C414yescomplemented
82#460—22no—
83#460C2×C414yes—
84#446—22no—
85#446—22no—
86#446—22no—
87#446C2×C414yescomplemented
88#440—44no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#106—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×#103—21no—
101C2×#103—21no—
102C2×#103—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×#97—21no—
111C2×#97—21no—
112C2×#97—21no—
113C2×#97—21no—
114C2×#97—21no—
115C2×#97—21no—
116C2×C4×C8—21no—
117C2×C4×C8D411yescharacteristic
118C2×C4×C8D411yescharacteristic
119C2×C4×C8D411yescharacteristic
120C2×C4×C8D411yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123#391—21no—
124#390—21no—
125#390—21no—
126#390—21no—
127#389—21no—
128#389—21no—
129#384C2311yescharacteristic
130#384C2311yescharacteristic
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4D411yescharacteristic
137C4×C4⋊C4D411yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139C43—21no—
140C43C2311yescharacteristic
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×C414yescomplemented
158#333—22no—
159#333C2×C414yescomplemented
160#331—22no—
161#331C2×C414yes—
162C22×Q8—21no—
163C22×Q8C2×D411yescharacteristic
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C23×C4—41no—
168C2×SD16—24no—
169C2×SD16—42no—
170C2×SD16—42no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8C22×C412yes—
185#127—21no—
186#127—21no—
187#127—21no—
188#127C22×C412yes—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—41no—
194C4×Q8—41no—
195C4×Q8—82no—
196C4×Q8C22×C412yes—
197C4×Q8#4112yes—
198C4×D4—42no—
199C4×D4—42no—
200C4×D4—42no—
201C4×D4—42no—
202C4×D4—82no—
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—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—41no—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42#4111yescharacteristic
233C2×C42#4111yescharacteristic
234C2×C42C2411yescharacteristic
235#107—24no—
236#107—42no—
237#106—42no—
238C4⋊C8—24no—
239C4⋊C8—24no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8C22×C412yes—
249C4⋊C8C22×C412yes—
250C4⋊C8C22×C412yes—
251C4⋊C8#4112yes—
252#103—24no—
253#103—42no—
254#103—42no—
255#102—24no—
256#102—24no—
257#102—24no—
258#102—42no—
259#102—42no—
260#102—42no—
261#102—42no—
262#97—24no—
263#97—24no—
264#97—42no—
265#97—42no—
266#97—42no—
267#97—42no—
268C4×C8—22no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—41no—
272C4×C8—41no—
273C4×C8—41no—
274C4×C8—42no—
275C4×C8C22×C412yes—
276C4×C8#4112yes—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—41no—
281#95—41no—
282C24—41no—
283C2×Q8—21no—
284C2×Q8—22no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—41no—
290C2×Q8—81no—
291C2×Q8—81no—
292C2×Q8C4×D412yes—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—81no—
308C2×D4—81no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C2×#4111yescharacteristic
326SD16—48no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—21no—
337C2×C8—22no—
338C2×C8—21no—
339C2×C8—21no—
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×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8C4×D412yes—
355C2×C8C4×D412yes—
356C2×C8#13712yes—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—81no—
379C4⋊C4—81no—
380C4⋊C4—82no—
381C4⋊C4C4×D412yes—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C42—22no—
386C42—22no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—22no—
394C42—41no—
395C42—42no—
396C42—42no—
397C42—41no—
398C42—41no—
399C42—42no—
400C42—42no—
401C42—82no—
402C42AGL(1,ℤ/8)12yes—
403C42#13712yes—
404C42C23×C411yescharacteristic
405C42C2×#4111yescharacteristic
406C42C2×#4111yescharacteristic
407C42C2×#4111yescharacteristic
408C23—42no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23C2×#12111yescharacteristic
416Q8—22no—
417Q8—22no—
418Q8—41no—
419Q8—41no—
420Q8—82no—
421Q8—82no—
422D4—42no—
423D4—42no—
424D4—42no—
425D4—42no—
426D4—82no—
427D4—82no—
428D4—81no—
429D4—81no—
430D4—81no—
431D4—81no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—164no—
465C2×C4#46812yes—
466C2×C4#46812yes—
467C2×C4#47212yes—
468C2×C4C2×C4×D411yescharacteristic
469C2×C4C2×C4×D411yescharacteristic
470C2×C4C4×#4111yescharacteristic
471C2×C4#52511yescharacteristic
472C2×C4#52911yescharacteristic
473C2×C4#53211yescharacteristic
474C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
475C2×C4C2×#13711yescharacteristic
476C2×C4C2×#13711yescharacteristic
477C2×C4#57511yescharacteristic
478C2×C4#57511yescharacteristic
479C2×C4#57511yescharacteristic
480C2×C4#57511yescharacteristic
481C2×C4#57511yescharacteristic
482C8—22no—
483C8—22no—
484C8—22no—
485C8—41no—
486C8—42no—
487C8—42no—
488C8—42no—
489C8—42no—
490C8—81no—
491C8—81no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—161no—
496C22—161no—
497C22—161no—
498C22—161no—
499C22C4×#12111yescharacteristic
500C22#303711yescharacteristic
501C22#304011yescharacteristic
502C22C2×#46811yescharacteristic
503C22#305411yescharacteristic
504C22C2×#47211yescharacteristic
505C22#306711yescharacteristic
506C4—21no—
507C4—21no—
508C4—22no—
509C4—22no—
510C4—22no—
511C4—42no—
512C4—42no—
513C4—41no—
514C4—82no—
515C4—81no—
516C4—81no—
517C4—161no—
518C4—161no—
519C4#294311yescharacteristic
520C4C4×AGL(1,ℤ/8)11yescharacteristic
521C4C4×#13711yescharacteristic
522C4C4×#13711yescharacteristic
523C4#311111yescharacteristic
524C4#311711yescharacteristic
525C4#317111yescharacteristic
526C4#317211yescharacteristic
527C2—162no—
528C2#1928511yescharacteristic
529C2#1928911yescharacteristic
530C2#1929911yescharacteristic
531C2#1930311yescharacteristic
532C2#2150811yescharacteristic
533C2C4×#46811yescharacteristic
534C2C4×#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. #21616C2semidirect
4. C4×#446C2semidirect
7. #9619C2semidirect
8. #9576C2semidirect
11. #9224C2semidirect
12. #8655C2semidirect
13. #8274C2semidirect
14. #8235C2semidirect
55. #1698C4semidirect
81. #477C2×C4semidirect
87. #446C2×C4semidirect
157. #335C2×C4semidirect
159. #333C2×C4semidirect

Maximal subgroups

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

subgroupindex
C4×#4862
#216162
#215052
C4×#4462
C2×#16982
#97472
#96192
#95762
#95722
#92302
#92242
#86552
#82742
#82352
#82342