Finite Group Databasebeta

← #218042

Subgroups — #218042

Order 512.

638 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21596C211yesmaximal, characteristic
2C4×#484C211yesmaximal, characteristic
3#21146C211yesmaximal, characteristic
4#9598C211yesmaximal, characteristic, complemented
5#9594C211yesmaximal, characteristic, complemented
6#9557C211yesmaximal, characteristic, complemented
7#9553C211yesmaximal, characteristic, complemented
8#9540C211yesmaximal, characteristic, complemented
9#9540C211yesmaximal, characteristic, complemented
10#8942C211yesmaximal, characteristic
11#8844C211yesmaximal, characteristic, complemented
12#8844C211yesmaximal, characteristic, complemented
13#8832C211yesmaximal, characteristic
14#8707C211yesmaximal, characteristic
15#8146C211yesmaximal, characteristic
16C2×#484C411yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×#423—21no—
19C2×#423C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21C4×C22⋊Q8C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1984C2211yescharacteristic
24#1940—21no—
25#1940—21no—
26#1939C2211yescharacteristic
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1928C411yescharacteristic
32#1928C411yescharacteristic
33#1921C411yescharacteristic
34#1921C411yescharacteristic
35#1917—21no—
36#1872C411yescharacteristic
37#1852C411yescharacteristic
38#1851—21no—
39#1824C411yescharacteristic
40#1797C2211yescharacteristic
41C4×#107C2211yescharacteristic
42C4×#107C2211yescharacteristic
43#1771—21no—
44#1770C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#103C2211yescharacteristic
47#1753—21no—
48C4×C22⋊C8C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50#1728—21no—
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×C82C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×M16—21no—
68C22×M16D411yescharacteristic
69C22×M16D411yescharacteristic
70C2×C22⋊Q8C2×C411yescharacteristic
71C2×C4×Q8C2311yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#484—24no—
77C8×Q8—41no—
78C8×Q8—41no—
79C2×#107C2×C411yescharacteristic
80C2×#107C2×C411yescharacteristic
81C2×#107C2×C411yescharacteristic
82C2×#107C2×C411yescharacteristic
83#423—41no—
84#423—41no—
85#423—44no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×#103C2×C411yescharacteristic
98C2×#103C2×C411yescharacteristic
99C2×#103C2×C411yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8C2×C411yescharacteristic
104C2×C22⋊C8C2×C411yescharacteristic
105C4×M16—41no—
106C4×M16—41no—
107C2×#97—21no—
108C2×#97D411yescharacteristic
109C2×#97D411yescharacteristic
110C2×C4×C8—21no—
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117#391C2×C411yescharacteristic
118#389C2×C411yescharacteristic
119#387—21no—
120#385—21no—
121#380C2×C411yescharacteristic
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C22⋊C4—21no—
131C2×#95—21no—
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336C2×C411yescharacteristic
151#336C2×C411yescharacteristic
152#335—41no—
153#335—41no—
154Q8⋊C8—41no—
155Q8⋊C8—41no—
156#322—41no—
157C82—41no—
158C22×Q8C22×C411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4#4111yescharacteristic
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—41no—
167C2×M16—41no—
168C2×M16—41no—
169C2×M16—42no—
170C2×M16—42no—
171C2×M16—42no—
172C2×M16Q1614yes—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8C22×C411yescharacteristic
180C22×C8C22×C411yescharacteristic
181C22×C8C22×C411yescharacteristic
182C22×C8C22×C411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C22⋊Q8—22no—
188C22⋊Q8—41no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—42no—
222C2×C42—42no—
223C2×C42C22×C411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42#4111yescharacteristic
226C2×C42C2411yescharacteristic
227#107—22no—
228#107—22no—
229#107—22no—
230#107—41no—
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⋊C8—42no—
240C4⋊C8—42no—
241C4⋊C8—42no—
242C4⋊C8—42no—
243C4⋊C8—42no—
244C4⋊C8—42no—
245C4⋊C8—42no—
246#103—22no—
247#103—22no—
248#103—41no—
249#103M1614yescomplemented
250C22⋊C8—22no—
251C22⋊C8—41no—
252C22⋊C8—42no—
253C22⋊C8—42no—
254#97—41no—
255#97—41no—
256#97—42no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—42no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—21no—
279C24C4×D411yescharacteristic
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8C2×M1612yes—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—81no—
313C22×C4—81no—
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4C2×Q1612yes—
318C22×C4#13512yes—
319C22×C4C23×C411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C2×#4111yescharacteristic
322M16—44no—
323M16—44no—
324M16—84no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
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×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8C2×M1612yes—
364C2×C8C2×Q1612yes—
365C2×C8C2×Q1612yes—
366C2×C8AGL(1,ℤ/8)12yes—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—81no—
385C4⋊C4C2×M1612yes—
386C4⋊C4#13112yes—
387C22⋊C4—42no—
388C22⋊C4—81no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—42no—
396C42—42no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42H(5,𝔽2)11yescharacteristic
406C42H(5,𝔽2)11yescharacteristic
407C422−1+411yescharacteristic
408C422−1+411yescharacteristic
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23C4×Q1612yes—
416C23C2×C4×D411yescharacteristic
417Q8—22no—
418Q8—22no—
419Q8—41no—
420Q8—41no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
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—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—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—84no—
466C2×C4—84no—
467C2×C4C4×Q1612yes—
468C2×C4#52011yescharacteristic
469C2×C4#52011yescharacteristic
470C2×C4#53411yescharacteristic
471C2×C4#53611yescharacteristic
472C2×C4#53911yescharacteristic
473C2×C4#53911yescharacteristic
474C2×C4C22×M1611yescharacteristic
475C2×C4C2×#13111yescharacteristic
476C2×C4#56811yescharacteristic
477C2×C4#56811yescharacteristic
478C2×C4C22×Q1611yescharacteristic
479C2×C4C2×#13511yescharacteristic
480C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
481C2×C4#57511yescharacteristic
482C8—22no—
483C8—22no—
484C8—22no—
485C8—22no—
486C8—42no—
487C8—41no—
488C8—41no—
489C8—41no—
490C8—41no—
491C8—81no—
492C8—81no—
493C8—81no—
494C8—81no—
495C8—81no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22#232211yescharacteristic
504C22#293311yescharacteristic
505C22D4×M1611yescharacteristic
506C22C2×C4×Q1611yescharacteristic
507C22C4×AGL(1,ℤ/8)11yescharacteristic
508C22#295611yescharacteristic
509C22#295711yescharacteristic
510C4—22no—
511C4—21no—
512C4—21no—
513C4—41no—
514C4—42no—
515C4—41no—
516C4—42no—
517C4—42no—
518C4—81no—
519C4—81no—
520C4—81no—
521C4—81no—
522C4—81no—
523C4#298911yescharacteristic
524C4#299011yescharacteristic
525C4#299311yescharacteristic
526C4#299511yescharacteristic
527C4#318811yescharacteristic
528C4#318911yescharacteristic
529C4#323011yescharacteristic
530C4#323211yescharacteristic
531C2—42no—
532C2#1799811yescharacteristic
533C2#1800611yescharacteristic
534C2M16×Q1611yescharacteristic
535C2#1804811yescharacteristic
536C2#2207111yescharacteristic
537C2#2207911yescharacteristic
538C2#2209711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9598C2semidirect
5. #9594C2semidirect
6. #9557C2semidirect
7. #9553C2semidirect
8. #9540C2semidirect
9. #9540C2semidirect
11. #8844C2semidirect
12. #8844C2semidirect
249. #103M16semidirect

Maximal subgroups

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

subgroupindex
#215962
C4×#4842
#211462
#95982
#95942
#95572
#95532
#95402
#89422
#88442
#88322
#87072
#81462