Finite Group Databasebeta

← #228418

Subgroups — #228418

Order 512.

626 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23756C211yesmaximal, characteristic
2#22953C211yesmaximal, characteristic
3#22600C211yesmaximal, characteristic
4#9593C211yesmaximal, characteristic, complemented
5#9553C211yesmaximal, characteristic, complemented
6#9547C211yesmaximal, characteristic, complemented
7#9538C211yesmaximal, characteristic, complemented
8#9535C211yesmaximal, characteristic, complemented
9#9071C211yesmaximal, characteristic
10#8846C211yesmaximal, characteristic, complemented
11#8833C211yesmaximal, characteristic
12#8283C211yesmaximal, characteristic, complemented
13C8×#103C211yesmaximal, characteristic, complemented
14C8×C22⋊C8C211yesmaximal, characteristic
15#7897C211yesmaximal, characteristic
16C2×C8×Q8C2211yescharacteristic
17C2×#433—21no—
18C2×#423C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C4×C22⋊Q8C2211yescharacteristic
21#2285C2211yescharacteristic
22#2099C411yescharacteristic
23#2096C411yescharacteristic
24#2084C411yescharacteristic
25#2077C411yescharacteristic
26#2077C411yescharacteristic
27#2072C411yescharacteristic
28#2035C411yescharacteristic
29#2035C411yescharacteristic
30#1984C2211yescharacteristic
31#1940—21no—
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1939C2211yescharacteristic
35#1917—21no—
36#1916—21no—
37#1916—21no—
38#1851—21no—
39#1818—21no—
40#1797C2211yescharacteristic
41C4×#106C2211yescharacteristic
42C4×#106C2211yescharacteristic
43#1770C2211yescharacteristic
44#1770C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C4×#103C2211yescharacteristic
47C4×#103C2211yescharacteristic
48#1753—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#334C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×C82C2211yescharacteristic
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—
68C23×C8—21no—
69C2×C22⋊Q8C2×C411yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×#117—21no—
72C22×C42C2311yescharacteristic
73C8×Q8—22no—
74C8×Q8—41no—
75#433—44no—
76C2×#106C2×C411yescharacteristic
77C2×#106C2×C411yescharacteristic
78C2×#106C2×C411yescharacteristic
79C2×#106C2×C411yescharacteristic
80#423—22no—
81#423—41no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×#103C2×C411yescharacteristic
92C2×#103C2×C411yescharacteristic
93C2×#103C2×C411yescharacteristic
94C2×#103C2×C411yescharacteristic
95C2×C22⋊C8—21no—
96C2×C22⋊C8C2×C411yescharacteristic
97C2×C22⋊C8C2×C411yescharacteristic
98C2×C22⋊C8D411yescharacteristic
99C2×C22⋊C8D411yescharacteristic
100C2×#97—21no—
101C2×#97—21no—
102C2×C4×C8—22no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—41no—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#391C2×C411yescharacteristic
114#389C2×C411yescharacteristic
115#387—21no—
116#385—21no—
117#384C2×C411yescharacteristic
118#384C2×C411yescharacteristic
119#382—21no—
120#380—21no—
121#380C2×C411yescharacteristic
122#380C2×C411yescharacteristic
123#380C2×C411yescharacteristic
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C22⋊C4—21no—
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#336—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#336C2×C411yescharacteristic
148#336C2×C411yescharacteristic
149#336D411yescharacteristic
150#336D411yescharacteristic
151#334—22no—
152#334—41no—
153Q8⋊C8—22no—
154Q8⋊C8—41no—
155C82—22no—
156C82—41no—
157C22×Q8C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4#4111yescharacteristic
161C2×M16—42no—
162C2×M16—42no—
163C2×M16—81no—
164C22×C8—21no—
165C22×C8—24no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—42no—
172C22×C8—42no—
173C22×C8C22×C411yescharacteristic
174C22×C8C22×C411yescharacteristic
175C22×C8C22×C411yescharacteristic
176C22×C8C22×C411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22⋊Q8—22no—
182C22⋊Q8—41no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187#117—42no—
188#117—42no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42C22×C411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42#4111yescharacteristic
218C2×C42C2411yescharacteristic
219#106—22no—
220#106—22no—
221#106—41no—
222#106—41no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—42no—
231C4⋊C8—42no—
232C4⋊C8—42no—
233C4⋊C8—42no—
234C4⋊C8—81no—
235C4⋊C8#4112yes—
236#103—22no—
237#103—22no—
238#103—22no—
239#103—22no—
240C22⋊C8—22no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8—42no—
244C22⋊C8SD1614yes—
245#97—42no—
246#97—81no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—42no—
259C4×C8—42no—
260C4×C8—42no—
261C4×C8#4112yes—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271C24C4×D411yescharacteristic
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8#13112yes—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4C4×D411yescharacteristic
301C22×C4C4×D411yescharacteristic
302C22×C4C4×D411yescharacteristic
303C22×C4C2×SD1612yes—
304C22×C4#13512yes—
305C22×C4C23×C411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C2×#4111yescharacteristic
308M16—84no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—22no—
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—22no—
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—21no—
334C2×C8—48no—
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#13112yes—
349C2×C8#13112yes—
350C2×C8C2×SD1612yes—
351C2×C8C2×SD1612yes—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
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—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—81no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—81no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—41no—
392C42C2×#4111yescharacteristic
393C42C2×#4111yescharacteristic
394C42H(5,𝔽2)11yescharacteristic
395C422−1+411yescharacteristic
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23C4×SD1612yes—
403C23C2×C4×D411yescharacteristic
404Q8—22no—
405Q8—22no—
406Q8—41no—
407Q8—41no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
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—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×C4C4×SD1612yes—
452C2×C4#44312yes—
453C2×C4C4×#4111yescharacteristic
454C2×C4#52011yescharacteristic
455C2×C4#54611yescharacteristic
456C2×C4#54711yescharacteristic
457C2×C4#55111yescharacteristic
458C2×C4#55211yescharacteristic
459C2×C4C2×#13111yescharacteristic
460C2×C4C2×#13111yescharacteristic
461C2×C4C2×#13111yescharacteristic
462C2×C4#56811yescharacteristic
463C2×C4C22×SD1611yescharacteristic
464C2×C4C2×#13511yescharacteristic
465C2×C4#57711yescharacteristic
466C2×C4#57811yescharacteristic
467C8—24no—
468C8—22no—
469C8—22no—
470C8—22no—
471C8—22no—
472C8—42no—
473C8—41no—
474C8—41no—
475C8—41no—
476C8—41no—
477C8—42no—
478C8—81no—
479C8—81no—
480C8—81no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22#235911yescharacteristic
489C22#293111yescharacteristic
490C22#293311yescharacteristic
491C22C2×C4×SD1611yescharacteristic
492C22#294711yescharacteristic
493C22C2×#44311yescharacteristic
494C22#295611yescharacteristic
495C4—22no—
496C4—21no—
497C4—21no—
498C4—22no—
499C4—41no—
500C4—42no—
501C4—81no—
502C4—81no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4—81no—
507C4—81no—
508C4#296611yescharacteristic
509C4#296811yescharacteristic
510C4#299211yescharacteristic
511C4#299411yescharacteristic
512C4#329611yescharacteristic
513C4#329811yescharacteristic
514C4#333511yescharacteristic
515C4#333611yescharacteristic
516C2—42no—
517C2#1795711yescharacteristic
518C2#1796911yescharacteristic
519C2#1799711yescharacteristic
520C2#1801911yescharacteristic
521C2#2290511yescharacteristic
522C2#2292011yescharacteristic
523C2#2294811yescharacteristic
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 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9593C2semidirect
5. #9553C2semidirect
6. #9547C2semidirect
7. #9538C2semidirect
8. #9535C2semidirect
10. #8846C2semidirect
12. #8283C2semidirect
13. C8×#103C2semidirect

Maximal subgroups

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

subgroupindex
#237562
#229532
#226002
#95932
#95532
#95472
#95382
#95352
#90712
#88462
#88332
#82832
C8×#1032
C8×C22⋊C82
#78972