Finite Group Databasebeta

← #231124

Subgroups — #231124

Order 512.

620 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23055C211yesmaximal, characteristic
2#22957C211yesmaximal, characteristic
3C4×#470C211yesmaximal, characteristic
4#9559C211yesmaximal, characteristic, complemented
5#9557C211yesmaximal, characteristic, complemented
6#9535C211yesmaximal, characteristic, complemented
7#9535C211yesmaximal, characteristic, complemented
8#8950C211yesmaximal, characteristic, complemented
9#8950C211yesmaximal, characteristic, complemented
10#8942C211yesmaximal, characteristic
11#8941C211yesmaximal, characteristic
12#8213C211yesmaximal, characteristic, complemented
13#8211C211yesmaximal, characteristic, complemented
14#8147C211yesmaximal, characteristic
15#7876C211yesmaximal, characteristic
16C2×#470C411yescharacteristic
17C2×#446C2211yescharacteristic
18C2×#446C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23C4×C22⋊Q8C2211yescharacteristic
24C4×C22⋊Q8C2211yescharacteristic
25#1984C2211yescharacteristic
26#1984C2211yescharacteristic
27#1971C411yescharacteristic
28#1944C411yescharacteristic
29#1939—21no—
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1935C411yescharacteristic
33#1935C411yescharacteristic
34#1924—21no—
35#1923—21no—
36#1917—21no—
37#1917—21no—
38#1894C411yescharacteristic
39#1894C411yescharacteristic
40#1851—21no—
41#1809C411yescharacteristic
42C4×#106C2211yescharacteristic
43#1770—21no—
44C8×C4⋊C4C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C4×#103C2211yescharacteristic
47C4×#103C2211yescharacteristic
48#1753—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×Q8⋊C8C2211yescharacteristic
52C2×Q8⋊C8C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×Q16C2×C411yescharacteristic
68C22×M16—21no—
69C22×M16C2×C411yescharacteristic
70C22×M16C2×C411yescharacteristic
71C2×C22⋊Q8C2×C411yescharacteristic
72C2×C22⋊Q8C2×C411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C22×C42C2311yescharacteristic
76#470—24no—
77#446—22no—
78#446—22no—
79#446—41no—
80#446—41no—
81C4×Q16—41no—
82C4×Q16—41no—
83#432—22no—
84#432—41no—
85C2×#107C2×C411yescharacteristic
86C2×#106C2×C411yescharacteristic
87C2×#106C2×C411yescharacteristic
88#423—22no—
89#423—41no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#103C2×C411yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8D411yescharacteristic
108C2×C22⋊C8D411yescharacteristic
109C4×M16—41no—
110C4×M16—41no—
111C2×#97—21no—
112C2×#97—21no—
113C2×#97—21no—
114C2×#97C2×C411yescharacteristic
115C2×#97C2×C411yescharacteristic
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#391C2×C411yescharacteristic
125#391C2×C411yescharacteristic
126#389C2×C411yescharacteristic
127#389C2×C411yescharacteristic
128#387—21no—
129#387—21no—
130#385—21no—
131#385—21no—
132#380C2×C411yescharacteristic
133#380C2×C411yescharacteristic
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4C2311yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336D411yescharacteristic
156#336D411yescharacteristic
157Q8⋊C8—22no—
158Q8⋊C8—22no—
159Q8⋊C8—41no—
160Q8⋊C8—41no—
161#322—41no—
162#322—41no—
163C22×Q8C22×C411yescharacteristic
164C22×Q8C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4#4111yescharacteristic
168C2×Q16—22no—
169C2×Q16—41no—
170C2×Q16—41no—
171C2×M16—22no—
172C2×M16—41no—
173C2×M16—41no—
174C2×M16—41no—
175C2×M16—41no—
176C2×M16—41no—
177C2×M16—42no—
178C2×M16—42no—
179C2×M16—81no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8C22×C411yescharacteristic
187C22×C8C22×C411yescharacteristic
188C22×C8C22×C411yescharacteristic
189C22×C8C22×C411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C22⋊Q8—41no—
197C22⋊Q8—41no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×Q8—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⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42C22×C411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42#4111yescharacteristic
233C2×C42C2411yescharacteristic
234#107—41no—
235#106—41no—
236#106M1614yescomplemented
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—42no—
244C4⋊C8—42no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247C4⋊C8—81no—
248C4⋊C8#4112yes—
249C4⋊C8#4112yes—
250#103—22no—
251#103—22no—
252#103—41no—
253#103—41no—
254#103—41no—
255#103—41no—
256C22⋊C8—22no—
257C22⋊C8—41no—
258C22⋊C8—42no—
259C22⋊C8—42no—
260#97—41no—
261#97—41no—
262#97—42no—
263#97—42no—
264#97—42no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—42no—
278C4×C8—81no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289C24C4×D411yescharacteristic
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4C4×D411yescharacteristic
327C22×C4C4×D411yescharacteristic
328C22×C4C4×D411yescharacteristic
329C22×C4C23×C411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4111yescharacteristic
332Q16—44no—
333M16—44no—
334M16—44no—
335M16—84no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—22no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8C2×M1612yes—
377C2×C8#13712yes—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—81no—
396C4⋊C4C2×M1612yes—
397C4⋊C4C2×M1612yes—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—81no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42C2×#4111yescharacteristic
417C42C2×#4111yescharacteristic
418C42H(5,𝔽2)11yescharacteristic
419C42H(5,𝔽2)11yescharacteristic
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23C2×C4×D411yescharacteristic
428Q8—41no—
429Q8—41no—
430Q8—41no—
431Q8—41no—
432Q8—41no—
433Q8—41no—
434Q8—41no—
435Q8—41no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
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—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4C4×#4111yescharacteristic
482C2×C4#51811yescharacteristic
483C2×C4#54611yescharacteristic
484C2×C4#54611yescharacteristic
485C2×C4#55311yescharacteristic
486C2×C4#55311yescharacteristic
487C2×C4C22×M1611yescharacteristic
488C2×C4#56811yescharacteristic
489C2×C4#56811yescharacteristic
490C2×C4#56811yescharacteristic
491C2×C4C2×#13711yescharacteristic
492C2×C4#57511yescharacteristic
493C2×C4#57711yescharacteristic
494C2×C4#57811yescharacteristic
495C8—22no—
496C8—22no—
497C8—42no—
498C8—41no—
499C8—42no—
500C8—41no—
501C8—41no—
502C8—41no—
503C8—41no—
504C8—42no—
505C8—41no—
506C8—81no—
507C8—81no—
508C8—81no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22#236411yescharacteristic
517C22#293211yescharacteristic
518C22D4×M1611yescharacteristic
519C22C4×#13711yescharacteristic
520C22#294711yescharacteristic
521C22#295511yescharacteristic
522C22#295611yescharacteristic
523C4—21no—
524C4—21no—
525C4—41no—
526C4—41no—
527C4—41no—
528C4—42no—
529C4—42no—
530C4—81no—
531C4—81no—
532C4—81no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4#296711yescharacteristic
537C4#297111yescharacteristic
538C4#297511yescharacteristic
539C4#297811yescharacteristic
540C4#331111yescharacteristic
541C4#331211yescharacteristic
542C4#332711yescharacteristic
543C4#333111yescharacteristic
544C2—81no—
545C2#1798611yescharacteristic
546C2#1798811yescharacteristic
547C2#1804711yescharacteristic
548C2#1805111yescharacteristic
549C2#2312111yescharacteristic
550C2#2314411yescharacteristic
551C2#2314811yescharacteristic
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. #9559C2semidirect
5. #9557C2semidirect
6. #9535C2semidirect
7. #9535C2semidirect
8. #8950C2semidirect
9. #8950C2semidirect
12. #8213C2semidirect
13. #8211C2semidirect
236. #106M16semidirect

Maximal subgroups

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

subgroupindex
#230552
#229572
C4×#4702
#95592
#95572
#95352
#89502
#89422
#89412
#82132
#82112
#81472
#78762