Finite Group Databasebeta

← #228271

Subgroups — #228271

Order 512.

658 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23050C211yesmaximal, characteristic
2#22632C211yesmaximal, characteristic
3C4×#485C211yesmaximal, characteristic
4#9593C211yesmaximal, characteristic, complemented
5#9553C211yesmaximal, characteristic, complemented
6#9543C211yesmaximal, characteristic, complemented
7#9535C211yesmaximal, characteristic, complemented
8#9533C211yesmaximal, characteristic, complemented
9#9071C211yesmaximal, characteristic
10#8844C211yesmaximal, characteristic, complemented
11#8831C211yesmaximal, characteristic
12C8×#107C211yesmaximal, characteristic, complemented
13C8×#103C211yesmaximal, characteristic, complemented
14C8×C22⋊C8C211yesmaximal, characteristic
15#7890C211yesmaximal, characteristic
16C2×#485C411yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C4×C22⋊Q8C2211yescharacteristic
21C4×#117C2211yescharacteristic
22#1984C2211yescharacteristic
23#1939C2211yescharacteristic
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1928C411yescharacteristic
27#1927C411yescharacteristic
28#1923—21no—
29#1923—21no—
30#1923—21no—
31#1923—21no—
32#1919C411yescharacteristic
33#1918C411yescharacteristic
34#1916—21no—
35#1873C411yescharacteristic
36#1854C411yescharacteristic
37#1831—21no—
38#1825C411yescharacteristic
39#1797C2211yescharacteristic
40C4×#107C2211yescharacteristic
41C4×#106C2211yescharacteristic
42C8×C4⋊C4—21no—
43C8×C4⋊C4C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C4×#103C2211yescharacteristic
47C4×#103C2211yescharacteristic
48C4×C22⋊C8C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50#1744—21no—
51C2×#335C2211yescharacteristic
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
67C23×C8—21no—
68C2×C22⋊Q8C2×C411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×#117C2×C411yescharacteristic
71C2×#117C2×C411yescharacteristic
72C22×C42C2311yescharacteristic
73#485C818yescomplemented
74C8×Q8—22no—
75C8×Q8—41no—
76#431—22no—
77#431—41no—
78C2×#107C2×C411yescharacteristic
79C2×#107C2×C411yescharacteristic
80C2×#106C2×C411yescharacteristic
81C2×#106C2×C411yescharacteristic
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#103C2×C411yescharacteristic
91C2×#103C2×C411yescharacteristic
92C2×#103C2×C411yescharacteristic
93C2×#103C2×C411yescharacteristic
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8C2×C411yescharacteristic
97C2×C22⋊C8C2×C411yescharacteristic
98C2×C22⋊C8D411yescharacteristic
99C2×C22⋊C8D411yescharacteristic
100C2×C4×C8—22no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—41no—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#391C2×C411yescharacteristic
115#389C2×C411yescharacteristic
116#387—21no—
117#385—21no—
118#380C2×C411yescharacteristic
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2311yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128#376—21no—
129C43—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#335—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
161C22×C8—21no—
162C22×C8—24no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—42no—
169C22×C8—42no—
170C22×C8C22×C411yescharacteristic
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22⋊Q8—22no—
179C22⋊Q8C2×C814yes—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184#117—22no—
185#117—22no—
186#117—41no—
187#117C2×C814yes—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42C22×C411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42#4111yescharacteristic
217C2×C42C2411yescharacteristic
218#107—22no—
219#107C2×C814yescomplemented
220#106—41no—
221#106C2×C814yescomplemented
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—42no—
230C4⋊C8—42no—
231C4⋊C8—81no—
232C4⋊C8—81no—
233C4⋊C8#4112yes—
234#103—22no—
235#103—41no—
236#103C2×C814yescomplemented
237#103C2×C814yescomplemented
238C22⋊C8—22no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241C22⋊C8—42no—
242C22⋊C8—81no—
243C22⋊C8C2×C814yes—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
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—42no—
256C4×C8—42no—
257C4×C8—42no—
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×Q8C22×C812yes—
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—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4C4×D411yescharacteristic
301C22×C4C4×D411yescharacteristic
302C22×C4C4×D411yescharacteristic
303C22×C4C22×C812yes—
304C22×C4#13112yes—
305C22×C4C23×C411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—22no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—22no—
324C2×C8—21no—
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—48no—
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×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8C22×C812yes—
348C2×C8C22×C812yes—
349C2×C8#13512yes—
350C2×C8#13512yes—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—24no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—42no—
365C4⋊C4C22×C812yes—
366C4⋊C4C22×C812yes—
367C4⋊C4C22×C812yes—
368C4⋊C4#13112yes—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—42no—
372C22⋊C4—81no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—24no—
380C42—42no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—42no—
390C42—81no—
391C42C2×#4111yescharacteristic
392C42C2×#4111yescharacteristic
393C42H(5,𝔽2)11yescharacteristic
394C422−1+411yescharacteristic
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23C8×D412yes—
402C23C2×C4×D411yescharacteristic
403Q8—22no—
404Q8—22no—
405Q8—41no—
406Q8—41no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—42no—
426C2×C4—42no—
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×C4C8×D412yes—
451C2×C4#44312yes—
452C2×C4C4×#4111yescharacteristic
453C2×C4#52011yescharacteristic
454C2×C4#54611yescharacteristic
455C2×C4#54711yescharacteristic
456C2×C4#55111yescharacteristic
457C2×C4#55211yescharacteristic
458C2×C4C23×C811yescharacteristic
459C2×C4C2×#13111yescharacteristic
460C2×C4C2×#13111yescharacteristic
461C2×C4#56811yescharacteristic
462C2×C4C2×#13511yescharacteristic
463C2×C4C2×#13511yescharacteristic
464C2×C4#57711yescharacteristic
465C2×C4#57811yescharacteristic
466C8—24no—
467C8—22no—
468C8—22no—
469C8—22no—
470C8—22no—
471C8—42no—
472C8—41no—
473C8—41no—
474C8—41no—
475C8—41no—
476C8—42no—
477C8—81no—
478C8—81no—
479C8—81no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22#235911yescharacteristic
488C22C2×C8×D411yescharacteristic
489C22#293311yescharacteristic
490C22C4×#13511yescharacteristic
491C22#294711yescharacteristic
492C22C2×#44311yescharacteristic
493C22#295611yescharacteristic
494C4—22no—
495C4—21no—
496C4—21no—
497C4—22no—
498C4—42no—
499C4—42no—
500C4—41no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—81no—
505C4—81no—
506C4—81no—
507C4C8×#4111yescharacteristic
508C4#296811yescharacteristic
509C4#298811yescharacteristic
510C4#299411yescharacteristic
511C4#330011yescharacteristic
512C4#330311yescharacteristic
513C4#333611yescharacteristic
514C4#334011yescharacteristic
515C2—42no—
516C2C8×#13511yescharacteristic
517C2#1785311yescharacteristic
518C2#1799611yescharacteristic
519C2#1801911yescharacteristic
520C2#2290011yescharacteristic
521C2#2291711yescharacteristic
522C2#2294811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C2×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. #9543C2semidirect
7. #9535C2semidirect
8. #9533C2semidirect
10. #8844C2semidirect
12. C8×#107C2semidirect
13. C8×#103C2semidirect
73. #485C8semidirect
219. #107C2×C8semidirect
221. #106C2×C8semidirect
236. #103C2×C8semidirect
237. #103C2×C8semidirect

Maximal subgroups

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

subgroupindex
#230502
#226322
C4×#4852
#95932
#95532
#95432
#95352
#95332
#90712
#88442
#88312
C8×#1072
C8×#1032
C8×C22⋊C82
#78902