Finite Group Databasebeta

← #6343644

Subgroups — #6343644

Order 512.

2897 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3507C211yesmaximal, characteristic
2C2×#2982C211yesmaximal, characteristic, complemented
3#34187C211yesmaximal, characteristic, complemented
4#17616C2116yesmaximal, complemented
5#17575C212yesmaximal, complemented
6#17574C212yesmaximal
7#17567C218yesmaximal, complemented
8#3507—24no—
9#3507—24no—
10#3507C418yescomplemented
11#3507C2218yes—
12C2×#559C412yescomplemented
13C2×#557C412yescomplemented
14C2×#556—22no—
15C2×#556C412yescomplemented
16C2×#554C412yescomplemented
17C2×#552—22no—
18C2×#539—22no—
19C2×#536—21no—
20C2×#533C2211yescharacteristic
21C2×#532C2211yescharacteristic
22C2×#529C2214yes—
23C2×#520—21no—
24C22×C4×Q8C2211yescharacteristic
25#2982—22no—
26#2982—22no—
27#2982C2218yescomplemented
28C2×C8×Q8—24no—
29C2×#431C2214yes—
30C2×#431C2212yes—
31#2356C2218yescomplemented
32#2355—24no—
33#2313—22no—
34#2310—21no—
35#2285C2212yes—
36#2283C2214yes—
37#2274—21no—
38#1521C22132yescomplemented
39#1520C2218yes—
40#1519C2218yes—
41#1518C2218yescomplemented
42#1517C2218yes—
43#1503C2218yescomplemented
44#1502C2218yes—
45#1500C22132yescomplemented
46C22×#41—21no—
47C23×Q8D411yescharacteristic
48#559—28no—
49#559C2×C4116yes—
50#557—28no—
51#557C2×C4116yes—
52#556—216no—
53#556—28no—
54#556—216no—
55#556C2×C4116yes—
56#554—28no—
57#554C2×C4116yes—
58#552—216no—
59#552—216no—
60#539—216no—
61#539—216no—
62#536—28no—
63#536—28no—
64#533—22no—
65#533—22no—
66#533C2×C414yes—
67#533C2314yes—
68#532—22no—
69#532—22no—
70#532C2×C414yes—
71#532C2314yes—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8C2×C411yescharacteristic
74C2×C4⋊Q8C2×C411yescharacteristic
75C2×C4⋊Q8C2×C411yescharacteristic
76C2×#127C2×C411yescharacteristic
77#529—28no—
78#529—28no—
79#529C2×C4116yes—
80#529C23116yes—
81C2×#126—24no—
82C2×#126—24no—
83C2×#125—24no—
84C2×#125—21no—
85C2×#125C2×C414yes—
86C2×#124—22no—
87C2×#124—21no—
88C2×#124C2×C414yes—
89C2×#123—24no—
90C2×#123—24no—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8C2312yes—
96C2×#121—21no—
97C2×#121C2312yes—
98#520—28no—
99#520—28no—
100C2×C4×Q8—22no—
101C2×C4×Q8—21no—
102C2×C4×Q8C2×C414yes—
103C2×C4×Q8C2×C412yes—
104C2×C4×Q8C2×C412yes—
105C2×C4×Q8C2×C412yes—
106C2×C4×Q8C2×C414yes—
107C2×C4×Q8D414yes—
108C2×C4×Q8D412yes—
109C2×C4×Q8D412yes—
110C2×C4×Q8D414yes—
111C2×C4×Q8C2314yes—
112C2×C4×D4—24no—
113C2×C4×D4—22no—
114C2×C4×D4—21no—
115C2×C4×D4C2×C414yes—
116C2×C4×D4C2×C412yes—
117C2×#117—21no—
118C2×#117—22no—
119C2×#117—22no—
120C2×#117—24no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4D412yes—
123C22×C4⋊C4D411yescharacteristic
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C8×Q8—216no—
127C8×Q8—216no—
128#431—28no—
129#431—22no—
130#431—22no—
131#431C23116yes—
132#431C2314yes—
133#431C2314yes—
134C2×C4⋊C8—24no—
135C2×C4⋊C8—22no—
136C2×C22⋊C8C2314yes—
137C2×C4×C8—24no—
138C2×C4×C8—22no—
139#391—28no—
140#387—216no—
141#384—22no—
142#383C2318yes—
143#382—24no—
144#382C2314yes—
145#381—24no—
146#380—22no—
147#380C2314yes—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—22no—
151C4×C4⋊C4C2314yes—
152C4×C22⋊C4—24no—
153#376C2318yes—
154#328C23132yes—
155#327C23132yes—
156C2×#41—24no—
157C2×#41—24no—
158C2×#41—48no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—28no—
162C22×Q8C22⋊C412yes—
163C22×Q8C22⋊C412yes—
164C22×Q8C22⋊C414yes—
165C22×Q8C22×C412yes—
166C22×Q8C2×D414yes—
167C22×D4—22no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4C22×C412yes—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2411yescharacteristic
178C22×C8—24no—
179C4⋊Q8—216no—
180C4⋊Q8—22no—
181C4⋊Q8—22no—
182C4⋊Q8—22no—
183C4⋊Q8C22×C414yes—
184C4⋊Q8C22×C414yes—
185C4⋊Q8C22×C414yes—
186#127—22no—
187#127C22×C414yes—
188#126—232no—
189#126—216no—
190#126—216no—
191#125—232no—
192#125—24no—
193#125—28no—
194#125—24no—
195#125C22×C4116yes—
196#124—216no—
197#124—24no—
198#124—28no—
199#124—24no—
200#124C22×C4116yes—
201#123—216no—
202#123—216no—
203#123—232no—
204C22⋊Q8—216no—
205C22⋊Q8—216no—
206C22⋊Q8—24no—
207C22⋊Q8—24no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—232no—
211C22⋊Q8C22×C414yes—
212C22⋊Q8C2414yes—
213#121—24no—
214#121—24no—
215#121—22no—
216#121—22no—
217#121C22×C414yes—
218#121C2414yes—
219C4×Q8—216no—
220C4×Q8—24no—
221C4×Q8—28no—
222C4×Q8—24no—
223C4×Q8—24no—
224C4×Q8C22⋊C418yes—
225C4×Q8C22⋊C418yes—
226C4×Q8C22⋊C418yes—
227C4×Q8C22⋊C418yes—
228C4×Q8C22×C4116yes—
229C4×Q8C22×C418yes—
230C4×Q8C22×C418yes—
231C4×Q8C22×C418yes—
232C4×Q8C2×D418yes—
233C4×Q8C2×D418yes—
234C4×D4—216no—
235C4×D4—216no—
236C4×D4—216no—
237C4×D4—24no—
238C4×D4—28no—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4C22×C4116yes—
242C4×D4C22×C418yes—
243#117—24no—
244#117—24no—
245#117—216no—
246#117—216no—
247#117—216no—
248#117—48no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—216no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4C22⋊C412yes—
263C2×C4⋊C4C22⋊C412yes—
264C2×C4⋊C4C22⋊C414yes—
265C2×C4⋊C4C22⋊C414yes—
266C2×C4⋊C4C22⋊C414yes—
267C2×C4⋊C4C22⋊C418yes—
268C2×C4⋊C4C22×C412yes—
269C2×C4⋊C4C22×C412yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×D418yes—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—28no—
283C2×C42C22×C411yescharacteristic
284C2×C42C22×C414yes—
285C2×C42C22×C411yescharacteristic
286C2×C42C22×C414yes—
287C2×C42C22×C412yes—
288C2×C42C2×D414yes—
289C2×C42C2×D412yes—
290C2×C42C2×D414yes—
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D414yes—
294C2×C42C2×D412yes—
295C2×C42C2414yes—
296C2×C42C2412yes—
297C4⋊C8—216no—
298C4⋊C8—24no—
299C4⋊C8—24no—
300C22⋊C8—24no—
301C22⋊C8C2418yes—
302C4×C8—216no—
303C4×C8—24no—
304C4×C8—24no—
305#95—28no—
306#95—28no—
307#95—24no—
308#95—28no—
309#95C2418yes—
310C24—22no—
311C24C22×D411yescharacteristic
312#41—48no—
313#41—48no—
314#41—416no—
315C2×Q8—22no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—28no—
319C2×Q8—28no—
320C2×Q8—28no—
321C2×Q8—28no—
322C2×Q8—216no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8—82no—
326C2×Q8C2×C22⋊C418yes—
327C2×Q8C2×C22⋊C418yes—
328C2×Q8C23×C414yes—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—22no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—44no—
335C2×D4—48no—
336C2×D4—44no—
337C2×D4—82no—
338C2×D4C23×C414yes—
339C22×C4—28no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—28no—
344C22×C4—216no—
345C22×C4—28no—
346C22×C4—42no—
347C22×C4—48no—
348C22×C4—82no—
349C22×C4—81no—
350C22×C4—88no—
351C22×C4C2×C22⋊C414yes—
352C22×C4C2×C22⋊C414yes—
353C22×C4C2×C22⋊C414yes—
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C414yes—
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C412yes—
361C22×C4#13518yes—
362C22×C4C23×C411yescharacteristic
363C22×C4C23×C411yescharacteristic
364C22×C4C22×D414yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D414yes—
368C22×C4C22×D412yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D412yes—
371C22×C4C2511yescharacteristic
372C2×C8—28no—
373C2×C8—48no—
374C4⋊C4—24no—
375C4⋊C4—216no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—216no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—28no—
382C4⋊C4—216no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—48no—
387C4⋊C4—48no—
388C4⋊C4—84no—
389C4⋊C4C2×C22⋊C418yes—
390C4⋊C4C2×C22⋊C418yes—
391C4⋊C4C2×C22⋊C4116yes—
392C4⋊C4C2×C22⋊C4116yes—
393C4⋊C4C23×C414yes—
394C4⋊C4C23×C414yes—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—216no—
399C22⋊C4—216no—
400C22⋊C4—28no—
401C22⋊C4—216no—
402C22⋊C4—24no—
403C22⋊C4—48no—
404C42—216no—
405C42—28no—
406C42—48no—
407C42—88no—
408C42C2×C22⋊C418yes—
409C42C2×C22⋊C412yes—
410C42C2×C22⋊C412yes—
411C42C2×C22⋊C418yes—
412C42C2×C22⋊C412yes—
413C42C2×C22⋊C412yes—
414C42C23×C412yes—
415C42C23×C418yes—
416C42C23×C412yes—
417C42C22×D418yes—
418C42C22×D412yes—
419C42C22×D412yes—
420C23—24no—
421C23—41no—
422C23—41no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23C22×C22⋊C411yescharacteristic
427C23C22×C22⋊C411yescharacteristic
428C23C2×#13514yes—
429C23C23×D411yescharacteristic
430Q8—216no—
431Q8—216no—
432Q8—44no—
433Q8—44no—
434Q8—48no—
435Q8—84no—
436D4—44no—
437D4—48no—
438D4—48no—
439D4—44no—
440D4—48no—
441D4—84no—
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—416no—
465C2×C4—48no—
466C2×C4—48no—
467C2×C4—48no—
468C2×C4—816no—
469C2×C4#41614yes—
470C2×C4#41614yes—
471C2×C4#41618yes—
472C2×C4C22×C22⋊C418yes—
473C2×C4C22×C22⋊C411yescharacteristic
474C2×C4C22×C22⋊C418yes—
475C2×C4C22×C22⋊C412yes—
476C2×C4C22×C22⋊C412yes—
477C2×C4C22×C22⋊C411yescharacteristic
478C2×C4C22×C22⋊C412yes—
479C2×C4C22×C22⋊C412yes—
480C2×C4C2×#13514yes—
481C2×C4C2×#13514yes—
482C2×C4C24×C411yescharacteristic
483C2×C4C23×D411yescharacteristic
484C8—48no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—44no—
492C22—44no—
493C22—44no—
494C22—44no—
495C22—88no—
496C22C2×#41612yes—
497C22C2×#41612yes—
498C22C23×C22⋊C411yescharacteristic
499C22C22×#13512yes—
500C4—21no—
501C4—21no—
502C4—21no—
503C4—28no—
504C4—28no—
505C4—22no—
506C4—22no—
507C4—44no—
508C4—44no—
509C4—44no—
510C4—88no—
511C4#288412yes—
512C4C2×#41614yes—
513C4C2×#41614yes—
514C2—22no—
515C2—22no—
516C2—22no—
517C2—84no—
518C2C2×#288411yescharacteristic
519C2C22×#41611yescharacteristic
520C2#6009311yescharacteristic
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2982C2semidirect
3. #34187C2semidirect
4. #17616C2semidirect
5. #17575C2semidirect
7. #17567C2semidirect
10. #3507C4semidirect
12. C2×#559C4semidirect
13. C2×#557C4semidirect
15. C2×#556C4semidirect
16. C2×#554C4semidirect
27. #2982C22semidirect
31. #2356C22semidirect
38. #1521C22semidirect
41. #1518C22semidirect
43. #1503C22semidirect
45. #1500C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35072
C2×#29822
#341872
#176162
#175752
#175742
#175672