Finite Group Databasebeta

← #6344009

Subgroups — #6344009

Order 512.

2840 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3509C211yesmaximal, characteristic
2#60276C211yesmaximal, characteristic
3#60125C211yesmaximal, characteristic
4#17725C2116yesmaximal, complemented
5#17718C214yesmaximal, complemented
6#17715C214yesmaximal, complemented
7#17708C212yesmaximal, complemented
8C2×#1546C212yesmaximal, complemented
9#3509—24no—
10#3509—24no—
11#3509C418yes—
12#3509C2218yes—
13C2×#557—21no—
14C2×#557C412yes—
15C2×#556—21no—
16C2×#556C412yes—
17C2×#552—22no—
18C2×#552C414yes—
19C2×#541—22no—
20C2×#536—21no—
21C2×#533C2211yescharacteristic
22C2×#533C2212yes—
23C2×#533C2212yes—
24C22×C4⋊Q8C2211yescharacteristic
25C2×#519—21no—
26C2×C4×#41C2211yescharacteristic
27#2988C2218yes—
28#2982—21no—
29#2972—22no—
30#2972—21no—
31C8×#41—24no—
32#2925—24no—
33#2925C2218yes—
34#2921—22no—
35#2920C2214yes—
36C2×#431C2212yes—
37C2×#431C2214yes—
38C2×#423C2212yes—
39#2879—21no—
40#2873—21no—
41#1554C22116yescomplemented
42#1552C22116yescomplemented
43#1550C2218yescomplemented
44#1549C2214yes—
45#1548C2214yes—
46#1546C2218yes—
47#1546C2218yescomplemented
48#1544C22116yescomplemented
49#1542C2218yescomplemented
50#1542C22116yescomplemented
51C2×#332C2218yescomplemented
52C22×#41D411yescharacteristic
53C23×Q8—21no—
54C22×M16—22no—
55#557—28no—
56#557—28no—
57#557—28no—
58#557C2×C4116yes—
59#556—28no—
60#556—28no—
61#556—28no—
62#556C2×C4116yes—
63#552—216no—
64#552—216no—
65#552—216no—
66#552C2×C4132yes—
67#541—216no—
68#541—216no—
69#536—28no—
70#536—28no—
71#533—22no—
72#533—22no—
73#533—24no—
74#533—24no—
75#533—24no—
76#533—24no—
77#533C2×C414yes—
78#533C2×C418yes—
79#533C2×C418yes—
80#533C2314yes—
81#533C2318yes—
82#533C2318yes—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—22no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8C2×C412yes—
89C2×C4⋊Q8C2×C414yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2314yes—
92C2×C4⋊Q8C2312yes—
93C2×C4⋊Q8C2311yescharacteristic
94C2×#126—24no—
95C2×#126—24no—
96C2×#125—24no—
97C2×#125—24no—
98C2×#125—21no—
99C2×#125—22no—
100C2×#125C2×C414yes—
101C2×#124—22no—
102C2×#123—24no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8C2×C412yes—
108C2×C22⋊Q8C2×C414yes—
109#519—28no—
110#519—28no—
111C4×#41C2×C414yes—
112C4×#41D414yes—
113C4×#41D414yes—
114C4×#41C2314yes—
115C2×C4×Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8—22no—
118C2×C4×Q8—22no—
119C2×C4×Q8—22no—
120C2×C4×Q8—21no—
121C2×C4×Q8C2×C414yes—
122C2×C4×Q8C2×C412yes—
123C2×C4×Q8C2×C411yescharacteristic
124C2×C4×Q8D411yescharacteristic
125C2×C4×D4C2×C412yes—
126C2×C4×D4C2×C411yescharacteristic
127C2×C4×D4D412yes—
128C2×C4×D4D411yescharacteristic
129C2×#117—21no—
130C2×#117—22no—
131C2×#117—24no—
132C2×#117C2×C414yes—
133C2×#117D411yescharacteristic
134C2×#117D412yes—
135C22×C4⋊C4—21no—
136C22×C4⋊C4C2×C412yes—
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139C8×Q8—28no—
140C8×D4—216no—
141C8×D4—28no—
142#433—216no—
143#432—24no—
144#432C2318yes—
145#431—22no—
146#431—24no—
147#431—24no—
148#431—22no—
149#431—24no—
150#431C2318yes—
151#431C2314yes—
152#431C2318yes—
153#431C2314yes—
154#423—22no—
155#423—22no—
156#423C2314yes—
157#423C2314yes—
158C2×C4⋊C8—22no—
159C2×C4⋊C8C2314yes—
160C2×C4⋊C8C2314yes—
161C2×C4⋊C8C2318yes—
162#408—28no—
163#407—24no—
164C2×C22⋊C8—22no—
165C2×C22⋊C8—24no—
166#405—28no—
167C4×M16—24no—
168C2×C4×C8—22no—
169C2×C4×C8—24no—
170#333C23116yes—
171#332C23132yes—
172#332C23116yes—
173C2×#41—28no—
174C2×#41C22⋊C414yes—
175C2×#41C2×D414yes—
176C22×Q8—22no—
177C22×Q8—24no—
178C22×Q8—21no—
179C22×Q8—22no—
180C22×Q8—24no—
181C22×Q8—48no—
182C22×Q8C22⋊C411yescharacteristic
183C22×Q8C22×C412yes—
184C22×D4C22⋊C412yes—
185C22×D4C22⋊C411yescharacteristic
186C23×C4—22no—
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C412yes—
189C23×C4C2×D412yes—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D412yes—
192C23×C4C2×D411yescharacteristic
193C2×M16—24no—
194C2×M16—24no—
195C2×M16—48no—
196C22×C8—24no—
197C22×C8—24no—
198C22×C8—28no—
199C4⋊Q8—216no—
200C4⋊Q8—24no—
201C4⋊Q8—24no—
202C4⋊Q8—24no—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—24no—
206C4⋊Q8—24no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8C22×C412yes—
212C4⋊Q8C22×C414yes—
213C4⋊Q8C22×C418yes—
214C4⋊Q8C22×C412yes—
215C4⋊Q8C2412yes—
216C4⋊Q8C2414yes—
217C4⋊Q8C2412yes—
218#126—232no—
219#126—232no—
220#125—232no—
221#125—232no—
222#125—24no—
223#125—28no—
224#125—28no—
225#125—24no—
226#125—28no—
227#125C22×C4116yes—
228#124—216no—
229#123—232no—
230C22⋊Q8—28no—
231C22⋊Q8—28no—
232C22⋊Q8—24no—
233C22⋊Q8—28no—
234C22⋊Q8—24no—
235C22⋊Q8—28no—
236C22⋊Q8—28no—
237C22⋊Q8—28no—
238C22⋊Q8—24no—
239C22⋊Q8—28no—
240C22⋊Q8C22×C418yes—
241C22⋊Q8C22×C4116yes—
242C4×Q8—216no—
243C4×Q8—216no—
244C4×Q8—28no—
245C4×Q8—28no—
246C4×Q8—28no—
247C4×Q8—24no—
248C4×Q8—28no—
249C4×Q8—24no—
250C4×Q8—28no—
251C4×Q8—28no—
252C4×Q8—28no—
253C4×Q8—24no—
254C4×Q8C22⋊C414yes—
255C4×Q8C22⋊C414yes—
256C4×Q8C22×C4116yes—
257C4×Q8C22×C418yes—
258C4×Q8C22×C414yes—
259C4×Q8C2×D414yes—
260C4×D4C22⋊C418yes—
261C4×D4C22⋊C414yes—
262C4×D4C22⋊C418yes—
263C4×D4C22⋊C414yes—
264C4×D4C22×C418yes—
265C4×D4C22×C414yes—
266C4×D4C2×D418yes—
267C4×D4C2×D414yes—
268#117—24no—
269#117—28no—
270#117—28no—
271#117—24no—
272#117—28no—
273#117—232no—
274#117C22⋊C414yes—
275#117C22⋊C418yes—
276#117C22×C4116yes—
277#117C2×D414yes—
278#117C2×D418yes—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4C22⋊C411yescharacteristic
295C2×C4⋊C4C22⋊C412yes—
296C2×C4⋊C4C22⋊C411yescharacteristic
297C2×C4⋊C4C22⋊C412yes—
298C2×C4⋊C4C22×C414yes—
299C2×C4⋊C4C22×C418yes—
300C2×C4⋊C4C22×C412yes—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4C22⋊C412yes—
305C2×C22⋊C4C22⋊C412yes—
306C2×C22⋊C4C22⋊C412yes—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42C22×C412yes—
311C2×C42C22×C411yescharacteristic
312C2×C42C22×C412yes—
313C2×C42C22×C414yes—
314C2×C42C2×D412yes—
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D412yes—
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D414yes—
321C2×C42C2×D414yes—
322C2×C42C2412yes—
323C2×C42C2414yes—
324C2×C42C2411yescharacteristic
325C4⋊C8—24no—
326C4⋊C8—24no—
327C4⋊C8—24no—
328C4⋊C8—28no—
329C4⋊C8—24no—
330C4⋊C8C2418yes—
331C4⋊C8C2418yes—
332C22⋊C8—24no—
333C22⋊C8—24no—
334C22⋊C8—28no—
335C22⋊C8—28no—
336C22⋊C8—28no—
337#97—28no—
338C4×C8—24no—
339C4×C8—28no—
340C4×C8—24no—
341C4×C8—28no—
342C24C2×C22⋊C412yes—
343C24C2×C22⋊C411yescharacteristic
344#41—28no—
345#41—28no—
346#41—216no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—24no—
350C2×Q8—28no—
351C2×Q8—22no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—416no—
361C2×Q8C2×C22⋊C414yes—
362C2×Q8C23×C414yes—
363C2×D4—28no—
364C2×D4—24no—
365C2×D4C2×C22⋊C418yes—
366C2×D4C2×C22⋊C414yes—
367C22×C4—28no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—28no—
372C22×C4—42no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—48no—
377C22×C4C2×C22⋊C412yes—
378C22×C4C2×C22⋊C414yes—
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C22⋊C414yes—
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×C22⋊C414yes—
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C412yes—
389C22×C4#13518yes—
390C22×C4C23×C411yescharacteristic
391C22×C4C23×C412yes—
392C22×C4C23×C414yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D414yes—
395C22×C4C22×D412yes—
396C22×C4C22×D414yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D412yes—
399M16—48no—
400M16—48no—
401C2×C8—28no—
402C2×C8—28no—
403C2×C8—48no—
404C2×C8—84no—
405C2×C8—88no—
406C4⋊C4—24no—
407C4⋊C4—28no—
408C4⋊C4—24no—
409C4⋊C4—28no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—216no—
414C4⋊C4—216no—
415C4⋊C4—216no—
416C4⋊C4—28no—
417C4⋊C4—216no—
418C4⋊C4—28no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4C2×C22⋊C414yes—
425C4⋊C4C2×C22⋊C418yes—
426C4⋊C4C2×C22⋊C414yes—
427C4⋊C4C2×C22⋊C418yes—
428C4⋊C4C23×C418yes—
429C4⋊C4C23×C414yes—
430C22⋊C4—28no—
431C22⋊C4—216no—
432C22⋊C4—216no—
433C22⋊C4C2×C22⋊C418yes—
434C22⋊C4C2×C22⋊C418yes—
435C22⋊C4C2×C22⋊C418yes—
436C42—216no—
437C42—216no—
438C42—28no—
439C42C2×C22⋊C414yes—
440C42C2×C22⋊C412yes—
441C42C2×C22⋊C414yes—
442C42C2×C22⋊C414yes—
443C42C2×C22⋊C412yes—
444C42C2×C22⋊C412yes—
445C42C23×C411yescharacteristic
446C42C23×C414yes—
447C42C23×C412yes—
448C42C22×D411yescharacteristic
449C42C22×D414yes—
450C42C22×D414yes—
451C42C22×D412yes—
452C42C22×D411yescharacteristic
453C42C22×D412yes—
454C42C2511yescharacteristic
455C23—24no—
456C23—41no—
457C23—41no—
458C23—42no—
459C23—42no—
460C23#41614yes—
461C23C22×C22⋊C414yes—
462C23C22×C22⋊C412yes—
463C23C22×C22⋊C411yescharacteristic
464Q8—28no—
465Q8—44no—
466Q8—44no—
467Q8—44no—
468Q8—44no—
469Q8—48no—
470Q8—416no—
471Q8—48no—
472D4—216no—
473D4—28no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—28no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—48no—
497C2×C4—416no—
498C2×C4#41614yes—
499C2×C4#41618yes—
500C2×C4C22×C22⋊C414yes—
501C2×C4C22×C22⋊C414yes—
502C2×C4C22×C22⋊C411yescharacteristic
503C2×C4C22×C22⋊C412yes—
504C2×C4C22×C22⋊C414yes—
505C2×C4C22×C22⋊C412yes—
506C2×C4C22×C22⋊C411yescharacteristic
507C2×C4C22×C22⋊C412yes—
508C2×C4C22×C22⋊C411yescharacteristic
509C2×C4C2×#13514yes—
510C2×C4C2×#13514yes—
511C2×C4C2×#13514yes—
512C2×C4C24×C411yescharacteristic
513C2×C4C23×D411yescharacteristic
514C2×C4C23×D411yescharacteristic
515C8—48no—
516C8—48no—
517C22—21no—
518C22—22no—
519C22—21no—
520C22—22no—
521C22—24no—
522C22—24no—
523C22—24no—
524C22—44no—
525C22—84no—
526C22C2×#41612yes—
527C22C2×#41614yes—
528C22C23×C22⋊C411yescharacteristic
529C4—21no—
530C4—24no—
531C4—22no—
532C4—24no—
533C4—21no—
534C4—24no—
535C4—22no—
536C4—44no—
537C4—48no—
538C4—44no—
539C4#288912yes—
540C4C2×#41612yes—
541C4C2×#41614yes—
542C4C22×#13512yes—
543C4#358612yes—
544C2—22no—
545C2—24no—
546C2—44no—
547C2C2×#288911yescharacteristic
548C2C22×#41611yescharacteristic
549C2#6008811yescharacteristic
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
prime radicalC22×#41D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17725C2semidirect
5. #17718C2semidirect
6. #17715C2semidirect
7. #17708C2semidirect
8. C2×#1546C2semidirect
41. #1554C22semidirect
42. #1552C22semidirect
43. #1550C22semidirect
47. #1546C22semidirect
48. #1544C22semidirect
49. #1542C22semidirect
50. #1542C22semidirect
51. C2×#332C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35092
#602762
#601252
#177252
#177182
#177152
#177082
C2×#15462