Finite Group Databasebeta

← #6344255

Subgroups — #6344255

Order 512.

2870 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
1C2×#3503C211yesmaximal, characteristic
2#60282C211yesmaximal, characteristic
3#60282C211yesmaximal, characteristic
4#17774C2116yesmaximal, complemented
5#17755C214yesmaximal, complemented
6#17742C212yesmaximal, complemented
7#17739C212yesmaximal, complemented
8C2×#1556C214yesmaximal, complemented
9#3503—24no—
10#3503—24no—
11#3503C418yes—
12#3503C2218yes—
13C2×Q82—21no—
14C2×#557—21no—
15C2×#554—21no—
16C2×#554—22no—
17C2×D4×Q8—21no—
18C2×D4×Q8—22no—
19C2×#544C414yes—
20C2×#544C412yes—
21C2×#533C2211yescharacteristic
22C2×#533C2212yes—
23C22×C4⋊Q8C2212yes—
24C22×C4⋊Q8C2211yescharacteristic
25C2×#520C412yes—
26C2×C4×#41C2211yescharacteristic
27#2990C2218yes—
28#2990C2218yes—
29#2983—21no—
30#2983—21no—
31#2973—22no—
32#2973—22no—
33#2972—21no—
34#2972—21no—
35#2970—24no—
36#2970—24no—
37#2920C2212yes—
38#2920C2212yes—
39C2×#432C2214yes—
40C2×#432C2214yes—
41#1561C2218yescomplemented
42#1556C2214yes—
43#1556C2218yes—
44#1556C2214yes—
45#1556C22116yescomplemented
46#1552C22116yescomplemented
47#1552C22116yescomplemented
48#1549C2218yescomplemented
49#1546C2218yescomplemented
50#1544C2218yescomplemented
51C2×#332C22116yescomplemented
52C22×#41C2×C411yescharacteristic
53C23×Q8—22no—
54C23×Q8—21no—
55Q82—28no—
56Q82—28no—
57#557—28no—
58#557—28no—
59#554—216no—
60#554—28no—
61#554—28no—
62#554—216no—
63D4×Q8—216no—
64D4×Q8—28no—
65D4×Q8—28no—
66D4×Q8—216no—
67#544—216no—
68#544—28no—
69#544C2×C4132yes—
70#544C2×C4116yes—
71#533—24no—
72#533—24no—
73#533—22no—
74#533—22no—
75#533C2×C414yes—
76#533C2×C418yes—
77#533C2314yes—
78#533C2318yes—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—22no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—22no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8C2×C414yes—
90C2×C4⋊Q8C2×C412yes—
91C2×C4⋊Q8C2×C418yes—
92C2×C4⋊Q8C2×C414yes—
93C2×C4⋊Q8C2318yes—
94C2×C4⋊Q8C2314yes—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2312yes—
97C2×C4⋊Q8C2311yescharacteristic
98C2×#125—24no—
99C2×#125—22no—
100C2×#125C2×C412yes—
101C2×#125C2×C414yes—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—24no—
109#520—28no—
110#520C2×C4116yes—
111C4×#41C2×C414yes—
112C4×#41D414yes—
113C4×#41D414yes—
114C4×#41C2314yes—
115C2×C4×Q8—22no—
116C2×C4×Q8—21no—
117C2×C4×Q8—22no—
118C2×C4×Q8—21no—
119C2×C4×Q8—22no—
120C2×C4×Q8D411yescharacteristic
121C2×C4×Q8D411yescharacteristic
122C2×C4×D4—24no—
123C2×C4×D4—22no—
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D412yes—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4D412yes—
128C2×#117C2×C412yes—
129C2×#117C2×C414yes—
130C2×#117C2×C411yescharacteristic
131C2×#117C2×C412yes—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2×C414yes—
135C22×C4⋊C4C2×C412yes—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138#446—28no—
139#446—28no—
140#435—216no—
141#435—216no—
142C8×D4—28no—
143C8×D4—28no—
144#432—24no—
145#432—24no—
146#432—24no—
147#432—24no—
148#432—22no—
149#432—22no—
150#432C2318yes—
151#432C2318yes—
152#432C2314yes—
153#432C2314yes—
154#431—22no—
155#431—22no—
156#431C2314yes—
157#431C2314yes—
158#423C2314yes—
159#423C2314yes—
160C2×C4⋊C8C2318yes—
161C2×C4⋊C8C2318yes—
162#407—22no—
163#407—22no—
164C2×C22⋊C8—24no—
165C2×C22⋊C8—24no—
166C4×M16—22no—
167C4×M16—22no—
168C2×#97—24no—
169C2×#97—24no—
170#332C23116yes—
171#332C23132yes—
172#332C23116yes—
173C2×#41—28no—
174C2×#41C22⋊C414yes—
175C2×#41C22×C414yes—
176C22×Q8—22no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—28no—
180C22×Q8—21no—
181C22×Q8—21no—
182C22×Q8—22no—
183C22×Q8—28no—
184C22×Q8—24no—
185C22×Q8—88no—
186C22×Q8—84no—
187C22×Q8C22⋊C411yescharacteristic
188C22×D4C22⋊C412yes—
189C22×D4C22⋊C411yescharacteristic
190C23×C4—22no—
191C23×C4—24no—
192C23×C4C22×C412yes—
193C23×C4C22×C411yescharacteristic
194C23×C4C2×D412yes—
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D412yes—
197C23×C4C2×D411yescharacteristic
198C2×M16—24no—
199C2×M16—24no—
200C22×C8—28no—
201C22×C8—28no—
202C4⋊Q8—232no—
203C4⋊Q8—216no—
204C4⋊Q8—28no—
205C4⋊Q8—24no—
206C4⋊Q8—28no—
207C4⋊Q8—24no—
208C4⋊Q8—28no—
209C4⋊Q8—24no—
210C4⋊Q8—21no—
211C4⋊Q8—22no—
212C4⋊Q8—21no—
213C4⋊Q8—22no—
214C4⋊Q8—24no—
215C4⋊Q8—28no—
216C4⋊Q8—24no—
217C4⋊Q8—28no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8C22×C412yes—
221C4⋊Q8C22×C414yes—
222C4⋊Q8C22×C4116yes—
223C4⋊Q8C22×C418yes—
224C4⋊Q8C22×C412yes—
225C4⋊Q8C2412yes—
226C4⋊Q8C2414yes—
227C4⋊Q8C2412yes—
228#125—232no—
229#125—216no—
230#125—28no—
231#125—24no—
232#125C22×C418yes—
233#125C22×C4116yes—
234C22⋊Q8—24no—
235C22⋊Q8—28no—
236C22⋊Q8—24no—
237C22⋊Q8—28no—
238C22⋊Q8—28no—
239C22⋊Q8—24no—
240C22⋊Q8—28no—
241C22⋊Q8—24no—
242C22⋊Q8—232no—
243C22⋊Q8—232no—
244C22⋊Q8—232no—
245C4×Q8—216no—
246C4×Q8—28no—
247C4×Q8—24no—
248C4×Q8—28no—
249C4×Q8—24no—
250C4×Q8—24no—
251C4×Q8—28no—
252C4×Q8—24no—
253C4×Q8—28no—
254C4×Q8C22⋊C414yes—
255C4×Q8C22⋊C414yes—
256C4×Q8C2×D414yes—
257C4×Q8C2×D414yes—
258C4×D4—232no—
259C4×D4—216no—
260C4×D4C22⋊C418yes—
261C4×D4C22⋊C414yes—
262C4×D4C22⋊C418yes—
263C4×D4C22⋊C414yes—
264C4×D4C2×D414yes—
265C4×D4C2×D418yes—
266C4×D4C2×D414yes—
267C4×D4C2×D418yes—
268#117—28no—
269#117—24no—
270#117C22⋊C414yes—
271#117C22⋊C418yes—
272#117C22×C418yes—
273#117C22×C4116yes—
274#117C22×C414yes—
275#117C22×C418yes—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4C22⋊C412yes—
294C2×C4⋊C4C22⋊C411yescharacteristic
295C2×C4⋊C4C22⋊C412yes—
296C2×C4⋊C4C22⋊C411yescharacteristic
297C2×C4⋊C4C22×C412yes—
298C2×C4⋊C4C22×C414yes—
299C2×C4⋊C4C22×C4116yes—
300C2×C4⋊C4C22×C418yes—
301C2×C4⋊C4C22×C412yes—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4C22⋊C412yes—
305C2×C22⋊C4C22⋊C412yes—
306C2×C22⋊C4C22⋊C412yes—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42C22×C412yes—
310C2×C42C22×C411yescharacteristic
311C2×C42C22×C414yes—
312C2×C42C22×C412yes—
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2×D414yes—
318C2×C42C2×D414yes—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2414yes—
322C2×C42C2412yes—
323C2×C42C2411yescharacteristic
324C4⋊C8—28no—
325C4⋊C8—28no—
326C4⋊C8—24no—
327C4⋊C8—24no—
328C4⋊C8C2418yes—
329C4⋊C8C2418yes—
330C22⋊C8—24no—
331C22⋊C8—24no—
332C22⋊C8—28no—
333C22⋊C8—28no—
334C22⋊C8—24no—
335C22⋊C8—24no—
336#97—24no—
337#97—24no—
338#97—28no—
339#97—28no—
340C4×C8—24no—
341C4×C8—24no—
342C24C2×C22⋊C411yescharacteristic
343C24C2×C22⋊C412yes—
344#41—216no—
345#41—216no—
346C2×Q8—22no—
347C2×Q8—24no—
348C2×Q8—22no—
349C2×Q8—28no—
350C2×Q8—216no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—48no—
358C2×Q8—816no—
359C2×Q8—88no—
360C2×Q8—88no—
361C2×Q8—84no—
362C2×Q8—816no—
363C2×Q8—88no—
364C2×Q8C2×C22⋊C414yes—
365C2×D4—28no—
366C2×D4#10118yes—
367C2×D4C2×C22⋊C418yes—
368C2×D4C2×C22⋊C414yes—
369C22×C4—28no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—28no—
375C22×C4—216no—
376C22×C4—44no—
377C22×C4—42no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—88no—
381C22×C4—84no—
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C22⋊C414yes—
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C412yes—
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C412yes—
389C22×C4C2×C22⋊C412yes—
390C22×C4C2×C22⋊C414yes—
391C22×C4C2×C22⋊C412yes—
392C22×C4C2×C22⋊C414yes—
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C23×C411yescharacteristic
395C22×C4C23×C414yes—
396C22×C4C23×C412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D414yes—
400C22×C4C22×D412yes—
401C22×C4C22×D414yes—
402C22×C4C22×D412yes—
403M16—84no—
404M16—84no—
405C2×C8—28no—
406C2×C8—28no—
407C2×C8—88no—
408C2×C8—88no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—28no—
414C4⋊C4—216no—
415C4⋊C4—216no—
416C4⋊C4—28no—
417C4⋊C4—22no—
418C4⋊C4—28no—
419C4⋊C4—216no—
420C4⋊C4—28no—
421C4⋊C4—216no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4C2×C22⋊C418yes—
427C4⋊C4C2×C22⋊C414yes—
428C4⋊C4C2×C22⋊C418yes—
429C4⋊C4C2×C22⋊C414yes—
430C4⋊C4C23×C414yes—
431C4⋊C4C23×C418yes—
432C4⋊C4C23×C414yes—
433C22⋊C4—28no—
434C22⋊C4—216no—
435C22⋊C4C2×C22⋊C418yes—
436C22⋊C4C2×C22⋊C418yes—
437C22⋊C4C2×C22⋊C418yes—
438C42—216no—
439C42—28no—
440C42C2×C22⋊C414yes—
441C42C2×C22⋊C412yes—
442C42C2×C22⋊C414yes—
443C42C2×C22⋊C414yes—
444C42C2×C22⋊C412yes—
445C42C2×C22⋊C412yes—
446C42C23×C411yescharacteristic
447C42C23×C414yes—
448C42C23×C412yes—
449C42C22×D412yes—
450C42C22×D412yes—
451C42C22×D411yescharacteristic
452C42C22×D411yescharacteristic
453C42C22×D414yes—
454C42C22×D414yes—
455C42C2511yescharacteristic
456C23—22no—
457C23—42no—
458C23—41no—
459C23—42no—
460C23—41no—
461C23C2×#10118yes—
462C23C22×C22⋊C412yes—
463C23C22×C22⋊C414yes—
464C23C22×C22⋊C411yescharacteristic
465Q8—28no—
466Q8—44no—
467Q8—44no—
468Q8—48no—
469Q8—44no—
470Q8—48no—
471Q8—44no—
472Q8—816no—
473Q8—88no—
474D4—216no—
475D4C2×#101116yes—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—28no—
480C2×C4—28no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—28no—
490C2×C4—24no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—44no—
500C2×C4—48no—
501C2×C4—44no—
502C2×C4—816no—
503C2×C4—88no—
504C2×C4C2×#10114yes—
505C2×C4C22×C22⋊C414yes—
506C2×C4C22×C22⋊C411yescharacteristic
507C2×C4C22×C22⋊C411yescharacteristic
508C2×C4C22×C22⋊C411yescharacteristic
509C2×C4C22×C22⋊C412yes—
510C2×C4C22×C22⋊C414yes—
511C2×C4C22×C22⋊C412yes—
512C2×C4C22×C22⋊C414yes—
513C2×C4C22×C22⋊C412yes—
514C2×C4C24×C411yescharacteristic
515C2×C4C23×D411yescharacteristic
516C2×C4C23×D411yescharacteristic
517C8—48no—
518C8—48no—
519C22—21no—
520C22—24no—
521C22—21no—
522C22—24no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—48no—
527C22—44no—
528C22C22×#10112yes—
529C22C22×#10118yes—
530C22C23×C22⋊C411yescharacteristic
531C4—21no—
532C4—21no—
533C4—24no—
534C4—21no—
535C4—24no—
536C4—22no—
537C4—21no—
538C4—24no—
539C4—22no—
540C4—44no—
541C4—48no—
542C4—44no—
543C4C22×#10114yes—
544C4#358512yes—
545C4#358612yes—
546C2—22no—
547C2—28no—
548C2—42no—
549C2C23×#10111yescharacteristic
550C2#6008711yescharacteristic
551C2#6008811yescharacteristic
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
prime radicalC22×#41C2×C4
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. #17774C2semidirect
5. #17755C2semidirect
6. #17742C2semidirect
7. #17739C2semidirect
8. C2×#1556C2semidirect
41. #1561C22semidirect
45. #1556C22semidirect
46. #1552C22semidirect
47. #1552C22semidirect
48. #1549C22semidirect
49. #1546C22semidirect
50. #1544C22semidirect
51. C2×#332C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35032
#602822
#177742
#177552
#177422
#177392
C2×#15562