Finite Group Databasebeta

← #6344081

Subgroups — #6344081

Order 512.

2888 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3503C211yesmaximal, characteristic
2#60277C211yesmaximal, characteristic
3#60125C211yesmaximal, characteristic
4#17761C2116yesmaximal, complemented
5#17747C214yesmaximal, complemented
6#17742C214yesmaximal, complemented
7#17708C212yesmaximal, complemented
8C2×#1549C212yesmaximal, complemented
9#3503—24no—
10#3503—24no—
11#3503C418yes—
12#3503C2218yes—
13C2×Q82C412yes—
14C2×#557—21no—
15C2×#554C414yes—
16C2×#554C412yes—
17C2×D4×Q8—22no—
18C2×D4×Q8—21no—
19C2×#544—21no—
20C2×#544—22no—
21C2×#533C2211yescharacteristic
22C2×#533C2212yes—
23C22×C4⋊Q8C2211yescharacteristic
24C22×C4⋊Q8C2212yes—
25C2×#520—21no—
26C2×C4×#41C2211yescharacteristic
27#2989C2218yes—
28#2983—21no—
29#2975—21no—
30#2973—22no—
31#2965—24no—
32#2925—24no—
33#2925C2218yes—
34#2921—22no—
35#2920C2214yes—
36#2920C2214yes—
37C2×#431C2212yes—
38C2×#423C2212yes—
39#2879—21no—
40#2873—21no—
41#1559C22116yescomplemented
42#1556C22116yescomplemented
43#1554C22116yescomplemented
44#1550C2218yescomplemented
45#1549C2218yes—
46#1549C2214yes—
47#1549C2218yescomplemented
48#1548C2214yes—
49#1546C22116yescomplemented
50#1542C2218yescomplemented
51C2×#332C2218yescomplemented
52C22×#41D411yescharacteristic
53C23×Q8—21no—
54C23×Q8—22no—
55C22×M16—22no—
56Q82—28no—
57Q82C2×C4116yes—
58#557—28no—
59#557—28no—
60#554—28no—
61#554—216no—
62#554C2×C4132yes—
63#554C2×C4116yes—
64D4×Q8—216no—
65D4×Q8—28no—
66D4×Q8—216no—
67D4×Q8—28no—
68#544—216no—
69#544—28no—
70#544—28no—
71#544—216no—
72#533—22no—
73#533—24no—
74#533—22no—
75#533—24no—
76#533C2×C414yes—
77#533C2×C418yes—
78#533C2314yes—
79#533C2318yes—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—22no—
82C2×C4⋊Q8—22no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8—22no—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8C2×C412yes—
91C2×C4⋊Q8C2×C414yes—
92C2×C4⋊Q8C2×C414yes—
93C2×C4⋊Q8C2×C418yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2314yes—
96C2×C4⋊Q8C2318yes—
97C2×C4⋊Q8C2312yes—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#125—24no—
100C2×#125—22no—
101C2×#125—22no—
102C2×#125—21no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8C2×C412yes—
109C2×C22⋊Q8C2×C414yes—
110#520—28no—
111#520—28no—
112C4×#41C2×C414yes—
113C4×#41D414yes—
114C4×#41D414yes—
115C4×#41C2314yes—
116C2×C4×Q8—22no—
117C2×C4×Q8—21no—
118C2×C4×Q8—22no—
119C2×C4×Q8C2×C412yes—
120C2×C4×Q8C2×C414yes—
121C2×C4×Q8C2×C411yescharacteristic
122C2×C4×Q8D411yescharacteristic
123C2×C4×D4—24no—
124C2×C4×D4—22no—
125C2×C4×D4C2×C412yes—
126C2×C4×D4C2×C411yescharacteristic
127C2×C4×D4D412yes—
128C2×C4×D4D411yescharacteristic
129C2×#117—22no—
130C2×#117—21no—
131C2×#117D411yescharacteristic
132C2×#117D412yes—
133C22×C4⋊C4—22no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4C2×C412yes—
136C22×C4⋊C4C2×C414yes—
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139#446—28no—
140#436—28no—
141#435—216no—
142#433—216no—
143#432—24no—
144#432—24no—
145#432C2318yes—
146#432C2318yes—
147#431—22no—
148#431—22no—
149#431—24no—
150#431—24no—
151#431C2314yes—
152#431C2318yes—
153#431C2314yes—
154#423—22no—
155#423—22no—
156#423C2318yes—
157#423C2314yes—
158#423C2314yes—
159C2×C4⋊C8—22no—
160C2×C4⋊C8C2314yes—
161C2×C4⋊C8C2314yes—
162#408—28no—
163#407—24no—
164#407—24no—
165C2×C22⋊C8—22no—
166#405—28no—
167C4×M16—24no—
168C4×M16—24no—
169C2×C4×C8—22no—
170#333C23116yes—
171#332C23132yes—
172#332C23116yes—
173C2×#41—28no—
174C2×#41C22⋊C414yes—
175C2×#41C2×D414yes—
176C22×Q8—24no—
177C22×Q8—22no—
178C22×Q8—28no—
179C22×Q8—24no—
180C22×Q8—21no—
181C22×Q8—22no—
182C22×Q8—24no—
183C22×Q8—28no—
184C22×Q8—84no—
185C22×Q8—88no—
186C22×Q8C22⋊C411yescharacteristic
187C22×Q8C22×C412yes—
188C22×D4C22⋊C412yes—
189C22×D4C22⋊C411yescharacteristic
190C23×C4—24no—
191C23×C4—22no—
192C23×C4C22×C412yes—
193C23×C4C22×C411yescharacteristic
194C23×C4C2×D412yes—
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D412yes—
197C23×C4C2×D411yescharacteristic
198C2×M16—24no—
199C2×M16—24no—
200C2×M16—28no—
201C2×M16—48no—
202C22×C8—24no—
203C22×C8—24no—
204C4⋊Q8—232no—
205C4⋊Q8—216no—
206C4⋊Q8—28no—
207C4⋊Q8—24no—
208C4⋊Q8—28no—
209C4⋊Q8—24no—
210C4⋊Q8—28no—
211C4⋊Q8—24no—
212C4⋊Q8—22no—
213C4⋊Q8—21no—
214C4⋊Q8—22no—
215C4⋊Q8—21no—
216C4⋊Q8—28no—
217C4⋊Q8—24no—
218C4⋊Q8—24no—
219C4⋊Q8—28no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8C22×C412yes—
223C4⋊Q8C22×C414yes—
224C4⋊Q8C22×C412yes—
225C4⋊Q8C22×C418yes—
226C4⋊Q8C22×C4116yes—
227C4⋊Q8C2412yes—
228C4⋊Q8C2414yes—
229C4⋊Q8C2412yes—
230#125—232no—
231#125—216no—
232#125—28no—
233#125—24no—
234#125—28no—
235#125—24no—
236C22⋊Q8—28no—
237C22⋊Q8—24no—
238C22⋊Q8—28no—
239C22⋊Q8—24no—
240C22⋊Q8—24no—
241C22⋊Q8—28no—
242C22⋊Q8—232no—
243C22⋊Q8—232no—
244C22⋊Q8—232no—
245C22⋊Q8C22×C418yes—
246C22⋊Q8C22×C4116yes—
247C4×Q8—216no—
248C4×Q8—28no—
249C4×Q8—24no—
250C4×Q8—28no—
251C4×Q8—24no—
252C4×Q8—24no—
253C4×Q8—28no—
254C4×Q8C22⋊C414yes—
255C4×Q8C22⋊C414yes—
256C4×Q8C22×C418yes—
257C4×Q8C22×C4116yes—
258C4×Q8C22×C414yes—
259C4×Q8C2×D414yes—
260C4×D4—232no—
261C4×D4—216no—
262C4×D4C22⋊C418yes—
263C4×D4C22⋊C414yes—
264C4×D4C22⋊C418yes—
265C4×D4C22⋊C414yes—
266C4×D4C22×C418yes—
267C4×D4C22×C414yes—
268C4×D4C2×D418yes—
269C4×D4C2×D414yes—
270#117—28no—
271#117—24no—
272#117—28no—
273#117—24no—
274#117C22⋊C418yes—
275#117C22⋊C414yes—
276#117C2×D414yes—
277#117C2×D418yes—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4C22⋊C412yes—
297C2×C4⋊C4C22⋊C411yescharacteristic
298C2×C4⋊C4C22⋊C412yes—
299C2×C4⋊C4C22⋊C411yescharacteristic
300C2×C4⋊C4C22×C414yes—
301C2×C4⋊C4C22×C412yes—
302C2×C4⋊C4C22×C418yes—
303C2×C4⋊C4C22×C4116yes—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4C22⋊C412yes—
307C2×C22⋊C4C22⋊C412yes—
308C2×C22⋊C4C22⋊C412yes—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42C22×C412yes—
312C2×C42C22×C411yescharacteristic
313C2×C42C22×C414yes—
314C2×C42C22×C412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D414yes—
321C2×C42C2×D412yes—
322C2×C42C2×D414yes—
323C2×C42C2412yes—
324C2×C42C2414yes—
325C2×C42C2411yescharacteristic
326C4⋊C8—24no—
327C4⋊C8—24no—
328C4⋊C8—24no—
329C4⋊C8—24no—
330C4⋊C8—28no—
331C4⋊C8C2418yes—
332C4⋊C8C2418yes—
333C22⋊C8—24no—
334C22⋊C8—24no—
335C22⋊C8—28no—
336C22⋊C8—28no—
337C22⋊C8—28no—
338#97—28no—
339#97—28no—
340C4×C8—24no—
341C4×C8—24no—
342C4×C8—28no—
343C24C2×C22⋊C412yes—
344C24C2×C22⋊C411yescharacteristic
345#41—28no—
346#41—28no—
347#41—216no—
348C2×Q8—22no—
349C2×Q8—24no—
350C2×Q8—22no—
351C2×Q8—216no—
352C2×Q8—28no—
353C2×Q8—22no—
354C2×Q8—24no—
355C2×Q8—24no—
356C2×Q8—44no—
357C2×Q8—48no—
358C2×Q8—44no—
359C2×Q8—88no—
360C2×Q8—816no—
361C2×Q8—84no—
362C2×Q8—88no—
363C2×Q8—88no—
364C2×Q8—816no—
365C2×Q8C2×C22⋊C414yes—
366C2×Q8C23×C414yes—
367C2×D4—28no—
368C2×D4—24no—
369C2×D4C2×C22⋊C418yes—
370C2×D4C2×C22⋊C414yes—
371C22×C4—28no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—216no—
376C22×C4—28no—
377C22×C4—42no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—44no—
381C22×C4—84no—
382C22×C4—88no—
383C22×C4C2×C22⋊C412yes—
384C22×C4C2×C22⋊C414yes—
385C22×C4C2×C22⋊C414yes—
386C22×C4C2×C22⋊C412yes—
387C22×C4C2×C22⋊C412yes—
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C414yes—
390C22×C4C2×C22⋊C412yes—
391C22×C4C2×C22⋊C412yes—
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C412yes—
395C22×C4#13718yes—
396C22×C4C23×C411yescharacteristic
397C22×C4C23×C414yes—
398C22×C4C23×C412yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D414yes—
401C22×C4C22×D412yes—
402C22×C4C22×D414yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D412yes—
405M16—48no—
406M16—48no—
407M16—88no—
408C2×C8—28no—
409C2×C8—28no—
410C2×C8—48no—
411C2×C8—84no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—216no—
417C4⋊C4—28no—
418C4⋊C4—216no—
419C4⋊C4—28no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—216no—
423C4⋊C4—28no—
424C4⋊C4—216no—
425C4⋊C4—28no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4C2×C22⋊C418yes—
431C4⋊C4C2×C22⋊C414yes—
432C4⋊C4C2×C22⋊C418yes—
433C4⋊C4C2×C22⋊C414yes—
434C4⋊C4C23×C418yes—
435C4⋊C4C23×C414yes—
436C22⋊C4—216no—
437C22⋊C4—28no—
438C22⋊C4C2×C22⋊C418yes—
439C22⋊C4C2×C22⋊C418yes—
440C22⋊C4C2×C22⋊C418yes—
441C42—216no—
442C42—28no—
443C42C2×C22⋊C414yes—
444C42C2×C22⋊C412yes—
445C42C2×C22⋊C414yes—
446C42C2×C22⋊C414yes—
447C42C2×C22⋊C412yes—
448C42C2×C22⋊C412yes—
449C42C23×C411yescharacteristic
450C42C23×C414yes—
451C42C23×C412yes—
452C42C22×D411yescharacteristic
453C42C22×D414yes—
454C42C22×D412yes—
455C42C22×D414yes—
456C42C22×D411yescharacteristic
457C42C22×D412yes—
458C42C2511yescharacteristic
459C23—24no—
460C23—41no—
461C23—42no—
462C23—41no—
463C23—42no—
464C23#41914yes—
465C23C22×C22⋊C414yes—
466C23C22×C22⋊C412yes—
467C23C22×C22⋊C411yescharacteristic
468Q8—28no—
469Q8—44no—
470Q8—48no—
471Q8—44no—
472Q8—48no—
473Q8—44no—
474Q8—44no—
475Q8—88no—
476Q8—816no—
477D4—216no—
478D4—28no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—28no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—24no—
489C2×C4—28no—
490C2×C4—24no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—48no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—88no—
503C2×C4—816no—
504C2×C4#41914yes—
505C2×C4#41918yes—
506C2×C4C22×C22⋊C414yes—
507C2×C4C22×C22⋊C411yescharacteristic
508C2×C4C22×C22⋊C414yes—
509C2×C4C22×C22⋊C412yes—
510C2×C4C22×C22⋊C414yes—
511C2×C4C22×C22⋊C412yes—
512C2×C4C22×C22⋊C411yescharacteristic
513C2×C4C22×C22⋊C412yes—
514C2×C4C22×C22⋊C411yescharacteristic
515C2×C4C2×#13714yes—
516C2×C4C2×#13714yes—
517C2×C4C2×#13714yes—
518C2×C4C24×C411yescharacteristic
519C2×C4C23×D411yescharacteristic
520C2×C4C23×D411yescharacteristic
521C8—48no—
522C8—48no—
523C22—21no—
524C22—22no—
525C22—21no—
526C22—22no—
527C22—24no—
528C22—24no—
529C22—24no—
530C22—44no—
531C22—48no—
532C22C2×#41912yes—
533C22C2×#41914yes—
534C22C23×C22⋊C411yescharacteristic
535C4—21no—
536C4—22no—
537C4—24no—
538C4—24no—
539C4—21no—
540C4—22no—
541C4—24no—
542C4—44no—
543C4—48no—
544C4—44no—
545C4#288912yes—
546C4C2×#41912yes—
547C4C2×#41914yes—
548C4C22×#13712yes—
549C4#358512yes—
550C2—22no—
551C2—24no—
552C2—44no—
553C2C2×#288911yescharacteristic
554C2C22×#41911yescharacteristic
555C2#6008711yescharacteristic
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. #17761C2semidirect
5. #17747C2semidirect
6. #17742C2semidirect
7. #17708C2semidirect
8. C2×#1549C2semidirect
41. #1559C22semidirect
42. #1556C22semidirect
43. #1554C22semidirect
44. #1550C22semidirect
47. #1549C22semidirect
49. #1546C22semidirect
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×#35032
#602772
#601252
#177612
#177472
#177422
#177082
C2×#15492