Finite Group Databasebeta

← #6487772

Subgroups — #6487772

Order 512.

2210 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3255C211yesmaximal, characteristic, complemented
2C22×#423C211yesmaximal, characteristic
3#36625C212yesmaximal
4#32937C214yesmaximal
5#29783C214yesmaximal, complemented
6#24845C214yesmaximal, complemented
7#24838C214yesmaximal, complemented
8#24760C214yesmaximal, complemented
9#24752C214yesmaximal, complemented
10#21964C211yesmaximal, characteristic, complemented
11#21894C212yesmaximal, complemented
12C23×M16C2212yes—
13C2×#533C2212yes—
14C23×C42C2211yescharacteristic
15#3255—28no—
16C2×#475C2214yescomplemented
17C2×#470C2214yes—
18C2×#470C2214yes—
19C2×#467C2214yes—
20C2×#462C2214yescomplemented
21C2×#423—28no—
22C2×#423C2214yes—
23C2×#423C2212yes—
24C2×#423C2211yescharacteristic
25C2×#423C2211yescharacteristic
26C22×C4⋊C8C2214yes—
27#2857C2212yes—
28#2857C2212yes—
29#2857C2212yes—
30#2857C2212yes—
31#2855—22no—
32#2855C2214yes—
33#2851—22no—
34#2851—22no—
35#2848—22no—
36#2848C2214yes—
37C4×C22⋊Q8—22no—
38C4×C22⋊Q8—22no—
39C4×C22⋊Q8C2214yes—
40C4×C22⋊Q8C2214yes—
41#2296—22no—
42#2285C2212yes—
43#2280—22no—
44#2088C2218yescomplemented
45#2068C2218yescomplemented
46#2066C2218yescomplemented
47#2056C2218yescomplemented
48#1988C2214yescomplemented
49#1981C2214yescomplemented
50#1948C2214yescomplemented
51#1947C2214yescomplemented
52#1894C2218yes—
53#1894C2218yes—
54#1892C2218yes—
55#1809C2214yes—
56#1809C2214yes—
57#1807C2214yes—
58#1803C2214yescomplemented
59#1803C2214yescomplemented
60C24×C4D412yes—
61C24×C4C2311yescharacteristic
62C22×Q16C2314yes—
63C22×M16—28no—
64C22×M16—44no—
65C22×M16—44no—
66C22×M16C2314yes—
67C22×M16C2312yes—
68C22×M16C2312yes—
69C23×C8C2314yes—
70#533—28no—
71#533—28no—
72C2×C4⋊Q8C2312yes—
73C2×C4⋊Q8C2312yes—
74C2×#125—22no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8C2314yes—
79C2×C22⋊Q8C2314yes—
80C2×C22⋊Q8C2314yes—
81C2×C4×Q8—22no—
82C2×C4×Q8C2314yes—
83C2×#117—22no—
84C2×#117—22no—
85C22×C42—24no—
86C22×C42—24no—
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D412yes—
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2312yes—
94C22×C42C2312yes—
95C22×C42C2311yescharacteristic
96#475—216no—
97#470—216no—
98#470—48no—
99#467—48no—
100#462—216no—
101C2×#107C2314yes—
102C2×#106C2314yes—
103C2×#106C2314yes—
104#423—28no—
105#423—24no—
106#423—216no—
107#423D418yes—
108C2×C4⋊C8—216no—
109C2×C4⋊C8C2318yes—
110C2×C4⋊C8C2314yes—
111C2×C4⋊C8C2314yes—
112C2×#103C2318yes—
113C2×#103C2318yes—
114C2×#103C2318yes—
115#391—22no—
116#391—22no—
117#391C2314yes—
118#391C2314yes—
119#389—22no—
120#389—22no—
121#389—22no—
122#389—22no—
123#389—22no—
124#389—22no—
125#389C2314yes—
126#389C2314yes—
127#387—24no—
128#387—24no—
129#387—24no—
130#387—24no—
131#385—24no—
132#385—24no—
133#385—24no—
134#385—24no—
135#385—24no—
136#385—24no—
137#384C2312yes—
138#384C2312yes—
139#382—22no—
140#381—24no—
141#380—22no—
142#380—22no—
143#380—22no—
144#380—22no—
145#380—22no—
146#380—22no—
147#380—22no—
148#380C2314yes—
149#380C2314yes—
150#380C2314yes—
151#379—24no—
152C4×C4⋊C4—22no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4C2314yes—
156C4×C22⋊C4—24no—
157C4×C22⋊C4—24no—
158#340C2318yes—
159#340C2318yes—
160#340C2318yes—
161C25C2×D411yescharacteristic
162C22×Q8—22no—
163C22×Q8C2414yes—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—28no—
171C23×C4—28no—
172C23×C4—44no—
173C23×C4C2×D412yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4C2411yescharacteristic
183C23×C4C2411yescharacteristic
184C23×C4C2411yescharacteristic
185C2×Q16—28no—
186C2×Q16—44no—
187C2×Q16—44no—
188C2×M16—24no—
189C2×M16—28no—
190C2×M16—28no—
191C2×M16—42no—
192C2×M16—42no—
193C2×M16—44no—
194C2×M16—42no—
195C2×M16—42no—
196C2×M16—432no—
197C2×M16C2×D418yes—
198C22×C8—28no—
199C22×C8—28no—
200C22×C8—44no—
201C22×C8C2414yes—
202C22×C8C2414yes—
203C4⋊Q8—24no—
204C4⋊Q8—24no—
205C4⋊Q8—24no—
206C4⋊Q8—24no—
207#125—48no—
208C22⋊Q8—28no—
209C22⋊Q8—28no—
210C22⋊Q8—28no—
211C22⋊Q8—28no—
212C22⋊Q8—28no—
213C22⋊Q8—28no—
214C22⋊Q8—48no—
215C22⋊Q8—48no—
216C22⋊Q8—48no—
217C4×Q8—28no—
218C4×Q8—28no—
219C4×Q8—48no—
220#117—48no—
221#117—48no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4C2414yes—
236C2×C4⋊C4C2414yes—
237C2×C4⋊C4C2414yes—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C42—28no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D414yes—
266C2×C42#4114yes—
267C2×C42C2412yes—
268C2×C42C2412yes—
269#107—44no—
270#106—28no—
271#106—44no—
272C4⋊C8—216no—
273C4⋊C8—28no—
274C4⋊C8C2×D4116yes—
275#103—216no—
276#103—48no—
277#103—48no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—24no—
292C24—42no—
293C24C22×D411yescharacteristic
294C24C22×D411yescharacteristic
295C24C22×D411yescharacteristic
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—44no—
303C2×Q8—44no—
304C2×Q8—44no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—28no—
308C22×C4—28no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—432no—
351C22×C4#12118yes—
352C22×C4#13718yes—
353C22×C4C22×D412yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D412yes—
356C22×C4C22×D412yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D412yes—
360C22×C4C22×D412yes—
361C22×C4C2×#4112yes—
362C22×C4H(5,𝔽2)12yes—
363C22×C4H(5,𝔽2)12yes—
364C22×C42−1+412yes—
365C22×C42−1+412yes—
366C22×C4C2511yescharacteristic
367Q16—88no—
368M16—48no—
369M16—416no—
370M16—48no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—416no—
377C2×C8C22×D418yes—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—48no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—44no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—44no—
395C4⋊C4—48no—
396C4⋊C4—44no—
397C22⋊C4—48no—
398C22⋊C4—48no—
399C22⋊C4—48no—
400C22⋊C4—48no—
401C22⋊C4—48no—
402C22⋊C4—48no—
403C42—22no—
404C42—24no—
405C42—24no—
406C42—24no—
407C42—24no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—48no—
412C42—48no—
413C42#12114yes—
414C42#12114yes—
415C42C22×D414yes—
416C42C2×#4114yes—
417C23—21no—
418C23—21no—
419C23—22no—
420C23—28no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—48no—
430C23—48no—
431C23—44no—
432C23—44no—
433C23C2×#12112yes—
434C23#53712yes—
435C23#53712yes—
436C23C2×#13714yes—
437C23C23×D411yescharacteristic
438Q8—44no—
439Q8—44no—
440Q8—44no—
441Q8—44no—
442Q8—88no—
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—28no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—216no—
469C2×C4—216no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—48no—
474C2×C4—48no—
475C2×C4—48no—
476C2×C4—48no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4C2×#12112yes—
490C2×C4C2×#12114yes—
491C2×C4C2×#12114yes—
492C2×C4#53712yes—
493C2×C4#53712yes—
494C2×C4C2×#13714yes—
495C2×C4C2×#13714yes—
496C2×C4#57514yes—
497C2×C4C23×D411yescharacteristic
498C2×C4C22×#4111yescharacteristic
499C2×C4C2×H(5,𝔽2)11yescharacteristic
500C2×C4C2×H(5,𝔽2)11yescharacteristic
501C2×C4C2×2−1+411yescharacteristic
502C2×C4C2×2−1+411yescharacteristic
503C8—28no—
504C8—44no—
505C22—24no—
506C22—22no—
507C22—22no—
508C22—24no—
509C22—24no—
510C22—22no—
511C22—22no—
512C22—24no—
513C22—416no—
514C22—42no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—42no—
520C22#325514yescomplemented
521C22C22×#12111yescharacteristic
522C22C2×#53711yescharacteristic
523C22C2×#53711yescharacteristic
524C22C22×#13712yes—
525C22C2×#57512yes—
526C4—24no—
527C4—24no—
528C4—22no—
529C4—22no—
530C4—22no—
531C4—42no—
532C4—84no—
533C4—84no—
534C4—84no—
535C4—84no—
536C4#305618yes—
537C4#348411yescharacteristic
538C4#348611yescharacteristic
539C4#349211yescharacteristic
540C4#349211yescharacteristic
541C2—24no—
542C2—42no—
543C2—42no—
544C2#3522311yescharacteristic
545C2C2×#305612yes—
546C2C2×#325511yescharacteristic
547C2#6090711yescharacteristic
548C2#6091812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3255C2semidirect
5. #29783C2semidirect
6. #24845C2semidirect
7. #24838C2semidirect
8. #24760C2semidirect
9. #24752C2semidirect
10. #21964C2semidirect
11. #21894C2semidirect
16. C2×#475C22semidirect
20. C2×#462C22semidirect
44. #2088C22semidirect
45. #2068C22semidirect
46. #2066C22semidirect
47. #2056C22semidirect
48. #1988C22semidirect
49. #1981C22semidirect
50. #1948C22semidirect
51. #1947C22semidirect
58. #1803C22semidirect
59. #1803C22semidirect
520. C22#3255semidirect

Maximal subgroups

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

subgroupindex
C2×#32552
C22×#4232
#366252
#329372
#297832
#248452
#248382
#247602
#247522
#219642
#218942