Finite Group Databasebeta

← #2924152

Subgroups — #2924152

Order 512.

2386 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2811C211yesmaximal, characteristic, complemented
2C2×#2502C211yesmaximal, characteristic
3#16522C214yesmaximal, complemented
4#16522C212yesmaximal, complemented
5#16110C212yesmaximal, complemented
6#15511C214yesmaximal, complemented
7#15466C212yesmaximal
8#15430C214yesmaximal
9#15199C212yesmaximal, complemented
10#14720C212yesmaximal
11#14575C214yesmaximal
12#14243C212yesmaximal
13#14162C211yesmaximal, characteristic, complemented
14C22×C22⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C23×C4⋊C4C2211yescharacteristic
17#2811—28no—
18#2502—28no—
19C2×#399C2212yes—
20C2×#398C2214yes—
21C2×#398C2214yes—
22C2×#398C2212yes—
23C2×#395C2214yes—
24C2×#395C2212yes—
25C2×#395C2214yes—
26C2×#395C2212yes—
27C2×#393C2212yes—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2214yes—
31C2×#391C2211yescharacteristic
32C2×#391C2214yes—
33C2×#391C2211yescharacteristic
34C2×#385C2214yes—
35C2×#385C2214yes—
36C2×#384C2211yescharacteristic
37C2×#384C2211yescharacteristic
38C2×#384C2214yes—
39C2×C4×C22⋊C4C2211yescharacteristic
40#1446C2212yes—
41#1446C2212yes—
42#1445C2214yes—
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1442C2212yes—
51#1442C2214yes—
52#1442C2214yes—
53#1441C2212yes—
54#1441C2214yes—
55#1441C2212yes—
56#1439C2212yes—
57#1439C2214yes—
58#1437C2214yes—
59#1437C2212yes—
60#1436C2214yes—
61#1436C2212yes—
62#1434C2214yes—
63#1434C2214yes—
64#1434C2214yes—
65#1434C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2212yes—
68C4×#95C2212yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2312yes—
71C23×Q8C2312yes—
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C2×C22⋊Q8—24no—
75C2×C22⋊Q8—42no—
76C2×C22⋊Q8—44no—
77C2×C22⋊Q8—48no—
78C2×C22⋊Q8—42no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—42no—
81C22×C4⋊C4—42no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C42C2311yescharacteristic
102C22×C42C2314yes—
103C22×C42C2312yes—
104C22×C42C2311yescharacteristic
105#399—44no—
106#398—216no—
107#398—48no—
108#398—44no—
109#395—28no—
110#395—216no—
111#395—48no—
112#395—44no—
113#393Q8116yescomplemented
114#391—24no—
115#391—28no—
116#391—28no—
117#391—216no—
118#391—48no—
119#391D418yes—
120#385—216no—
121#385—48no—
122#384—24no—
123#384—24no—
124#384—48no—
125C4×C22⋊C4—24no—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C25C2411yescharacteristic
152C22×Q8—24no—
153C22×Q8—24no—
154C22×Q8—24no—
155C22×Q8—24no—
156C22×Q8—41no—
157C22×Q8—41no—
158C22×Q8—44no—
159C22×Q8—42no—
160C22×Q8—44no—
161C22×Q8—42no—
162C22×Q8—44no—
163C22×Q8—44no—
164C22×Q8—42no—
165C22×Q8C2×D414yes—
166C22×Q8C2×Q818yes—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4C2412yes—
178C23×C4C2414yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2414yes—
186C23×C4C2411yescharacteristic
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C22⋊Q8—416no—
190C22⋊Q8—432no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—416no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4C2×D414yes—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4C2×Q818yes—
246C2×C22⋊C4C2×Q8116yes—
247C2×C42—22no—
248C2×C42—28no—
249C2×C42—28no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—42no—
253C2×C42—44no—
254C2×C42—44no—
255C2×C42C2×D414yes—
256#95—24no—
257#95—28no—
258#95—24no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—24no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—42no—
274#95—44no—
275#95—42no—
276#95—44no—
277#95—42no—
278#95C2×D4116yes—
279#95C2×Q8116yes—
280#95C2×Q818yes—
281C24—21no—
282C24—24no—
283C24—21no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24C22×Q814yes—
288C24H(5,𝔽2)12yes—
289C24C2511yescharacteristic
290C2×Q8—28no—
291C2×Q8—216no—
292C2×Q8—44no—
293C2×Q8—44no—
294C2×Q8—416no—
295C2×Q8—48no—
296C2×Q8—84no—
297C2×Q8—84no—
298C2×Q8—88no—
299C2×Q8—84no—
300C2×Q8—84no—
301C2×Q8—84no—
302C2×Q8—88no—
303C2×Q8—84no—
304C2×Q8—88no—
305C22×C4—24no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—28no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—44no—
365C22×C4—48no—
366C22×C4—44no—
367C22×C4—48no—
368C22×C4—88no—
369C22×C4—84no—
370C22×C4—88no—
371C22×C4C22×D414yes—
372C22×C4C22×D414yes—
373C22×C4C22×D412yes—
374C22×C4C22×D414yes—
375C22×C4C22×Q814yes—
376C22×C4C22×Q818yes—
377C22×C4C22×Q818yes—
378C22×C4C22×Q814yes—
379C22×C4H(5,𝔽2)18yes—
380C22×C4H(5,𝔽2)12yes—
381C22×C42−1+414yes—
382C22×C42−1+414yes—
383C4⋊C4—44no—
384C4⋊C4—48no—
385C4⋊C4—48no—
386C4⋊C4—48no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—416no—
390C4⋊C4—88no—
391C4⋊C4—84no—
392C4⋊C4—84no—
393C4⋊C4—88no—
394C4⋊C4—88no—
395C4⋊C4—88no—
396C4⋊C4—88no—
397C4⋊C4—88no—
398C22⋊C4—216no—
399C22⋊C4—416no—
400C22⋊C4—48no—
401C22⋊C4—416no—
402C42—44no—
403C42—416no—
404C42—44no—
405C42—84no—
406C23—24no—
407C23—22no—
408C23—28no—
409C23—24no—
410C23—28no—
411C23—24no—
412C23—24no—
413C23—22no—
414C23—41no—
415C23—41no—
416C23—42no—
417C23—42no—
418C23—44no—
419C23—44no—
420C23—42no—
421C23Q8214yes—
422C23C23×D411yescharacteristic
423C23C23×Q812yes—
424C23C2×H(5,𝔽2)11yescharacteristic
425C23C2×H(5,𝔽2)11yescharacteristic
426C23C2×H(5,𝔽2)14yes—
427C23C2×2−1+412yes—
428C23C2×2−1+412yes—
429C23#58512yes—
430Q8—88no—
431Q8—816no—
432Q8—816no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—28no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—28no—
449C2×C4—28no—
450C2×C4—28no—
451C2×C4—28no—
452C2×C4—28no—
453C2×C4—28no—
454C2×C4—28no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—24no—
458C2×C4—28no—
459C2×C4—416no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—44no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—816no—
501C2×C4#53614yes—
502C2×C4D4×Q818yes—
503C22—24no—
504C22—28no—
505C22—42no—
506C22—44no—
507C22—48no—
508C22—42no—
509C22—44no—
510C22—44no—
511C22C2×#53611yescharacteristic
512C22C2×D4×Q812yes—
513C22C2×Q8211yescharacteristic
514C22#348411yescharacteristic
515C22#348511yescharacteristic
516C22#349812yes—
517C22#350114yes—
518C22#350312yes—
519C22#350314yes—
520C22#353212yes—
521C22#353512yes—
522C22#353512yes—
523C22#353514yes—
524C22#354411yescharacteristic
525C22#354414yes—
526C22#354912yes—
527C4—24no—
528C4—28no—
529C4—42no—
530C4—44no—
531C4—48no—
532C4—48no—
533C4—44no—
534C4—44no—
535C4—48no—
536C4—82no—
537C4—82no—
538C4—84no—
539C2—44no—
540C2#3672011yescharacteristic
541C2#3961712yes—
542C2#4349412yes—
543C2#4355312yes—
544C2#5160114yes—
545C2#5190211yescharacteristic
546C2#5875012yes—
547C2#5915111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2811C2semidirect
3. #16522C2semidirect
4. #16522C2semidirect
5. #16110C2semidirect
6. #15511C2semidirect
9. #15199C2semidirect
13. #14162C2semidirect
113. #393Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28112
C2×#25022
#165222
#161102
#155112
#154662
#154302
#151992
#147202
#145752
#142432
#141622