Finite Group Databasebeta

← #2808538

Subgroups — #2808538

Order 512.

2282 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic, complemented
2C2×#2502C211yesmaximal, characteristic
3#16631C214yesmaximal, complemented
4#16629C212yesmaximal, complemented
5#15790C212yesmaximal, complemented
6#15766C214yesmaximal, complemented
7#15430C212yesmaximal
8#15292C214yesmaximal
9#14881C212yesmaximal, complemented
10#14821C212yesmaximal
11#14553C214yesmaximal
12#14454C211yesmaximal, characteristic, complemented
13#14172C212yesmaximal
14C22×C22⋊Q8C2211yescharacteristic
15C22×C4×Q8C2212yes—
16C23×C4⋊C4C2211yescharacteristic
17#2582—28no—
18#2502—28no—
19C2×#399C2212yes—
20C2×#398C2212yes—
21C2×#398C2212yes—
22C2×#398C2214yes—
23C2×#398C2212yes—
24C2×#395C2214yes—
25C2×#393C2212yes—
26C2×#391C2211yescharacteristic
27C2×#391C2214yes—
28C2×#391C2211yescharacteristic
29C2×#389C2214yes—
30C2×#385C2214yes—
31C2×#385C2214yes—
32C2×#384C2211yescharacteristic
33C2×#384C2212yes—
34C2×#384C2211yescharacteristic
35C2×#380C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37#1446C2214yes—
38#1446C2212yes—
39#1445C2212yes—
40#1444C2214yes—
41#1444C2214yes—
42#1444C2212yes—
43#1444C2214yes—
44#1443C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2212yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2212yes—
55#1439C2214yes—
56#1439C2212yes—
57#1437C2212yes—
58#1437C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2212yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1434C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2312yes—
71C24×C4C2311yescharacteristic
72C24×C4C2312yes—
73C2×C22⋊Q8—24no—
74C2×C22⋊Q8—42no—
75C2×C22⋊Q8—42no—
76C2×C4×Q8—28no—
77C2×C4×Q8—48no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4—42no—
80C22×C4⋊C4—42no—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C42C2311yescharacteristic
101C22×C42C2314yes—
102C22×C42C2311yescharacteristic
103C22×C42C2312yes—
104#399—44no—
105#398—28no—
106#398—28no—
107#398—216no—
108#398—44no—
109#395—48no—
110#393Q8116yescomplemented
111#391—216no—
112#391—24no—
113#391—24no—
114#389—216no—
115#385—216no—
116#385—48no—
117#384—24no—
118#384—44no—
119#384D418yes—
120#380—44no—
121C4×C22⋊C4—24no—
122C2×#95C2314yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C25C2411yescharacteristic
149C22×Q8—24no—
150C22×Q8—22no—
151C22×Q8—24no—
152C22×Q8—24no—
153C22×Q8—44no—
154C22×Q8—44no—
155C22×Q8—41no—
156C22×Q8—41no—
157C22×Q8—42no—
158C22×Q8C2×Q818yes—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4C2414yes—
170C23×C4C2412yes—
171C23×C4C2414yes—
172C23×C4C2412yes—
173C23×C4C2412yes—
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2414yes—
177C23×C4C2414yes—
178C23×C4C2411yescharacteristic
179C23×C4C2412yes—
180C23×C4C2411yescharacteristic
181C22⋊Q8—416no—
182C4×Q8—432no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—416no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4C2×D414yes—
226C2×C4⋊C4C2×D4116yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4C2×Q818yes—
238C2×C22⋊C4C2×Q8116yes—
239C2×C42—24no—
240C2×C42—28no—
241C2×C42—22no—
242C2×C42—44no—
243C2×C42—44no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—44no—
247C2×C42C2×D414yes—
248#95—24no—
249#95—28no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—24no—
254#95—24no—
255#95—28no—
256#95—28no—
257#95—24no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—42no—
270#95—44no—
271#95—42no—
272#95C2×Q8116yes—
273#95C2×Q818yes—
274C24—21no—
275C24—22no—
276C24—21no—
277C24—22no—
278C24—22no—
279C24—24no—
280C24C22×Q814yes—
281C24H(5,𝔽2)12yes—
282C24C2511yescharacteristic
283C2×Q8—216no—
284C2×Q8—416no—
285C2×Q8—48no—
286C2×Q8—48no—
287C2×Q8—416no—
288C2×Q8—44no—
289C2×Q8—44no—
290C2×Q8—44no—
291C2×Q8—84no—
292C2×Q8—84no—
293C22×C4—24no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—28no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
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—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—44no—
354C22×C4—48no—
355C22×C4—44no—
356C22×C4—48no—
357C22×C4—88no—
358C22×C4—84no—
359C22×C4—88no—
360C22×C4C22×D414yes—
361C22×C4C22×D414yes—
362C22×C4C22×D412yes—
363C22×C4C22×D414yes—
364C22×C4C22×Q814yes—
365C22×C4C22×Q818yes—
366C22×C4C22×Q818yes—
367C22×C4C22×Q814yes—
368C22×C4H(5,𝔽2)12yes—
369C22×C4H(5,𝔽2)14yes—
370C22×C42−1+414yes—
371C4⋊C4—44no—
372C4⋊C4—416no—
373C4⋊C4—44no—
374C4⋊C4—48no—
375C4⋊C4—48no—
376C4⋊C4—44no—
377C4⋊C4—48no—
378C4⋊C4—416no—
379C4⋊C4—84no—
380C4⋊C4—84no—
381C4⋊C4—88no—
382C4⋊C4—88no—
383C4⋊C4—84no—
384C4⋊C4—88no—
385C4⋊C4—88no—
386C22⋊C4—216no—
387C22⋊C4—416no—
388C22⋊C4—48no—
389C22⋊C4—88no—
390C42—28no—
391C42—44no—
392C42—416no—
393C42—48no—
394C23—24no—
395C23—22no—
396C23—28no—
397C23—24no—
398C23—28no—
399C23—24no—
400C23—24no—
401C23—22no—
402C23—41no—
403C23—42no—
404C23—41no—
405C23—42no—
406C23—42no—
407C23—44no—
408C23—44no—
409C23Q8214yes—
410C23C23×D411yescharacteristic
411C23C23×Q812yes—
412C23C2×H(5,𝔽2)12yes—
413C23C2×H(5,𝔽2)11yescharacteristic
414C23C2×H(5,𝔽2)11yescharacteristic
415C23C2×2−1+412yes—
416C23#58514yes—
417C23#58512yes—
418Q8—432no—
419Q8—88no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—28no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—28no—
428C2×C4—24no—
429C2×C4—28no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—28no—
434C2×C4—28no—
435C2×C4—28no—
436C2×C4—28no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—28no—
440C2×C4—28no—
441C2×C4—24no—
442C2×C4—416no—
443C2×C4—44no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
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—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—816no—
488C2×C4#53514yes—
489C2×C4D4×Q818yes—
490C22—24no—
491C22—28no—
492C22—44no—
493C22—44no—
494C22—42no—
495C22—42no—
496C22—48no—
497C22—44no—
498C22C2×#53511yescharacteristic
499C22C2×D4×Q812yes—
500C22C2×Q8211yescharacteristic
501C22#348411yescharacteristic
502C22#348512yes—
503C22#348711yescharacteristic
504C22#350114yes—
505C22#350312yes—
506C22#350414yes—
507C22#353312yes—
508C22#353412yes—
509C22#353412yes—
510C22#354012yes—
511C22#354114yes—
512C22#354411yescharacteristic
513C22#355714yes—
514C4—24no—
515C4—28no—
516C4—44no—
517C4—44no—
518C4—44no—
519C4—42no—
520C4—48no—
521C4—48no—
522C4—84no—
523C4—82no—
524C4—84no—
525C4—82no—
526C2—44no—
527C2#3671011yescharacteristic
528C2#3959412yes—
529C2#4312612yes—
530C2#4513912yes—
531C2#5108314yes—
532C2#5205411yescharacteristic
533C2#5660012yes—
534C2#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×#2582C2semidirect
3. #16631C2semidirect
4. #16629C2semidirect
5. #15790C2semidirect
6. #15766C2semidirect
9. #14881C2semidirect
12. #14454C2semidirect
110. #393Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
C2×#25022
#166312
#166292
#157902
#157662
#154302
#152922
#148812
#148212
#145532
#144542
#141722