Finite Group Databasebeta

← #2809061

Subgroups — #2809061

Order 512.

2526 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic, complemented
2C2×#2566C211yesmaximal, characteristic
3#16686C212yesmaximal, complemented
4#16595C214yesmaximal, complemented
5#15849C212yesmaximal, complemented
6#15662C212yesmaximal
7#15322C214yesmaximal, complemented
8#15265C214yesmaximal, complemented
9#15112C212yesmaximal
10#14723C212yesmaximal
11#14526C214yesmaximal, complemented
12#14454C211yesmaximal, characteristic, complemented
13#14239C212yesmaximal, complemented
14C22×C4⋊Q8C2214yes—
15C22×#125C2211yescharacteristic
16C22×C22⋊Q8C2212yes—
17C22×C4×D4C2211yescharacteristic
18#2566—28no—
19#2551—28no—
20C2×#400C2212yes—
21C2×#397C2214yes—
22C2×#396C2214yes—
23C2×#395C2212yes—
24C2×#395C2212yes—
25C2×#395C2212yes—
26C2×#393C2214yes—
27C2×#393C2212yes—
28C2×#391C2212yes—
29C2×#390C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#389C2214yes—
32C2×#388C2214yes—
33C2×#385C2214yes—
34C2×#385C2214yes—
35C2×#384C2211yescharacteristic
36C2×#382C2211yescharacteristic
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39#1446C2214yes—
40#1445C2212yes—
41#1445C2212yes—
42#1444C2212yes—
43#1444C2214yes—
44#1443C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1441C2212yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2212yes—
51#1441C2214yes—
52#1440C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2212yes—
56#1439C2214yes—
57#1438C2214yes—
58#1437C2214yes—
59#1437C2214yes—
60#1436C2212yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2212yes—
66#1434C2212yes—
67#1434C2214yes—
68C22⋊#95C2212yes—
69C23×Q8C2312yes—
70C23×Q8C2312yes—
71C23×D4C2311yescharacteristic
72C24×C4C2312yes—
73C2×C4⋊Q8—216no—
74C2×C4⋊Q8—48no—
75C2×C4⋊Q8—48no—
76C2×#125—24no—
77C2×#125—42no—
78C2×#125—42no—
79C2×C22⋊Q8—48no—
80C2×C22⋊Q8—44no—
81C2×C4×D4—24no—
82C2×C4×D4—42no—
83C2×C4×D4—42no—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2314yes—
108C22×C42C2312yes—
109#400—28no—
110#397—216no—
111#396—216no—
112#395—28no—
113#395—28no—
114#395—44no—
115#393—44no—
116#393—48no—
117#391—44no—
118#390—24no—
119#390—24no—
120#389—216no—
121#388—48no—
122#385—216no—
123#385—48no—
124#384D418yes—
125#382—24no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4—28no—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C25C2412yes—
153C22×Q8—24no—
154C22×Q8—24no—
155C22×Q8—28no—
156C22×Q8—28no—
157C22×Q8—42no—
158C22×Q8—44no—
159C22×Q8—44no—
160C22×Q8—42no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×D4—24no—
164C22×D4—22no—
165C22×D4—24no—
166C22×D4—41no—
167C22×D4—41no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—42no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4C2412yes—
174C23×C4C2414yes—
175C23×C4C2412yes—
176C23×C4C2414yes—
177C23×C4C2412yes—
178C23×C4C2412yes—
179C23×C4C2414yes—
180C23×C4C2414yes—
181C23×C4C2411yescharacteristic
182C23×C4C2411yescharacteristic
183C23×C4C2412yes—
184C23×C4C2411yescharacteristic
185C4⋊Q8—464no—
186#125—416no—
187C22⋊Q8—432no—
188C4×D4—416no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
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—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4C2×D414yes—
223C2×C4⋊C4C2×D414yes—
224C2×C4⋊C4C2×D4116yes—
225C2×C4⋊C4#4118yes—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—24no—
251C2×C42—28no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42—44no—
257C2×C42—42no—
258C2×C42C2×D414yes—
259#95—24no—
260#95—28no—
261#95—28no—
262#95—24no—
263#95—24no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—24no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—24no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—42no—
280#95—44no—
281#95—44no—
282#95—42no—
283C24—22no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—22no—
288C24—22no—
289C24—24no—
290C24—22no—
291C24—22no—
292C24—22no—
293C242−1+414yes—
294C24C2511yescharacteristic
295C2×Q8—48no—
296C2×Q8—416no—
297C2×Q8—416no—
298C2×Q8—48no—
299C2×Q8—84no—
300C2×Q8—84no—
301C2×Q8—88no—
302C2×Q8—88no—
303C2×Q8—84no—
304C2×Q8—84no—
305C2×D4—28no—
306C2×D4—48no—
307C2×D4—48no—
308C2×D4—44no—
309C2×D4—44no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—21no—
317C22×C4—24no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—48no—
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×C4C2×#4118yes—
376C22×C4C2×#4114yes—
377C22×C4H(5,𝔽2)12yes—
378C22×C4H(5,𝔽2)18yes—
379C22×C4H(5,𝔽2)12yes—
380C22×C42−1+414yes—
381C22×C42−1+414yes—
382C4⋊C4—216no—
383C4⋊C4—216no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—416no—
387C4⋊C4—416no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—416no—
391C4⋊C4—416no—
392C4⋊C4—44no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C22⋊C4—48no—
396C22⋊C4—416no—
397C22⋊C4—416no—
398C22⋊C4—416no—
399C22⋊C4—416no—
400C22⋊C4—88no—
401C22⋊C4—88no—
402C42—28no—
403C42—28no—
404C42—416no—
405C42—44no—
406C42—84no—
407C23—24no—
408C23—28no—
409C23—24no—
410C23—24no—
411C23—28no—
412C23—42no—
413C23—44no—
414C23—42no—
415C23—42no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—42no—
422C23—42no—
423C23—44no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23C23×D411yescharacteristic
431C23C22×#4112yes—
432C23C2×H(5,𝔽2)11yescharacteristic
433C23C2×H(5,𝔽2)14yes—
434C23C2×H(5,𝔽2)11yescharacteristic
435C23C2×2−1+412yes—
436C23C2×2−1+412yes—
437C23C2×2−1+412yes—
438Q8—816no—
439Q8—816no—
440D4—416no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—28no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—28no—
449C2×C4—28no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—28no—
454C2×C4—28no—
455C2×C4—24no—
456C2×C4—28no—
457C2×C4—28no—
458C2×C4—28no—
459C2×C4—24no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—28no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—41no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—816no—
507C2×C4#53614yes—
508C2×C4#54818yes—
509C22—48no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—48no—
514C22—82no—
515C22—84no—
516C22—82no—
517C22—82no—
518C22—84no—
519C22—82no—
520C22C2×#53611yescharacteristic
521C22C2×#54812yes—
522C22#348311yescharacteristic
523C22#348412yes—
524C22#348411yescharacteristic
525C22#348611yescharacteristic
526C22#351114yes—
527C22#352014yes—
528C22#352212yes—
529C22#353212yes—
530C22#353712yes—
531C22#353812yes—
532C22#353812yes—
533C22#355011yescharacteristic
534C22#355214yes—
535C22#355214yes—
536C4—28no—
537C4—24no—
538C4—48no—
539C4—42no—
540C4—44no—
541C4—44no—
542C4—48no—
543C4—42no—
544C4—82no—
545C4—84no—
546C4—84no—
547C4—82no—
548C2—84no—
549C2#3670811yescharacteristic
550C2#3972012yes—
551C2#4437712yes—
552C2#4533712yes—
553C2#5082314yes—
554C2#5200411yescharacteristic
555C2#5894312yes—
556C2#5908411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2551C2semidirect
3. #16686C2semidirect
4. #16595C2semidirect
5. #15849C2semidirect
7. #15322C2semidirect
8. #15265C2semidirect
11. #14526C2semidirect
12. #14454C2semidirect
13. #14239C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#25662
#166862
#165952
#158492
#156622
#153222
#152652
#151122
#147232
#145262
#144542
#142392