Finite Group Databasebeta

← #2037244

Subgroups — #2037244

Order 512.

2254 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2724C211yesmaximal, characteristic
2C2×#2861C211yesmaximal, characteristic
3#17078C212yesmaximal, complemented
4#16427C214yesmaximal, complemented
5#15983C212yesmaximal
6#15881C212yesmaximal, complemented
7#15856C212yesmaximal, complemented
8#15840C214yesmaximal, complemented
9#15357C214yesmaximal, complemented
10#15262C212yesmaximal
11#14699C212yesmaximal, complemented
12#14445C211yesmaximal, characteristic
13#14403C214yesmaximal, complemented
14C22×#125C2211yescharacteristic
15C22×#124C2211yescharacteristic
16C22×#123C2212yes—
17#2861—28no—
18#2724—28no—
19C2×#400C2212yes—
20C2×#398C2212yes—
21C2×#397C2214yes—
22C2×#395C2214yes—
23C2×#391C2211yescharacteristic
24C2×#391C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#388C2214yes—
30C2×#388C2214yes—
31C2×#388C2214yes—
32C2×#387C2214yes—
33C2×#385C2214yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#382C2212yes—
37C2×C43C2211yescharacteristic
38#1445C2212yes—
39#1445C2214yes—
40#1445C2214yes—
41#1445C2214yes—
42#1444C2214yes—
43#1444C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1440C2212yes—
50#1440C2214yes—
51#1440C2212yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1438C2212yes—
58#1436C2214yes—
59#1436C2212yes—
60#1436C2212yes—
61#1436C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67#1434C2214yes—
68C22⋊#95C2214yes—
69C23×Q8C2311yescharacteristic
70C23×D4C2311yescharacteristic
71C24×C4C2312yes—
72C2×#125—24no—
73C2×#125—42no—
74C2×#125—42no—
75C2×#124—24no—
76C2×#124—42no—
77C2×#124—42no—
78C2×#123—28no—
79C2×#123—48no—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C42C2311yescharacteristic
103C22×C42C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108#400—44no—
109#398—28no—
110#397—216no—
111#395—48no—
112#391—24no—
113#391—24no—
114#390—24no—
115#390D418yes—
116#389—28no—
117#389—28no—
118#388—216no—
119#388—216no—
120#388—216no—
121#387—48no—
122#385—216no—
123#385—44no—
124#385—44no—
125#382—44no—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C43—24no—
149C25C2412yes—
150C22×Q8—22no—
151C22×Q8—24no—
152C22×Q8—41no—
153C22×Q8—41no—
154C22×Q8—42no—
155C22×D4—24no—
156C22×D4—41no—
157C22×D4—41no—
158C22×D4—42no—
159C22×D4C2×D414yes—
160C23×C4—24no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4C2412yes—
166C23×C4C2414yes—
167C23×C4C2412yes—
168C23×C4C2414yes—
169C23×C4C2414yes—
170C23×C4C2412yes—
171C23×C4C2414yes—
172C23×C4C2411yescharacteristic
173C23×C4C2411yescharacteristic
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2411yescharacteristic
177#125—416no—
178#124—416no—
179#123—432no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4#4118yes—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4C2×D4116yes—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—28no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42C2×D414yes—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—24no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—42no—
279#95—44no—
280C24—24no—
281C24—22no—
282C24—22no—
283C24—24no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24C22×D414yes—
289C24C2511yescharacteristic
290C2×Q8—44no—
291C2×Q8—44no—
292C2×Q8—44no—
293C2×Q8—84no—
294C2×Q8—84no—
295C2×D4—28no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—84no—
299C2×D4—84no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
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—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—48no—
363C22×C4—84no—
364C22×C4—84no—
365C22×C4—88no—
366C22×C4—88no—
367C22×C4C22×D414yes—
368C22×C4C22×D412yes—
369C22×C4C22×D414yes—
370C22×C4C2×#4118yes—
371C22×C4C2×#4114yes—
372C22×C4H(5,𝔽2)12yes—
373C22×C4H(5,𝔽2)12yes—
374C22×C4H(5,𝔽2)14yes—
375C22×C42−1+414yes—
376C22×C42−1+414yes—
377C4⋊C4—44no—
378C4⋊C4—48no—
379C4⋊C4—416no—
380C4⋊C4—48no—
381C4⋊C4—48no—
382C4⋊C4—44no—
383C4⋊C4—416no—
384C4⋊C4—48no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C4⋊C4—84no—
388C4⋊C4—88no—
389C4⋊C4—88no—
390C22⋊C4—232no—
391C22⋊C4—48no—
392C22⋊C4—416no—
393C22⋊C4—48no—
394C22⋊C4—48no—
395C22⋊C4—48no—
396C22⋊C4—416no—
397C22⋊C4—48no—
398C22⋊C4—88no—
399C42—28no—
400C42—44no—
401C42—48no—
402C42—48no—
403C42—44no—
404C42—48no—
405C23—24no—
406C23—28no—
407C23—24no—
408C23—24no—
409C23—28no—
410C23—44no—
411C23—42no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—42no—
417C23—44no—
418C23—42no—
419C23—42no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23C23×D411yescharacteristic
427C23C22×#4112yes—
428C23C2×H(5,𝔽2)12yes—
429C23C2×H(5,𝔽2)11yescharacteristic
430C23C2×H(5,𝔽2)11yescharacteristic
431C23C2×2−1+412yes—
432C23C2×2−1+412yes—
433C23#58512yes—
434C23#58512yes—
435Q8—88no—
436D4—88no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—28no—
448C2×C4—28no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—28no—
454C2×C4—24no—
455C2×C4—28no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—816no—
508C2×C4#53514yes—
509C2×C4#54718yes—
510C22—48no—
511C22—44no—
512C22—48no—
513C22—44no—
514C22—44no—
515C22—84no—
516C22—84no—
517C22—84no—
518C22—84no—
519C22C2×#53511yescharacteristic
520C22#346112yes—
521C22C2×#54712yes—
522C22#348111yescharacteristic
523C22#348411yescharacteristic
524C22#348511yescharacteristic
525C22#348912yes—
526C22#349112yes—
527C22#351712yes—
528C22#352012yes—
529C22#352212yes—
530C22#352911yescharacteristic
531C22#354712yes—
532C22#355011yescharacteristic
533C22#355011yescharacteristic
534C22#355212yes—
535C22#355812yes—
536C22#355812yes—
537C22#355912yes—
538C22#356312yes—
539C22#356412yes—
540C4—24no—
541C4—28no—
542C4—44no—
543C4—48no—
544C4—42no—
545C4—44no—
546C4—42no—
547C4—84no—
548C4—84no—
549C4—82no—
550C4—84no—
551C4—82no—
552C2—84no—
553C2#3696711yescharacteristic
554C2#3725612yes—
555C2#3776312yes—
556C2#3877012yes—
557C2#3892012yes—
558C2#4594012yes—
559C2#4622811yescharacteristic
560C2#5335612yes—
561C2#5907011yescharacteristic
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
3. #17078C2semidirect
4. #16427C2semidirect
6. #15881C2semidirect
7. #15856C2semidirect
8. #15840C2semidirect
9. #15357C2semidirect
11. #14699C2semidirect
13. #14403C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27242
C2×#28612
#170782
#164272
#159832
#158812
#158562
#158402
#153572
#152622
#146992
#144452
#144032