Finite Group Databasebeta

← #4458118

Subgroups — #4458118

Order 512.

1956 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16996C214yesmaximal, complemented
2#16612C214yesmaximal
3#16550C214yesmaximal
4#16332C214yesmaximal, complemented
5#15135C212yesmaximal
6#15133C212yesmaximal, complemented
7#15133C212yesmaximal, complemented
8#15131C212yesmaximal
9#14343C211yesmaximal, characteristic
10#14029C212yesmaximal, complemented
11#14005C212yesmaximal, complemented
12#13852C211yesmaximal, characteristic
13#13777C211yesmaximal, characteristic
14C22×#117C2211yescharacteristic
15C2×#399C2214yes—
16C2×#398C2214yes—
17C2×#398C2214yes—
18C2×#396C2214yes—
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#389C2214yes—
24C2×#387C2214yes—
25C2×#384C2212yes—
26C2×#384C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C4⋊C4C2212yes—
31C2×C4×C22⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#1445C2214yes—
34#1445C2214yes—
35#1445C2214yes—
36#1444C2214yes—
37#1444C2214yes—
38#1441C2214yes—
39#1441C2214yes—
40#1441C2212yes—
41#1441C2212yes—
42#1441C2214yes—
43#1441C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1440C2214yes—
51#1439C2214yes—
52#1439C2212yes—
53#1439C2212yes—
54#1439C2214yes—
55#1437C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59C4⋊#95C2212yes—
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2211yescharacteristic
62C4⋊#95C2212yes—
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2212yes—
66C4×#95C2212yes—
67C24×C4C2311yescharacteristic
68C2×#117—44no—
69C2×#117Q818yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C22⋊C4C2311yescharacteristic
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2311yescharacteristic
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2312yes—
91C22×C42C2312yes—
92C22×C42C2312yes—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2314yes—
96#399—48no—
97#398—48no—
98#398—48no—
99#396—48no—
100#389—28no—
101#389—48no—
102#389—44no—
103#389—44no—
104#389—44no—
105#387—48no—
106#384—28no—
107#384—44no—
108#380—28no—
109#380—44no—
110C4×C4⋊C4—44no—
111C4×C4⋊C4—44no—
112C4×C22⋊C4—42no—
113C4×C22⋊C4—42no—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2312yes—
117C2×#95C2312yes—
118C2×#95C2312yes—
119C2×#95C2312yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2311yescharacteristic
138C2×#95C2314yes—
139C2×#95C2311yescharacteristic
140C25C2411yescharacteristic
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4C2×Q814yes—
146C23×C4C2414yes—
147C23×C4C2414yes—
148C23×C4C2414yes—
149C23×C4C2412yes—
150C23×C4C2412yes—
151C23×C4C2411yescharacteristic
152C23×C4C2414yes—
153C23×C4C2412yes—
154C23×C4C2412yes—
155C23×C4C2412yes—
156C23×C4C2412yes—
157C23×C4C2411yescharacteristic
158#117—416no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4C2×Q818yes—
203C2×C4⋊C4#4118yes—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4C2×Q818yes—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—44no—
229C2×C42—44no—
230C2×C42—44no—
231C2×C42C2×Q814yes—
232C2×C42C2×Q814yes—
233#95—28no—
234#95—28no—
235#95—24no—
236#95—28no—
237#95—24no—
238#95—28no—
239#95—24no—
240#95—24no—
241#95—42no—
242#95—41no—
243#95—41no—
244#95—42no—
245#95—42no—
246#95—44no—
247#95—44no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—21no—
264C24—22no—
265C24—22no—
266C24—21no—
267C24C22×Q812yes—
268C24C2511yescharacteristic
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—28no—
316C22×C4—28no—
317C22×C4—28no—
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—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—48no—
340C22×C4—48no—
341C22×C4C22×Q814yes—
342C22×C4C22×Q814yes—
343C22×C4C22×Q812yes—
344C22×C4C22×Q812yes—
345C22×C4C2×#4118yes—
346C22×C4C2×#4114yes—
347C22×C4H(5,𝔽2)14yes—
348C22×C4H(5,𝔽2)14yes—
349C22×C4H(5,𝔽2)14yes—
350C22×C4H(5,𝔽2)12yes—
351C22×C42−1+414yes—
352C22×C42−1+414yes—
353C22×C42−1+412yes—
354C4⋊C4—216no—
355C4⋊C4—48no—
356C4⋊C4—416no—
357C4⋊C4—416no—
358C4⋊C4—48no—
359C4⋊C4—48no—
360C4⋊C4—48no—
361C4⋊C4—48no—
362C4⋊C4—48no—
363C4⋊C4—88no—
364C4⋊C4—88no—
365C4⋊C4—88no—
366C4⋊C4—88no—
367C4⋊C4—88no—
368C4⋊C4—88no—
369C22⋊C4—48no—
370C22⋊C4—48no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—88no—
374C22⋊C4—88no—
375C42—28no—
376C42—28no—
377C42—48no—
378C42—48no—
379C42—88no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—24no—
384C23—24no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—41no—
393C23—42no—
394C23—41no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—41no—
399C23—42no—
400C23—41no—
401C23C23×Q811yescharacteristic
402C23C22×#4112yes—
403C23C2×H(5,𝔽2)12yes—
404C23C2×H(5,𝔽2)11yescharacteristic
405C23C2×H(5,𝔽2)12yes—
406C23C2×H(5,𝔽2)12yes—
407C23C2×2−1+411yescharacteristic
408C23C2×2−1+412yes—
409C23C2×2−1+412yes—
410C2×C4—22no—
411C2×C4—28no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—28no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—28no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—416no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
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—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4#53314yes—
487C2×C4#53918yes—
488C2×C4#54018yes—
489C2×C4#54314yes—
490C2×C4#54414yes—
491C22—42no—
492C22—44no—
493C22—42no—
494C22—42no—
495C22—44no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22C2×#53311yescharacteristic
501C22C2×#53912yes—
502C22C2×#54012yes—
503C22C2×#54311yescharacteristic
504C22C2×#54411yescharacteristic
505C22#350312yes—
506C22#350312yes—
507C22#351912yes—
508C22#352012yes—
509C22#352212yes—
510C22#352312yes—
511C22#353612yes—
512C22#353712yes—
513C22#353711yescharacteristic
514C22#353812yes—
515C22#353811yescharacteristic
516C22#354811yescharacteristic
517C22#354812yes—
518C22#354812yes—
519C22#355012yes—
520C22#355611yescharacteristic
521C4—42no—
522C4—48no—
523C4—44no—
524C4—44no—
525C4—44no—
526C4—44no—
527C4—44no—
528C4—44no—
529C4—84no—
530C4—84no—
531C4—84no—
532C4#266718yes—
533C2—82no—
534C2C2×#266711yescharacteristic
535C2#3557812yes—
536C2#3730712yes—
537C2#3730811yescharacteristic
538C2#3730911yescharacteristic
539C2#4255612yes—
540C2#4260812yes—
541C2#4263312yes—
542C2#4284012yes—
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. #16996C2semidirect
4. #16332C2semidirect
6. #15133C2semidirect
7. #15133C2semidirect
10. #14029C2semidirect
11. #14005C2semidirect

Maximal subgroups

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

subgroupindex
#169962
#166122
#165502
#163322
#151352
#151332
#151312
#143432
#140292
#140052
#138522
#137772