Finite Group Databasebeta

← #4131208

Subgroups — #4131208

Order 512.

2316 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17221C211yesmaximal, characteristic, complemented
2#16244C211yesmaximal, characteristic, complemented
3#15782C212yesmaximal, complemented
4#15780C212yesmaximal, complemented
5#15399C214yesmaximal, complemented
6#15366C214yesmaximal, complemented
7#15340C212yesmaximal
8#14246C212yesmaximal, complemented
9#14230C212yesmaximal, complemented
10#14158C214yesmaximal, complemented
11#14152C211yesmaximal, characteristic
12#14128C212yesmaximal, complemented
13#14071C214yesmaximal, complemented
14C22×#121C2211yescharacteristic
15C22×#117C2212yes—
16C2×#399C2212yes—
17C2×#399C2212yes—
18C2×#398C2212yes—
19C2×#394C2214yes—
20C2×#390C2212yes—
21C2×#389C2212yes—
22C2×#388C2214yes—
23C2×#388C2214yes—
24C2×#388C2214yes—
25C2×#387C2214yes—
26C2×#386C2214yes—
27C2×#385C2214yes—
28C2×#381C2212yes—
29C2×C4×C22⋊C4C2211yescharacteristic
30C22×#95C2211yescharacteristic
31#1445C2211yescharacteristic
32#1445C2211yescharacteristic
33#1445C2212yes—
34#1444C2212yes—
35#1444C2214yes—
36#1444C2212yes—
37#1444C2212yes—
38#1443C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2212yes—
42#1442C2212yes—
43#1440C2214yes—
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2212yes—
49#1438C2212yes—
50#1438C2214yes—
51#1438C2211yescharacteristic
52#1438C2211yescharacteristic
53#1437C2214yes—
54#1437C2214yes—
55#1437C2212yes—
56#1437C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2212yes—
63C4⋊#95C2214yes—
64C22⋊#95C2214yes—
65C4×#95C2212yes—
66C4×#95C2214yes—
67C23×D4C2312yes—
68C23×D4C2311yescharacteristic
69C24×C4C2312yes—
70C24×C4C2311yescharacteristic
71C2×#121—24no—
72C2×#121—42no—
73C2×#121—42no—
74C2×#117—48no—
75C2×#117—44no—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2312yes—
95C22×C42C2314yes—
96C22×C42C2314yes—
97C22×C42C2312yes—
98C22×C42C2312yes—
99#399—28no—
100#399—44no—
101#398—28no—
102#394—48no—
103#390—44no—
104#389—44no—
105#388—216no—
106#388—48no—
107#388—48no—
108#387—48no—
109#386—216no—
110#385—48no—
111#381—28no—
112C4×C22⋊C4—42no—
113C2×#95—42no—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2312yes—
117C2×#95C2312yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2312yes—
127C2×#95C2312yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2311yescharacteristic
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2311yescharacteristic
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2311yescharacteristic
138C2×#95C2311yescharacteristic
139C2×#95C2312yes—
140C25C2412yes—
141C25C2411yescharacteristic
142C22×D4—28no—
143C22×D4—24no—
144C22×D4—24no—
145C22×D4—22no—
146C22×D4—41no—
147C22×D4—41no—
148C22×D4—42no—
149C22×D4—44no—
150C22×D4—44no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—28no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4C2414yes—
161C23×C4C2414yes—
162C23×C4C2414yes—
163C23×C4C2412yes—
164C23×C4C2412yes—
165C23×C4C2412yes—
166C23×C4C2412yes—
167C23×C4C2411yescharacteristic
168C23×C4C2414yes—
169C23×C4C2411yescharacteristic
170C23×C4C2412yes—
171#121—416no—
172#117—432no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4#4114yes—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4#4118yes—
227C2×C42—28no—
228C2×C42—28no—
229C2×C42—24no—
230C2×C42—44no—
231C2×C42—44no—
232C2×C42—44no—
233C2×C42—44no—
234C2×C42—42no—
235C2×C42—44no—
236C2×C42—44no—
237#95—28no—
238#95—24no—
239#95—28no—
240#95—28no—
241#95—28no—
242#95—24no—
243#95—28no—
244#95—24no—
245#95—28no—
246#95—28no—
247#95—28no—
248#95—28no—
249#95—28no—
250#95—41no—
251#95—42no—
252#95—41no—
253#95—44no—
254#95—42no—
255#95—41no—
256#95—41no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—42no—
262#95#4118yes—
263C24—22no—
264C24—21no—
265C24—24no—
266C24—21no—
267C24—22no—
268C24—22no—
269C24—24no—
270C24—24no—
271C24—22no—
272C24—24no—
273C24—24no—
274C24—22no—
275C24—22no—
276C24—24no—
277C24—24no—
278C24C2×#4114yes—
279C24H(5,𝔽2)12yes—
280C24C2511yescharacteristic
281C2×D4—48no—
282C2×D4—416no—
283C2×D4—44no—
284C2×D4—48no—
285C2×D4—84no—
286C2×D4—88no—
287C2×D4—84no—
288C2×D4—84no—
289C2×D4—84no—
290C22×C4—216no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—21no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—28no—
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—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—416no—
345C22×C4—416no—
346C22×C4—48no—
347C22×C4—48no—
348C22×C4—84no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4C2×#4114yes—
352C22×C4C2×#4114yes—
353C22×C4C2×#4114yes—
354C22×C4C2×#4114yes—
355C22×C4C2×#4118yes—
356C22×C4C2×#4112yes—
357C22×C4H(5,𝔽2)18yes—
358C22×C4H(5,𝔽2)12yes—
359C22×C42−1+414yes—
360C4⋊C4—44no—
361C4⋊C4—416no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—84no—
366C4⋊C4—88no—
367C4⋊C4—88no—
368C4⋊C4—84no—
369C22⋊C4—48no—
370C22⋊C4—416no—
371C22⋊C4—416no—
372C22⋊C4—416no—
373C22⋊C4—48no—
374C22⋊C4—416no—
375C22⋊C4—416no—
376C22⋊C4—88no—
377C22⋊C4—88no—
378C22⋊C4—88no—
379C42—416no—
380C42—88no—
381C42—84no—
382C42—84no—
383C23—24no—
384C23—28no—
385C23—22no—
386C23—24no—
387C23—24no—
388C23—28no—
389C23—24no—
390C23—22no—
391C23—42no—
392C23—41no—
393C23—44no—
394C23—42no—
395C23—42no—
396C23—41no—
397C23—44no—
398C23—44no—
399C23—42no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—42no—
404C23—42no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—42no—
410C23—42no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—42no—
416C23—42no—
417C23—44no—
418C23—44no—
419C23#55914yes—
420C23C22×#4111yescharacteristic
421C23C22×#4112yes—
422C23C22×#4112yes—
423C23C2×H(5,𝔽2)11yescharacteristic
424C23C2×H(5,𝔽2)14yes—
425C23C2×H(5,𝔽2)11yescharacteristic
426C23C2×2−1+412yes—
427C23#58512yes—
428D4—816no—
429D4—88no—
430C2×C4—28no—
431C2×C4—28no—
432C2×C4—28no—
433C2×C4—28no—
434C2×C4—28no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—28no—
441C2×C4—28no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—28no—
446C2×C4—28no—
447C2×C4—28no—
448C2×C4—24no—
449C2×C4—28no—
450C2×C4—432no—
451C2×C4—44no—
452C2×C4—42no—
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—42no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
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—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—88no—
493C22—24no—
494C22—28no—
495C22—44no—
496C22—48no—
497C22—44no—
498C22—42no—
499C22—42no—
500C22—44no—
501C22—84no—
502C22—84no—
503C22—84no—
504C22—84no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—84no—
509C22—84no—
510C22#347512yes—
511C22C2×#55911yescharacteristic
512C22#348114yes—
513C22#348212yes—
514C22#348311yescharacteristic
515C22#349011yescharacteristic
516C22#349212yes—
517C22#349414yes—
518C22#350512yes—
519C22#350812yes—
520C22#352014yes—
521C22#353212yes—
522C22#354312yes—
523C22#354511yescharacteristic
524C22#355011yescharacteristic
525C22#355914yes—
526C4—48no—
527C4—48no—
528C4—44no—
529C4—42no—
530C4—42no—
531C4—44no—
532C4—44no—
533C4—44no—
534C4—44no—
535C4—84no—
536C4—84no—
537C2—44no—
538C2—162no—
539C2#4657812yes—
540C2#4703111yescharacteristic
541C2#4769512yes—
542C2#5962412yes—
543C2#5986111yescharacteristic
544C2#5999414yes—
545C2#6003812yes—
546C2#6004711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17221C2semidirect
2. #16244C2semidirect
3. #15782C2semidirect
4. #15780C2semidirect
5. #15399C2semidirect
6. #15366C2semidirect
8. #14246C2semidirect
9. #14230C2semidirect
10. #14158C2semidirect
12. #14128C2semidirect
13. #14071C2semidirect

Maximal subgroups

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

subgroupindex
#172212
#162442
#157822
#157802
#153992
#153662
#153402
#142462
#142302
#141582
#141522
#141282
#140712