Finite Group Databasebeta

← #4474248

Subgroups — #4474248

Order 512.

2402 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17090C214yesmaximal, complemented
2#17027C214yesmaximal, complemented
3#16903C211yesmaximal, characteristic, complemented
4#16537C214yesmaximal, complemented
5#16252C211yesmaximal, characteristic
6#15732C214yesmaximal, complemented
7#15450C214yesmaximal, complemented
8#15299C214yesmaximal, complemented
9#14719C212yesmaximal, complemented
10#14418C212yesmaximal, complemented
11#14342C211yesmaximal, characteristic, complemented
12C22×#121C2212yes—
13C23×C4⋊C4C2211yescharacteristic
14C2×#397C2214yes—
15C2×#394C2212yes—
16C2×#394C2212yes—
17C2×#394C2214yes—
18C2×#392C2214yes—
19C2×#392C2214yes—
20C2×#390C2214yes—
21C2×#388C2214yes—
22C2×#387C2214yes—
23C2×#386C2214yes—
24C2×#386C2214yes—
25C2×#381C2212yes—
26#1445C2214yes—
27#1445C2214yes—
28#1442C2214yes—
29#1442C2214yes—
30#1442C2214yes—
31#1441C2212yes—
32#1441C2214yes—
33#1441C2212yes—
34#1440C2214yes—
35#1440C2214yes—
36#1440C2214yes—
37#1439C2214yes—
38#1439C2214yes—
39#1439C2214yes—
40#1439C2214yes—
41#1439C2214yes—
42#1439C2214yes—
43#1439C2214yes—
44#1438C2214yes—
45#1438C2214yes—
46#1438C2214yes—
47#1438C2214yes—
48#1438C2214yes—
49#1437C2214yes—
50#1437C2214yes—
51#1434C2214yes—
52#1434C2214yes—
53C22⋊#95C2211yescharacteristic
54C22⋊#95C2211yescharacteristic
55C22⋊#95C2212yes—
56C22⋊#95C2212yes—
57C23×D4C2314yes—
58C23×D4C2312yes—
59C23×D4C2314yes—
60C24×C4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C2×#121—28no—
64C2×#121—48no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—44no—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2314yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2312yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2312yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86C22×C42C2314yes—
87#397—48no—
88#394—48no—
89#394—44no—
90#394—44no—
91#392—48no—
92#392—48no—
93#390—48no—
94#388—48no—
95#387—48no—
96#386—48no—
97#386—48no—
98#381—44no—
99C2×#95C2314yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2312yes—
106C2×#95C2312yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C25C2414yes—
119C25C2411yescharacteristic
120C22×D4—28no—
121C22×D4—28no—
122C22×D4—28no—
123C22×D4—24no—
124C22×D4—28no—
125C22×D4—44no—
126C22×D4—44no—
127C22×D4—44no—
128C22×D4—44no—
129C22×D4—44no—
130C22×D4—44no—
131C22×D4—44no—
132C22×D4—44no—
133C22×D4—44no—
134C22×D4—44no—
135C22×D4—44no—
136C22×D4—42no—
137C22×D4—42no—
138C22×D4—44no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—22no—
143C23×C4—41no—
144C23×C4—41no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—44no—
148C23×C4—44no—
149C23×C4#4114yes—
150C23×C4C2414yes—
151C23×C4C2414yes—
152C23×C4C2414yes—
153C23×C4C2411yescharacteristic
154C23×C4C2411yescharacteristic
155C23×C4C2414yes—
156C23×C4C2412yes—
157C23×C4C2412yes—
158C23×C4C2414yes—
159#121—432no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—416no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C42—28no—
218C2×C42—44no—
219C2×C42—44no—
220#95—28no—
221#95—28no—
222#95—28no—
223#95—28no—
224#95—44no—
225#95—44no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—42no—
237#95—42no—
238#95—44no—
239C24—22no—
240C24—24no—
241C24—21no—
242C24—21no—
243C24—24no—
244C24—24no—
245C24—24no—
246C24—24no—
247C24—24no—
248C24—24no—
249C24—24no—
250C24—24no—
251C24—24no—
252C24—24no—
253C24—24no—
254C24—24no—
255C24—24no—
256C24—24no—
257C24—24no—
258C24—24no—
259C24C2×#4112yes—
260C24H(5,𝔽2)14yes—
261C24C2511yescharacteristic
262C2×D4—416no—
263C2×D4—416no—
264C2×D4—416no—
265C2×D4—48no—
266C2×D4—416no—
267C2×D4—84no—
268C2×D4—84no—
269C2×D4—88no—
270C2×D4—88no—
271C2×D4—88no—
272C2×D4—88no—
273C2×D4—88no—
274C2×D4—88no—
275C2×D4—88no—
276C2×D4—88no—
277C2×D4—88no—
278C2×D4—88no—
279C2×D4—88no—
280C2×D4—88no—
281C22×C4—28no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
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—44no—
333C22×C4—48no—
334C22×C4—48no—
335C22×C4—48no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—84no—
340C22×C4—88no—
341C22×C4—88no—
342C22×C4C2×#4114yes—
343C22×C4H(5,𝔽2)18yes—
344C22×C4H(5,𝔽2)14yes—
345C4⋊C4—216no—
346C4⋊C4—48no—
347C4⋊C4—48no—
348C4⋊C4—88no—
349C4⋊C4—88no—
350C22⋊C4—416no—
351C22⋊C4—416no—
352C22⋊C4—88no—
353C22⋊C4—88no—
354C22⋊C4—88no—
355C22⋊C4—88no—
356C22⋊C4—84no—
357C22⋊C4—88no—
358C22⋊C4—84no—
359C22⋊C4—88no—
360C22⋊C4—88no—
361C22⋊C4—88no—
362C22⋊C4—88no—
363C22⋊C4—88no—
364C42—88no—
365C23—22no—
366C23—24no—
367C23—22no—
368C23—24no—
369C23—24no—
370C23—24no—
371C23—28no—
372C23—28no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23—44no—
379C23—44no—
380C23—44no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23—44no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23—44no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23#54014yes—
413C23C22×#4111yescharacteristic
414C23C2×H(5,𝔽2)12yes—
415C23C2×H(5,𝔽2)14yes—
416C23C2×H(5,𝔽2)12yes—
417C23#58511yescharacteristic
418C23#58511yescharacteristic
419C23#58514yes—
420D4—832no—
421D4—816no—
422D4—832no—
423C2×C4—24no—
424C2×C4—28no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—28no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—28no—
432C2×C4—416no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
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—88no—
478C2×C4—88no—
479C22—28no—
480C22—24no—
481C22—44no—
482C22—42no—
483C22—48no—
484C22—48no—
485C22—42no—
486C22—84no—
487C22—84no—
488C22—84no—
489C22—84no—
490C22—84no—
491C22—84no—
492C22—84no—
493C22—84no—
494C22—84no—
495C22—84no—
496C22—84no—
497C22—84no—
498C22—84no—
499C22—84no—
500C22—84no—
501C22C2×#54011yescharacteristic
502C22#349012yes—
503C22#349011yescharacteristic
504C22#349311yescharacteristic
505C22#351012yes—
506C22#353414yes—
507C22#353414yes—
508C22#353412yes—
509C22#353412yes—
510C22#353914yes—
511C22#354914yes—
512C22#356214yes—
513C22#356214yes—
514C4—48no—
515C4—44no—
516C4—44no—
517C4—84no—
518C4—84no—
519C4—84no—
520C4—84no—
521C4—81no—
522C4—81no—
523C2—44no—
524C2—164no—
525C2#3673111yescharacteristic
526C2#4054612yes—
527C2#4645611yescharacteristic
528C2#4756311yescharacteristic
529C2#4788612yes—
530C2#5516214yes—
531C2#5728814yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#16252C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17090C2semidirect
2. #17027C2semidirect
3. #16903C2semidirect
4. #16537C2semidirect
6. #15732C2semidirect
7. #15450C2semidirect
8. #15299C2semidirect
9. #14719C2semidirect
10. #14418C2semidirect
11. #14342C2semidirect

Maximal subgroups

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

subgroupindex
#170902
#170272
#169032
#165372
#162522
#157322
#154502
#152992
#147192
#144182
#143422