Finite Group Databasebeta

← #4464001

Subgroups — #4464001

Order 512.

1720 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17287C214yesmaximal, complemented
2#17202C214yesmaximal, complemented
3#17180C214yesmaximal
4#17158C214yesmaximal
5#15218C212yesmaximal, complemented
6#15216C211yesmaximal, characteristic
7#15200C212yesmaximal, complemented
8#15183C212yesmaximal
9#15150C211yesmaximal, characteristic
10#15146C212yesmaximal, complemented
11#15139C212yesmaximal, complemented
12#14264C212yesmaximal
13#13844C211yesmaximal, characteristic
14C22×#125C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#400C2214yes—
17C2×#400C2214yes—
18C2×#397C2214yes—
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#384C2211yescharacteristic
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28#1445C2214yes—
29#1445C2214yes—
30#1445C2214yes—
31#1444C2214yes—
32#1444C2212yes—
33#1444C2214yes—
34#1444C2214yes—
35#1443C2214yes—
36#1443C2214yes—
37#1442C2214yes—
38#1442C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1442C2212yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1440C2214yes—
51#1439C2212yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2212yes—
56#1439C2214yes—
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2211yescharacteristic
66C4×#95C2212yes—
67C24×C4C2311yescharacteristic
68C2×#125—42no—
69C2×#125—44no—
70C2×#117—44no—
71C2×#117—42no—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C22⋊C4C2312yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2311yescharacteristic
88C22×C22⋊C4C2311yescharacteristic
89C22×C22⋊C4C2314yes—
90C22×C42C2312yes—
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2312yes—
94#400—48no—
95#400—48no—
96#397—48no—
97#389—44no—
98#389—44no—
99#389—44no—
100#384—42no—
101#380—28no—
102#380—44no—
103C4×C4⋊C4—44no—
104C4×C22⋊C4—24no—
105C4×C22⋊C4—42no—
106C2×#95C2314yes—
107C2×#95C2312yes—
108C2×#95C2312yes—
109C2×#95C2314yes—
110C2×#95C2312yes—
111C2×#95C2314yes—
112C2×#95C2311yescharacteristic
113C2×#95C2314yes—
114C2×#95C2311yescharacteristic
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2312yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C25C2411yescharacteristic
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4#4114yes—
142C23×C4C2414yes—
143C23×C4C2414yes—
144C23×C4C2412yes—
145C23×C4C2412yes—
146C23×C4C2412yes—
147C23×C4C2414yes—
148C23×C4C2414yes—
149C23×C4C2412yes—
150C23×C4C2411yescharacteristic
151C23×C4C2412yes—
152C23×C4C2412yes—
153C23×C4C2411yescharacteristic
154#125—416no—
155#117—416no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C42—24no—
209C2×C42—24no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—22no—
213C2×C42—24no—
214C2×C42—22no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—24no—
218#95—22no—
219#95—22no—
220#95—28no—
221#95—28no—
222#95—24no—
223#95—28no—
224#95—24no—
225#95—24no—
226#95—28no—
227#95—42no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—42no—
235#95—42no—
236#95—42no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—44no—
248C24—22no—
249C24—22no—
250C24—21no—
251C24—22no—
252C24—22no—
253C24—21no—
254C24—22no—
255C24—22no—
256C24C2×#4112yes—
257C24C2511yescharacteristic
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—216no—
275C22×C4—28no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—48no—
335C22×C4C2×#4112yes—
336C22×C4C2×#4112yes—
337C22×C4H(5,𝔽2)14yes—
338C22×C4H(5,𝔽2)14yes—
339C22×C4H(5,𝔽2)12yes—
340C22×C42−1+414yes—
341C22×C42−1+414yes—
342C22×C42−1+412yes—
343C4⋊C4—48no—
344C4⋊C4—48no—
345C4⋊C4—48no—
346C4⋊C4—416no—
347C4⋊C4—48no—
348C4⋊C4—48no—
349C4⋊C4—48no—
350C4⋊C4—48no—
351C4⋊C4—88no—
352C4⋊C4—88no—
353C4⋊C4—88no—
354C4⋊C4—88no—
355C22⋊C4—48no—
356C22⋊C4—48no—
357C22⋊C4—44no—
358C22⋊C4—44no—
359C22⋊C4—88no—
360C22⋊C4—88no—
361C42—48no—
362C42—48no—
363C42—44no—
364C42—44no—
365C23—22no—
366C23—24no—
367C23—24no—
368C23—22no—
369C23—22no—
370C23—41no—
371C23—41no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—41no—
382C23—41no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23C22×#4111yescharacteristic
387C23C2×H(5,𝔽2)12yes—
388C23C2×H(5,𝔽2)11yescharacteristic
389C23C2×H(5,𝔽2)12yes—
390C23C2×2−1+412yes—
391C23C2×2−1+411yescharacteristic
392C23C2×2−1+412yes—
393C23#58512yes—
394C23#58512yes—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—416no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4#53918yes—
472C2×C4#53914yes—
473C2×C4H(3,𝔽4)14yes—
474C2×C4#56314yes—
475C2×C4#56318yes—
476C22—42no—
477C22—44no—
478C22—42no—
479C22—42no—
480C22—44no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22C2×#53911yescharacteristic
486C22C2×#53912yes—
487C22#349512yes—
488C22#350811yescharacteristic
489C22#351211yescharacteristic
490C22C2×H(3,𝔽4)11yescharacteristic
491C22C2×#56311yescharacteristic
492C22C2×#56312yes—
493C22#354012yes—
494C22#354112yes—
495C22#354212yes—
496C22#354412yes—
497C22#354612yes—
498C22#354712yes—
499C22#354812yes—
500C22#355412yes—
501C22#355611yescharacteristic
502C22#355711yescharacteristic
503C22#355912yes—
504C22#356312yes—
505C22#356512yes—
506C4—44no—
507C4—44no—
508C4—42no—
509C4—82no—
510C4—82no—
511C4—82no—
512C4—84no—
513C4—84no—
514C4—82no—
515C4—84no—
516C4—84no—
517C4#283518yes—
518C2—82no—
519C2C2×#283511yescharacteristic
520C2#3731811yescharacteristic
521C2#3733512yes—
522C2#3738211yescharacteristic
523C2#3740712yes—
524C2#5298312yes—
525C2#5302312yes—
526C2#5304612yes—
527C2#5310712yes—
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. #17287C2semidirect
2. #17202C2semidirect
5. #15218C2semidirect
7. #15200C2semidirect
10. #15146C2semidirect
11. #15139C2semidirect

Maximal subgroups

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

subgroupindex
#172872
#172022
#171802
#171582
#152182
#152162
#152002
#151832
#151502
#151462
#151392
#142642
#138442