Finite Group Databasebeta

← #4458136

Subgroups — #4458136

Order 512.

1780 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17001C214yesmaximal, complemented
2#16970C214yesmaximal, complemented
3#16954C214yesmaximal
4#16725C214yesmaximal
5#15235C212yesmaximal, complemented
6#15220C212yesmaximal
7#15218C212yesmaximal, complemented
8#15216C212yesmaximal
9#14343C211yesmaximal, characteristic
10#14054C212yesmaximal, complemented
11#13982C212yesmaximal, complemented
12#13852C211yesmaximal, characteristic
13#13777C211yesmaximal, characteristic
14C22×#125C2212yes—
15C22×#117C2211yescharacteristic
16C2×#389C2212yes—
17C2×#389C2212yes—
18C2×#382C2212yes—
19C2×#380C2212yes—
20C2×#380C2212yes—
21C2×C4×C4⋊C4C2212yes—
22C2×C4×C4⋊C4C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25#1445C2214yes—
26#1445C2214yes—
27#1445C2214yes—
28#1444C2214yes—
29#1444C2214yes—
30#1443C2214yes—
31#1443C2214yes—
32#1443C2214yes—
33#1443C2214yes—
34#1442C2214yes—
35#1442C2214yes—
36#1442C2214yes—
37#1442C2214yes—
38#1441C2212yes—
39#1441C2212yes—
40#1441C2214yes—
41#1441C2214yes—
42#1441C2214yes—
43#1441C2214yes—
44#1440C2214yes—
45#1439C2212yes—
46#1439C2212yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1437C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55C4⋊#95C2212yes—
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2214yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2212yes—
64C4⋊#95C2211yescharacteristic
65#1434C2214yes—
66C4×#95C2212yes—
67C24×C4C2311yescharacteristic
68C2×#125—28no—
69C2×#125—44no—
70C2×#125—44no—
71C2×#117—44no—
72C2×#117Q818yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2311yescharacteristic
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2311yescharacteristic
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2312yes—
90C22×C42C2312yes—
91C22×C42C2312yes—
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2314yes—
95#389—28no—
96#389—44no—
97#382—44no—
98#380—28no—
99#380—44no—
100C4×C4⋊C4—44no—
101C4×C4⋊C4—44no—
102C4×C22⋊C4—42no—
103C4×C22⋊C4—42no—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2312yes—
107C2×#95C2312yes—
108C2×#95C2312yes—
109C2×#95C2312yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2311yescharacteristic
132C2×#95C2311yescharacteristic
133C2×#95C2314yes—
134C25C2411yescharacteristic
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4C2×Q814yes—
140C23×C4C2414yes—
141C23×C4C2412yes—
142C23×C4C2412yes—
143C23×C4C2412yes—
144C23×C4C2412yes—
145C23×C4C2414yes—
146C23×C4C2414yes—
147C23×C4C2412yes—
148C23×C4C2412yes—
149C23×C4C2411yescharacteristic
150C23×C4C2414yes—
151C23×C4C2411yescharacteristic
152#125—432no—
153#117—416no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—28no—
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—28no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4C2×Q818yes—
186C2×C4⋊C4#4118yes—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4C2×Q818yes—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42—24no—
207C2×C42—24no—
208C2×C42—24no—
209C2×C42—24no—
210C2×C42—24no—
211C2×C42—44no—
212C2×C42—44no—
213C2×C42—44no—
214C2×C42C2×Q814yes—
215C2×C42C2×Q814yes—
216#95—28no—
217#95—28no—
218#95—44no—
219#95—42no—
220#95—44no—
221#95—42no—
222#95—44no—
223#95—44no—
224#95—44no—
225#95—44no—
226#95—44no—
227#95—42no—
228#95—42no—
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—41no—
238#95—41no—
239#95—44no—
240#95—42no—
241#95—42no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—44no—
246C24—22no—
247C24—22no—
248C24—21no—
249C24—21no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24C22×Q812yes—
255C24C2511yescharacteristic
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—28no—
279C22×C4—28no—
280C22×C4—28no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—48no—
331C22×C4—48no—
332C22×C4C22×Q814yes—
333C22×C4C22×Q814yes—
334C22×C4C22×Q812yes—
335C22×C4C22×Q812yes—
336C22×C4C2×#4118yes—
337C22×C4C2×#4114yes—
338C22×C4H(5,𝔽2)12yes—
339C22×C4H(5,𝔽2)14yes—
340C22×C42−1+412yes—
341C4⋊C4—216no—
342C4⋊C4—48no—
343C4⋊C4—416no—
344C4⋊C4—416no—
345C4⋊C4—48no—
346C4⋊C4—48no—
347C4⋊C4—48no—
348C4⋊C4—48no—
349C4⋊C4—48no—
350C4⋊C4—88no—
351C4⋊C4—88no—
352C22⋊C4—48no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—48no—
356C22⋊C4—88no—
357C22⋊C4—88no—
358C42—28no—
359C42—28no—
360C42—48no—
361C42—48no—
362C42—88no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—24no—
367C23—24no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—41no—
375C23—41no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—41no—
383C23—41no—
384C23C23×Q811yescharacteristic
385C23C22×#4112yes—
386C23C2×H(5,𝔽2)11yescharacteristic
387C23C2×H(5,𝔽2)12yes—
388C23C2×2−1+411yescharacteristic
389C23#58512yes—
390C23#58512yes—
391C23#58512yes—
392C23#58512yes—
393C2×C4—22no—
394C2×C4—28no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—28no—
404C2×C4—28no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—416no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4#53314yes—
470C2×C4#53818yes—
471C2×C4#54218yes—
472C2×C4#54314yes—
473C2×C4#54414yes—
474C22—42no—
475C22—44no—
476C22—42no—
477C22—42no—
478C22—44no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22C2×#53311yescharacteristic
484C22C2×#53812yes—
485C22C2×#54212yes—
486C22C2×#54311yescharacteristic
487C22C2×#54411yescharacteristic
488C22#350412yes—
489C22#350412yes—
490C22#352112yes—
491C22#352112yes—
492C22#352412yes—
493C22#352412yes—
494C22#353911yescharacteristic
495C22#353912yes—
496C22#353912yes—
497C22#353911yescharacteristic
498C22#354012yes—
499C22#355112yes—
500C22#355112yes—
501C22#355712yes—
502C22#355711yescharacteristic
503C22#355711yescharacteristic
504C4—48no—
505C4—42no—
506C4—44no—
507C4—44no—
508C4—82no—
509C4—82no—
510C4—82no—
511C4—82no—
512C4—84no—
513C4—84no—
514C4—84no—
515C4#266518yes—
516C2—82no—
517C2C2×#266511yescharacteristic
518C2#3564112yes—
519C2#3742512yes—
520C2#3742711yescharacteristic
521C2#3742811yescharacteristic
522C2#4272512yes—
523C2#4274512yes—
524C2#4276012yes—
525C2#4290612yes—
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. #17001C2semidirect
2. #16970C2semidirect
5. #15235C2semidirect
7. #15218C2semidirect
10. #14054C2semidirect
11. #13982C2semidirect

Maximal subgroups

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

subgroupindex
#170012
#169702
#169542
#167252
#152352
#152202
#152182
#152162
#143432
#140542
#139822
#138522
#137772