Finite Group Databasebeta

← #4458115

Subgroups — #4458115

Order 512.

2068 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16845C214yesmaximal
2#16630C214yesmaximal, complemented
3#16627C214yesmaximal, complemented
4#16296C214yesmaximal
5#15152C212yesmaximal, complemented
6#15131C212yesmaximal
7#15128C212yesmaximal, complemented
8#15125C212yesmaximal
9#14343C211yesmaximal, characteristic
10#14053C212yesmaximal, complemented
11#13981C212yesmaximal, complemented
12#13852C211yesmaximal, characteristic
13#13777C211yesmaximal, characteristic
14C22×C4⋊Q8C2212yes—
15C22×#117C2211yescharacteristic
16C2×#400C2214yes—
17C2×#396C2214yes—
18C2×#395C2214yes—
19C2×#393C2214yes—
20C2×#391C2214yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#387C2214yes—
24C2×#384C2212yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#1446C2214yes—
32#1446C2214yes—
33#1446C2214yes—
34#1445C2214yes—
35#1445C2214yes—
36#1444C2214yes—
37#1442C2214yes—
38#1442C2214yes—
39#1442C2214yes—
40#1441C2214yes—
41#1441C2214yes—
42#1441C2214yes—
43#1441C2212yes—
44#1441C2212yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1440C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2212yes—
52#1439C2212yes—
53#1437C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57C4⋊#95C2212yes—
58C4⋊#95C2211yescharacteristic
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2212yes—
65C4⋊#95C2211yescharacteristic
66C4×#95C2212yes—
67C23×Q8C2314yes—
68C24×C4C2311yescharacteristic
69C2×C4⋊Q8—28no—
70C2×C4⋊Q8—44no—
71C2×C4⋊Q8—44no—
72C2×#117—44no—
73C2×#117Q818yescomplemented
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2311yescharacteristic
88C22×C22⋊C4C2311yescharacteristic
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2312yes—
92C22×C42C2312yes—
93C22×C42C2312yes—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2314yes—
97#400—48no—
98#396—48no—
99#395—48no—
100#393—48no—
101#391—48no—
102#389—28no—
103#389—44no—
104#387—48no—
105#384—44no—
106#380—28no—
107#380—44no—
108C4×C4⋊C4—44no—
109C4×C4⋊C4—44no—
110C4×C22⋊C4—42no—
111C4×C22⋊C4—42no—
112C2×#95C2314yes—
113C2×#95C2314yes—
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
141C22×Q8—28no—
142C22×Q8—28no—
143C22×Q8—44no—
144C22×Q8—44no—
145C22×Q8—44no—
146C22×Q8—44no—
147C22×Q8—44no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4C2×Q814yes—
153C23×C4C2414yes—
154C23×C4C2414yes—
155C23×C4C2414yes—
156C23×C4C2412yes—
157C23×C4C2412yes—
158C23×C4C2411yescharacteristic
159C23×C4C2414yes—
160C23×C4C2412yes—
161C23×C4C2412yes—
162C23×C4C2412yes—
163C23×C4C2412yes—
164C23×C4C2411yescharacteristic
165C4⋊Q8—432no—
166#117—416no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
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⋊C4C2×Q818yes—
202C2×C4⋊C4#4118yes—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4C2×Q818yes—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—44no—
228C2×C42—44no—
229C2×C42—44no—
230C2×C42C2×Q814yes—
231C2×C42C2×Q814yes—
232#95—28no—
233#95—28no—
234#95—24no—
235#95—28no—
236#95—28no—
237#95—24no—
238#95—28no—
239#95—28no—
240#95—24no—
241#95—24no—
242#95—42no—
243#95—41no—
244#95—41no—
245#95—42no—
246#95—42no—
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—
259#95—44no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—21no—
265C24—22no—
266C24—22no—
267C24—21no—
268C24C22×Q812yes—
269C24C2511yescharacteristic
270C2×Q8—416no—
271C2×Q8—416no—
272C2×Q8—88no—
273C2×Q8—88no—
274C2×Q8—88no—
275C2×Q8—88no—
276C2×Q8—88no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—24no—
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—28no—
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—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4C22×Q814yes—
350C22×C4C22×Q814yes—
351C22×C4C22×Q812yes—
352C22×C4C22×Q812yes—
353C22×C4C2×#4118yes—
354C22×C4C2×#4114yes—
355C22×C4H(5,𝔽2)14yes—
356C22×C4H(5,𝔽2)14yes—
357C22×C4H(5,𝔽2)12yes—
358C22×C4H(5,𝔽2)14yes—
359C22×C42−1+414yes—
360C22×C42−1+414yes—
361C22×C42−1+412yes—
362C4⋊C4—216no—
363C4⋊C4—48no—
364C4⋊C4—416no—
365C4⋊C4—48no—
366C4⋊C4—48no—
367C4⋊C4—416no—
368C4⋊C4—48no—
369C4⋊C4—48no—
370C4⋊C4—48no—
371C4⋊C4—88no—
372C4⋊C4—88no—
373C4⋊C4—88no—
374C22⋊C4—48no—
375C22⋊C4—48no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—88no—
379C22⋊C4—88no—
380C42—28no—
381C42—28no—
382C42—48no—
383C42—48no—
384C42—88no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—24no—
389C23—24no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—41no—
406C23C23×Q811yescharacteristic
407C23C22×#4112yes—
408C23C2×H(5,𝔽2)11yescharacteristic
409C23C2×H(5,𝔽2)12yes—
410C23C2×H(5,𝔽2)12yes—
411C23C2×H(5,𝔽2)12yes—
412C23C2×2−1+411yescharacteristic
413C23C2×2−1+412yes—
414C23C2×2−1+412yes—
415Q8—832no—
416C2×C4—22no—
417C2×C4—28no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—28no—
427C2×C4—24no—
428C2×C4—28no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—416no—
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—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4#53314yes—
493C2×C4#53818yes—
494C2×C4#54218yes—
495C2×C4#54314yes—
496C2×C4#54414yes—
497C22—42no—
498C22—44no—
499C22—42no—
500C22—42no—
501C22—44no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22C2×#53311yescharacteristic
507C22C2×#53812yes—
508C22C2×#54212yes—
509C22C2×#54311yescharacteristic
510C22C2×#54411yescharacteristic
511C22#350312yes—
512C22#350312yes—
513C22#351912yes—
514C22#352012yes—
515C22#352212yes—
516C22#352312yes—
517C22#353612yes—
518C22#353612yes—
519C22#353612yes—
520C22#353711yescharacteristic
521C22#353811yescharacteristic
522C22#354811yescharacteristic
523C22#354912yes—
524C22#355012yes—
525C22#355012yes—
526C22#355611yescharacteristic
527C4—42no—
528C4—44no—
529C4—48no—
530C4—44no—
531C4—44no—
532C4—44no—
533C4—44no—
534C4—44no—
535C4—84no—
536C4—84no—
537C4—84no—
538C4#266518yes—
539C2—82no—
540C2C2×#266511yescharacteristic
541C2#3557812yes—
542C2#3730612yes—
543C2#3730811yescharacteristic
544C2#3730911yescharacteristic
545C2#4255812yes—
546C2#4258512yes—
547C2#4260612yes—
548C2#4286212yes—
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
2. #16630C2semidirect
3. #16627C2semidirect
5. #15152C2semidirect
7. #15128C2semidirect
10. #14053C2semidirect
11. #13981C2semidirect
73. C2×#117Q8semidirect

Maximal subgroups

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

subgroupindex
#168452
#166302
#166272
#162962
#151522
#151312
#151282
#151252
#143432
#140532
#139812
#138522
#137772