Finite Group Databasebeta

← #4464092

Subgroups — #4464092

Order 512.

2130 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2638C212yesmaximal
2#17221C211yesmaximal, characteristic, complemented
3#17193C212yesmaximal
4#17189C214yesmaximal, complemented
5#17143C212yesmaximal, complemented
6#17118C212yesmaximal
7#17097C214yesmaximal, complemented
8#17085C211yesmaximal, characteristic, complemented
9#15206C212yesmaximal, complemented
10#15122C212yesmaximal, complemented
11#14837C214yesmaximal, complemented
12#14763C214yesmaximal, complemented
13#13847C211yesmaximal, characteristic
14C22×C22⋊Q8C2212yes—
15C22×C4×D4C2211yescharacteristic
16#2638—216no—
17C2×#398C2212yes—
18C2×#397C2212yes—
19C2×#397C2214yes—
20C2×#397C2214yes—
21C2×#395C2212yes—
22C2×#395C2214yes—
23C2×#394C2214yes—
24C2×#393C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#387C2214yes—
28C2×#385C2214yes—
29C2×#381C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2214yes—
32C2×#376C2211yescharacteristic
33#1446C2214yes—
34#1445C2212yes—
35#1445C2212yes—
36#1445C2211yescharacteristic
37#1445C2211yescharacteristic
38#1445C2214yes—
39#1444C2212yes—
40#1444C2214yes—
41#1444C2212yes—
42#1443C2214yes—
43#1443C2214yes—
44#1442C2212yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2212yes—
48#1441C2212yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2212yes—
52#1441C2214yes—
53#1441C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2211yescharacteristic
63#1438C2211yescharacteristic
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2212yes—
67#1434C2212yes—
68C23×Q8C2312yes—
69C23×D4C2311yescharacteristic
70C24×C4C2312yes—
71C2×C22⋊Q8—28no—
72C2×C22⋊Q8—44no—
73C2×C22⋊Q8—44no—
74C2×C4×D4—42no—
75C2×C4×D4—42no—
76C2×C4×D4D418yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2312yes—
95C22×C42C2312yes—
96C22×C42C2311yescharacteristic
97C22×C42C2312yes—
98#398—28no—
99#397—28no—
100#397—216no—
101#397—48no—
102#395—28no—
103#395—48no—
104#394—216no—
105#393—28no—
106#389—44no—
107#389—44no—
108#387—216no—
109#385—216no—
110#381—28no—
111C4×C4⋊C4—42no—
112C4×C22⋊C4—48no—
113#376—42no—
114C2×#95C2314yes—
115C2×#95C2312yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2312yes—
119C2×#95C2314yes—
120C2×#95C2312yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2312yes—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2311yescharacteristic
136C2×#95C2311yescharacteristic
137C2×#95C2311yescharacteristic
138C2×#95C2311yescharacteristic
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C25C2412yes—
144C22×Q8—24no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×Q8—44no—
148C22×Q8—44no—
149C22×D4—41no—
150C22×D4—41no—
151C22×D4—44no—
152C22×D4C2×D414yes—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4C2×D418yes—
158C23×C4C2414yes—
159C23×C4C2414yes—
160C23×C4C2414yes—
161C23×C4C2414yes—
162C23×C4C2412yes—
163C23×C4C2411yescharacteristic
164C23×C4C2412yes—
165C23×C4C2411yescharacteristic
166C23×C4C2412yes—
167C23×C4C2412yes—
168C23×C4C2412yes—
169C23×C4C2411yescharacteristic
170C22⋊Q8—432no—
171C4×D4—416no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4C2×D414yes—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—44no—
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⋊C4C2×D418yes—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—42no—
229C2×C42—44no—
230C2×C42C2×D414yes—
231#95—28no—
232#95—24no—
233#95—24no—
234#95—28no—
235#95—28no—
236#95—28no—
237#95—28no—
238#95—28no—
239#95—28no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—42no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—44no—
248#95—44no—
249#95—44no—
250#95—42no—
251#95—41no—
252#95—41no—
253#95—44no—
254#95—41no—
255#95—41no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260C24—24no—
261C24—22no—
262C24—22no—
263C24—24no—
264C24—24no—
265C24—24no—
266C24—24no—
267C24—24no—
268C24C22×D414yes—
269C24C2511yescharacteristic
270C2×Q8—48no—
271C2×Q8—88no—
272C2×Q8—88no—
273C2×Q8—84no—
274C2×Q8—84no—
275C2×D4—44no—
276C2×D4—48no—
277C2×D4—48no—
278C2×D4—48no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—21no—
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—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—216no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
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—416no—
339C22×C4—88no—
340C22×C4—84no—
341C22×C4—84no—
342C22×C4C22×D412yes—
343C22×C4C22×D412yes—
344C22×C4C22×D414yes—
345C22×C4C22×D412yes—
346C22×C4H(5,𝔽2)14yes—
347C22×C4H(5,𝔽2)12yes—
348C22×C4H(5,𝔽2)12yes—
349C22×C42−1+414yes—
350C22×C42−1+414yes—
351C4⋊C4—44no—
352C4⋊C4—48no—
353C4⋊C4—416no—
354C4⋊C4—48no—
355C4⋊C4—48no—
356C4⋊C4—48no—
357C4⋊C4—416no—
358C4⋊C4—88no—
359C4⋊C4—84no—
360C4⋊C4—88no—
361C4⋊C4—84no—
362C22⋊C4—48no—
363C22⋊C4—416no—
364C22⋊C4—416no—
365C22⋊C4—416no—
366C22⋊C4—416no—
367C22⋊C4—88no—
368C22⋊C4—88no—
369C42—28no—
370C42—48no—
371C42—84no—
372C23—24no—
373C23—24no—
374C23—24no—
375C23—28no—
376C23—28no—
377C23—44no—
378C23—44no—
379C23—44no—
380C23—44no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23—44no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23—44no—
393C23C23×D411yescharacteristic
394C23C2×H(5,𝔽2)11yescharacteristic
395C23C2×H(5,𝔽2)11yescharacteristic
396C23C2×H(5,𝔽2)12yes—
397C23C2×2−1+412yes—
398C23C2×2−1+412yes—
399C23#58514yes—
400C23#58512yes—
401Q8—816no—
402D4—416no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—28no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—28no—
415C2×C4—28no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—28no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
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—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—816no—
470C2×C4#53414yes—
471C22—44no—
472C22—44no—
473C22—44no—
474C22—48no—
475C22—48no—
476C22—84no—
477C22—84no—
478C22—84no—
479C22—84no—
480C22C2×#53411yescharacteristic
481C22#344912yes—
482C22#349612yes—
483C22#349912yes—
484C22#352912yes—
485C22#352912yes—
486C22#353412yes—
487C22#353514yes—
488C22#353812yes—
489C22#354114yes—
490C22#354311yescharacteristic
491C22#354511yescharacteristic
492C22#355011yescharacteristic
493C22#355011yescharacteristic
494C22#355714yes—
495C22#356514yes—
496C4—24no—
497C4—44no—
498C4—42no—
499C4—42no—
500C4—44no—
501C4—44no—
502C4—44no—
503C4—84no—
504C4—84no—
505C4—84no—
506C4—84no—
507C4—82no—
508C2—84no—
509C2#3665211yescharacteristic
510C2#3674314yes—
511C2#3712911yescharacteristic
512C2#3738611yescharacteristic
513C2#5262312yes—
514C2#5267212yes—
515C2#5271412yes—
516C2#5289212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17221C2semidirect
4. #17189C2semidirect
5. #17143C2semidirect
7. #17097C2semidirect
8. #17085C2semidirect
9. #15206C2semidirect
10. #15122C2semidirect
11. #14837C2semidirect
12. #14763C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26382
#172212
#171932
#171892
#171432
#171182
#170972
#170852
#152062
#151222
#148372
#147632
#138472