Finite Group Databasebeta

← #4430414

Subgroups — #4430414

Order 512.

2865 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×#2840C211yesmaximal, characteristic, complemented
2C2×#2622C211yesmaximal, characteristic
3#16831C214yesmaximal, complemented
4#16368C214yesmaximal, complemented
5#15468C214yesmaximal, complemented
6#15371C214yesmaximal, complemented
7#15050C214yesmaximal, complemented
8#14817C214yesmaximal, complemented
9#14255C212yesmaximal
10#13869C212yesmaximal, complemented
11C22⋊C42C211yesmaximal, characteristic, complemented
12C22×#120C2211yescharacteristic
13C23×C22⋊C4C2211yescharacteristic
14C23×C22⋊C4C2212yes—
15#2840—28no—
16#2622—28no—
17C2×#397C2214yes—
18C2×#394C2214yes—
19C2×#393C2214yes—
20C2×#393C2214yes—
21C2×#393C2214yes—
22C2×#393C2214yescomplemented
23C2×#391C2214yescomplemented
24C2×#386C2212yes—
25C2×#386C2214yes—
26C2×#385C2212yes—
27C2×#381C2214yes—
28C2×#379C2212yes—
29C2×#379C2211yescharacteristic
30C2×C4×C22⋊C4C2214yes—
31#1446C2214yescomplemented
32#1446C2214yescomplemented
33#1445C2214yescomplemented
34#1443C2214yescomplemented
35#1443C2214yescomplemented
36#1443C2214yescomplemented
37#1443C2214yescomplemented
38#1442C2214yescomplemented
39#1442C2214yescomplemented
40#1440C2214yes—
41#1440C2214yes—
42#1440C2214yes—
43#1440C2212yes—
44#1440C2212yes—
45#1440C2214yescomplemented
46#1440C2214yescomplemented
47#1438C2214yes—
48#1438C2212yes—
49#1438C2212yes—
50#1437C2214yes—
51#1437C2214yescomplemented
52#1434C2214yescomplemented
53#1434C2214yescomplemented
54C22⋊#95C2214yes—
55C22⋊#95C2214yes—
56C22⋊#95C2214yes—
57C22⋊#95C2214yes—
58C22⋊#95C2214yes—
59C26C2311yescharacteristic
60C23×Q8C2314yes—
61C23×D4C2312yes—
62C23×D4C2311yescharacteristic
63C24×C4C2312yes—
64C24×C4C2314yes—
65C2×#120—24no—
66C2×#120—28no—
67C22×C4⋊C4C2314yes—
68C22×C22⋊C4—24no—
69C22×C22⋊C4—28no—
70C22×C22⋊C4—28no—
71C22×C22⋊C4—48no—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2312yes—
74C22×C22⋊C4C2312yes—
75C22×C22⋊C4C2311yescharacteristic
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2312yes—
78C22×C22⋊C4C2312yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2312yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2312yes—
87C22×C42C2314yes—
88#397—216no—
89#394—216no—
90#393—216no—
91#393—216no—
92#393—216no—
93#393—48no—
94#391—48no—
95#386—28no—
96#386—48no—
97#385—28no—
98#381—48no—
99#379—28no—
100#379D418yes—
101C4×C22⋊C4—48no—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
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—
124C25—44no—
125C25—44no—
126C25—44no—
127C25C2×D414yes—
128C25#4118yes—
129C25C2411yescharacteristic
130C25C2411yescharacteristic
131C25C2412yes—
132C22×Q8—28no—
133C22×Q8—44no—
134C22×Q8—44no—
135C22×Q8—44no—
136C22×Q8—44no—
137C22×Q8—44no—
138C22×Q8—44no—
139C22×D4—22no—
140C22×D4—24no—
141C22×D4—24no—
142C22×D4—24no—
143C22×D4—24no—
144C22×D4—44no—
145C22×D4—44no—
146C22×D4—44no—
147C23×C4—24no—
148C23×C4—28no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—28no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4C2414yes—
160C23×C4C2414yes—
161C23×C4C2414yes—
162C23×C4C2414yes—
163C23×C4C2412yes—
164C23×C4C2411yescharacteristic
165C23×C4C2414yes—
166C23×C4C2414yes—
167#120—416no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—432no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—416no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
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⋊C4C2×D418yes—
211C2×C22⋊C4C2×D4116yes—
212C2×C42—28no—
213C2×C42—44no—
214C2×C42—44no—
215#95—28no—
216#95—28no—
217#95—28no—
218#95—28no—
219#95—28no—
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—44no—
237C24—24no—
238C24—24no—
239C24—24no—
240C24—24no—
241C24—24no—
242C24—24no—
243C24—24no—
244C24—24no—
245C24—24no—
246C24—21no—
247C24—22no—
248C24—24no—
249C24—24no—
250C24—24no—
251C24—48no—
252C24—48no—
253C24—416no—
254C24—416no—
255C24—416no—
256C24—48no—
257C24—88no—
258C24—88no—
259C24—88no—
260C24—88no—
261C24#124116yes—
262C24C22×D414yes—
263C24C22×D412yes—
264C24C2×#4114yes—
265C24C2×#4114yes—
266C24C2×#4114yes—
267C24C2511yescharacteristic
268C2×Q8—416no—
269C2×Q8—88no—
270C2×Q8—88no—
271C2×Q8—88no—
272C2×Q8—88no—
273C2×Q8—88no—
274C2×Q8—88no—
275C2×D4—416no—
276C2×D4—48no—
277C2×D4—48no—
278C2×D4—44no—
279C2×D4—84no—
280C2×D4—88no—
281C2×D4—84no—
282C2×D4—88no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—21no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
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—24no—
319C22×C4—24no—
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—416no—
332C22×C4—416no—
333C22×C4—416no—
334C22×C4—48no—
335C22×C4—88no—
336C22×C4—88no—
337C22×C4—88no—
338C22×C4—84no—
339C22×C4—84no—
340C22×C4—88no—
341C22×C4—88no—
342C22×C4—88no—
343C22×C4C22×D418yes—
344C22×C4H(5,𝔽2)18yes—
345C4⋊C4—88no—
346C22⋊C4—216no—
347C22⋊C4—48no—
348C22⋊C4—416no—
349C22⋊C4—416no—
350C22⋊C4—48no—
351C22⋊C4—416no—
352C22⋊C4—48no—
353C22⋊C4—44no—
354C22⋊C4—88no—
355C22⋊C4—88no—
356C22⋊C4—84no—
357C22⋊C4—84no—
358C22⋊C4—88no—
359C22⋊C4—88no—
360C22⋊C4—88no—
361C42—88no—
362C23—28no—
363C23—28no—
364C23—28no—
365C23—24no—
366C23—24no—
367C23—24no—
368C23—24no—
369C23—24no—
370C23—24no—
371C23—28no—
372C23—24no—
373C23—28no—
374C23—24no—
375C23—28no—
376C23—24no—
377C23—28no—
378C23—28no—
379C23—28no—
380C23—416no—
381C23—432no—
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—41no—
395C23—44no—
396C23—42no—
397C23—42no—
398C23—44no—
399C23—42no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23—44no—
405C23—832no—
406C23—832no—
407C23—832no—
408C23C2×#12418yes—
409C23C2×#12414yes—
410C23C23×D411yescharacteristic
411C23C22×#4112yes—
412C23C2×H(5,𝔽2)14yes—
413C23#58514yes—
414C23#58514yes—
415Q8—832no—
416D4—88no—
417D4—816no—
418C2×C4—28no—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—28no—
423C2×C4—28no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
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—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
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—832no—
469C2×C4—816no—
470C22—24no—
471C22—28no—
472C22—28no—
473C22—28no—
474C22—28no—
475C22—48no—
476C22—48no—
477C22—48no—
478C22—48no—
479C22—48no—
480C22—48no—
481C22—48no—
482C22—48no—
483C22—48no—
484C22—832no—
485C22—81no—
486C22—84no—
487C22—82no—
488C22—84no—
489C22—84no—
490C22C22×#12411yescharacteristic
491C22#345914yes—
492C22#348512yes—
493C22#349012yes—
494C22#349114yes—
495C22#349112yes—
496C22#349614yes—
497C22#354014yes—
498C22#354514yes—
499C22#355814yes—
500C22#356814yes—
501C4—48no—
502C4—48no—
503C4—81no—
504C4—84no—
505C4—82no—
506C4—84no—
507C4—84no—
508C4—84no—
509C2—44no—
510C2—48no—
511C2—161no—
512C2#3499312yes—
513C2#3545511yescharacteristic
514C2#4093214yes—
515C2#4161614yes—
516C2#4785214yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2840C2semidirect
3. #16831C2semidirect
4. #16368C2semidirect
5. #15468C2semidirect
6. #15371C2semidirect
7. #15050C2semidirect
8. #14817C2semidirect
10. #13869C2semidirect
11. C22⋊C42C2semidirect
22. C2×#393C22semidirect
23. C2×#391C22semidirect
31. #1446C22semidirect
32. #1446C22semidirect
33. #1445C22semidirect
34. #1443C22semidirect
35. #1443C22semidirect
36. #1443C22semidirect
37. #1443C22semidirect
38. #1442C22semidirect
39. #1442C22semidirect
45. #1440C22semidirect
46. #1440C22semidirect
51. #1437C22semidirect
52. #1434C22semidirect
53. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28402
C2×#26222
#168312
#163682
#154682
#153712
#150502
#148172
#142552
#138692
C22⋊C422