Finite Group Databasebeta

← #16854968

Subgroups — #16854968

Order 3968.

754 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D31C211yesmaximal, characteristic
2C2×#11758495C211yesmaximal, characteristic, complemented
3C2×#11758492C211yesmaximal, characteristic, complemented
4C22⋊Q8×D31C211yesmaximal, characteristic, complemented
5#14521213C211yesmaximal, characteristic, complemented
6#14521211C211yesmaximal, characteristic, complemented
7#14521209C211yesmaximal, characteristic, complemented
8#14521206C211yesmaximal, characteristic, complemented
9#14521203C211yesmaximal, characteristic, complemented
10#14521201C211yesmaximal, characteristic, complemented
11#14521198C211yesmaximal, characteristic, complemented
12#14521049C211yesmaximal, characteristic, complemented
13#14521047C211yesmaximal, characteristic, complemented
14#14521011C211yesmaximal, characteristic, complemented
15C31×#452C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17#41×D31—24no—
18#41×D31—24no—
19C2×#92499C2211yescharacteristic
20C2×Q8×D31C2211yescharacteristic
21C2×Q8×D31C2211yescharacteristic, complemented
22C2×#92497—21no—
23C2×#92497C2211yescharacteristic
24C2×D4×D31—21no—
25C2×D4×D31C2211yescharacteristic
26C2×#92495—21no—
27C2×#92495C2211yescharacteristic
28C22×C4×D31—21no—
29C22×C4×D31C2211yescharacteristic
30C2×Q16×C31C2211yescharacteristic, complemented
31C2×SD16×C31C2211yescharacteristic, complemented
32C31×C22⋊Q8C2211yescharacteristic, complemented
33#11758523C2211yescharacteristic
34C2×#92482C2211yescharacteristic, complemented
35C2×#92482C2211yescharacteristic, complemented
36C2×#92481C2211yescharacteristic, complemented
37C2×#92480C2211yescharacteristic, complemented
38#11758501C2211yescharacteristic, complemented
39#11758495—24no—
40#11758492—24no—
41C2×Dic124C2211yescharacteristic, complemented
42C2×#92470C2211yescharacteristic, complemented
43C2×#92469C2211yescharacteristic, complemented
44C2×#92469C2211yescharacteristic, complemented
45#11758474C2211yescharacteristic
46#11758473—21no—
47C4⋊C4×D31—21no—
48C4⋊C4×D31C2211yescharacteristic, complemented
49C22⋊C4×D31—21no—
50#11758455—21no—
51C31×#103C2211yescharacteristic, complemented
52C31×#102C2211yescharacteristic, complemented
53C31×C22⋊C8C2211yescharacteristic, complemented
54#11758423C2211yescharacteristic, complemented
55#11758420C2211yescharacteristic, complemented
56C22⋊(C31⋊C8)C2211yescharacteristic, complemented
57#11758409C2211yescharacteristic, complemented
58(C2×D31)⋊C8C2211yescharacteristic
59(C2×D31)⋊C8C2211yescharacteristic
60#11758404C2211yescharacteristic, complemented
61#11758398C2211yescharacteristic, complemented
62#11758397C2211yescharacteristic, complemented
63C23×D31—21no—
64C23×D31D411yescharacteristic
65#41×C31—22no—
66#41×C31—22no—
67C2×Q8×C31C2311yescharacteristic
68C2×Q8×C31C2311yescharacteristic
69C2×D4×C31—21no—
70C2×D4×C31C2311yescharacteristic
71C22×C4×C31—21no—
72C22×C4×C31C2311yescharacteristic
73C2×C22⋊D31—21no—
74C2×C22⋊D31—21no—
75C2×C22⋊D31—21no—
76C22×Dic31—21no—
77C22×Dic31D411yescharacteristic
78#92499—22no—
79#92499D414yes—
80Q8×D31—22no—
81Q8×D31—22no—
82Q8×D31—22no—
83Q8×D31—22no—
84#92497—28no—
85#92497—22no—
86#92497D414yes—
87D4×D31—28no—
88D4×D31—22no—
89D4×D31—22no—
90#92495—28no—
91#92495—22no—
92#92495—22no—
93C2×D124—21no—
94C2×D124C2311yescharacteristic
95C2×C4×D31—28no—
96C2×C4×D31—21no—
97C2×C4×D31—21no—
98C2×C4×D31—21no—
99C2×C4×D31—22no—
100C2×C4×D31—22no—
101C2×C4×D31—21no—
102C2×C4×D31—21no—
103C2×C4×D31D411yescharacteristic
104C2×C4×D31D411yescharacteristic
105C2×C4×D31C2311yescharacteristic
106C2×C4×D31C2311yescharacteristic
107C2×Dic62—21no—
108C2×Dic62—21no—
109C2×Dic62C2311yescharacteristic
110C2×Dic62C2311yescharacteristic
111Q16×C31—22no—
112SD16×C31—22no—
113C2×C8×C31C2311yescharacteristic
114C2×C8×C31C2311yescharacteristic
115C4⋊C4×C31—21no—
116C4⋊C4×C31C2311yescharacteristic
117C22⋊C4×C31—21no—
118C22⋊Dic31—21no—
119#92482—22no—
120#92482—22no—
121#92481—22no—
122#92480—22no—
123(C2×D31)⋊C4—21no—
124(C2×D31)⋊C4—21no—
125C4⋊Dic31—21no—
126C4⋊Dic31C2311yescharacteristic
127Dic31⋊C4—21no—
128Dic31⋊C4—21no—
129Dic31⋊C4—21no—
130C2×C31⋊C8C2311yescharacteristic
131C2×C31⋊C8C2311yescharacteristic
132Dic124—22no—
133#92470—22no—
134#92469—22no—
135#92469—22no—
136C23×C31—21no—
137C23×C31C2×D411yescharacteristic
138C22×D31—24no—
139C22×D31—21no—
140C22×D31—22no—
141C22×D31—22no—
142C22×D31—41no—
143C22×D31—42no—
144C22×D31—42no—
145C22×D31C2×D411yescharacteristic
146C22×D31C2×D411yescharacteristic
147Q8×C31—21no—
148Q8×C31—21no—
149Q8×C31—21no—
150Q8×C31C2×D412yes—
151D4×C31—24no—
152D4×C31—21no—
153D4×C31C2×D412yes—
154C2×C4×C31—24no—
155C2×C4×C31—21no—
156C2×C4×C31—21no—
157C2×C4×C31—21no—
158C2×C4×C31—21no—
159C2×C4×C31—21no—
160C2×C4×C31C2×D411yescharacteristic
161C2×C4×C31C2411yescharacteristic
162C22⋊D31—24no—
163C22⋊D31—24no—
164C22⋊D31—24no—
165C2×Dic31—21no—
166C2×Dic31—22no—
167C2×Dic31—24no—
168C2×Dic31—21no—
169C2×Dic31—21no—
170C2×Dic31C2×D411yescharacteristic
171C2×Dic31C2×D411yescharacteristic
172D124—24no—
173D124—21no—
174D124C2×D412yes—
175C4×D31—24no—
176C4×D31—24no—
177C4×D31—22no—
178C4×D31—21no—
179C4×D31—21no—
180C4×D31—22no—
181C4×D31—24no—
182C4×D31—42no—
183C4×D31—42no—
184C4×D31C2×D412yes—
185C4×D31C2×D412yes—
186Dic62—24no—
187Dic62—21no—
188Dic62—21no—
189Dic62—21no—
190Dic62—42no—
191Dic62C2×D412yes—
192C8×C31—21no—
193C8×C31—21no—
194C31⋊C8—21no—
195C31⋊C8—21no—
196#3007—311nomaximal, Sylow
197C22×C31—21no—
198C22×C31—22no—
199C22×C31—21no—
200C22×C31—21no—
201C22×C31—41no—
202C22×C31—41no—
203C22×C31C22×D411yescharacteristic
204C2×D31—21no—
205C2×D31—21no—
206C2×D31—21no—
207C2×D31—21no—
208C2×D31—22no—
209C2×D31—44no—
210C2×D31—48no—
211C2×D31—41no—
212C2×D31—41no—
213C2×D31#12012yes—
214C2×D31#13712yescomplemented
215C4×C31—22no—
216C4×C31—21no—
217C4×C31—41no—
218C4×C31—41no—
219C4×C31C22×D411yescharacteristic
220C4×C31C22×D411yescharacteristic
221Dic31—21no—
222Dic31—22no—
223Dic31—41no—
224Dic31—41no—
225Dic31#12012yes—
226C22×#41—311no—
227C2×#137—311no—
228C2×#137—311no—
229C2×C22⋊Q8—311no—
230#452—311no—
231#452—311no—
232#452—311no—
233#452D3111yescharacteristic, complemented
234#451—311no—
235#451—311no—
236#450—311no—
237#450—311no—
238#417—311no—
239#417—311no—
240#407—311no—
241C2×C31—41no—
242C2×C31—42no—
243C2×C31C2×#12011yescharacteristic
244C2×C31C2×#13711yescharacteristic
245C2×C31#57511yescharacteristic
246D31—22no—
247D31—22no—
248D31—42no—
249C2×#41—311no—
250C2×#41—311no—
251C2×#41—311no—
252C2×#41—621no—
253C2×#41—621no—
254C2×#41—624no—
255C2×#41—624no—
256C2×#41C2×D3111yescharacteristic
257C22×Q8—311no—
258C22×Q8—311no—
259C22×D4—311no—
260C22×D4—621no—
261C23×C4—311no—
262C23×C4—621no—
263#137—624no—
264#137—624no—
265C2×Q16—311no—
266C2×Q16—311no—
267C2×Q16—311no—
268C2×Q16C2×D3111yescharacteristic, complemented
269C2×SD16—311no—
270C2×SD16—311no—
271C2×SD16—311no—
272C2×SD16C2×D3111yescharacteristic, complemented
273C2×M16—311no—
274C2×M16—311no—
275C22⋊Q8—311no—
276C22⋊Q8—311no—
277C22⋊Q8—311no—
278C22⋊Q8—621no—
279C22⋊Q8—621no—
280C22⋊Q8C2×D3111yescharacteristic, complemented
281C2×C4⋊C4—311no—
282C2×C4⋊C4—621no—
283C2×C22⋊C4—621no—
284#103—311no—
285#103—311no—
286#103—311no—
287#103C2×D3111yescharacteristic, complemented
288#102—311no—
289#102—311no—
290#102—311no—
291#102C2×D3111yescharacteristic, complemented
292C22⋊C8—311no—
293C22⋊C8—311no—
294C22⋊C8—311no—
295C22⋊C8C2×D3111yescharacteristic, complemented
296C31#300711yescharacteristic, complemented, Sylow
297C24—311no—
298C24—621no—
299#41—22no—
300#41—22no—
301#41—314no—
302#41—314no—
303#41—622no—
304#41—622no—
305#41—628no—
306#41—622no—
307#41—622no—
308#41—628no—
309C2×Q8—311no—
310C2×Q8—311no—
311C2×Q8—621no—
312C2×Q8—621no—
313C2×Q8—622no—
314C2×Q8—622no—
315C2×Q8—622no—
316C2×Q8—622no—
317C2×Q8C22×D3111yescharacteristic
318C2×Q8C22×D3111yescharacteristic
319C2×D4—21no—
320C2×D4—311no—
321C2×D4—621no—
322C2×D4—621no—
323C2×D4—622no—
324C2×D4—628no—
325C2×D4—621no—
326C2×D4—621no—
327C2×D4—622no—
328C2×D4C22×D3111yescharacteristic
329C22×C4—21no—
330C22×C4—311no—
331C22×C4—311no—
332C22×C4—311no—
333C22×C4—311no—
334C22×C4—311no—
335C22×C4—622no—
336C22×C4—622no—
337C22×C4—621no—
338C22×C4—621no—
339C22×C4—628no—
340C22×C4—621no—
341C22×C4—621no—
342C22×C4—621no—
343C22×C4—621no—
344C22×C4C22×D3111yescharacteristic
345Q16—22no—
346Q16—622no—
347Q16—622no—
348Q16—622no—
349SD16—22no—
350SD16—622no—
351SD16—622no—
352SD16—622no—
353M16—622no—
354M16—622no—
355C2×C8—311no—
356C2×C8—311no—
357C2×C8C22×D3111yescharacteristic
358C2×C8C22×D3111yescharacteristic
359C4⋊C4—21no—
360C4⋊C4—311no—
361C4⋊C4—621no—
362C4⋊C4—621no—
363C4⋊C4—621no—
364C4⋊C4—621no—
365C4⋊C4C22×D3111yescharacteristic
366C22⋊C4—21no—
367C22⋊C4—621no—
368C22⋊C4—621no—
369C22⋊C4—621no—
370C23—21no—
371C23—311no—
372C23—311no—
373C23—622no—
374C23—622no—
375C23—624no—
376C23—621no—
377C23—1241no—
378C23—1242no—
379C23—1242no—
380C23D4×D3111yescharacteristic
381Q8—21no—
382Q8—21no—
383Q8—21no—
384Q8—312no—
385Q8—624no—
386Q8—621no—
387Q8—621no—
388Q8—621no—
389Q8—1242no—
390Q8D4×D3112yes—
391D4—24no—
392D4—21no—
393D4—312no—
394D4—624no—
395D4—624no—
396D4—624no—
397D4—624no—
398D4—621no—
399D4D4×D3112yes—
400C2×C4—24no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—311no—
407C2×C4—312no—
408C2×C4—312no—
409C2×C4—311no—
410C2×C4—622no—
411C2×C4—624no—
412C2×C4—622no—
413C2×C4—624no—
414C2×C4—624no—
415C2×C4—622no—
416C2×C4—624no—
417C2×C4—621no—
418C2×C4—621no—
419C2×C4—621no—
420C2×C4—621no—
421C2×C4—621no—
422C2×C4—1242no—
423C2×C4—1242no—
424C2×C4D4×D3111yescharacteristic
425C2×C4C23×D3111yescharacteristic
426C8—21no—
427C8—21no—
428C8—621no—
429C8—621no—
430C22—21no—
431C22—22no—
432C22—21no—
433C22—21no—
434C22—41no—
435C22—41no—
436C22—312no—
437C22—312no—
438C22—621no—
439C22—621no—
440C22—621no—
441C22—621no—
442C22—622no—
443C22—1244no—
444C22—1248no—
445C22—1241no—
446C22—1241no—
447C22C2×D4×D3111yescharacteristic
448C4—21no—
449C4—22no—
450C4—41no—
451C4—41no—
452C4—312no—
453C4—621no—
454C4—622no—
455C4—1241no—
456C4—1241no—
457C4C2×D4×D3111yescharacteristic
458C4C2×D4×D3111yescharacteristic
459C2—41no—
460C2—42no—
461C2—622no—
462C2—622no—
463C2—1242no—
464C2#120×D3111yescharacteristic
465C2#137×D3111yescharacteristic
466C2#1452184611yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#452C2
socleC22×C31C22×D4
prime radicalC2×#41×D31C2
hypercenter#452D31
hypocenterC31#3007
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. C2×#11758495C2semidirect
3. C2×#11758492C2semidirect
4. C22⋊Q8×D31C2semidirect
5. #14521213C2semidirect
6. #14521211C2semidirect
7. #14521209C2semidirect
8. #14521206C2semidirect
9. #14521203C2semidirect
10. #14521201C2semidirect
11. #14521198C2semidirect
12. #14521049C2semidirect
13. #14521047C2semidirect
14. #14521011C2semidirect
15. C31×#452C2semidirect
21. C2×Q8×D31C22semidirect
30. C2×Q16×C31C22semidirect
31. C2×SD16×C31C22semidirect
32. C31×C22⋊Q8C22semidirect
34. C2×#92482C22semidirect
35. C2×#92482C22semidirect
36. C2×#92481C22semidirect
37. C2×#92480C22semidirect
38. #11758501C22semidirect
41. C2×Dic124C22semidirect
42. C2×#92470C22semidirect
43. C2×#92469C22semidirect
44. C2×#92469C22semidirect
48. C4⋊C4×D31C22semidirect
51. C31×#103C22semidirect
52. C31×#102C22semidirect
53. C31×C22⋊C8C22semidirect
54. #11758423C22semidirect
55. #11758420C22semidirect
56. C22⋊(C31⋊C8)C22semidirect
57. #11758409C22semidirect
60. #11758404C22semidirect
61. #11758398C22semidirect
62. #11758397C22semidirect
214. C2×D31#137semidirect
233. #452D31semidirect
268. C2×Q16C2×D31semidirect
272. C2×SD16C2×D31semidirect
280. C22⋊Q8C2×D31semidirect
287. #103C2×D31semidirect
291. #102C2×D31semidirect
295. C22⋊C8C2×D31semidirect
296. C31#3007semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D312
C2×#117584952
C2×#117584922
C22⋊Q8×D312
#145212132
#145212112
#145212092
#145212062
#145212032
#145212012
#145211982
#145210492
#145210472
#145210112
C31×#4522
#300731