Finite Group Databasebeta

← #16854992

Subgroups — #16854992

Order 3968.

732 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758564C211yesmaximal, characteristic, complemented
2C2×#11758491C211yesmaximal, characteristic, complemented
3C2×SD16×D31C211yesmaximal, characteristic, complemented
4#14521615C211yesmaximal, characteristic, complemented
5#14521197C211yesmaximal, characteristic, complemented
6#14521196C211yesmaximal, characteristic, complemented
7#14521193C211yesmaximal, characteristic, complemented
8#14521192C211yesmaximal, characteristic, complemented
9#14521189C211yesmaximal, characteristic, complemented
10#14521188C211yesmaximal, characteristic, complemented
11#14521186C211yesmaximal, characteristic, complemented
12#14521056C211yesmaximal, characteristic, complemented
13#14521044C211yesmaximal, characteristic, complemented
14#14521014C211yesmaximal, characteristic, complemented
15C31×#448C211yesmaximal, characteristic, complemented
16C22×Q8×C31C2211yescharacteristic, complemented
17#11758564—24no—
18#11758564—24no—
19C2×#92499—21no—
20C2×#92499C2211yescharacteristic
21C2×#92499C2211yescharacteristic, complemented
22C2×Q8×D31—21no—
23C2×Q8×D31C2211yescharacteristic
24C2×Q8×D31C2211yescharacteristic, complemented
25C2×D4×D31C2211yescharacteristic
26C2×#92495—21no—
27C2×#92495—21no—
28C2×#92495C2211yescharacteristic
29C2×#92495C2211yescharacteristic, complemented
30C2×SD16×C31C2211yescharacteristic
31C2×SD16×C31C2211yescharacteristic
32C31×#121C2211yescharacteristic
33C2×#92481C2211yescharacteristic
34C2×#92481C2211yescharacteristic
35#11758517—21no—
36#11758515C2211yescharacteristic
37#11758514—21no—
38C2×#92480C2211yescharacteristic
39C2×#92479C2211yescharacteristic
40#11758507C2211yescharacteristic
41#11758491—24no—
42SD16×D31—24no—
43C2×D248C2211yescharacteristic
44C2×#92470C2211yescharacteristic
45C2×#92469C2211yescharacteristic
46C2×C8×D31C2211yescharacteristic
47#11758472C2211yescharacteristic
48#11758469C2211yescharacteristic
49#11758462—21no—
50#11758459—21no—
51C31×#103C2211yescharacteristic, complemented
52C31×#103C2211yescharacteristic, complemented
53C31×C22⋊C8C2211yescharacteristic, complemented
54#11758423C2211yescharacteristic, complemented
55#11758423C2211yescharacteristic, complemented
56C22⋊(C31⋊C8)C2211yescharacteristic, complemented
57#11758409C2211yescharacteristic, complemented
58(C2×D31)⋊C8C2211yescharacteristic, complemented
59(C2×D31)⋊C8C2211yescharacteristic, complemented
60#11758404C2211yescharacteristic, complemented
61#11758398C2211yescharacteristic, complemented
62#11758397C2211yescharacteristic, complemented
63C23×D31—21no—
64C2×Q8×C31—22no—
65C2×Q8×C31—22no—
66C2×Q8×C31—21no—
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—
76C2×C22⋊D31—21no—
77C2×C22⋊D31D411yescharacteristic
78C2×C22⋊D31D411yescharacteristic
79#92499—28no—
80#92499—22no—
81#92499—22no—
82#92499—22no—
83#92499D414yescomplemented
84Q8×D31—28no—
85Q8×D31—22no—
86Q8×D31—22no—
87Q8×D31—22no—
88Q8×D31D414yescomplemented
89D4×D31—22no—
90D4×D31—22no—
91#92495—28no—
92#92495—28no—
93#92495—22no—
94#92495—22no—
95#92495—22no—
96#92495—22no—
97C2×D124—21no—
98C2×D124—21no—
99C2×D124—21no—
100C2×D124D411yescharacteristic
101C2×D124C2311yescharacteristic
102C2×D124C2311yescharacteristic
103C2×C4×D31—21no—
104C2×C4×D31—21no—
105C2×C4×D31—21no—
106C2×C4×D31—21no—
107C2×C4×D31—21no—
108C2×C4×D31C2311yescharacteristic
109C2×C4×D31C2311yescharacteristic
110C2×Dic62—21no—
111C2×Dic62—21no—
112C2×Dic62D411yescharacteristic
113C2×Dic62C2311yescharacteristic
114SD16×C31—22no—
115SD16×C31—22no—
116C2×C8×C31C2311yescharacteristic
117C2×C8×C31C2311yescharacteristic
118C4⋊C4×C31C2311yescharacteristic
119C22⋊C4×C31—21no—
120C22⋊Dic31—21no—
121#92481—22no—
122#92481—22no—
123#92480—22no—
124#92479—22no—
125(C2×D31)⋊C4—21no—
126(C2×D31)⋊C4—21no—
127(C2×D31)⋊C4—21no—
128C4⋊Dic31C2311yescharacteristic
129C4×Dic31—21no—
130C2×C31⋊C8C2311yescharacteristic
131C2×C31⋊C8C2311yescharacteristic
132D248—22no—
133#92470—22no—
134#92469—22no—
135C8×D31—22no—
136C23×C31—21no—
137C23×C31C2×D411yescharacteristic
138C22×D31—21no—
139C22×D31—21no—
140C22×D31—42no—
141C22×D31—81no—
142C22×D31—81no—
143C22×D31C2×D411yescharacteristic
144C22×D31C2×D411yescharacteristic
145Q8×C31—24no—
146Q8×C31—21no—
147Q8×C31—21no—
148Q8×C31C2×D412yes—
149Q8×C31C2×D412yes—
150D4×C31—21no—
151D4×C31—21no—
152D4×C31—81no—
153C2×C4×C31—24no—
154C2×C4×C31—21no—
155C2×C4×C31—21no—
156C2×C4×C31—21no—
157C2×C4×C31—21no—
158C2×C4×C31—21no—
159C2×C4×C31C2×D411yescharacteristic
160C2×C4×C31C2411yescharacteristic
161C22⋊D31—24no—
162C22⋊D31—22no—
163C22⋊D31—22no—
164C22⋊D31—42no—
165C22⋊D31—81no—
166C22⋊D31—81no—
167C2×Dic31—21no—
168C2×Dic31—21no—
169C2×Dic31C2×D411yescharacteristic
170C2×Dic31C2×D411yescharacteristic
171D124—24no—
172D124—24no—
173D124—22no—
174D124—21no—
175D124—21no—
176D124—21no—
177D124—81no—
178D124C2×D412yes—
179C4×D31—24no—
180C4×D31—24no—
181C4×D31—24no—
182C4×D31—24no—
183C4×D31—21no—
184C4×D31—21no—
185C4×D31—24no—
186C4×D31C2×D412yes—
187C4×D31C2×D412yes—
188Dic62—24no—
189Dic62—21no—
190Dic62—24no—
191Dic62—22no—
192Dic62C2×D412yes—
193C8×C31—21no—
194C8×C31—21no—
195C31⋊C8—21no—
196C31⋊C8—21no—
197#3010—311nomaximal, Sylow
198C22×C31—21no—
199C22×C31—21no—
200C22×C31—21no—
201C22×C31—41no—
202C22×C31—41no—
203C22×C31—41no—
204C22×C31C22×D411yescharacteristic
205C2×D31—21no—
206C2×D31—21no—
207C2×D31—21no—
208C2×D31—22no—
209C2×D31—21no—
210C2×D31—21no—
211C2×D31—41no—
212C2×D31—41no—
213C2×D31—41no—
214C2×D31—41no—
215C2×D31—41no—
216C2×D31—84no—
217C2×D31#12012yes—
218C4×C31—22no—
219C4×C31—22no—
220C4×C31—21no—
221C4×C31—41no—
222C4×C31C22×D411yescharacteristic
223C4×C31C22×D411yescharacteristic
224Dic31—21no—
225Dic31—22no—
226Dic31—41no—
227Dic31#12012yes—
228C2×2−1+4—311no—
229C2×AGL(1,ℤ/8)—311no—
230C22×SD16—311no—
231#535—311no—
232#449—311no—
233#449—311no—
234#449—311no—
235#449—311no—
236#448—311no—
237#448—311no—
238#448—311no—
239#448D3111yescharacteristic, complemented
240#419—311no—
241#416—311no—
242#408—311no—
243C2×C31—41no—
244C2×C31—81no—
245C2×C31C2×#12011yescharacteristic
246C2×C31#57711yescharacteristic
247C2×C31#57711yescharacteristic
248D31—22no—
249D31—41no—
250D31—42no—
251D31—81no—
2522−1+4—624no—
2532−1+4—624no—
254C2×#41—311no—
255C2×#41—311no—
256C2×#41—311no—
257C2×#41—311no—
258C2×#41—621no—
259C2×#41—621no—
260C2×#41—621no—
261C22×Q8—311no—
262C22×Q8—311no—
263C22×Q8—621no—
264C22×Q8C2×D3111yescharacteristic, complemented
265C22×D4—311no—
266AGL(1,ℤ/8)—624no—
267C2×SD16—311no—
268C2×SD16—311no—
269C2×SD16—311no—
270C2×SD16—311no—
271C2×SD16—624no—
272C2×SD16C2×D3111yescharacteristic
273C2×SD16C2×D3111yescharacteristic
274C2×D8—311no—
275C2×D8—311no—
276C2×M16—311no—
277C22×C8—311no—
278#127—621no—
279#124—621no—
280#121—311no—
281#121—311no—
282#121—311no—
283#121C2×D3111yescharacteristic
284#120—621no—
285#120—621no—
286#117—311no—
287#103—311no—
288#103—311no—
289#103—311no—
290#103—311no—
291#103C2×D3111yescharacteristic, complemented
292#103C2×D3111yescharacteristic, complemented
293#102—311no—
294#102—311no—
295C22⋊C8—311no—
296C22⋊C8—311no—
297C22⋊C8—311no—
298C22⋊C8C2×D3111yescharacteristic, complemented
299C31#301011yescharacteristic, complemented, Sylow
300C24—621no—
301#41—314no—
302#41—622no—
303#41—622no—
304#41—622no—
305#41—622no—
306#41—628no—
307#41—628no—
308#41—622no—
309#41—622no—
310#41—628no—
311#41—622no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—21no—
315C2×Q8—311no—
316C2×Q8—314no—
317C2×Q8—311no—
318C2×Q8—621no—
319C2×Q8—621no—
320C2×Q8—622no—
321C2×Q8—622no—
322C2×Q8—628no—
323C2×Q8—622no—
324C2×Q8C22×D3111yescharacteristic
325C2×Q8C22×D3111yescharacteristic
326C2×D4—21no—
327C2×D4—311no—
328C2×D4—311no—
329C2×D4—311no—
330C2×D4—311no—
331C2×D4—311no—
332C2×D4—621no—
333C2×D4—621no—
334C2×D4—621no—
335C2×D4—622no—
336C2×D4—622no—
337C2×D4—621no—
338C2×D4—621no—
339C2×D4—621no—
340C2×D4—621no—
341C2×D4C22×D3111yescharacteristic
342C22×C4—21no—
343C22×C4—311no—
344C22×C4—311no—
345C22×C4—621no—
346C22×C4—621no—
347C22×C4—621no—
348C22×C4—621no—
349C22×C4—621no—
350C22×C4C22×D3111yescharacteristic
351SD16—22no—
352SD16—22no—
353SD16—622no—
354SD16—622no—
355SD16—622no—
356SD16—622no—
357D8—622no—
358D8—622no—
359M16—622no—
360C2×C8—311no—
361C2×C8—311no—
362C2×C8—622no—
363C2×C8C22×D3111yescharacteristic
364C2×C8C22×D3111yescharacteristic
365C4⋊C4—311no—
366C4⋊C4C22×D3111yescharacteristic
367C22⋊C4—21no—
368C22⋊C4—621no—
369C22⋊C4—621no—
370C22⋊C4—621no—
371C22⋊C4—621no—
372C42—621no—
373C23—21no—
374C23—311no—
375C23—311no—
376C23—621no—
377C23—621no—
378C23—1242no—
379C23—2481no—
380C23—2481no—
381C23D4×D3111yescharacteristic
382Q8—24no—
383Q8—21no—
384Q8—21no—
385Q8—312no—
386Q8—624no—
387Q8—622no—
388Q8—624no—
389Q8—621no—
390Q8D4×D3112yes—
391Q8D4×D3112yes—
392D4—21no—
393D4—21no—
394D4—81no—
395D4—312no—
396D4—622no—
397D4—624no—
398D4—624no—
399D4—622no—
400D4—622no—
401D4—624no—
402D4—621no—
403D4—621no—
404D4—621no—
405D4—1242no—
406D4—2481no—
407D4—2481no—
408D4—2481no—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—311no—
416C2×C4—312no—
417C2×C4—312no—
418C2×C4—311no—
419C2×C4—624no—
420C2×C4—624no—
421C2×C4—624no—
422C2×C4—624no—
423C2×C4—624no—
424C2×C4—621no—
425C2×C4—621no—
426C2×C4—621no—
427C2×C4—621no—
428C2×C4D4×D3111yescharacteristic
429C2×C4C23×D3111yescharacteristic
430C8—21no—
431C8—21no—
432C8—621no—
433C8—621no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—312no—
441C22—621no—
442C22—621no—
443C22—621no—
444C22—622no—
445C22—621no—
446C22—621no—
447C22—1241no—
448C22—1241no—
449C22—1241no—
450C22—1241no—
451C22—1241no—
452C22—2484no—
453C22C2×D4×D3111yescharacteristic
454C4—22no—
455C4—22no—
456C4—21no—
457C4—41no—
458C4—312no—
459C4—621no—
460C4—622no—
461C4—1241no—
462C4C2×D4×D3111yescharacteristic
463C4C2×D4×D3111yescharacteristic
464C2—41no—
465C2—81no—
466C2—622no—
467C2—1241no—
468C2—1242no—
469C2—2481no—
470C2#120×D3111yescharacteristic
471C2#1452185511yescharacteristic
472C2#1452185811yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#448C2
socleC22×C31C22×D4
hypercenter#448D31
hypocenterC31#3010
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#11758564C2semidirect
2. C2×#11758491C2semidirect
3. C2×SD16×D31C2semidirect
4. #14521615C2semidirect
5. #14521197C2semidirect
6. #14521196C2semidirect
7. #14521193C2semidirect
8. #14521192C2semidirect
9. #14521189C2semidirect
10. #14521188C2semidirect
11. #14521186C2semidirect
12. #14521056C2semidirect
13. #14521044C2semidirect
14. #14521014C2semidirect
15. C31×#448C2semidirect
16. C22×Q8×C31C22semidirect
21. C2×#92499C22semidirect
24. C2×Q8×D31C22semidirect
29. C2×#92495C22semidirect
51. C31×#103C22semidirect
52. C31×#103C22semidirect
53. C31×C22⋊C8C22semidirect
54. #11758423C22semidirect
55. #11758423C22semidirect
56. C22⋊(C31⋊C8)C22semidirect
57. #11758409C22semidirect
58. (C2×D31)⋊C8C22semidirect
59. (C2×D31)⋊C8C22semidirect
60. #11758404C22semidirect
61. #11758398C22semidirect
62. #11758397C22semidirect
83. #92499D4semidirect
88. Q8×D31D4semidirect
239. #448D31semidirect
264. C22×Q8C2×D31semidirect
291. #103C2×D31semidirect
292. #103C2×D31semidirect
298. C22⋊C8C2×D31semidirect
299. C31#3010semidirect

Maximal subgroups

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

subgroupindex
C2×#117585642
C2×#117584912
C2×SD16×D312
#145216152
#145211972
#145211962
#145211932
#145211922
#145211892
#145211882
#145211862
#145210562
#145210442
#145210142
C31×#4482
#301031