Finite Group Databasebeta

← #27151878

Subgroups — #27151878

Order 12416.

450 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261272C211yesmaximal, characteristic, complemented
2C4×D4×D97C211yesmaximal, characteristic, complemented
3#20261064C211yesmaximal, characteristic, complemented
4#20261050C211yesmaximal, characteristic, complemented
5#20261038C211yesmaximal, characteristic, complemented
6#20261037C211yesmaximal, characteristic, complemented
7#20260953C211yesmaximal, characteristic, complemented
8#20260952C211yesmaximal, characteristic, complemented
9#20260884C211yesmaximal, characteristic, complemented
10#20260870C211yesmaximal, characteristic
11#20260848C211yesmaximal, characteristic, complemented
12#20260845C211yesmaximal, characteristic, complemented
13#20260833C211yesmaximal, characteristic, complemented
14#20260790C211yesmaximal, characteristic, complemented
15#440×C97C211yesmaximal, characteristic, complemented
16C2×D4×D97C2211yescharacteristic
17C22×C4×D97—21no—
18C2×#13095498C2211yescharacteristic
19C2×D776C2211yescharacteristic
20C2×#13095496C2211yescharacteristic
21#15690805—21no—
22#15690801C2211yescharacteristic, complemented
23#15690799C2211yescharacteristic, complemented
24#15690795—21no—
25#15690786C2211yescharacteristic
26#15690785C2211yescharacteristic
27C4×D388—21no—
28C4×D388C2211yescharacteristic
29C4×D388C2211yescharacteristic, complemented
30D4×Dic97C2211yescharacteristic
31#15690772C2211yescharacteristic
32C4×C22⋊D97—21no—
33#15690770—21no—
34#15690767—21no—
35C4⋊C4×D97C2211yescharacteristic
36C22⋊C4×D97—21no—
37C42×D97C2211yescharacteristic
38#15690743C2211yescharacteristic, complemented
39#15690742C2211yescharacteristic, complemented
40#15690741C2211yescharacteristic, complemented
41#15690739C2211yescharacteristic, complemented
42#15690739C2211yescharacteristic, complemented
43#15690734C2211yescharacteristic, complemented
44#15690732C2211yescharacteristic
45#15690732C2211yescharacteristic, complemented
46#15690731C2211yescharacteristic
47#15690730C2211yescharacteristic
48#15690730C2211yescharacteristic, complemented
49#15690722C2211yescharacteristic
50#15690722C2211yescharacteristic
51#15690721C2211yescharacteristic, complemented
52C2×SD16×C97C2211yescharacteristic
53C4×Q8×C97C2211yescharacteristic, complemented
54C4×D4×C97C2211yescharacteristic
55#107×C97C2211yescharacteristic, complemented
56#103×C97C2211yescharacteristic, complemented
57#102×C97C2211yescharacteristic
58#97×C97C2211yescharacteristic, complemented
59C23×D97—21no—
60C2×D388—21no—
61C2×D388C2311yescharacteristic
62C2×D388C2311yescharacteristic
63C2×C22⋊D97—21no—
64D4×D97—22no—
65D4×D97—22no—
66C22×Dic97—21no—
67C2×C4×D97—21no—
68C2×C4×D97—21no—
69C2×C4×D97—21no—
70C2×C4×D97—21no—
71C2×C4×D97—44no—
72C2×C4×D97D411yescharacteristic
73C2×C4×D97D411yescharacteristic
74C2×C4×D97C2311yescharacteristic
75#13095498—22no—
76D776—22no—
77#13095496—22no—
78C2×#11715895C2311yescharacteristic
79C2×#11715895C2311yescharacteristic
80C4⋊Dic97—21no—
81C4⋊Dic97C2311yescharacteristic
82C4⋊Dic97C2311yescharacteristic
83#13095490—21no—
84#13095490—21no—
85#13095490—21no—
86C22⋊Dic97—21no—
87#13095488—21no—
88#13095488—21no—
89#13095488—21no—
90#13095488—21no—
91#13095488—21no—
92#13095488—21no—
93#13095488—21no—
94C4×Dic97D411yescharacteristic
95C4×Dic97D411yescharacteristic
96C4×Dic97C2311yescharacteristic
97C2×Q8×C97C2311yescharacteristic
98C2×D4×C97C2311yescharacteristic
99C22×C4×C97—21no—
100SD16×C97—22no—
101C2×C8×C97C2311yescharacteristic
102C2×C8×C97C2311yescharacteristic
103C4⋊C4×C97—21no—
104C4⋊C4×C97C2311yescharacteristic
105C4⋊C4×C97C2311yescharacteristic
106C22⋊C4×C97—21no—
107C42×C97—21no—
108C42×C97C2311yescharacteristic
109C23×C97—21no—
110C22×D97—24no—
111C22×D97—21no—
112C22×D97—21no—
113C22×D97—42no—
114C22×D97—42no—
115C22×D97C2×D411yescharacteristic
116Q8×C97—21no—
117Q8×C97—21no—
118D4×C97—21no—
119D4×C97#4112yes—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—42no—
124C2×C4×C97C2×D411yescharacteristic
125C2×C4×C97C2×D411yescharacteristic
126C2×C4×C97C2411yescharacteristic
127C22⋊D97—24no—
128C2×Dic97—21no—
129C2×Dic97—21no—
130C2×Dic97—42no—
131C2×Dic97C2×D411yescharacteristic
132C2×Dic97C2×D411yescharacteristic
133C2×Dic97C2×D411yescharacteristic
134D388—21no—
135D388—21no—
136D388—21no—
137D388—42no—
138D388#4112yes—
139C4×D97—22no—
140C4×D97—21no—
141C4×D97—21no—
142C4×D97—41no—
143C4×D97—42no—
144C4×D97—42no—
145C4×D97—42no—
146C4×D97—42no—
147C8×C97—21no—
148C8×C97—21no—
149#11715895—21no—
150#11715895—21no—
151C22×C97—22no—
152C22×C97—41no—
153C22×C97—41no—
154C22×C97C22×D411yescharacteristic
155C2×D97—21no—
156C2×D97—21no—
157C2×D97—22no—
158C2×D97—48no—
159C2×D97—41no—
160C2×D97—41no—
161C2×D97—41no—
162C2×D97—41no—
163C2×D97—41no—
164C2×D97AGL(1,ℤ/8)12yes—
165C4×C97—21no—
166C4×C97—21no—
167C4×C97—41no—
168C4×C97—41no—
169C4×C97—41no—
170C4×C97C2×#4111yescharacteristic
171C4×C97C2×#4111yescharacteristic
172Dic97—21no—
173Dic97—21no—
174Dic97—21no—
175Dic97—41no—
176Dic97—41no—
177C2×C97—42no—
178C2×C97#52511yescharacteristic
179C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
180C2×C97#57511yescharacteristic
181D97—22no—
182D97—42no—
183D97—81no—
184#3114—971nomaximal, Sylow
185C97#311411yescharacteristic, complemented, Sylow
186#529—971no—
187C2×C4×D4—971no—
188#483—971no—
189#480—971no—
190#478—971no—
191#478—971no—
192#463—971no—
193#463—971no—
194#450—971no—
195#447—971no—
196#442—971no—
197#442—971no—
198#440—971no—
199#440D9711yescharacteristic, complemented
200#432—971no—
201C22×D4—971no—
202C23×C4—1941no—
203C2×SD16—971no—
204C2×SD16C2×D9711yescharacteristic
205C2×D8—971no—
206C2×D8—971no—
207#126—1941no—
208#125—971no—
209#124—971no—
210#123—1941no—
211C22⋊Q8—971no—
212#121—971no—
213C4×Q8C2×D9711yescharacteristic, complemented
214C4×D4—971no—
215C4×D4—971no—
216C4×D4—971no—
217C4×D4—971no—
218C4×D4—1941no—
219C4×D4—1941no—
220C4×D4—1941no—
221C4×D4C2×D9711yescharacteristic
222#117—1941no—
223C2×C4⋊C4—971no—
224C2×C22⋊C4—1941no—
225C2×C42—971no—
226#107—971no—
227#107C2×D9711yescharacteristic, complemented
228#106—971no—
229#106—971no—
230C4⋊C8—971no—
231C4⋊C8—971no—
232#103—971no—
233#103C2×D9711yescharacteristic, complemented
234#102—971no—
235#102—971no—
236#102—971no—
237#102—971no—
238#102—971no—
239#102C2×D9711yescharacteristic
240C22⋊C8—971no—
241C22⋊C8—971no—
242#97—971no—
243#97C2×D9711yescharacteristic, complemented
244C24—1941no—
245C2×Q8C22×D9711yescharacteristic
246C2×D4—971no—
247C2×D4—971no—
248C2×D4—1941no—
249C2×D4—1941no—
250C2×D4—1942no—
251C2×D4—1942no—
252C2×D4C22×D9711yescharacteristic
253C22×C4—21no—
254C22×C4—971no—
255C22×C4—971no—
256C22×C4—971no—
257C22×C4—3884no—
258C22×C4—1941no—
259C22×C4—1941no—
260C22×C4—1941no—
261C22×C4—1941no—
262C22×C4—1941no—
263SD16—22no—
264SD16—1942no—
265D8—1942no—
266D8—1942no—
267C2×C8—971no—
268C2×C8—971no—
269C2×C8C22×D9711yescharacteristic
270C2×C8C22×D9711yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—971no—
273C4⋊C4—971no—
274C4⋊C4—1941no—
275C4⋊C4—1941no—
276C4⋊C4—1941no—
277C4⋊C4—1941no—
278C4⋊C4C22×D9711yescharacteristic
279C4⋊C4C22×D9711yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—1941no—
282C22⋊C4—1941no—
283C22⋊C4—1941no—
284C22⋊C4—1941no—
285C22⋊C4—1941no—
286C22⋊C4—1941no—
287C22⋊C4—1941no—
288C22⋊C4—1941no—
289C42—21no—
290C42—971no—
291C42—971no—
292C42—971no—
293C42C22×D9711yescharacteristic
294C23—21no—
295C23—971no—
296C23—3882no—
297C23—3882no—
298C23—1944no—
299C23—1941no—
300C23—1941no—
301Q8—21no—
302Q8—21no—
303D4—21no—
304D4—972no—
305D4—3882no—
306D4—1944no—
307D4—1941no—
308D4—1941no—
309D4—1941no—
310D4#1309551112yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—971no—
316C2×C4—971no—
317C2×C4—971no—
318C2×C4—3882no—
319C2×C4—3882no—
320C2×C4—3881no—
321C2×C4—3882no—
322C2×C4—3882no—
323C2×C4—3882no—
324C2×C4—1942no—
325C2×C4—1941no—
326C2×C4—1941no—
327C2×C4—1941no—
328C2×C4—1941no—
329C2×C4D4×D9711yescharacteristic
330C2×C4D4×D9711yescharacteristic
331C2×C4C23×D9711yescharacteristic
332C8—21no—
333C8—21no—
334C8—1941no—
335C8—1941no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—972no—
340C22—3888no—
341C22—3881no—
342C22—3881no—
343C22—3881no—
344C22—3881no—
345C22—3881no—
346C22—1942no—
347C22—1941no—
348C22—1941no—
349C22C2×D4×D9711yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—3881no—
356C4—3881no—
357C4—1941no—
358C4—1941no—
359C4—1941no—
360C4#41×D9711yescharacteristic
361C4C2×#1309551111yescharacteristic
362C2—42no—
363C2—7761no—
364C2—3882no—
365C2—1942no—
366C2#2026124411yescharacteristic
367C2AGL(1,ℤ/8)×D9711yescharacteristic
368C2#2026153411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#440×C97C2
socleC22×C97C22×D4
prime radical#20260870C2
hypercenter#440D97
hypocenterC97#3114
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #20261272C2semidirect
2. C4×D4×D97C2semidirect
3. #20261064C2semidirect
4. #20261050C2semidirect
5. #20261038C2semidirect
6. #20261037C2semidirect
7. #20260953C2semidirect
8. #20260952C2semidirect
9. #20260884C2semidirect
11. #20260848C2semidirect
12. #20260845C2semidirect
13. #20260833C2semidirect
14. #20260790C2semidirect
15. #440×C97C2semidirect
22. #15690801C22semidirect
23. #15690799C22semidirect
29. C4×D388C22semidirect
38. #15690743C22semidirect
39. #15690742C22semidirect
40. #15690741C22semidirect
41. #15690739C22semidirect
42. #15690739C22semidirect
43. #15690734C22semidirect
45. #15690732C22semidirect
48. #15690730C22semidirect
51. #15690721C22semidirect
53. C4×Q8×C97C22semidirect
55. #107×C97C22semidirect
56. #103×C97C22semidirect
58. #97×C97C22semidirect
185. C97#3114semidirect
199. #440D97semidirect
213. C4×Q8C2×D97semidirect
227. #107C2×D97semidirect
233. #103C2×D97semidirect
243. #97C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612722
C4×D4×D972
#202610642
#202610502
#202610382
#202610372
#202609532
#202609522
#202608842
#202608702
#202608482
#202608452
#202608332
#202607902
#440×C972
#311497