Finite Group Databasebeta

← #440×D97 #27150511

Subgroups — #440×D97 #27150511

Order 12416.

544 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D97C211yesmaximal, characteristic
2C4×Q8×D97C211yesmaximal, characteristic, complemented
3C4×D4×D97C211yesmaximal, characteristic
4#20260838C211yesmaximal, characteristic, complemented
5#20260837C211yesmaximal, characteristic, complemented
6#20260836C211yesmaximal, characteristic, complemented
7#20260835C211yesmaximal, characteristic, complemented
8#20260834C211yesmaximal, characteristic, complemented
9#20260833C211yesmaximal, characteristic, complemented
10#20260832C211yesmaximal, characteristic, complemented
11#107×D97C211yesmaximal, characteristic, complemented
12#103×D97C211yesmaximal, characteristic, complemented
13#102×D97C211yesmaximal, characteristic
14#97×D97C211yesmaximal, characteristic, complemented
15#440×C97C211yesmaximal, characteristic, complemented
16C2×Q8×D97C2211yescharacteristic
17C2×D4×D97C2211yescharacteristic
18C22×C4×D97—21no—
19C2×#13095500C2211yescharacteristic
20C2×#13095499C2211yescharacteristic
21C2×#13095498C2211yescharacteristic
22SD16×D97C418yescomplemented
23C2×C8×D97C2211yescharacteristic
24C2×C8×D97C2211yescharacteristic
25Q8×Dic97C2211yescharacteristic, complemented
26C4×Dic194—21no—
27C4×Dic194C2211yescharacteristic, complemented
28#15690775—21no—
29#15690775C2211yescharacteristic, complemented
30C4×D388C2211yescharacteristic
31D4×Dic97C2211yescharacteristic
32#15690772C2211yescharacteristic
33C4×C22⋊D97—21no—
34#15690770—21no—
35C4⋊C4×D97—21no—
36C4⋊C4×D97C2211yescharacteristic
37C4⋊C4×D97C2211yescharacteristic
38C22⋊C4×D97—21no—
39C42×D97—21no—
40C42×D97C2211yescharacteristic
41#15690744C2211yescharacteristic, complemented
42#15690743C2211yescharacteristic, complemented
43#15690743C2211yescharacteristic, complemented
44#15690735C2211yescharacteristic, complemented
45#15690734C2211yescharacteristic, complemented
46#15690733C2211yescharacteristic, complemented
47#15690732C2211yescharacteristic
48#15690731C2211yescharacteristic
49#15690730C2211yescharacteristic
50#15690721C2211yescharacteristic, complemented
51#15690720C2211yescharacteristic, complemented
52#15690720C2211yescharacteristic, complemented
53C2×SD16×C97C2211yescharacteristic
54C4×Q8×C97C2211yescharacteristic, complemented
55C4×D4×C97C2211yescharacteristic
56#107×C97C2211yescharacteristic, complemented
57#103×C97C2211yescharacteristic, complemented
58#102×C97C2211yescharacteristic
59#97×C97C2211yescharacteristic, complemented
60C23×D97—21no—
61C2×Dic194—21no—
62C2×Dic194C2311yescharacteristic
63Q8×D97—22no—
64Q8×D97C2×C414yescomplemented
65C2×D388C2311yescharacteristic
66C2×C22⋊D97—21no—
67D4×D97—22no—
68D4×D97C2×C414yes—
69C22×Dic97—21no—
70C2×C4×D97—21no—
71C2×C4×D97—21no—
72C2×C4×D97—21no—
73C2×C4×D97—21no—
74C2×C4×D97—21no—
75C2×C4×D97—44no—
76C2×C4×D97D411yescharacteristic
77C2×C4×D97D411yescharacteristic
78C2×C4×D97C2311yescharacteristic
79#13095500C2×C414yescomplemented
80#13095499C2×C414yescomplemented
81#13095498C2×C414yescomplemented
82C2×#11715895C2311yescharacteristic
83C2×#11715895C2311yescharacteristic
84C8×D97—22no—
85C8×D97C2×C414yescomplemented
86C4⋊Dic97—21no—
87C4⋊Dic97C2311yescharacteristic
88C4⋊Dic97C2311yescharacteristic
89#13095490—21no—
90#13095490—21no—
91#13095490—21no—
92#13095490—21no—
93C22⋊Dic97—21no—
94#13095488—21no—
95#13095488—21no—
96C4×Dic97—21no—
97C4×Dic97—21no—
98C4×Dic97—21no—
99C4×Dic97D411yescharacteristic
100C4×Dic97D411yescharacteristic
101C4×Dic97C2311yescharacteristic
102C2×Q8×C97C2311yescharacteristic
103C2×D4×C97C2311yescharacteristic
104C22×C4×C97—21no—
105SD16×C97C2×C414yescomplemented
106C2×C8×C97C2311yescharacteristic
107C2×C8×C97C2311yescharacteristic
108C4⋊C4×C97—21no—
109C4⋊C4×C97C2311yescharacteristic
110C4⋊C4×C97C2311yescharacteristic
111C22⋊C4×C97—21no—
112C42×C97—21no—
113C42×C97C2311yescharacteristic
114C23×C97—21no—
115C22×D97—21no—
116C22×D97—24no—
117C22×D97—42no—
118C22×D97—42no—
119C22×D97C2×D411yescharacteristic
120Q8×C97—21no—
121Q8×C97C22×C412yescomplemented
122D4×C97—21no—
123D4×C97C22×C412yes—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—21no—
127C2×C4×C97—42no—
128C2×C4×C97C2×D411yescharacteristic
129C2×C4×C97C2×D411yescharacteristic
130C2×C4×C97C2411yescharacteristic
131C22⋊D97—24no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97—42no—
136C2×Dic97C2×D411yescharacteristic
137C2×Dic97C2×D411yescharacteristic
138C2×Dic97C2×D411yescharacteristic
139D388—21no—
140D388C22×C412yes—
141C4×D97—24no—
142C4×D97—22no—
143C4×D97—22no—
144C4×D97—42no—
145C4×D97—42no—
146C4×D97—42no—
147C4×D97—42no—
148C4×D97C22×C412yes—
149C4×D97#4112yes—
150Dic194—21no—
151Dic194—24no—
152Dic194C22×C412yescomplemented
153C8×C97—21no—
154C8×C97C22×C412yescomplemented
155#11715895—21no—
156#11715895C22×C412yescomplemented
157C22×C97—22no—
158C22×C97—41no—
159C22×C97—41no—
160C22×C97C22×D411yescharacteristic
161C2×D97—22no—
162C2×D97—41no—
163C2×D97—41no—
164C2×D97—48no—
165C2×D97C4×D412yes—
166C2×D97AGL(1,ℤ/8)12yes—
167C2×D97#13712yes—
168C4×C97—22no—
169C4×C97—21no—
170C4×C97—21no—
171C4×C97—41no—
172C4×C97—41no—
173C4×C97C23×C411yescharacteristic
174C4×C97C2×#4111yescharacteristic
175Dic97—22no—
176Dic97—21no—
177Dic97—21no—
178Dic97—41no—
179Dic97—41no—
180Dic97C4×D412yes—
181C2×C97—42no—
182C2×C97C2×C4×D411yescharacteristic
183C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C97C2×#13711yescharacteristic
185D97—42no—
186D97#44014yesdirect product factor
187C2×#440—971nomaximal, Sylow
188C97C2×#44011yescharacteristic, complemented, Sylow
189C22×SD16—971no—
190C2×C4×Q8—971no—
191C2×C4×D4—971no—
192#440—971no—
193#440—971no—
194#440—971no—
195#440—971no—
196#440—971no—
197#440—971no—
198#440—971no—
199#440D9711yescharacteristic, direct product factor
200C2×#107—971no—
201C2×#103—971no—
202C2×#102—971no—
203C2×#97—971no—
204C22×Q8—971no—
205C22×D4—971no—
206C23×C4—1941no—
207C2×SD16—971no—
208C2×SD16—971no—
209C2×SD16—978no—
210C2×SD16—971no—
211C2×SD16C2×D9711yescharacteristic
212C22×C8—971no—
213C22×C8—971no—
214C4×Q8—971no—
215C4×Q8—971no—
216C4×Q8—971no—
217C4×Q8—1941no—
218C4×Q8—1941no—
219C4×Q8C2×D9711yescharacteristic, complemented
220C4×D4—971no—
221C4×D4—971no—
222C4×D4—971no—
223C4×D4—1941no—
224C4×D4—1941no—
225C4×D4C2×D9711yescharacteristic
226C2×C4⋊C4—971no—
227C2×C4⋊C4—971no—
228C2×C4⋊C4—1941no—
229C2×C22⋊C4—1941no—
230C2×C42—971no—
231C2×C42—1941no—
232#107—971no—
233#107—971no—
234#107—971no—
235#107C2×D9711yescharacteristic, complemented
236#103—971no—
237#103—971no—
238#103—971no—
239#103C2×D9711yescharacteristic, complemented
240#102—971no—
241#102—971no—
242#102—971no—
243#102C2×D9711yescharacteristic
244#97—971no—
245#97—971no—
246#97—971no—
247#97C2×D9711yescharacteristic, complemented
248C24—1941no—
249C2×Q8—974no—
250C2×Q8—971no—
251C2×Q8—1941no—
252C2×Q8—1942no—
253C2×Q8C22×D9711yescharacteristic
254C2×D4—974no—
255C2×D4—971no—
256C2×D4—1942no—
257C2×D4—1941no—
258C2×D4C22×D9711yescharacteristic
259C22×C4—21no—
260C22×C4—971no—
261C22×C4—971no—
262C22×C4—971no—
263C22×C4—3884no—
264C22×C4—1941no—
265C22×C4—1941no—
266C22×C4—1941no—
267C22×C4—1941no—
268C22×C4—1941no—
269C22×C4—1941no—
270SD16—974no—
271SD16—974no—
272SD16—974no—
273SD16C4×D9714yescomplemented
274C2×C8—971no—
275C2×C8—974no—
276C2×C8—971no—
277C2×C8—1942no—
278C2×C8C22×D9711yescharacteristic
279C2×C8C22×D9711yescharacteristic
280C4⋊C4—21no—
281C4⋊C4—971no—
282C4⋊C4—971no—
283C4⋊C4—1941no—
284C4⋊C4—1941no—
285C4⋊C4—1941no—
286C4⋊C4—1941no—
287C4⋊C4—1941no—
288C4⋊C4C22×D9711yescharacteristic
289C4⋊C4C22×D9711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—1941no—
292C22⋊C4—1941no—
293C22⋊C4—1941no—
294C42—21no—
295C42—971no—
296C42—971no—
297C42—971no—
298C42—1941no—
299C42—1941no—
300C42—1941no—
301C42C22×D9711yescharacteristic
302C23—21no—
303C23—971no—
304C23—3882no—
305C23—3882no—
306C23—1944no—
307C23—1941no—
308Q8—21no—
309Q8—972no—
310Q8—1944no—
311Q8—1941no—
312Q8C2×C4×D9712yescomplemented
313D4—21no—
314D4—972no—
315D4—1944no—
316D4—1941no—
317D4C2×C4×D9712yes—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—971no—
323C2×C4—972no—
324C2×C4—972no—
325C2×C4—971no—
326C2×C4—971no—
327C2×C4—3882no—
328C2×C4—3882no—
329C2×C4—3882no—
330C2×C4—3882no—
331C2×C4—3882no—
332C2×C4—1942no—
333C2×C4—1942no—
334C2×C4—1944no—
335C2×C4—1941no—
336C2×C4—1941no—
337C2×C4—1941no—
338C2×C4D4×D9711yescharacteristic
339C2×C4D4×D9711yescharacteristic
340C2×C4C23×D9711yescharacteristic
341C8—21no—
342C8—972no—
343C8—1941no—
344C8C2×C4×D9712yescomplemented
345C22—22no—
346C22—41no—
347C22—41no—
348C22—972no—
349C22—972no—
350C22—972no—
351C22—3888no—
352C22—3881no—
353C22—3881no—
354C22—1942no—
355C22C2×D4×D9711yescharacteristic
356C4—22no—
357C4—21no—
358C4—21no—
359C4—41no—
360C4—41no—
361C4—972no—
362C4—3881no—
363C4—3881no—
364C4—1942no—
365C4—1941no—
366C4—1941no—
367C4C22×C4×D9711yescharacteristic
368C4#41×D9711yescharacteristic
369C2—42no—
370C2—974no—
371C2—3882no—
372C2C4×D4×D9711yescharacteristic
373C2AGL(1,ℤ/8)×D9711yescharacteristic
374C2#137×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutatorC4×C97C23×C4
FrattiniC2×C4C23×D97
Fitting#440×C97C2
socleC22×C97C22×D4
prime radicalC2×D4×D97C22
cosocleC2×C4×C97C24
hypercenter#440D97
hypocenterC97C2×#440
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. C4×Q8×D97C2semidirect
4. #20260838C2semidirect
5. #20260837C2semidirect
6. #20260836C2semidirect
7. #20260835C2semidirect
8. #20260834C2semidirect
9. #20260833C2semidirect
10. #20260832C2semidirect
11. #107×D97C2semidirect
12. #103×D97C2semidirect
14. #97×D97C2semidirect
15. #440×C97C2semidirect
22. SD16×D97C4semidirect
25. Q8×Dic97C22semidirect
27. C4×Dic194C22semidirect
29. #15690775C22semidirect
41. #15690744C22semidirect
42. #15690743C22semidirect
43. #15690743C22semidirect
44. #15690735C22semidirect
45. #15690734C22semidirect
46. #15690733C22semidirect
50. #15690721C22semidirect
51. #15690720C22semidirect
52. #15690720C22semidirect
54. C4×Q8×C97C22semidirect
56. #107×C97C22semidirect
57. #103×C97C22semidirect
59. #97×C97C22semidirect
64. Q8×D97C2×C4semidirect
79. #13095500C2×C4semidirect
80. #13095499C2×C4semidirect
81. #13095498C2×C4semidirect
85. C8×D97C2×C4semidirect
105. SD16×C97C2×C4semidirect
121. Q8×C97C22×C4semidirect
152. Dic194C22×C4semidirect
154. C8×C97C22×C4semidirect
156. #11715895C22×C4semidirect
186. D97#440direct (N × H)
188. C97C2×#440semidirect
199. #440D97direct (N × H)
219. C4×Q8C2×D97semidirect
235. #107C2×D97semidirect
239. #103C2×D97semidirect
247. #97C2×D97semidirect
273. SD16C4×D97semidirect
312. Q8C2×C4×D97semidirect
344. C8C2×C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D972
C4×Q8×D972
C4×D4×D972
#202608382
#202608372
#202608362
#202608352
#202608342
#202608332
#202608322
#107×D972
#103×D972
#102×D972
#97×D972
#440×C972
C2×#44097