Finite Group Databasebeta

← C4×#15690837 #27150545

Subgroups — C4×#15690837 #27150545

Order 12416.

548 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690837C211yesmaximal, characteristic
2C4×#13095512C211yesmaximal, characteristic, complemented
3C4×D4×D97C211yesmaximal, characteristic, complemented
4#20260846C211yesmaximal, characteristic, complemented
5#20260844C211yesmaximal, characteristic, complemented
6#20260838C211yesmaximal, characteristic, complemented
7#20260836C211yesmaximal, characteristic, complemented
8C4×#13095498C211yesmaximal, characteristic, complemented
9C4×D776C211yesmaximal, characteristic, complemented
10C4×#13095496C211yesmaximal, characteristic, complemented
11#20260774C211yesmaximal, characteristic, complemented
12#20260738C211yesmaximal, characteristic, complemented
13#20260735C211yesmaximal, characteristic, complemented
14C4×#13095494C211yesmaximal, characteristic, complemented
15C4×SD16×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×D4×D97C2211yescharacteristic
18C22×C4×D97—21no—
19#15690837C418yesdirect product factor
20C2×#13095498C2211yescharacteristic
21C2×D776C2211yescharacteristic
22C2×#13095496C2211yescharacteristic
23C2×#13095494C2211yescharacteristic
24C2×#13095494C2211yescharacteristic
25Q8×Dic97C2211yescharacteristic, complemented
26C4×D388—21no—
27C4×D388C2211yescharacteristic
28C4×D388C2211yescharacteristic, complemented
29D4×Dic97C2211yescharacteristic
30#15690772—21no—
31#15690772C2211yescharacteristic
32#15690772C2211yescharacteristic, complemented
33C4×C22⋊D97—21no—
34#15690770—21no—
35#15690768—21no—
36#15690768C2211yescharacteristic
37C4⋊C4×D97C2211yescharacteristic
38C22⋊C4×D97—21no—
39C42×D97—21no—
40C42×D97C2211yescharacteristic
41#15690744C2211yescharacteristic, complemented
42#15690743C2211yescharacteristic, complemented
43#15690741C2211yescharacteristic, complemented
44#15690734C2211yescharacteristic, complemented
45#15690732C2211yescharacteristic
46#15690732C2211yescharacteristic, complemented
47#15690731C2211yescharacteristic
48#15690730C2211yescharacteristic
49#15690730C2211yescharacteristic, complemented
50#15690720C2211yescharacteristic, complemented
51#15690720C2211yescharacteristic, complemented
52C4×#11715895C2211yescharacteristic, complemented
53C2×SD16×C97C2211yescharacteristic
54C4×Q8×C97C2211yescharacteristic, complemented
55C4×D4×C97C2211yescharacteristic
56#106×C97C2211yescharacteristic, complemented
57#103×C97C2211yescharacteristic, complemented
58#102×C97C2211yescharacteristic
59C4×C8×C97C2211yescharacteristic, complemented
60C23×D97—21no—
61#13095512—22no—
62#13095512C2×C414yescomplemented
63C2×D388—21no—
64C2×D388C2311yescharacteristic
65C2×D388C2311yescharacteristic
66C2×C22⋊D97—21no—
67D4×D97—22no—
68D4×D97C2×C414yescomplemented
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—21no—
76C2×C4×D97—44no—
77C2×C4×D97D411yescharacteristic
78C2×C4×D97D411yescharacteristic
79C2×C4×D97C2311yescharacteristic
80#13095498C2×C414yescomplemented
81D776C2×C414yescomplemented
82#13095496C2×C414yescomplemented
83#13095494—22no—
84#13095494C2×C414yescomplemented
85C2×#11715895C2311yescharacteristic
86C2×#11715895C2311yescharacteristic
87C4⋊Dic97—21no—
88C4⋊Dic97C2311yescharacteristic
89C4⋊Dic97C2311yescharacteristic
90#13095490—21no—
91C22⋊Dic97—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95#13095488—21no—
96#13095488—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—21no—
117C22×D97—24no—
118C22×D97—42no—
119C22×D97—42no—
120C22×D97C2×D411yescharacteristic
121Q8×C97—21no—
122Q8×C97C22×C412yescomplemented
123D4×C97—21no—
124D4×C97C22×C412yes—
125C2×C4×C97—21no—
126C2×C4×C97—21no—
127C2×C4×C97—21no—
128C2×C4×C97—42no—
129C2×C4×C97C2×D411yescharacteristic
130C2×C4×C97C2×D411yescharacteristic
131C2×C4×C97C2411yescharacteristic
132C22⋊D97—24no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97—42no—
136C2×Dic97C2×D411yescharacteristic
137C2×Dic97C2×D411yescharacteristic
138C2×Dic97C2×D411yescharacteristic
139D388—24no—
140D388—21no—
141D388—21no—
142D388C22×C412yes—
143D388C22×C412yescomplemented
144C4×D97—22no—
145C4×D97—24no—
146C4×D97—41no—
147C4×D97—42no—
148C4×D97—42no—
149C4×D97—42no—
150C4×D97—42no—
151C4×D97—42no—
152C4×D97C22×C412yes—
153C4×D97#4112yes—
154C8×C97—21no—
155C8×C97C22×C412yescomplemented
156#11715895—21no—
157#11715895C22×C412yescomplemented
158C22×C97—22no—
159C22×C97—41no—
160C22×C97—41no—
161C22×C97C22×D411yescharacteristic
162C2×D97—21no—
163C2×D97—21no—
164C2×D97—22no—
165C2×D97—22no—
166C2×D97—48no—
167C2×D97—41no—
168C2×D97—41no—
169C2×D97—41no—
170C2×D97—41no—
171C2×D97C4×D412yes—
172C4×C97—22no—
173C4×C97—21no—
174C4×C97—41no—
175C4×C97—41no—
176C4×C97AGL(1,ℤ/8)12yescomplemented
177C4×C97C23×C411yescharacteristic
178C4×C97C2×#4111yescharacteristic
179Dic97—21no—
180Dic97—21no—
181Dic97—41no—
182Dic97—41no—
183Dic97C4×D412yes—
184C2×C97—42no—
185C2×C97C2×C4×D411yescharacteristic
186C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
187C2×C97#57511yescharacteristic
188D97—22no—
189D97—42no—
190D97—42no—
191C4×AGL(1,ℤ/8)—971nomaximal, Sylow
192C97C4×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
193C2×AGL(1,ℤ/8)—971no—
194C4×#41—971no—
195C2×C4×D4—971no—
196#442—971no—
197#442—971no—
198#440—971no—
199#440—971no—
200C4×SD16—971no—
201C4×SD16D9711yescharacteristic, complemented
202C4×D8—971no—
203C4×D8—971no—
204#428—971no—
205#418—971no—
206#417—971no—
207C4×M16—971no—
208C2×#41—971no—
209C22×D4—971no—
210C23×C4—1941no—
211AGL(1,ℤ/8)—978no—
212C2×SD16—971no—
213C2×SD16C2×D9711yescharacteristic
214C2×D8—971no—
215C2×D8—971no—
216C2×M16—971no—
217C2×M16—971no—
218C4×Q8—971no—
219C4×Q8C2×D9711yescharacteristic, complemented
220C4×D4—971no—
221C4×D4—971no—
222C4×D4—971no—
223C4×D4—971no—
224C4×D4—971no—
225C4×D4—1941no—
226C4×D4—1941no—
227C4×D4—1941no—
228C4×D4—1941no—
229C4×D4C2×D9711yescharacteristic
230#117—971no—
231#117—1941no—
232C2×C4⋊C4—971no—
233C2×C22⋊C4—1941no—
234C2×C42—971no—
235C2×C42—1941no—
236#107—971no—
237#107—971no—
238#106—971no—
239#106C2×D9711yescharacteristic, complemented
240#103—971no—
241#103C2×D9711yescharacteristic, complemented
242#102—971no—
243#102—971no—
244#102—971no—
245#102—971no—
246#102—971no—
247#102C2×D9711yescharacteristic
248#97—971no—
249#97—971no—
250C4×C8—971no—
251C4×C8C2×D9711yescharacteristic, complemented
252C24—1941no—
253#41—974no—
254#41—1942no—
255C2×Q8C22×D9711yescharacteristic
256C2×D4—974no—
257C2×D4—971no—
258C2×D4—971no—
259C2×D4—1941no—
260C2×D4—1942no—
261C2×D4—1941no—
262C2×D4C22×D9711yescharacteristic
263C22×C4—21no—
264C22×C4—971no—
265C22×C4—971no—
266C22×C4—971no—
267C22×C4—3884no—
268C22×C4—1941no—
269C22×C4—1941no—
270C22×C4—1941no—
271C22×C4—1941no—
272C22×C4—1941no—
273C22×C4—1941no—
274C22×C4—1941no—
275SD16—974no—
276SD16C4×D9714yescomplemented
277D8—974no—
278D8—974no—
279M16—974no—
280M16—1942no—
281C2×C8—971no—
282C2×C8—971no—
283C2×C8C22×D9711yescharacteristic
284C2×C8C22×D9711yescharacteristic
285C4⋊C4—21no—
286C4⋊C4—971no—
287C4⋊C4—971no—
288C4⋊C4—1941no—
289C4⋊C4—1941no—
290C4⋊C4C22×D9711yescharacteristic
291C4⋊C4C22×D9711yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—1941no—
294C22⋊C4—1941no—
295C22⋊C4—1941no—
296C22⋊C4—1941no—
297C22⋊C4—1941no—
298C22⋊C4—1941no—
299C42—21no—
300C42—971no—
301C42—971no—
302C42—971no—
303C42—1941no—
304C42—1941no—
305C42C22×D9711yescharacteristic
306C23—21no—
307C23—971no—
308C23—3882no—
309C23—3882no—
310C23—1944no—
311C23—1941no—
312C23—1941no—
313Q8—21no—
314Q8C2×C4×D9712yescomplemented
315D4—21no—
316D4—972no—
317D4—972no—
318D4—1944no—
319D4—1944no—
320D4—1941no—
321D4—1941no—
322D4C2×C4×D9712yes—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—971no—
328C2×C4—972no—
329C2×C4—972no—
330C2×C4—971no—
331C2×C4—971no—
332C2×C4—3881no—
333C2×C4—3882no—
334C2×C4—3882no—
335C2×C4—3882no—
336C2×C4—3882no—
337C2×C4—3882no—
338C2×C4—3882no—
339C2×C4—1944no—
340C2×C4—1941no—
341C2×C4—1941no—
342C2×C4—1942no—
343C2×C4D4×D9711yescharacteristic
344C2×C4D4×D9711yescharacteristic
345C2×C4C23×D9711yescharacteristic
346C8—21no—
347C8—972no—
348C8—1941no—
349C8C2×C4×D9712yescomplemented
350C22—22no—
351C22—41no—
352C22—41no—
353C22—972no—
354C22—3888no—
355C22—3881no—
356C22—3881no—
357C22—3881no—
358C22—3881no—
359C22—1942no—
360C22—1942no—
361C22—1941no—
362C22—1941no—
363C22C2×D4×D9711yescharacteristic
364C4—22no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—972no—
369C4—3881no—
370C4—3881no—
371C4—1941no—
372C4—1941no—
373C4#1569083712yesdirect product factor
374C4C22×C4×D9711yescharacteristic
375C4#41×D9711yescharacteristic
376C2—42no—
377C2—3882no—
378C2—3882no—
379C2—1942no—
380C2C4×D4×D9711yescharacteristic
381C2C2×#1569083711yescharacteristic
382C2#2026153211yescharacteristic
subgroupG/H
centerC2×C4D4×D97
commutatorC4×C97C23×C4
FrattiniC2×C4C23×D97
FittingC4×SD16×C97C2
socleC22×C97C22×D4
prime radicalC2×#15690837C2
cosocleC2×C4×C97C24
hypercenterC4×SD16D97
hypocenterC97C4×AGL(1,ℤ/8)
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. C4×#13095512C2semidirect
3. C4×D4×D97C2semidirect
4. #20260846C2semidirect
5. #20260844C2semidirect
6. #20260838C2semidirect
7. #20260836C2semidirect
8. C4×#13095498C2semidirect
9. C4×D776C2semidirect
10. C4×#13095496C2semidirect
11. #20260774C2semidirect
12. #20260738C2semidirect
13. #20260735C2semidirect
14. C4×#13095494C2semidirect
15. C4×SD16×C97C2semidirect
19. #15690837C4direct (N × H)
25. Q8×Dic97C22semidirect
28. C4×D388C22semidirect
32. #15690772C22semidirect
41. #15690744C22semidirect
42. #15690743C22semidirect
43. #15690741C22semidirect
44. #15690734C22semidirect
46. #15690732C22semidirect
49. #15690730C22semidirect
50. #15690720C22semidirect
51. #15690720C22semidirect
52. C4×#11715895C22semidirect
54. C4×Q8×C97C22semidirect
56. #106×C97C22semidirect
57. #103×C97C22semidirect
59. C4×C8×C97C22semidirect
62. #13095512C2×C4semidirect
68. D4×D97C2×C4semidirect
80. #13095498C2×C4semidirect
81. D776C2×C4semidirect
82. #13095496C2×C4semidirect
84. #13095494C2×C4semidirect
105. SD16×C97C2×C4semidirect
122. Q8×C97C22×C4semidirect
143. D388C22×C4semidirect
155. C8×C97C22×C4semidirect
157. #11715895C22×C4semidirect
176. C4×C97AGL(1,ℤ/8)semidirect
192. C97C4×AGL(1,ℤ/8)semidirect
201. C4×SD16D97semidirect
219. C4×Q8C2×D97semidirect
239. #106C2×D97semidirect
241. #103C2×D97semidirect
251. C4×C8C2×D97semidirect
276. SD16C4×D97semidirect
314. Q8C2×C4×D97semidirect
349. C8C2×C4×D97semidirect
373. C4#15690837direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156908372
C4×#130955122
C4×D4×D972
#202608462
#202608442
#202608382
#202608362
C4×#130954982
C4×D7762
C4×#130954962
#202607742
#202607382
#202607352
C4×#130954942
C4×SD16×C972
C4×AGL(1,ℤ/8)97