Finite Group Databasebeta

← C4×#15690834 #27150539

Subgroups — C4×#15690834 #27150539

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×#15690834C211yesmaximal, characteristic
2C4×#13095510C211yesmaximal, characteristic, complemented
3C4×D4×D97C211yesmaximal, characteristic, complemented
4#20260847C211yesmaximal, characteristic, complemented
5#20260844C211yesmaximal, characteristic, complemented
6#20260834C211yesmaximal, characteristic, complemented
7#20260832C211yesmaximal, characteristic, complemented
8C4×#13095500C211yesmaximal, characteristic, complemented
9C4×#13095499C211yesmaximal, characteristic, complemented
10C4×#13095496C211yesmaximal, characteristic, complemented
11#20260776C211yesmaximal, characteristic, complemented
12#20260738C211yesmaximal, characteristic, complemented
13#20260733C211yesmaximal, characteristic, complemented
14C4×#13095494C211yesmaximal, characteristic, complemented
15C4×D8×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C2211yescharacteristic
17C2×D4×D97C2211yescharacteristic
18C22×C4×D97—21no—
19#15690834C418yesdirect product factor
20C2×#13095500C2211yescharacteristic
21C2×#13095499C2211yescharacteristic
22C2×#13095496C2211yescharacteristic
23C2×#13095494C2211yescharacteristic
24C2×#13095494C2211yescharacteristic
25C4×Dic194C2211yescharacteristic, complemented
26#15690775C2211yescharacteristic, complemented
27C4×D388C2211yescharacteristic
28D4×Dic97C2211yescharacteristic
29D4×Dic97C2211yescharacteristic, complemented
30#15690772C2211yescharacteristic
31C4×C22⋊D97—21no—
32C4×C22⋊D97—21no—
33#15690770—21no—
34#15690770—21no—
35#15690768C2211yescharacteristic
36#15690766—21no—
37C4⋊C4×D97C2211yescharacteristic
38C22⋊C4×D97—21no—
39C2×C4×Dic97—21no—
40C42×D97C2211yescharacteristic
41#15690743C2211yescharacteristic, complemented
42#15690742C2211yescharacteristic, complemented
43#15690741C2211yescharacteristic, complemented
44#15690735C2211yescharacteristic, complemented
45#15690733C2211yescharacteristic, complemented
46#15690732C2211yescharacteristic
47#15690731C2211yescharacteristic
48#15690731C2211yescharacteristic, complemented
49#15690730C2211yescharacteristic
50#15690720C2211yescharacteristic, complemented
51#15690720C2211yescharacteristic, complemented
52C4×#11715895C2211yescharacteristic, complemented
53C2×D8×C97C2211yescharacteristic
54C4×D4×C97C2211yescharacteristic
55C4×D4×C97C2211yescharacteristic, complemented
56#107×C97C2211yescharacteristic, complemented
57#102×C97C2211yescharacteristic
58#102×C97C2211yescharacteristic, complemented
59C4×C8×C97C2211yescharacteristic, complemented
60C23×D97—21no—
61#13095510—22no—
62#13095510C2×C414yescomplemented
63C2×Dic194C2311yescharacteristic
64C2×D388C2311yescharacteristic
65C2×C22⋊D97—21no—
66C2×C22⋊D97—21no—
67D4×D97—22no—
68D4×D97C2×C414yescomplemented
69C22×Dic97—21no—
70C22×Dic97—21no—
71C22×Dic97—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#13095496C2×C414yescomplemented
82#13095494—22no—
83#13095494C2×C414yescomplemented
84C2×#11715895C2311yescharacteristic
85C2×#11715895C2311yescharacteristic
86C4⋊Dic97C2311yescharacteristic
87C4⋊Dic97C2311yescharacteristic
88#13095490—21no—
89#13095490—21no—
90#13095490—21no—
91C22⋊Dic97—21no—
92C22⋊Dic97—21no—
93#13095488—21no—
94#13095488—21no—
95#13095488—21no—
96C4×Dic97—21no—
97C4×Dic97—21no—
98C4×Dic97—21no—
99C4×Dic97D411yescharacteristic
100C4×Dic97D411yescharacteristic
101C4×Dic97C2311yescharacteristic
102C2×D4×C97C2311yescharacteristic
103C2×D4×C97C2311yescharacteristic
104C22×C4×C97—21no—
105C22×C4×C97—21no—
106D8×C97C2×C414yescomplemented
107C2×C8×C97C2311yescharacteristic
108C2×C8×C97C2311yescharacteristic
109C4⋊C4×C97C2311yescharacteristic
110C4⋊C4×C97C2311yescharacteristic
111C22⋊C4×C97—21no—
112C22⋊C4×C97—21no—
113C42×C97C2311yescharacteristic
114C23×C97—21no—
115C23×C97—21no—
116C22×D97—21no—
117C22×D97—24no—
118C22×D97—42no—
119C22×D97—42no—
120C22×D97C2×D411yescharacteristic
121D4×C97—21no—
122D4×C97—21no—
123D4×C97C22×C412yes—
124D4×C97C22×C412yescomplemented
125C2×C4×C97—21no—
126C2×C4×C97—21no—
127C2×C4×C97—42no—
128C2×C4×C97—42no—
129C2×C4×C97C2×D411yescharacteristic
130C2×C4×C97C2×D411yescharacteristic
131C2×C4×C97C2411yescharacteristic
132C22⋊D97—24no—
133C22⋊D97—24no—
134C2×Dic97—21no—
135C2×Dic97—24no—
136C2×Dic97—21no—
137C2×Dic97—21no—
138C2×Dic97—42no—
139C2×Dic97—42no—
140C2×Dic97C2×D411yescharacteristic
141C2×Dic97C2×D411yescharacteristic
142C2×Dic97C2×D411yescharacteristic
143D388—21no—
144D388C22×C412yes—
145C4×D97—22no—
146C4×D97—41no—
147C4×D97—42no—
148C4×D97—42no—
149C4×D97—42no—
150C4×D97C22×C412yes—
151C4×D97#4112yes—
152Dic194—21no—
153Dic194C22×C412yescomplemented
154C8×C97—21no—
155C8×C97C22×C412yescomplemented
156#11715895—21no—
157#11715895C22×C412yescomplemented
158C22×C97—22no—
159C22×C97—22no—
160C22×C97—41no—
161C22×C97—41no—
162C22×C97—41no—
163C22×C97—41no—
164C22×C97C22×D411yescharacteristic
165C2×D97—21no—
166C2×D97—21no—
167C2×D97—22no—
168C2×D97—48no—
169C2×D97—41no—
170C2×D97—41no—
171C2×D97C4×D412yes—
172C4×C97—21no—
173C4×C97—41no—
174C4×C97—41no—
175C4×C97AGL(1,ℤ/8)12yescomplemented
176C4×C97C23×C411yescharacteristic
177C4×C97C2×#4111yescharacteristic
178Dic97—22no—
179Dic97—21no—
180Dic97—21no—
181Dic97—41no—
182Dic97—41no—
183Dic97C4×D412yes—
184C2×C97—42no—
185C2×C97—42no—
186C2×C97C2×C4×D411yescharacteristic
187C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
188C2×C97#57511yescharacteristic
189D97—22no—
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×SD16—971no—
202C4×D8—971no—
203C4×D8D9711yescharacteristic, complemented
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×SD16—971no—
214C2×D8—971no—
215C2×D8C2×D9711yescharacteristic
216C2×M16—971no—
217C2×M16—971no—
218C4×Q8—971no—
219C4×Q8—971no—
220C4×D4—971no—
221C4×D4—971no—
222C4×D4—971no—
223C4×D4—971no—
224C4×D4—1941no—
225C4×D4—1941no—
226C4×D4—1941no—
227C4×D4—1941no—
228C4×D4C2×D9711yescharacteristic
229C4×D4C2×D9711yescharacteristic, complemented
230#117—971no—
231#117—1941no—
232C2×C4⋊C4—971no—
233C2×C22⋊C4—1941no—
234C2×C42—971no—
235C2×C42—1941no—
236#107—971no—
237#107C2×D9711yescharacteristic, complemented
238#106—971no—
239#106—971no—
240#103—971no—
241#103—971no—
242#102—971no—
243#102—971no—
244#102—971no—
245#102—971no—
246#102C2×D9711yescharacteristic
247#102C2×D9711yescharacteristic, complemented
248#97—971no—
249#97—971no—
250C4×C8—971no—
251C4×C8C2×D9711yescharacteristic, complemented
252C24—1941no—
253#41—974no—
254#41—1942no—
255C2×Q8—971no—
256C2×D4—974no—
257C2×D4—971no—
258C2×D4—1942no—
259C2×D4—1941no—
260C2×D4—1941no—
261C2×D4C22×D9711yescharacteristic
262C2×D4C22×D9711yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—971no—
266C22×C4—971no—
267C22×C4—971no—
268C22×C4—3884no—
269C22×C4—1941no—
270C22×C4—1941no—
271C22×C4—1941no—
272C22×C4—1941no—
273C22×C4—1941no—
274C22×C4—1941no—
275SD16—974no—
276SD16—974no—
277D8—974no—
278D8C4×D9714yescomplemented
279M16—974no—
280M16—1942no—
281C2×C8—971no—
282C2×C8—971no—
283C2×C8C22×D9711yescharacteristic
284C2×C8C22×D9711yescharacteristic
285C4⋊C4—971no—
286C4⋊C4—971no—
287C4⋊C4—1941no—
288C4⋊C4—1941no—
289C4⋊C4—1941no—
290C4⋊C4C22×D9711yescharacteristic
291C4⋊C4C22×D9711yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—1941no—
295C22⋊C4—1941no—
296C22⋊C4—1941no—
297C22⋊C4—1941no—
298C22⋊C4—1941no—
299C42—971no—
300C42—971no—
301C42—971no—
302C42—1941no—
303C42—1941no—
304C42—1941no—
305C42C22×D9711yescharacteristic
306C23—21no—
307C23—21no—
308C23—971no—
309C23—3882no—
310C23—3882no—
311C23—1944no—
312C23—1941no—
313Q8—972no—
314Q8—1941no—
315D4—21no—
316D4—21no—
317D4—972no—
318D4—1944no—
319D4—1944no—
320D4—1941no—
321D4C2×C4×D9712yes—
322D4C2×C4×D9712yescomplemented
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
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—1944no—
339C2×C4—1941no—
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—22no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—972no—
357C22—3888no—
358C22—3881no—
359C22—3881no—
360C22—1942no—
361C22—1941no—
362C22—1941no—
363C22C2×D4×D9711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—972no—
368C4—3881no—
369C4—3881no—
370C4—1942no—
371C4—1941no—
372C4—1941no—
373C4#1569083412yesdirect product factor
374C4C22×C4×D9711yescharacteristic
375C4#41×D9711yescharacteristic
376C2—42no—
377C2—42no—
378C2—3882no—
379C2—1942no—
380C2C4×D4×D9711yescharacteristic
381C2C2×#1569083411yescharacteristic
382C2#2026153211yescharacteristic
subgroupG/H
centerC2×C4D4×D97
commutatorC4×C97C23×C4
FrattiniC2×C4C23×D97
FittingC4×D8×C97C2
socleC22×C97C22×D4
prime radicalC2×#15690834C2
cosocleC2×C4×C97C24
hypercenterC4×D8D97
hypocenterC97C4×AGL(1,ℤ/8)
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
2. C4×#13095510C2semidirect
3. C4×D4×D97C2semidirect
4. #20260847C2semidirect
5. #20260844C2semidirect
6. #20260834C2semidirect
7. #20260832C2semidirect
8. C4×#13095500C2semidirect
9. C4×#13095499C2semidirect
10. C4×#13095496C2semidirect
11. #20260776C2semidirect
12. #20260738C2semidirect
13. #20260733C2semidirect
14. C4×#13095494C2semidirect
15. C4×D8×C97C2semidirect
19. #15690834C4direct (N × H)
25. C4×Dic194C22semidirect
26. #15690775C22semidirect
29. D4×Dic97C22semidirect
41. #15690743C22semidirect
42. #15690742C22semidirect
43. #15690741C22semidirect
44. #15690735C22semidirect
45. #15690733C22semidirect
48. #15690731C22semidirect
50. #15690720C22semidirect
51. #15690720C22semidirect
52. C4×#11715895C22semidirect
55. C4×D4×C97C22semidirect
56. #107×C97C22semidirect
58. #102×C97C22semidirect
59. C4×C8×C97C22semidirect
62. #13095510C2×C4semidirect
68. D4×D97C2×C4semidirect
79. #13095500C2×C4semidirect
80. #13095499C2×C4semidirect
81. #13095496C2×C4semidirect
83. #13095494C2×C4semidirect
106. D8×C97C2×C4semidirect
124. D4×C97C22×C4semidirect
153. Dic194C22×C4semidirect
155. C8×C97C22×C4semidirect
157. #11715895C22×C4semidirect
175. C4×C97AGL(1,ℤ/8)semidirect
192. C97C4×AGL(1,ℤ/8)semidirect
203. C4×D8D97semidirect
229. C4×D4C2×D97semidirect
237. #107C2×D97semidirect
247. #102C2×D97semidirect
251. C4×C8C2×D97semidirect
278. D8C4×D97semidirect
322. D4C2×C4×D97semidirect
349. C8C2×C4×D97semidirect
373. C4#15690834direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156908342
C4×#130955102
C4×D4×D972
#202608472
#202608442
#202608342
#202608322
C4×#130955002
C4×#130954992
C4×#130954962
#202607762
#202607382
#202607332
C4×#130954942
C4×D8×C972
C4×AGL(1,ℤ/8)97