Finite Group Databasebeta

← C4×#16812136 #28345270

Subgroups — C4×#16812136 #28345270

Order 13696.

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×#16812136C211yesmaximal, characteristic
2C4×#13133819C211yesmaximal, characteristic, complemented
3C4×D4×D107C211yesmaximal, characteristic, complemented
4#20292738C211yesmaximal, characteristic, complemented
5#20292736C211yesmaximal, characteristic, complemented
6#20292730C211yesmaximal, characteristic, complemented
7#20292728C211yesmaximal, characteristic, complemented
8C4×#13133805C211yesmaximal, characteristic, complemented
9C4×D856C211yesmaximal, characteristic, complemented
10C4×#13133803C211yesmaximal, characteristic, complemented
11#20292666C211yesmaximal, characteristic, complemented
12#20292630C211yesmaximal, characteristic, complemented
13#20292627C211yesmaximal, characteristic, complemented
14C4×#13133801C211yesmaximal, characteristic, complemented
15C4×SD16×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×D4×D107C2211yescharacteristic
18C22×C4×D107—21no—
19#16812136C418yesdirect product factor
20C2×#13133805C2211yescharacteristic
21C2×D856C2211yescharacteristic
22C2×#13133803C2211yescharacteristic
23C2×#13133801C2211yescharacteristic
24C2×#13133801C2211yescharacteristic
25Q8×Dic107C2211yescharacteristic, complemented
26C4×D428—21no—
27C4×D428C2211yescharacteristic
28C4×D428C2211yescharacteristic, complemented
29D4×Dic107C2211yescharacteristic
30#16812071—21no—
31#16812071C2211yescharacteristic
32#16812071C2211yescharacteristic, complemented
33C4×C22⋊D107—21no—
34(C22⋊D107)⋊C4—21no—
35#16812067—21no—
36#16812067C2211yescharacteristic
37C4⋊C4×D107C2211yescharacteristic
38C22⋊C4×D107—21no—
39C42×D107—21no—
40C42×D107C2211yescharacteristic
41#16812043C2211yescharacteristic, complemented
42#16812042C2211yescharacteristic, complemented
43#16812040C2211yescharacteristic, complemented
44#16812033C2211yescharacteristic, complemented
45#16812031C2211yescharacteristic
46#16812031C2211yescharacteristic, complemented
47#16812030C2211yescharacteristic
48#16812029C2211yescharacteristic
49#16812029C2211yescharacteristic, complemented
50#16812019C2211yescharacteristic, complemented
51#16812019C2211yescharacteristic, complemented
52C4×C107⋊C8C2211yescharacteristic, complemented
53C2×SD16×C107C2211yescharacteristic
54C4×Q8×C107C2211yescharacteristic, complemented
55C4×D4×C107C2211yescharacteristic
56#106×C107C2211yescharacteristic, complemented
57#103×C107C2211yescharacteristic, complemented
58#102×C107C2211yescharacteristic
59C4×C8×C107C2211yescharacteristic, complemented
60C23×D107—21no—
61#13133819—22no—
62#13133819C2×C414yescomplemented
63C2×D428—21no—
64C2×D428C2311yescharacteristic
65C2×D428C2311yescharacteristic
66C2×C22⋊D107—21no—
67D4×D107—22no—
68D4×D107C2×C414yescomplemented
69C22×Dic107—21no—
70C2×C4×D107—21no—
71C2×C4×D107—21no—
72C2×C4×D107—21no—
73C2×C4×D107—21no—
74C2×C4×D107—21no—
75C2×C4×D107—21no—
76C2×C4×D107—44no—
77C2×C4×D107D411yescharacteristic
78C2×C4×D107D411yescharacteristic
79C2×C4×D107C2311yescharacteristic
80#13133805C2×C414yescomplemented
81D856C2×C414yescomplemented
82#13133803C2×C414yescomplemented
83#13133801—22no—
84#13133801C2×C414yescomplemented
85C2×C107⋊C8C2311yescharacteristic
86C2×C107⋊C8C2311yescharacteristic
87C4⋊Dic107—21no—
88C4⋊Dic107C2311yescharacteristic
89C4⋊Dic107C2311yescharacteristic
90Dic107⋊C4—21no—
91C22⋊Dic107—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95(C2×D107)⋊C4—21no—
96(C2×D107)⋊C4—21no—
97C4×Dic107—21no—
98C4×Dic107—21no—
99C4×Dic107D411yescharacteristic
100C4×Dic107D411yescharacteristic
101C4×Dic107C2311yescharacteristic
102C2×Q8×C107C2311yescharacteristic
103C2×D4×C107C2311yescharacteristic
104C22×C4×C107—21no—
105SD16×C107C2×C414yescomplemented
106C2×C8×C107C2311yescharacteristic
107C2×C8×C107C2311yescharacteristic
108C4⋊C4×C107—21no—
109C4⋊C4×C107C2311yescharacteristic
110C4⋊C4×C107C2311yescharacteristic
111C22⋊C4×C107—21no—
112C42×C107—21no—
113C42×C107C2311yescharacteristic
114C23×C107—21no—
115C22×D107—21no—
116C22×D107—21no—
117C22×D107—24no—
118C22×D107—42no—
119C22×D107—42no—
120C22×D107C2×D411yescharacteristic
121Q8×C107—21no—
122Q8×C107C22×C412yescomplemented
123D4×C107—21no—
124D4×C107C22×C412yes—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—21no—
128C2×C4×C107—42no—
129C2×C4×C107C2×D411yescharacteristic
130C2×C4×C107C2×D411yescharacteristic
131C2×C4×C107C2411yescharacteristic
132C22⋊D107—24no—
133C2×Dic107—21no—
134C2×Dic107—21no—
135C2×Dic107—42no—
136C2×Dic107C2×D411yescharacteristic
137C2×Dic107C2×D411yescharacteristic
138C2×Dic107C2×D411yescharacteristic
139D428—24no—
140D428—21no—
141D428—21no—
142D428C22×C412yes—
143D428C22×C412yescomplemented
144C4×D107—22no—
145C4×D107—24no—
146C4×D107—41no—
147C4×D107—42no—
148C4×D107—42no—
149C4×D107—42no—
150C4×D107—42no—
151C4×D107—42no—
152C4×D107C22×C412yes—
153C4×D107#4112yes—
154C8×C107—21no—
155C8×C107C22×C412yescomplemented
156C107⋊C8—21no—
157C107⋊C8C22×C412yescomplemented
158C22×C107—22no—
159C22×C107—41no—
160C22×C107—41no—
161C22×C107C22×D411yescharacteristic
162C2×D107—21no—
163C2×D107—21no—
164C2×D107—22no—
165C2×D107—22no—
166C2×D107—48no—
167C2×D107—41no—
168C2×D107—41no—
169C2×D107—41no—
170C2×D107—41no—
171C2×D107C4×D412yes—
172C4×C107—22no—
173C4×C107—21no—
174C4×C107—41no—
175C4×C107—41no—
176C4×C107AGL(1,ℤ/8)12yescomplemented
177C4×C107C23×C411yescharacteristic
178C4×C107C2×#4111yescharacteristic
179Dic107—21no—
180Dic107—21no—
181Dic107—41no—
182Dic107—41no—
183Dic107C4×D412yes—
184C2×C107—42no—
185C2×C107C2×C4×D411yescharacteristic
186C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
187C2×C107#57511yescharacteristic
188D107—22no—
189D107—42no—
190D107—42no—
191C4×AGL(1,ℤ/8)—1071nomaximal, Sylow
192C107C4×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
193C2×AGL(1,ℤ/8)—1071no—
194C4×#41—1071no—
195C2×C4×D4—1071no—
196#442—1071no—
197#442—1071no—
198#440—1071no—
199#440—1071no—
200C4×SD16—1071no—
201C4×SD16D10711yescharacteristic, complemented
202C4×D8—1071no—
203C4×D8—1071no—
204#428—1071no—
205#418—1071no—
206#417—1071no—
207C4×M16—1071no—
208C2×#41—1071no—
209C22×D4—1071no—
210C23×C4—2141no—
211AGL(1,ℤ/8)—1078no—
212C2×SD16—1071no—
213C2×SD16C2×D10711yescharacteristic
214C2×D8—1071no—
215C2×D8—1071no—
216C2×M16—1071no—
217C2×M16—1071no—
218C4×Q8—1071no—
219C4×Q8C2×D10711yescharacteristic, complemented
220C4×D4—1071no—
221C4×D4—1071no—
222C4×D4—1071no—
223C4×D4—1071no—
224C4×D4—1071no—
225C4×D4—2141no—
226C4×D4—2141no—
227C4×D4—2141no—
228C4×D4—2141no—
229C4×D4C2×D10711yescharacteristic
230#117—1071no—
231#117—2141no—
232C2×C4⋊C4—1071no—
233C2×C22⋊C4—2141no—
234C2×C42—1071no—
235C2×C42—2141no—
236#107—1071no—
237#107—1071no—
238#106—1071no—
239#106C2×D10711yescharacteristic, complemented
240#103—1071no—
241#103C2×D10711yescharacteristic, complemented
242#102—1071no—
243#102—1071no—
244#102—1071no—
245#102—1071no—
246#102—1071no—
247#102C2×D10711yescharacteristic
248#97—1071no—
249#97—1071no—
250C4×C8—1071no—
251C4×C8C2×D10711yescharacteristic, complemented
252C24—2141no—
253#41—1074no—
254#41—2142no—
255C2×Q8C22×D10711yescharacteristic
256C2×D4—1074no—
257C2×D4—1071no—
258C2×D4—1071no—
259C2×D4—2141no—
260C2×D4—2142no—
261C2×D4—2141no—
262C2×D4C22×D10711yescharacteristic
263C22×C4—21no—
264C22×C4—1071no—
265C22×C4—1071no—
266C22×C4—1071no—
267C22×C4—4284no—
268C22×C4—2141no—
269C22×C4—2141no—
270C22×C4—2141no—
271C22×C4—2141no—
272C22×C4—2141no—
273C22×C4—2141no—
274C22×C4—2141no—
275SD16—1074no—
276SD16C4×D10714yescomplemented
277D8—1074no—
278D8—1074no—
279M16—1074no—
280M16—2142no—
281C2×C8—1071no—
282C2×C8—1071no—
283C2×C8C22×D10711yescharacteristic
284C2×C8C22×D10711yescharacteristic
285C4⋊C4—21no—
286C4⋊C4—1071no—
287C4⋊C4—1071no—
288C4⋊C4—2141no—
289C4⋊C4—2141no—
290C4⋊C4C22×D10711yescharacteristic
291C4⋊C4C22×D10711yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—2141no—
294C22⋊C4—2141no—
295C22⋊C4—2141no—
296C22⋊C4—2141no—
297C22⋊C4—2141no—
298C22⋊C4—2141no—
299C42—21no—
300C42—1071no—
301C42—1071no—
302C42—1071no—
303C42—2141no—
304C42—2141no—
305C42C22×D10711yescharacteristic
306C23—21no—
307C23—1071no—
308C23—4282no—
309C23—4282no—
310C23—2144no—
311C23—2141no—
312C23—2141no—
313Q8—21no—
314Q8C2×C4×D10712yescomplemented
315D4—21no—
316D4—1072no—
317D4—1072no—
318D4—2144no—
319D4—2144no—
320D4—2141no—
321D4—2141no—
322D4C2×C4×D10712yes—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—1071no—
328C2×C4—1072no—
329C2×C4—1072no—
330C2×C4—1071no—
331C2×C4—1071no—
332C2×C4—4281no—
333C2×C4—4282no—
334C2×C4—4282no—
335C2×C4—4282no—
336C2×C4—4282no—
337C2×C4—4282no—
338C2×C4—4282no—
339C2×C4—2144no—
340C2×C4—2141no—
341C2×C4—2141no—
342C2×C4—2142no—
343C2×C4D4×D10711yescharacteristic
344C2×C4D4×D10711yescharacteristic
345C2×C4C23×D10711yescharacteristic
346C8—21no—
347C8—1072no—
348C8—2141no—
349C8C2×C4×D10712yescomplemented
350C22—22no—
351C22—41no—
352C22—41no—
353C22—1072no—
354C22—4288no—
355C22—4281no—
356C22—4281no—
357C22—4281no—
358C22—4281no—
359C22—2142no—
360C22—2142no—
361C22—2141no—
362C22—2141no—
363C22C2×D4×D10711yescharacteristic
364C4—21no—
365C4—22no—
366C4—41no—
367C4—41no—
368C4—1072no—
369C4—4281no—
370C4—4281no—
371C4—2141no—
372C4—2141no—
373C4#1681213612yesdirect product factor
374C4C22×C4×D10711yescharacteristic
375C4#41×D10711yescharacteristic
376C2—42no—
377C2—4282no—
378C2—4282no—
379C2—2142no—
380C2C4×D4×D10711yescharacteristic
381C2C2×#1681213611yescharacteristic
382C2#2029342411yescharacteristic
subgroupG/H
centerC2×C4D4×D107
commutatorC4×C107C23×C4
FrattiniC2×C4C23×D107
FittingC4×SD16×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812136C2
cosocleC2×C4×C107C24
hypercenterC4×SD16D107
hypocenterC107C4×AGL(1,ℤ/8)
LCSG ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. C4×#13133819C2semidirect
3. C4×D4×D107C2semidirect
4. #20292738C2semidirect
5. #20292736C2semidirect
6. #20292730C2semidirect
7. #20292728C2semidirect
8. C4×#13133805C2semidirect
9. C4×D856C2semidirect
10. C4×#13133803C2semidirect
11. #20292666C2semidirect
12. #20292630C2semidirect
13. #20292627C2semidirect
14. C4×#13133801C2semidirect
15. C4×SD16×C107C2semidirect
19. #16812136C4direct (N × H)
25. Q8×Dic107C22semidirect
28. C4×D428C22semidirect
32. #16812071C22semidirect
41. #16812043C22semidirect
42. #16812042C22semidirect
43. #16812040C22semidirect
44. #16812033C22semidirect
46. #16812031C22semidirect
49. #16812029C22semidirect
50. #16812019C22semidirect
51. #16812019C22semidirect
52. C4×C107⋊C8C22semidirect
54. C4×Q8×C107C22semidirect
56. #106×C107C22semidirect
57. #103×C107C22semidirect
59. C4×C8×C107C22semidirect
62. #13133819C2×C4semidirect
68. D4×D107C2×C4semidirect
80. #13133805C2×C4semidirect
81. D856C2×C4semidirect
82. #13133803C2×C4semidirect
84. #13133801C2×C4semidirect
105. SD16×C107C2×C4semidirect
122. Q8×C107C22×C4semidirect
143. D428C22×C4semidirect
155. C8×C107C22×C4semidirect
157. C107⋊C8C22×C4semidirect
176. C4×C107AGL(1,ℤ/8)semidirect
192. C107C4×AGL(1,ℤ/8)semidirect
201. C4×SD16D107semidirect
219. C4×Q8C2×D107semidirect
239. #106C2×D107semidirect
241. #103C2×D107semidirect
251. C4×C8C2×D107semidirect
276. SD16C4×D107semidirect
314. Q8C2×C4×D107semidirect
349. C8C2×C4×D107semidirect
373. C4#16812136direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168121362
C4×#131338192
C4×D4×D1072
#202927382
#202927362
#202927302
#202927282
C4×#131338052
C4×D8562
C4×#131338032
#202926662
#202926302
#202926272
C4×#131338012
C4×SD16×C1072
C4×AGL(1,ℤ/8)107