Finite Group Databasebeta

← C2×#14521215 #16855033

Subgroups — C2×#14521215 #16855033

Order 3968.

844 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 844 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758566C211yesmaximal, characteristic, complemented
2C2×#11758509C211yesmaximal, characteristic, complemented
3C2×#11758517C211yesmaximal, characteristic, complemented
4#14521215C218yesmaximal, direct product factor
5C2×#11758425C211yesmaximal, characteristic, complemented
6C2×#11758413C211yesmaximal, characteristic, complemented
7C2×#11758411C211yesmaximal, characteristic, complemented
8C2×C31×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
9C2×#41×C31C2211yescharacteristic
10C22×D4×C31C2211yescharacteristic, complemented
11C22×C22⋊D31—22no—
12#11758566—22no—
13#11758566—22no—
14#11758566—22no—
15#11758566C2214yescomplemented
16C2×#92499—21no—
17C2×D4×D31—22no—
18C2×D4×D31—21no—
19C2×#92495—21no—
20C2×#92495C2211yescharacteristic
21C22×D124—21no—
22C22×D124C2211yescharacteristic
23C31×AGL(1,ℤ/8)—22no—
24C31×AGL(1,ℤ/8)C2214yescomplemented
25C2×SD16×C31—21no—
26C2×D8×C31—21no—
27C2×M16×C31C2211yescharacteristic, complemented
28#11758517—22no—
29#11758517C2214yescomplemented
30C2×#92480—21no—
31#11758509—22no—
32#11758509C2214yescomplemented
33C2×#92479—21no—
34C2×C4×Dic31C2211yescharacteristic, complemented
35C2×#92474C2211yescharacteristic, complemented
36#11758425C2214yescomplemented
37#11758413C2214yescomplemented
38#11758411C2214yescomplemented
39C24×C31—21no—
40C23×D31—22no—
41C23×D31D412yes—
42#41×C31—21no—
43#41×C31—21no—
44#41×C31D412yescomplemented
45#41×C31C2312yes—
46C2×Q8×C31D411yescharacteristic, complemented
47C2×D4×C31—21no—
48C2×D4×C31—21no—
49C2×D4×C31—21no—
50C2×D4×C31—21no—
51C2×D4×C31—21no—
52C2×D4×C31—21no—
53C2×D4×C31D411yescharacteristic, complemented
54C2×D4×C31C2312yescomplemented
55C22×C4×C31—21no—
56C22×C4×C31C2311yescharacteristic
57C2×C22⋊D31—22no—
58C2×C22⋊D31—21no—
59C2×C22⋊D31—28no—
60C2×C22⋊D31—48no—
61C22×Dic31—21no—
62#92499—24no—
63#92499—24no—
64D4×D31—28no—
65D4×D31—24no—
66D4×D31—28no—
67D4×D31—24no—
68#92495—24no—
69#92495—21no—
70#92495—21no—
71#92495—24no—
72#92495D412yescomplemented
73#92495C2312yes—
74C2×D124—22no—
75C2×D124—24no—
76C2×D124—21no—
77C2×D124—21no—
78C2×D124—21no—
79C2×D124—21no—
80C2×D124—24no—
81C2×D124D412yes—
82C2×D124D411yescharacteristic, complemented
83C2×D124C2312yes—
84C2×C4×D31—22no—
85C2×C4×D31—21no—
86C2×Dic62D411yescharacteristic, complemented
87SD16×C31—24no—
88D8×C31—24no—
89M16×C31—21no—
90M16×C31C2312yescomplemented
91C2×C8×C31—21no—
92#92480—24no—
93#92479—24no—
94C4×Dic31—21no—
95C4×Dic31C2312yescomplemented
96#92474—21no—
97#92474C2312yescomplemented
98C2×C31⋊C8—21no—
99C23×C31—21no—
100C23×C31—22no—
101C23×C31—44no—
102C23×C31—41no—
103C23×C31—41no—
104C23×C31C2×D411yescharacteristic
105C22×D31—22no—
106C22×D31—21no—
107C22×D31—28no—
108C22×D31—22no—
109C22×D31—44no—
110C22×D31—84no—
111C22×D31C2×D414yes—
112Q8×C31—21no—
113Q8×C31C2×D412yescomplemented
114D4×C31—22no—
115D4×C31—22no—
116D4×C31—21no—
117D4×C31—22no—
118D4×C31—22no—
119D4×C31—41no—
120D4×C31—41no—
121D4×C31C2×D412yescomplemented
122C2×C4×C31—21no—
123C2×C4×C31—22no—
124C2×C4×C31—22no—
125C2×C4×C31C2×D411yescharacteristic
126C2×C4×C31C2×D411yescharacteristic
127C2×C4×C31C2×D411yescharacteristic
128C2×C4×C31C2×D411yescharacteristic
129C2×C4×C31C2×D411yescharacteristic
130C2×C4×C31C2411yescharacteristic
131C22⋊D31—28no—
132C22⋊D31—22no—
133C22⋊D31—22no—
134C22⋊D31—416no—
135C2×Dic31—24no—
136C2×Dic31—21no—
137C2×Dic31—22no—
138D124—28no—
139D124—22no—
140D124—21no—
141D124—22no—
142D124—22no—
143D124—22no—
144D124C2×D412yescomplemented
145C4×D31—28no—
146C4×D31—22no—
147C4×D31—22no—
148Dic62—21no—
149Dic62C2×D412yescomplemented
150C8×C31—22no—
151C31⋊C8—22no—
152C2×#453—311nomaximal, Sylow
153C22×C31—22no—
154C22×C31—22no—
155C22×C31—44no—
156C22×C31—42no—
157C22×C31—42no—
158C22×C31—82no—
159C22×C31—82no—
160C22×C31#12011yescharacteristic
161C22×C31#12011yescharacteristic
162C22×C31C22×D411yescharacteristic
163C2×D31—24no—
164C2×D31—22no—
165C2×D31—44no—
166C2×D31—42no—
167C2×D31—44no—
168C2×D31—82no—
169C2×D31—88no—
170C4×C31—22no—
171C4×C31#12011yescharacteristic
172C4×C31#12011yescharacteristic
173C4×C31C22×D411yescharacteristic
174C4×C31C22×D411yescharacteristic
175Dic31—24no—
176Dic31—22no—
177C2×H(5,𝔽2)—311no—
178C2×AGL(1,ℤ/8)—311no—
179C2×AGL(1,ℤ/8)D3111yescharacteristic, complemented
180C2×#127—311no—
181#453—318no—
182C2×#104—311no—
183C2×#104—311no—
184C2×#100—311no—
185C2×C31—21no—
186C2×C31—21no—
187C2×C31—42no—
188C2×C31—82no—
189C2×C31#45312yescomplemented
190C2×C31C2×#12011yescharacteristic
191D31—44no—
192D31—42no—
193H(5,𝔽2)—314no—
194H(5,𝔽2)—622no—
195H(5,𝔽2)—622no—
196H(5,𝔽2)—622no—
197C2×#41—311no—
198C2×#41—621no—
199C2×#41—621no—
200C2×#41C2×D3111yescharacteristic
201C22×D4—311no—
202C22×D4—621no—
203C22×D4—622no—
204C22×D4—621no—
205C22×D4—622no—
206C22×D4C2×D3111yescharacteristic, complemented
207AGL(1,ℤ/8)—22no—
208AGL(1,ℤ/8)—314no—
209AGL(1,ℤ/8)—622no—
210AGL(1,ℤ/8)C2×D3114yescomplemented
211C2×SD16—21no—
212C2×SD16—621no—
213C2×D8—21no—
214C2×D8—621no—
215C2×M16—311no—
216C2×M16C2×D3111yescharacteristic, complemented
217#127—314no—
218#127—622no—
219C2×C42—311no—
220#104—314no—
221#104—314no—
222#100—314no—
223C31C2×#45311yescharacteristic, complemented, Sylow
224C24—21no—
225C24—312no—
226C24—622no—
227#41—21no—
228#41—21no—
229#41—312no—
230#41—312no—
231#41—624no—
232#41—624no—
233#41—624no—
234#41—624no—
235#41—621no—
236#41—621no—
237#41C22⋊D3112yescomplemented
238#41C22×D3112yes—
239C2×Q8—311no—
240C2×Q8C22⋊D3111yescharacteristic, complemented
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—311no—
248C2×D4—312no—
249C2×D4—312no—
250C2×D4—624no—
251C2×D4—622no—
252C2×D4—624no—
253C2×D4—628no—
254C2×D4—628no—
255C2×D4—624no—
256C2×D4—622no—
257C2×D4—621no—
258C2×D4—624no—
259C2×D4—628no—
260C2×D4—621no—
261C2×D4—621no—
262C2×D4—621no—
263C2×D4—621no—
264C2×D4—1248no—
265C2×D4C22⋊D3111yescharacteristic, complemented
266C2×D4C22×D3112yescomplemented
267C22×C4—21no—
268C22×C4—622no—
269C22×C4—621no—
270C22×C4—621no—
271C22×C4C22×D3111yescharacteristic
272SD16—24no—
273SD16—624no—
274D8—24no—
275D8—624no—
276M16—21no—
277M16—312no—
278M16—621no—
279M16C22×D3112yescomplemented
280C2×C8—21no—
281C2×C8—621no—
282C42—312no—
283C42—621no—
284C23—21no—
285C23—22no—
286C23—44no—
287C23—41no—
288C23—41no—
289C23—314no—
290C23—628no—
291C23—621no—
292C23—622no—
293C23—622no—
294C23—1244no—
295C23—2484no—
296C23D4×D3111yescharacteristic
297Q8—21no—
298Q8—312no—
299Q8—621no—
300Q8C2×C22⋊D3112yescomplemented
301D4—22no—
302D4—22no—
303D4—21no—
304D4—22no—
305D4—22no—
306D4—41no—
307D4—41no—
308D4—312no—
309D4—628no—
310D4—628no—
311D4—622no—
312D4—622no—
313D4—621no—
314D4—622no—
315D4—622no—
316D4—622no—
317D4—622no—
318D4—12416no—
319D4C2×C22⋊D3112yescomplemented
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—628no—
324C2×C4—624no—
325C2×C4—621no—
326C2×C4—622no—
327C2×C4—622no—
328C2×C4—622no—
329C2×C4D4×D3111yescharacteristic
330C2×C4D4×D3111yescharacteristic
331C2×C4D4×D3111yescharacteristic
332C2×C4C2×C22⋊D3111yescharacteristic
333C2×C4C2×C22⋊D3111yescharacteristic
334C2×C4C23×D3111yescharacteristic
335C8—22no—
336C8—622no—
337C22—22no—
338C22—22no—
339C22—44no—
340C22—42no—
341C22—42no—
342C22—82no—
343C22—82no—
344C22—622no—
345C22—624no—
346C22—1244no—
347C22—1244no—
348C22—1242no—
349C22—2482no—
350C22—2488no—
351C22#1175851411yescharacteristic
352C22#1175851411yescharacteristic
353C22C2×D4×D3111yescharacteristic
354C4—22no—
355C4—624no—
356C4—622no—
357C4#1175851411yescharacteristic
358C4#1175851411yescharacteristic
359C4C2×D4×D3111yescharacteristic
360C4C22×C22⋊D3111yescharacteristic
361C2—21no—
362C2—21no—
363C2—42no—
364C2—82no—
365C2—1244no—
366C2—1242no—
367C2#1452121512yesdirect product factor
368C2C2×#1175851411yescharacteristic
subgroupG/H
centerC22#11758514
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×AGL(1,ℤ/8)C2
socleC22×C31#120
hypercenterC2×AGL(1,ℤ/8)D31
hypocenterC31C2×#453
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#11758566C2semidirect
2. C2×#11758509C2semidirect
3. C2×#11758517C2semidirect
4. #14521215C2direct (N × H)
5. C2×#11758425C2semidirect
6. C2×#11758413C2semidirect
7. C2×#11758411C2semidirect
8. C2×C31×AGL(1,ℤ/8)C2semidirect
10. C22×D4×C31C22semidirect
15. #11758566C22semidirect
24. C31×AGL(1,ℤ/8)C22semidirect
27. C2×M16×C31C22semidirect
29. #11758517C22semidirect
32. #11758509C22semidirect
34. C2×C4×Dic31C22semidirect
35. C2×#92474C22semidirect
36. #11758425C22semidirect
37. #11758413C22semidirect
38. #11758411C22semidirect
44. #41×C31D4semidirect
46. C2×Q8×C31D4semidirect
53. C2×D4×C31D4semidirect
54. C2×D4×C31C23semidirect
72. #92495D4semidirect
82. C2×D124D4semidirect
86. C2×Dic62D4semidirect
90. M16×C31C23semidirect
95. C4×Dic31C23semidirect
97. #92474C23semidirect
113. Q8×C31C2×D4semidirect
121. D4×C31C2×D4semidirect
144. D124C2×D4semidirect
149. Dic62C2×D4semidirect
179. C2×AGL(1,ℤ/8)D31semidirect
189. C2×C31#453semidirect
206. C22×D4C2×D31semidirect
210. AGL(1,ℤ/8)C2×D31semidirect
216. C2×M16C2×D31semidirect
223. C31C2×#453semidirect
237. #41C22⋊D31semidirect
240. C2×Q8C22⋊D31semidirect
265. C2×D4C22⋊D31semidirect
266. C2×D4C22×D31semidirect
279. M16C22×D31semidirect
300. Q8C2×C22⋊D31semidirect
319. D4C2×C22⋊D31semidirect
367. C2#14521215direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117585662
C2×#117585092
C2×#117585172
#145212152
C2×#117584252
C2×#117584132
C2×#117584112
C2×C31×AGL(1,ℤ/8)2
C2×#45331