Finite Group Databasebeta

← C2×#14521225 #16855039

Subgroups — C2×#14521225 #16855039

Order 3968.

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

normal subgroups only

This group has 760 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
2C2×#11758519C211yesmaximal, characteristic, complemented
3C2×#11758524C211yesmaximal, characteristic, complemented
4#14521225C218yesmaximal, direct product factor
5C2×#11758425C211yesmaximal, characteristic, complemented
6C2×#11758413C211yesmaximal, characteristic, complemented
7C2×#11758411C211yesmaximal, characteristic, complemented
8C2×C31×#137C211yesmaximal, characteristic, complemented
9C2×#41×C31C2211yescharacteristic
10C22×Q8×C31C2211yescharacteristic, complemented
11#11758566—22no—
12#11758566—22no—
13#11758566—22no—
14#11758566C2214yes—
15C2×#92499—21no—
16C2×D4×D31—22no—
17C2×D4×D31—21no—
18C2×#92495—21no—
19C2×#92495C2211yescharacteristic
20C22×D124—21no—
21C22×D124C2211yescharacteristic
22C31×#137—22no—
23C31×#137C2214yescomplemented
24C2×Q16×C31—21no—
25C2×SD16×C31—21no—
26C2×M16×C31C2211yescharacteristic, complemented
27#11758524—22no—
28#11758524C2214yescomplemented
29C2×#92482—21no—
30#11758519—22no—
31#11758519C2214yescomplemented
32C2×#92481—21no—
33C2×(C2×D31)⋊C4—22no—
34C2×C4×Dic31C2211yescharacteristic, complemented
35C2×#92474C2211yescharacteristic, complemented
36#11758425C2214yescomplemented
37#11758413C2214yescomplemented
38#11758411C2214yescomplemented
39C23×D31—22no—
40C23×D31D412yes—
41#41×C31—21no—
42#41×C31—21no—
43#41×C31D412yes—
44#41×C31C2312yes—
45C2×Q8×C31—21no—
46C2×Q8×C31—21no—
47C2×Q8×C31—21no—
48C2×Q8×C31—21no—
49C2×Q8×C31—21no—
50C2×Q8×C31D411yescharacteristic
51C2×Q8×C31C2312yescomplemented
52C2×D4×C31—21no—
53C2×D4×C31D411yescharacteristic
54C22×C4×C31—21no—
55C22×C4×C31—21no—
56C22×C4×C31C2311yescharacteristic
57C2×C22⋊D31—22no—
58C2×C22⋊D31—21no—
59C22×Dic31—21no—
60#92499—24no—
61#92499—24no—
62D4×D31—28no—
63D4×D31—24no—
64D4×D31—28no—
65D4×D31—24no—
66#92495—24no—
67#92495—21no—
68#92495—21no—
69#92495—24no—
70#92495D412yes—
71#92495C2312yes—
72C2×D124—22no—
73C2×D124—24no—
74C2×D124—21no—
75C2×D124—21no—
76C2×D124—21no—
77C2×D124—21no—
78C2×D124—24no—
79C2×D124D412yes—
80C2×D124D411yescharacteristic
81C2×D124C2312yes—
82C2×C4×D31—22no—
83C2×C4×D31—21no—
84C2×Dic62D411yescharacteristic
85Q16×C31—24no—
86SD16×C31—24no—
87M16×C31—21no—
88M16×C31C2312yescomplemented
89C2×C8×C31—21no—
90#92482—24no—
91#92481—24no—
92(C2×D31)⋊C4—28no—
93C4×Dic31—21no—
94C4×Dic31C2312yescomplemented
95#92474—21no—
96#92474C2312yescomplemented
97C2×C31⋊C8—21no—
98C23×C31—21no—
99C23×C31C2×D411yescharacteristic
100C22×D31—22no—
101C22×D31—21no—
102C22×D31—28no—
103C22×D31—22no—
104C22×D31—44no—
105C22×D31—84no—
106C22×D31C2×D414yes—
107Q8×C31—21no—
108Q8×C31—22no—
109Q8×C31—22no—
110Q8×C31—41no—
111Q8×C31—41no—
112Q8×C31C2×D412yes—
113D4×C31—22no—
114D4×C31—22no—
115D4×C31—21no—
116D4×C31C2×D412yes—
117C2×C4×C31—21no—
118C2×C4×C31—22no—
119C2×C4×C31—22no—
120C2×C4×C31—22no—
121C2×C4×C31—41no—
122C2×C4×C31—41no—
123C2×C4×C31C2×D411yescharacteristic
124C2×C4×C31C2×D411yescharacteristic
125C2×C4×C31C2×D411yescharacteristic
126C2×C4×C31C2×D411yescharacteristic
127C2×C4×C31C2×D411yescharacteristic
128C2×C4×C31C2411yescharacteristic
129C22⋊D31—28no—
130C22⋊D31—22no—
131C22⋊D31—22no—
132C2×Dic31—21no—
133C2×Dic31—22no—
134C2×Dic31—42no—
135D124—28no—
136D124—22no—
137D124—21no—
138D124—22no—
139D124—22no—
140D124—22no—
141D124C2×D412yes—
142C4×D31—28no—
143C4×D31—22no—
144C4×D31—22no—
145Dic62—21no—
146Dic62C2×D412yes—
147C8×C31—22no—
148C31⋊C8—22no—
149C2×#455—311nomaximal, Sylow
150C22×C31—22no—
151C22×C31—22no—
152C22×C31—42no—
153C22×C31#12011yescharacteristic
154C22×C31#12011yescharacteristic
155C22×C31C22×D411yescharacteristic
156C2×D31—24no—
157C2×D31—22no—
158C2×D31—44no—
159C2×D31—42no—
160C2×D31—44no—
161C2×D31—82no—
162C2×D31—88no—
163C4×C31—22no—
164C4×C31—42no—
165C4×C31#12011yescharacteristic
166C4×C31#12011yescharacteristic
167C4×C31C22×D411yescharacteristic
168C4×C31C22×D411yescharacteristic
169Dic31—22no—
170Dic31—42no—
171C2×H(5,𝔽2)—311no—
172C2×#137—311no—
173C2×#137D3111yescharacteristic, complemented
174C2×#124—311no—
175#455—318no—
176C2×#104—311no—
177C2×#104—311no—
178C2×#100—311no—
179C2×C31—21no—
180C2×C31—21no—
181C2×C31—42no—
182C2×C31#45512yescomplemented
183C2×C31C2×#12011yescharacteristic
184D31—44no—
185D31—42no—
186H(5,𝔽2)—314no—
187H(5,𝔽2)—622no—
188H(5,𝔽2)—622no—
189H(5,𝔽2)—622no—
190C2×#41—311no—
191C2×#41—621no—
192C2×#41—621no—
193C2×#41C2×D3111yescharacteristic
194C22×Q8C2×D3111yescharacteristic, complemented
195C22×D4—311no—
196C22×D4—621no—
197C22×D4—622no—
198C22×D4—621no—
199#137—22no—
200#137—314no—
201#137—622no—
202#137C2×D3114yescomplemented
203C2×Q16—21no—
204C2×Q16—621no—
205C2×SD16—21no—
206C2×SD16—621no—
207C2×M16—311no—
208C2×M16C2×D3111yescharacteristic, complemented
209#124—314no—
210#124—622no—
211C2×C22⋊C4—622no—
212C2×C42—311no—
213#104—314no—
214#104—314no—
215#100—314no—
216C31C2×#45511yescharacteristic, complemented, Sylow
217C24—312no—
218C24—622no—
219#41—21no—
220#41—21no—
221#41—312no—
222#41—312no—
223#41—624no—
224#41—624no—
225#41—624no—
226#41—624no—
227#41—621no—
228#41—621no—
229#41C22⋊D3112yes—
230#41C22×D3112yes—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—311no—
237C2×Q8C22⋊D3111yescharacteristic
238C2×Q8C22×D3112yescomplemented
239C2×D4—21no—
240C2×D4—311no—
241C2×D4—312no—
242C2×D4—312no—
243C2×D4—624no—
244C2×D4—622no—
245C2×D4—624no—
246C2×D4—628no—
247C2×D4—628no—
248C2×D4—624no—
249C2×D4—622no—
250C2×D4—621no—
251C2×D4—624no—
252C2×D4—621no—
253C2×D4—621no—
254C2×D4—621no—
255C2×D4—621no—
256C2×D4C22⋊D3111yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—622no—
260C22×C4—621no—
261C22×C4—621no—
262C22×C4C22×D3111yescharacteristic
263Q16—24no—
264Q16—624no—
265SD16—24no—
266SD16—624no—
267M16—21no—
268M16—312no—
269M16—621no—
270M16C22×D3112yescomplemented
271C2×C8—21no—
272C2×C8—621no—
273C22⋊C4—628no—
274C42—312no—
275C42—621no—
276C23—21no—
277C23—314no—
278C23—628no—
279C23—621no—
280C23—622no—
281C23—622no—
282C23—1244no—
283C23—2484no—
284C23D4×D3111yescharacteristic
285Q8—21no—
286Q8—22no—
287Q8—22no—
288Q8—41no—
289Q8—41no—
290Q8—312no—
291Q8—621no—
292Q8C2×C22⋊D3112yes—
293D4—22no—
294D4—22no—
295D4—21no—
296D4—312no—
297D4—628no—
298D4—628no—
299D4—622no—
300D4—622no—
301D4—621no—
302D4—622no—
303D4—622no—
304D4—622no—
305D4—622no—
306D4C2×C22⋊D3112yes—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—628no—
314C2×C4—621no—
315C2×C4—622no—
316C2×C4—622no—
317C2×C4—622no—
318C2×C4—1242no—
319C2×C4D4×D3111yescharacteristic
320C2×C4D4×D3111yescharacteristic
321C2×C4D4×D3111yescharacteristic
322C2×C4C2×C22⋊D3111yescharacteristic
323C2×C4C2×C22⋊D3111yescharacteristic
324C2×C4C23×D3111yescharacteristic
325C8—22no—
326C8—622no—
327C22—22no—
328C22—22no—
329C22—42no—
330C22—622no—
331C22—624no—
332C22—1244no—
333C22—1244no—
334C22—1242no—
335C22—2482no—
336C22—2488no—
337C22#1175851411yescharacteristic
338C22#1175851411yescharacteristic
339C22C2×D4×D3111yescharacteristic
340C4—22no—
341C4—42no—
342C4—622no—
343C4—1242no—
344C4#1175851411yescharacteristic
345C4#1175851411yescharacteristic
346C4C2×D4×D3111yescharacteristic
347C4C22×C22⋊D3111yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—1244no—
352C2—1242no—
353C2#1452122512yesdirect product factor
354C2C2×#1175851411yescharacteristic
subgroupG/H
centerC22#11758514
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×#137C2
socleC22×C31#120
prime radicalC2×#11758566C2
hypercenterC2×#137D31
hypocenterC31C2×#455
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#11758519C2semidirect
3. C2×#11758524C2semidirect
4. #14521225C2direct (N × H)
5. C2×#11758425C2semidirect
6. C2×#11758413C2semidirect
7. C2×#11758411C2semidirect
8. C2×C31×#137C2semidirect
10. C22×Q8×C31C22semidirect
23. C31×#137C22semidirect
26. C2×M16×C31C22semidirect
28. #11758524C22semidirect
31. #11758519C22semidirect
34. C2×C4×Dic31C22semidirect
35. C2×#92474C22semidirect
36. #11758425C22semidirect
37. #11758413C22semidirect
38. #11758411C22semidirect
51. C2×Q8×C31C23semidirect
88. M16×C31C23semidirect
94. C4×Dic31C23semidirect
96. #92474C23semidirect
173. C2×#137D31semidirect
182. C2×C31#455semidirect
194. C22×Q8C2×D31semidirect
202. #137C2×D31semidirect
208. C2×M16C2×D31semidirect
216. C31C2×#455semidirect
238. C2×Q8C22×D31semidirect
270. M16C22×D31semidirect
353. C2#14521225direct (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×#117585192
C2×#117585242
#145212252
C2×#117584252
C2×#117584132
C2×#117584112
C2×C31×#1372
C2×#45531