Finite Group Databasebeta

← #16854977

Subgroups — #16854977

Order 3968.

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

normal subgroups only

This group has 784 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
3C22×#92479C211yesmaximal, characteristic, complemented
4#14521192C214yesmaximal, complemented
5#14521182C214yesmaximal, complemented
6#14521053C211yesmaximal, characteristic, complemented
7#14521043C211yesmaximal, characteristic, complemented
8#14521016C211yesmaximal, characteristic, complemented
9C31×#535C211yesmaximal, characteristic, complemented
10C2×#41×C31C2211yescharacteristic
11C22×D4×C31C2211yescharacteristic, complemented
12#11758566—24no—
13#11758566—24no—
14C2×#92499—21no—
15C2×D4×D31—21no—
16C2×D4×D31—22no—
17C2×#92495C2211yescharacteristic
18C2×#92495C2212yescomplemented
19C22×D124C2211yescharacteristic
20C22×D124C2212yescomplemented
21C31×#127—21no—
22C31×#124—21no—
23C31×#121C2214yescomplemented
24C31×#120—22no—
25C31×#117C2211yescharacteristic, complemented
26C2×#92480C2212yescomplemented
27#11758509—24no—
28C2×#92479—24no—
29C2×#92479C2212yescomplemented
30C2×#92479C2214yescomplemented
31C2×#92474C2211yescharacteristic, complemented
32C22×C31⋊C8C2211yescharacteristic, complemented
33C22⋊(C31⋊C8)C2214yescomplemented
34#11758398C2212yescomplemented
35#11758397C2212yescomplemented
36#11758397C2214yescomplemented
37C24×C31—21no—
38C23×D31—21no—
39C23×D31—22no—
40#41×C31—22no—
41#41×C31—22no—
42C2×Q8×C31D411yescharacteristic
43C2×D4×C31—24no—
44C2×D4×C31—22no—
45C2×D4×C31—22no—
46C2×D4×C31—21no—
47C2×D4×C31D412yes—
48C2×D4×C31D411yescharacteristic
49C2×D4×C31C2312yes—
50C22×C4×C31C2312yes—
51C22×C4×C31C2311yescharacteristic
52C2×C22⋊D31—21no—
53C2×C22⋊D31—22no—
54#92499—28no—
55D4×D31—216no—
56D4×D31—28no—
57#92495—22no—
58#92495—24no—
59#92495—24no—
60#92495D414yescomplemented
61C2×D124—21no—
62C2×D124—22no—
63C2×D124—22no—
64C2×D124—24no—
65C2×D124—24no—
66C2×D124D414yescomplemented
67C2×D124C2312yes—
68C2×D124C2311yescharacteristic
69C2×C4×D31—21no—
70C2×C4×D31—22no—
71C2×Dic62C2311yescharacteristic
72C4⋊C4×C31C2312yes—
73C22⋊C4×C31—21no—
74C22⋊C4×C31—24no—
75C42×C31—21no—
76#92480—24no—
77#92479—28no—
78#92479—24no—
79#92474—22no—
80C2×C31⋊C8—22no—
81C2×C31⋊C8C2312yes—
82C2×C31⋊C8C2312yes—
83C23×C31—22no—
84C23×C31—42no—
85C23×C31—82no—
86C23×C31C2×D412yes—
87C23×C31C2×D411yescharacteristic
88C22×D31—21no—
89C22×D31—22no—
90C22×D31—28no—
91C22×D31—24no—
92C22×D31—82no—
93C22×D31—84no—
94Q8×C31—22no—
95D4×C31—24no—
96D4×C31—22no—
97D4×C31—22no—
98D4×C31—22no—
99D4×C31—42no—
100D4×C31—84no—
101C2×C4×C31—22no—
102C2×C4×C31—22no—
103C2×C4×C31—22no—
104C2×C4×C31C2×D412yes—
105C2×C4×C31C2×D412yes—
106C2×C4×C31C2×D412yes—
107C2×C4×C31C2×D411yescharacteristic
108C2×C4×C31C2411yescharacteristic
109C22⋊D31—28no—
110C22⋊D31—24no—
111C2×Dic31—21no—
112D124—28no—
113D124—24no—
114D124—21no—
115D124—22no—
116D124C2×D412yes—
117D124C2×D414yes—
118C4×D31—28no—
119C4×D31—24no—
120Dic62—21no—
121Dic62C2×D412yes—
122C31⋊C8—22no—
123C31⋊C8—22no—
124#3008—311nomaximal, Sylow
125C22×C31—22no—
126C22×C31—22no—
127C22×C31—24no—
128C22×C31—42no—
129C22×C31—44no—
130C22×C31—84no—
131C22×C31#12012yes—
132C22×C31C22×D411yescharacteristic
133C2×D31—24no—
134C2×D31—22no—
135C2×D31—44no—
136C2×D31—42no—
137C2×D31—84no—
138C2×D31—88no—
139C4×C31—22no—
140C4×C31—42no—
141C4×C31#12012yes—
142C4×C31C22×D411yescharacteristic
143C4×C31C22×D411yescharacteristic
144Dic31—22no—
145C2×H(5,𝔽2)—311no—
146C2×AGL(1,ℤ/8)—311no—
147C22×D8—311no—
148#535D3111yescharacteristic, complemented
149#449—314no—
150#447—314no—
151#418—311no—
152#416—311no—
153#408—311no—
154C2×C31—22no—
155C2×C31—42no—
156C2×C31—82no—
157C2×C31C2×#12011yescharacteristic
158C2×C31#57612yes—
159D31—44no—
160D31—42no—
161H(5,𝔽2)—624no—
162H(5,𝔽2)—624no—
163C2×#41—311no—
164C2×#41—312no—
165C2×#41—621no—
166C2×#41C2×D3111yescharacteristic
167C22×D4—311no—
168C22×D4—312no—
169C22×D4—622no—
170C22×D4—621no—
171C22×D4C2×D3111yescharacteristic, complemented
172AGL(1,ℤ/8)—624no—
173C2×SD16—312no—
174C2×D8—314no—
175C2×D8—312no—
176C2×D8—624no—
177C2×M16—311no—
178C22×C8—311no—
179#127—21no—
180#124—21no—
181#121C2×D3114yescomplemented
182#120—22no—
183#117C2×D3111yescharacteristic, complemented
184#103—312no—
185#102—312no—
186#102—314no—
187C22⋊C8—314no—
188C31#300811yescharacteristic, complemented, Sylow
189C24—21no—
190C24—622no—
191C24—621no—
192#41—22no—
193#41—22no—
194#41—314no—
195#41—624no—
196#41—624no—
197#41—628no—
198#41—622no—
199C2×Q8—311no—
200C2×Q8C22⋊D3111yescharacteristic
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—314no—
206C2×D4—312no—
207C2×D4—311no—
208C2×D4—624no—
209C2×D4—624no—
210C2×D4—6216no—
211C2×D4—628no—
212C2×D4—622no—
213C2×D4—621no—
214C2×D4—622no—
215C2×D4—622no—
216C2×D4—621no—
217C2×D4C22⋊D3112yes—
218C2×D4C22⋊D3111yescharacteristic
219C2×D4C22×D3112yes—
220C22×C4—622no—
221C22×C4—621no—
222C22×C4C22×D3112yes—
223C22×C4C22×D3111yescharacteristic
224SD16—624no—
225D8—628no—
226D8—624no—
227M16—622no—
228C2×C8—312no—
229C2×C8—312no—
230C2×C8—622no—
231C4⋊C4C22×D3112yes—
232C22⋊C4—21no—
233C22⋊C4—24no—
234C42—21no—
235C23—22no—
236C23—42no—
237C23—82no—
238C23—628no—
239C23—622no—
240C23—621no—
241C23—624no—
242C23—2482no—
243C23—2484no—
244C23C2×C22⋊D3112yes—
245C23C2×C22⋊D3111yescharacteristic
246Q8—22no—
247Q8—312no—
248Q8—621no—
249D4—24no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—84no—
255D4—314no—
256D4—312no—
257D4—628no—
258D4—628no—
259D4—624no—
260D4—624no—
261D4—621no—
262D4—622no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—628no—
267C2×C4—621no—
268C2×C4—624no—
269C2×C4D4×D3112yes—
270C2×C4D4×D3112yes—
271C2×C4C2×C22⋊D3112yes—
272C2×C4C2×C22⋊D3111yescharacteristic
273C2×C4C23×D3111yescharacteristic
274C8—622no—
275C8—622no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—42no—
280C22—44no—
281C22—84no—
282C22—624no—
283C22—622no—
284C22—1244no—
285C22—1242no—
286C22—2484no—
287C22—2488no—
288C22#1175851412yes—
289C22C22×C22⋊D3111yescharacteristic
290C4—22no—
291C4—42no—
292C4—622no—
293C4#1175851412yes—
294C4C2×D4×D3111yescharacteristic
295C4C2×D4×D3111yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—1244no—
300C2—1242no—
301C2C2×#1175851411yescharacteristic
302C2#1452185212yes—
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#535C2
socleC22×C31C22×D4
hypercenter#535D31
hypocenterC31#3008
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#11758566C2semidirect
2. C2×#11758509C2semidirect
3. C22×#92479C2semidirect
4. #14521192C2semidirect
5. #14521182C2semidirect
6. #14521053C2semidirect
7. #14521043C2semidirect
8. #14521016C2semidirect
9. C31×#535C2semidirect
11. C22×D4×C31C22semidirect
18. C2×#92495C22semidirect
20. C22×D124C22semidirect
23. C31×#121C22semidirect
25. C31×#117C22semidirect
26. C2×#92480C22semidirect
29. C2×#92479C22semidirect
30. C2×#92479C22semidirect
31. C2×#92474C22semidirect
32. C22×C31⋊C8C22semidirect
33. C22⋊(C31⋊C8)C22semidirect
34. #11758398C22semidirect
35. #11758397C22semidirect
36. #11758397C22semidirect
60. #92495D4semidirect
66. C2×D124D4semidirect
148. #535D31semidirect
171. C22×D4C2×D31semidirect
181. #121C2×D31semidirect
183. #117C2×D31semidirect
188. C31#3008semidirect

Maximal subgroups

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

subgroupindex
C2×#117585662
C2×#117585092
C22×#924792
#145211922
#145211822
#145210532
#145210432
#145210162
C31×#5352
#300831