Finite Group Databasebeta

← #16854986

Subgroups — #16854986

Order 3968.

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

normal subgroups only

This group has 752 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
3C22×#92481C211yesmaximal, characteristic, complemented
4#14521209C214yesmaximal, complemented
5#14521199C214yesmaximal, complemented
6#14521053C211yesmaximal, characteristic, complemented
7#14521043C211yesmaximal, characteristic, complemented
8#14521016C211yesmaximal, characteristic, complemented
9C31×#536C211yesmaximal, characteristic, complemented
10C2×#41×C31C2211yescharacteristic
11C22×Q8×C31C2211yescharacteristic, complemented
12#11758566—24no—
13#11758566—24no—
14C2×#92499—21no—
15C2×D4×D31—22no—
16C2×D4×D31—21no—
17C2×#92495C2212yes—
18C2×#92495C2211yescharacteristic
19C22×D124C2212yes—
20C22×D124C2211yescharacteristic
21C31×C4⋊Q8—21no—
22C31×#124—21no—
23C31×#123—22no—
24C31×C22⋊Q8C2214yescomplemented
25C31×#117C2211yescharacteristic, complemented
26C2×#92482C2212yescomplemented
27#11758519—24no—
28C2×#92481—24no—
29C2×#92481C2212yescomplemented
30C2×#92481C2214yescomplemented
31C2×#92474C2211yescharacteristic, complemented
32C22×C31⋊C8C2211yescharacteristic, complemented
33C22⋊(C31⋊C8)C2214yescomplemented
34#11758398C2212yescomplemented
35#11758397C2212yescomplemented
36#11758397C2214yescomplemented
37C23×D31—22no—
38C23×D31—21no—
39#41×C31—22no—
40#41×C31—22no—
41C2×Q8×C31—21no—
42C2×Q8×C31—22no—
43C2×Q8×C31—22no—
44C2×Q8×C31D411yescharacteristic
45C2×Q8×C31C2312yes—
46C2×D4×C31D412yes—
47C2×D4×C31D411yescharacteristic
48C22×C4×C31—21no—
49C22×C4×C31C2312yes—
50C22×C4×C31C2311yescharacteristic
51C2×C22⋊D31—22no—
52C2×C22⋊D31—21no—
53#92499—28no—
54D4×D31—216no—
55D4×D31—28no—
56#92495—22no—
57#92495—24no—
58#92495—24no—
59#92495D414yes—
60C2×D124—22no—
61C2×D124—21no—
62C2×D124—22no—
63C2×D124—24no—
64C2×D124—24no—
65C2×D124D414yes—
66C2×D124C2311yescharacteristic
67C2×D124C2312yes—
68C2×C4×D31—22no—
69C2×C4×D31—21no—
70C2×Dic62C2311yescharacteristic
71C4⋊C4×C31—24no—
72C4⋊C4×C31C2312yes—
73C22⋊C4×C31—21no—
74C22⋊C4×C31—24no—
75C42×C31—21no—
76#92482—24no—
77#92481—28no—
78#92481—24no—
79#92474—22no—
80C2×C31⋊C8—22no—
81C2×C31⋊C8C2312yes—
82C2×C31⋊C8C2312yes—
83C23×C31C2×D412yes—
84C23×C31C2×D411yescharacteristic
85C22×D31—22no—
86C22×D31—21no—
87C22×D31—28no—
88C22×D31—24no—
89C22×D31—82no—
90C22×D31—84no—
91Q8×C31—22no—
92Q8×C31—22no—
93Q8×C31—22no—
94Q8×C31—42no—
95D4×C31—24no—
96D4×C31—22no—
97C2×C4×C31—22no—
98C2×C4×C31—22no—
99C2×C4×C31—22no—
100C2×C4×C31—22no—
101C2×C4×C31—42no—
102C2×C4×C31C2×D412yes—
103C2×C4×C31C2×D412yes—
104C2×C4×C31C2×D412yes—
105C2×C4×C31C2×D411yescharacteristic
106C2×C4×C31C2411yescharacteristic
107C22⋊D31—28no—
108C22⋊D31—24no—
109C2×Dic31—21no—
110D124—28no—
111D124—24no—
112D124—21no—
113D124—22no—
114D124C2×D412yes—
115D124C2×D414yes—
116C4×D31—28no—
117C4×D31—24no—
118Dic62—21no—
119Dic62C2×D412yes—
120C31⋊C8—22no—
121C31⋊C8—22no—
122#3009—311nomaximal, Sylow
123C22×C31—22no—
124C22×C31—22no—
125C22×C31—24no—
126C22×C31#12012yes—
127C22×C31C22×D411yescharacteristic
128C2×D31—24no—
129C2×D31—22no—
130C2×D31—44no—
131C2×D31—42no—
132C2×D31—84no—
133C2×D31—88no—
134C4×C31—22no—
135C4×C31—42no—
136C4×C31—42no—
137C4×C31#12012yes—
138C4×C31C22×D411yescharacteristic
139C4×C31C22×D411yescharacteristic
140Dic31—22no—
141C2×H(5,𝔽2)—311no—
142C2×#137—311no—
143C22×SD16—311no—
144#536D3111yescharacteristic, complemented
145#452—314no—
146#450—314no—
147#418—311no—
148#416—311no—
149#408—311no—
150C2×C31—22no—
151C2×C31—42no—
152C2×C31C2×#12011yescharacteristic
153C2×C31#57712yes—
154D31—44no—
155D31—42no—
156H(5,𝔽2)—624no—
157H(5,𝔽2)—624no—
158C2×#41—311no—
159C2×#41—312no—
160C2×#41—621no—
161C2×#41C2×D3111yescharacteristic
162C22×Q8C2×D3111yescharacteristic, complemented
163C22×D4—311no—
164C22×D4—312no—
165C22×D4—622no—
166C22×D4—621no—
167#137—624no—
168C2×Q16—312no—
169C2×SD16—314no—
170C2×SD16—312no—
171C2×SD16—624no—
172C2×M16—311no—
173C22×C8—311no—
174C4⋊Q8—21no—
175#124—21no—
176#123—22no—
177C22⋊Q8C2×D3114yescomplemented
178#117C2×D3111yescharacteristic, complemented
179#103—312no—
180#102—312no—
181#102—314no—
182C22⋊C8—314no—
183C31#300911yescharacteristic, complemented, Sylow
184C24—622no—
185C24—621no—
186#41—22no—
187#41—22no—
188#41—314no—
189#41—624no—
190#41—624no—
191#41—628no—
192#41—622no—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—311no—
197C2×Q8C22⋊D3111yescharacteristic
198C2×Q8C22×D3112yes—
199C2×D4—314no—
200C2×D4—312no—
201C2×D4—311no—
202C2×D4—624no—
203C2×D4—624no—
204C2×D4—6216no—
205C2×D4—628no—
206C2×D4—622no—
207C2×D4—621no—
208C2×D4—622no—
209C2×D4—622no—
210C2×D4—621no—
211C2×D4C22⋊D3112yes—
212C2×D4C22⋊D3111yescharacteristic
213C22×C4—21no—
214C22×C4—622no—
215C22×C4—621no—
216C22×C4C22×D3112yes—
217C22×C4C22×D3111yescharacteristic
218Q16—624no—
219SD16—628no—
220SD16—624no—
221M16—622no—
222C2×C8—312no—
223C2×C8—312no—
224C2×C8—622no—
225C4⋊C4—24no—
226C4⋊C4C22×D3112yes—
227C22⋊C4—21no—
228C22⋊C4—24no—
229C42—21no—
230C23—628no—
231C23—622no—
232C23—621no—
233C23—624no—
234C23—2482no—
235C23—2484no—
236C23C2×C22⋊D3112yes—
237C23C2×C22⋊D3111yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—312no—
243Q8—621no—
244D4—24no—
245D4—22no—
246D4—314no—
247D4—312no—
248D4—628no—
249D4—628no—
250D4—624no—
251D4—624no—
252D4—621no—
253D4—622no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—628no—
260C2×C4—621no—
261C2×C4—624no—
262C2×C4D4×D3112yes—
263C2×C4D4×D3112yes—
264C2×C4C2×C22⋊D3112yes—
265C2×C4C2×C22⋊D3111yescharacteristic
266C2×C4C23×D3111yescharacteristic
267C8—622no—
268C8—622no—
269C22—22no—
270C22—24no—
271C22—22no—
272C22—624no—
273C22—622no—
274C22—1244no—
275C22—1242no—
276C22—2484no—
277C22—2488no—
278C22#1175851412yes—
279C22C22×C22⋊D3111yescharacteristic
280C4—22no—
281C4—42no—
282C4—42no—
283C4—622no—
284C4#1175851412yes—
285C4C2×D4×D3111yescharacteristic
286C4C2×D4×D3111yescharacteristic
287C2—22no—
288C2—42no—
289C2—1244no—
290C2—1242no—
291C2C2×#1175851411yescharacteristic
292C2#1452185712yes—
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#536C2
socleC22×C31C22×D4
prime radicalC2×#11758566C2
hypercenter#536D31
hypocenterC31#3009
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#11758519C2semidirect
3. C22×#92481C2semidirect
4. #14521209C2semidirect
5. #14521199C2semidirect
6. #14521053C2semidirect
7. #14521043C2semidirect
8. #14521016C2semidirect
9. C31×#536C2semidirect
11. C22×Q8×C31C22semidirect
24. C31×C22⋊Q8C22semidirect
25. C31×#117C22semidirect
26. C2×#92482C22semidirect
29. C2×#92481C22semidirect
30. C2×#92481C22semidirect
31. C2×#92474C22semidirect
32. C22×C31⋊C8C22semidirect
33. C22⋊(C31⋊C8)C22semidirect
34. #11758398C22semidirect
35. #11758397C22semidirect
36. #11758397C22semidirect
144. #536D31semidirect
162. C22×Q8C2×D31semidirect
177. C22⋊Q8C2×D31semidirect
178. #117C2×D31semidirect
183. C31#3009semidirect

Maximal subgroups

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

subgroupindex
C2×#117585662
C2×#117585192
C22×#924812
#145212092
#145211992
#145210532
#145210432
#145210162
C31×#5362
#300931