Finite Group Databasebeta

← #16856102

Subgroups — #16856102

Order 3968.

424 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#14521595C211yesmaximal, characteristic
2C2×#11758465C211yesmaximal, characteristic, complemented
3#14521410C212yesmaximal, complemented
4#14521407C212yesmaximal, complemented
5#14521398C212yesmaximal, complemented
6#14521394C212yesmaximal, complemented
7#14521051C211yesmaximal, characteristic, complemented
8#14521050C211yesmaximal, characteristic, complemented
9#14521048C211yesmaximal, characteristic, complemented
10M16×Dic31C211yesmaximal, characteristic, complemented
11C31×#417C211yesmaximal, characteristic, complemented
12C2×#41×C31C2211yescharacteristic
13C2×#92495C2211yescharacteristic
14C22×Dic62—21no—
15C2×M16×C31C2211yescharacteristic
16C2×C4⋊C4×C31C2211yescharacteristic
17#11758524C2211yescharacteristic
18#11758521C2211yescharacteristic
19#11758517C2211yescharacteristic
20#11758516—21no—
21#11758516—21no—
22#11758513C2211yescharacteristic
23C4×C22⋊D31—21no—
24C4×C22⋊D31—21no—
25C2×C4⋊Dic31C2211yescharacteristic
26C2×Dic31⋊C4—21no—
27C2×C4×Dic31C2211yescharacteristic
28C2×#92474C2211yescharacteristic
29#11758465—21no—
30#11758465—21no—
31#11758465C2212yes—
32#11758465C2212yes—
33C31×#103C2212yes—
34C31×#102C2212yes—
35#11758423C2212yes—
36#11758420C2212yes—
37#11758409C2212yes—
38#11758404C2212yes—
39#11758403C2212yes—
40C8×Dic31C2212yes—
41#11758398C2212yes—
42#11758397C2212yes—
43#41×C31—22no—
44#41×C31—22no—
45C2×Q8×C31C2311yescharacteristic
46C2×D4×C31—21no—
47C2×D4×C31C2311yescharacteristic
48C22×C4×C31—21no—
49C22×C4×C31—21no—
50C22×C4×C31C2311yescharacteristic
51C2×C22⋊D31—21no—
52C2×C22⋊D31—21no—
53C2×C22⋊D31—21no—
54C22×Dic31—21no—
55C22×Dic31D411yescharacteristic
56C22×Dic31D411yescharacteristic
57#92495—22no—
58#92495—22no—
59C2×D124C2311yescharacteristic
60C2×C4×D31—21no—
61C2×Dic62—22no—
62C2×Dic62—22no—
63C2×Dic62—44no—
64C2×Dic62C2311yescharacteristic
65M16×C31—22no—
66C2×C8×C31C2312yes—
67C4⋊C4×C31—21no—
68C4⋊C4×C31C2312yes—
69C22⋊Dic31—21no—
70C22⋊Dic31—21no—
71(C2×D31)⋊C4—21no—
72(C2×D31)⋊C4—21no—
73C4⋊Dic31—21no—
74C4⋊Dic31C2312yes—
75Dic31⋊C4—22no—
76Dic31⋊C4—22no—
77Dic31⋊C4—21no—
78Dic31⋊C4—21no—
79C4×Dic31D411yescharacteristic
80C4×Dic31D411yescharacteristic
81C4×Dic31C2311yescharacteristic
82C4×Dic31C2311yescharacteristic
83#92474—22no—
84C2×C31⋊C8C2312yes—
85C23×C31—21no—
86C23×C31C2×D411yescharacteristic
87C22×D31—21no—
88Q8×C31—21no—
89Q8×C31—21no—
90D4×C31—21no—
91D4×C31—21no—
92D4×C31—42no—
93C2×C4×C31—21no—
94C2×C4×C31—22no—
95C2×C4×C31—42no—
96C2×C4×C31—42no—
97C2×C4×C31C2×D411yescharacteristic
98C2×C4×C31#4112yes—
99C2×C4×C31#4112yes—
100C2×C4×C31C2411yescharacteristic
101C22⋊D31—42no—
102C22⋊D31—42no—
103C22⋊D31—42no—
104C2×Dic31—22no—
105C2×Dic31—22no—
106C2×Dic31—22no—
107C2×Dic31—21no—
108C2×Dic31—42no—
109C2×Dic31C2×D411yescharacteristic
110C2×Dic31C2×D411yescharacteristic
111C2×Dic31C2×D411yescharacteristic
112C2×Dic31C2×D411yescharacteristic
113D124—21no—
114D124—21no—
115C4×D31—42no—
116Dic62—21no—
117Dic62—21no—
118Dic62—44no—
119Dic62—44no—
120C8×C31—22no—
121C31⋊C8—22no—
122#3141—311nomaximal, Sylow
123C22×C31—21no—
124C22×C31—21no—
125C22×C31—41no—
126C22×C31—41no—
127C22×C31—41no—
128C22×C31#12412yes—
129C22×C31C22×D411yescharacteristic
130C2×D31—41no—
131C2×D31—41no—
132C2×D31—41no—
133C4×C31—41no—
134C4×C31—42no—
135C4×C31#12412yes—
136C4×C31C2×#4111yescharacteristic
137C4×C31C2×#4111yescharacteristic
138Dic31—21no—
139Dic31—21no—
140Dic31—41no—
141Dic31—42no—
142Dic31AGL(1,ℤ/8)12yes—
143Dic31#13712yes—
144#532—311no—
145C2×C4⋊Q8—311no—
146#489—312no—
147#489—312no—
148#487—312no—
149#486—312no—
150#417—311no—
151#417—311no—
152#417—311no—
153#417D3111yescharacteristic, complemented
154C4×M16—311no—
155C2×C31—22no—
156C2×C31—81no—
157C2×C31C2×#12411yescharacteristic
158C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
159C2×C31C2×#13711yescharacteristic
160D31—81no—
161C2×#41—311no—
162C2×#41C2×D3111yescharacteristic
163C22×Q8—621no—
164C2×M16—311no—
165C2×M16C2×D3111yescharacteristic
166C4⋊Q8—312no—
167C4⋊Q8—312no—
168C4⋊Q8—311no—
169C4⋊Q8—621no—
170C4⋊Q8—621no—
171#127—311no—
172#124—311no—
173#124—311no—
174#121—621no—
175#121—621no—
176C4×D4—621no—
177C4×D4—621no—
178C2×C4⋊C4—311no—
179C2×C4⋊C4—621no—
180C2×C4⋊C4C2×D3111yescharacteristic
181C2×C42—311no—
182#103—312no—
183#103—312no—
184#103—312no—
185#103C2×D3112yes—
186#102—312no—
187#102—312no—
188#102—312no—
189#102C2×D3112yes—
190#97—312no—
191C4×C8—312no—
192C31#314111yescharacteristic, complemented, Sylow
193#41—22no—
194#41—22no—
195#41—622no—
196#41—622no—
197C2×Q8—311no—
198C2×Q8—622no—
199C2×Q8—622no—
200C2×Q8—1244no—
201C2×Q8C22×D3111yescharacteristic
202C2×D4—21no—
203C2×D4—311no—
204C2×D4—621no—
205C2×D4—621no—
206C2×D4—621no—
207C2×D4C22×D3111yescharacteristic
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—311no—
211C22×C4—311no—
212C22×C4—621no—
213C22×C4—621no—
214C22×C4C22×D3111yescharacteristic
215M16—22no—
216M16—622no—
217C2×C8—312no—
218C2×C8C22×D3112yes—
219C4⋊C4—21no—
220C4⋊C4—312no—
221C4⋊C4—622no—
222C4⋊C4—622no—
223C4⋊C4—621no—
224C4⋊C4—621no—
225C4⋊C4—621no—
226C4⋊C4C22×D3112yes—
227C22⋊C4—621no—
228C22⋊C4—621no—
229C22⋊C4—621no—
230C22⋊C4—621no—
231C42—311no—
232C42—311no—
233C42—311no—
234C42—311no—
235C23—21no—
236C23—621no—
237C23D4×D3111yescharacteristic
238Q8—21no—
239Q8—21no—
240Q8—621no—
241Q8—621no—
242Q8—1244no—
243Q8—1244no—
244D4—21no—
245D4—21no—
246D4—42no—
247D4—621no—
248D4—621no—
249D4—1242no—
250D4—1242no—
251D4—1242no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—311no—
257C2×C4—311no—
258C2×C4—311no—
259C2×C4—311no—
260C2×C4—622no—
261C2×C4—622no—
262C2×C4—621no—
263C2×C4—622no—
264C2×C4—1242no—
265C2×C4—1242no—
266C2×C4#9249512yes—
267C2×C4D4×D3111yescharacteristic
268C2×C4#9249712yes—
269C2×C4C23×D3111yescharacteristic
270C8—22no—
271C8—622no—
272C22—21no—
273C22—21no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—1241no—
278C22—1241no—
279C22—1241no—
280C22#1175846212yes—
281C22C2×D4×D3111yescharacteristic
282C4—41no—
283C4—42no—
284C4—312no—
285C4—312no—
286C4—621no—
287C4—621no—
288C4—1241no—
289C4—1242no—
290C4#1175846212yes—
291C4C2×#9249511yescharacteristic
292C4C2×#9249711yescharacteristic
293C2—22no—
294C2—81no—
295C2—2481no—
296C2C2×#1175846211yescharacteristic
297C2AGL(1,ℤ/8)×D3111yescharacteristic
298C2#137×D3111yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#417C2
socleC22×C31C22×D4
prime radical#14521595C2
hypercenter#417D31
hypocenterC31#3141
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#11758465C2semidirect
3. #14521410C2semidirect
4. #14521407C2semidirect
5. #14521398C2semidirect
6. #14521394C2semidirect
7. #14521051C2semidirect
8. #14521050C2semidirect
9. #14521048C2semidirect
10. M16×Dic31C2semidirect
11. C31×#417C2semidirect
153. #417D31semidirect
192. C31#3141semidirect

Maximal subgroups

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

subgroupindex
#145215952
C2×#117584652
#145214102
#145214072
#145213982
#145213942
#145210512
#145210502
#145210482
M16×Dic312
C31×#4172
#314131