Finite Group Databasebeta

← #16857441

Subgroups — #16857441

Order 3968.

494 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 133 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×#11758527C212yesmaximal, complemented
2#14521704C211yesmaximal, characteristic, complemented
3#14521421C211yesmaximal, characteristic, complemented
4#14521276C212yesmaximal, complemented
5#14521259C211yesmaximal, characteristic, complemented
6#14521241C211yesmaximal, characteristic, complemented
7#14521211C212yesmaximal, complemented
8#14521192C212yesmaximal, complemented
9C4×#92481C211yesmaximal, characteristic, complemented
10D4×C31⋊C8C211yesmaximal, characteristic, complemented
11C31×#550C211yesmaximal, characteristic, complemented
12C2×#41×C31C2212yes—
13C2×#92495C2212yes—
14C31×#127—21no—
15C31×#127C2211yescharacteristic
16C31×#121—22no—
17C31×#121C2212yes—
18C4×Q8×C31C2211yescharacteristic
19C4×D4×C31—21no—
20C4×D4×C31C2211yescharacteristic
21#11758527—28no—
22C2×#92482C2212yes—
23C2×#92481C2211yescharacteristic
24#11758515—21no—
25C2×#92480C2212yes—
26C2×#92480C2212yes—
27C2×#92479C2212yes—
28#11758501C2212yes—
29C22×C31⋊C8C2212yes—
30C4⋊C4×D31—21no—
31#11758460—22no—
32C4×D124C2211yescharacteristic
33#11758446C2211yescharacteristic
34#11758423C2211yescharacteristic
35#11758420C2212yes—
36C22⋊(C31⋊C8)C2212yes—
37#11758398C2212yes—
38#11758397C2211yescharacteristic
39#11758396C2211yescharacteristic
40C4⋊(C31⋊C8)C2211yescharacteristic
41C4×C31⋊C8C2211yescharacteristic
42#41×C31—24no—
43#41×C31—42no—
44C2×Q8×C31C2311yescharacteristic
45C2×D4×C31—22no—
46C2×D4×C31—22no—
47C2×D4×C31—21no—
48C2×D4×C31—21no—
49C2×D4×C31D411yescharacteristic
50C2×D4×C31C2312yes—
51C22×C4×C31—22no—
52C22×C4×C31C2312yes—
53C2×C22⋊D31—22no—
54#92495—24no—
55#92495—42no—
56C2×D124C2311yescharacteristic
57C2×C4×D31—21no—
58C2×C4×D31—22no—
59C2×Dic62C2312yes—
60C4⋊C4×C31—21no—
61C4⋊C4×C31D411yescharacteristic
62C4⋊C4×C31C2311yescharacteristic
63C22⋊C4×C31—22no—
64C22⋊C4×C31D412yes—
65C42×C31—21no—
66C42×C31C2311yescharacteristic
67C22⋊Dic31—22no—
68#92482—24no—
69#92481D414yescomplemented
70#92480—24no—
71#92480—42no—
72#92479—24no—
73(C2×D31)⋊C4—21no—
74C4⋊Dic31—21no—
75C4⋊Dic31—21no—
76C4⋊Dic31C2311yescharacteristic
77Dic31⋊C4—22no—
78C2×C31⋊C8—24no—
79C2×C31⋊C8C2311yescharacteristic
80C2×C31⋊C8C2312yes—
81C2×C31⋊C8C2311yescharacteristic
82C23×C31—22no—
83C23×C31C2×D412yes—
84C22×D31—21no—
85Q8×C31—21no—
86Q8×C31C2×D412yes—
87D4×C31—22no—
88D4×C31—22no—
89D4×C31—22no—
90D4×C31—44no—
91D4×C31—42no—
92D4×C31—81no—
93D4×C31—82no—
94C2×C4×C31—21no—
95C2×C4×C31—22no—
96C2×C4×C31—22no—
97C2×C4×C31—21no—
98C2×C4×C31—44no—
99C2×C4×C31C2×D412yes—
100C2×C4×C31C2×D411yescharacteristic
101C2×C4×C31C2×D411yescharacteristic
102C2×C4×C31C2411yescharacteristic
103C22⋊D31—44no—
104C2×Dic31—21no—
105C2×Dic31—22no—
106D124—21no—
107D124C2×D412yes—
108C4×D31—44no—
109C4×D31—42no—
110Dic62—22no—
111Dic62—22no—
112C31⋊C8—22no—
113C31⋊C8—21no—
114C31⋊C8C2×D412yes—
115#3284—311nomaximal, Sylow
116C22×C31—22no—
117C22×C31—22no—
118C22×C31—22no—
119C22×C31—42no—
120C22×C31—42no—
121C22×C31—42no—
122C22×C31C22×D411yescharacteristic
123C2×D31—22no—
124C2×D31—41no—
125C2×D31—41no—
126C4×C31—22no—
127C4×C31—22no—
128C4×C31—21no—
129C4×C31—41no—
130C4×C31#13512yescomplemented
131C4×C31C22×D411yescharacteristic
132C4×C31H(5,𝔽2)11yescharacteristic
133Dic31—42no—
134Dic31—41no—
135C2×#135—312no—
136#550D3111yescharacteristic, complemented
137#547—311no—
138#492—311no—
139#465—312no—
140#462—311no—
141#459—311no—
142#452—312no—
143#449—312no—
144C4×SD16—311no—
145C8×D4—311no—
146C2×C31—42no—
147C2×C31—82no—
148C2×C31D4211yescharacteristic
149C2×C31C2×#13511yescharacteristic
150C2×C31#57711yescharacteristic
151D31—42no—
152C2×#41—312no—
153C2×#41C2×D3112yes—
154#135—628no—
155C2×Q16—312no—
156C2×SD16—311no—
157C2×SD16—312no—
158C2×SD16—312no—
159C2×D8—312no—
160C22×C8—312no—
161C4⋊Q8—311no—
162#127—21no—
163#127C2×D3111yescharacteristic
164#123—622no—
165C22⋊Q8—312no—
166#121—22no—
167#121—621no—
168#121C2×D3112yes—
169C4×Q8C2×D3111yescharacteristic
170C4×D4—21no—
171C4×D4—311no—
172C4×D4C2×D3111yescharacteristic
173C2×C4⋊C4—621no—
174#106—311no—
175C4⋊C8—311no—
176#103—311no—
177#103—312no—
178#102—312no—
179#102—311no—
180C22⋊C8—312no—
181C4×C8—311no—
182C31#328411yescharacteristic, complemented, Sylow
183#41—24no—
184#41—42no—
185#41—624no—
186#41—1242no—
187C2×Q8—312no—
188C2×Q8C22×D3111yescharacteristic
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—311no—
194C2×D4—622no—
195C2×D4C22⋊D3111yescharacteristic
196C2×D4C22×D3112yes—
197C22×C4—22no—
198C22×C4—621no—
199C22×C4—622no—
200C22×C4C22×D3112yes—
201Q16—624no—
202SD16—314no—
203SD16—624no—
204SD16—1242no—
205D8—624no—
206C2×C8—311no—
207C2×C8—311no—
208C2×C8—312no—
209C2×C8—624no—
210C4⋊C4—21no—
211C4⋊C4—311no—
212C4⋊C4—622no—
213C4⋊C4—621no—
214C4⋊C4—621no—
215C4⋊C4C22⋊D3111yescharacteristic
216C4⋊C4C22×D3111yescharacteristic
217C22⋊C4—22no—
218C22⋊C4—622no—
219C22⋊C4—621no—
220C22⋊C4C22⋊D3112yes—
221C42—21no—
222C42C22×D3111yescharacteristic
223C23—22no—
224C23—621no—
225C23C2×C22⋊D3112yes—
226Q8—21no—
227Q8—622no—
228Q8—622no—
229Q8D4×D3112yes—
230D4—22no—
231D4—22no—
232D4—22no—
233D4—44no—
234D4—42no—
235D4—81no—
236D4—82no—
237D4—312no—
238D4—621no—
239D4—1244no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—44no—
245C2×C4—621no—
246C2×C4—622no—
247C2×C4—1244no—
248C2×C4—1242no—
249C2×C4C2×C22⋊D3111yescharacteristic
250C2×C4C2×C22⋊D3112yes—
251C2×C4C2×C22⋊D3111yescharacteristic
252C2×C4C23×D3111yescharacteristic
253C8—312no—
254C8—622no—
255C8—621no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—622no—
263C22—1241no—
264C22—1241no—
265C22C22×C22⋊D3111yescharacteristic
266C4—21no—
267C4—22no—
268C4—22no—
269C4—41no—
270C4—1242no—
271C4—1241no—
272C4#1175852712yescomplemented
273C4C2×D4×D3111yescharacteristic
274C4#1175856111yescharacteristic
275C2—42no—
276C2—82no—
277C2—1242no—
278C2D4×C22⋊D3111yescharacteristic
279C2C2×#1175852711yescharacteristic
280C2#1452186011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#550C2
socleC22×C31C22×D4
hypercenter#550D31
hypocenterC31#3284
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#11758527C2semidirect
2. #14521704C2semidirect
3. #14521421C2semidirect
4. #14521276C2semidirect
5. #14521259C2semidirect
6. #14521241C2semidirect
7. #14521211C2semidirect
8. #14521192C2semidirect
9. C4×#92481C2semidirect
10. D4×C31⋊C8C2semidirect
11. C31×#550C2semidirect
69. #92481D4semidirect
130. C4×C31#135semidirect
136. #550D31semidirect
182. C31#3284semidirect
272. C4#11758527semidirect

Maximal subgroups

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

subgroupindex
C2×#117585272
#145217042
#145214212
#145212762
#145212592
#145212412
#145212112
#145211922
C4×#924812
D4×C31⋊C82
C31×#5502
#328431