Finite Group Databasebeta

← #16857437

Subgroups — #16857437

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#14521720C211yesmaximal, characteristic, complemented
3#14521423C211yesmaximal, characteristic, complemented
4#14521280C212yesmaximal, complemented
5#14521256C211yesmaximal, characteristic, complemented
6#14521244C211yesmaximal, characteristic, complemented
7#14521209C212yesmaximal, complemented
8#14521194C212yesmaximal, complemented
9C4×#92480C211yesmaximal, characteristic, complemented
10D4×C31⋊C8C211yesmaximal, characteristic, complemented
11C31×#547C211yesmaximal, characteristic, complemented
12C2×#41×C31C2212yes—
13C2×#92495C2212yes—
14C31×C4⋊Q8C2211yescharacteristic
15C31×#123—22no—
16C31×C22⋊Q8C2212yes—
17C31×#121—21no—
18C4×D4×C31C2211yescharacteristic
19C4×D4×C31C2211yescharacteristic
20C2×C4⋊C4×C31—21no—
21#11758527—28no—
22C2×#92482C2212yes—
23C2×#92481C2212yes—
24C2×#92481C2212yes—
25#11758517—21no—
26C2×#92480C2211yescharacteristic
27C2×#92479C2212yes—
28#11758507C2212yes—
29C22×C31⋊C8C2212yes—
30#11758470—21no—
31#11758461—22no—
32#11758451C2211yescharacteristic
33C4×Dic62C2211yescharacteristic
34#11758423C2212yes—
35#11758420C2211yescharacteristic
36C22⋊(C31⋊C8)C2212yes—
37#11758398C2211yescharacteristic
38#11758397C2212yes—
39#11758396C2211yescharacteristic
40C4⋊(C31⋊C8)C2211yescharacteristic
41C4×C31⋊C8C2211yescharacteristic
42#41×C31—24no—
43#41×C31—42no—
44C2×Q8×C31C2312yes—
45C2×D4×C31—22no—
46C2×D4×C31D411yescharacteristic
47C2×D4×C31C2311yescharacteristic
48C22×C4×C31—21no—
49C22×C4×C31—22no—
50C22×C4×C31C2312yes—
51C2×C22⋊D31—22no—
52C2×C22⋊D31—22no—
53#92495—24no—
54#92495—42no—
55C2×D124—21no—
56C2×D124—21no—
57C2×D124C2312yes—
58C2×C4×D31—22no—
59C2×Dic62C2311yescharacteristic
60C4⋊C4×C31—21no—
61C4⋊C4×C31—22no—
62C4⋊C4×C31—21no—
63C4⋊C4×C31D411yescharacteristic
64C4⋊C4×C31C2311yescharacteristic
65C22⋊C4×C31—22no—
66C22⋊C4×C31—21no—
67C22⋊C4×C31D412yes—
68C42×C31C2311yescharacteristic
69#92482—24no—
70#92481—24no—
71#92481—42no—
72#92480D414yescomplemented
73#92479—24no—
74(C2×D31)⋊C4—22no—
75C4⋊Dic31C2311yescharacteristic
76Dic31⋊C4—21no—
77C4×Dic31—21no—
78C2×C31⋊C8—24no—
79C2×C31⋊C8C2311yescharacteristic
80C2×C31⋊C8C2312yes—
81C2×C31⋊C8C2311yescharacteristic
82C23×C31—21no—
83C23×C31C2×D412yes—
84C22×D31—22no—
85Q8×C31—22no—
86Q8×C31—22no—
87D4×C31—22no—
88D4×C31—21no—
89D4×C31—44no—
90D4×C31C2×D412yes—
91C2×C4×C31—22no—
92C2×C4×C31—22no—
93C2×C4×C31—21no—
94C2×C4×C31—22no—
95C2×C4×C31—44no—
96C2×C4×C31—42no—
97C2×C4×C31C2×D411yescharacteristic
98C2×C4×C31C2×D412yes—
99C2×C4×C31C2×D411yescharacteristic
100C2×C4×C31C2411yescharacteristic
101C22⋊D31—44no—
102C22⋊D31—82no—
103C2×Dic31—21no—
104C2×Dic31—21no—
105D124—22no—
106D124—22no—
107D124—42no—
108D124—81no—
109C4×D31—44no—
110Dic62—21no—
111Dic62C2×D412yes—
112C31⋊C8—22no—
113C31⋊C8—21no—
114C31⋊C8C2×D412yes—
115#3284—311nomaximal, Sylow
116C22×C31—22no—
117C22×C31—22no—
118C22×C31—22no—
119C22×C31—22no—
120C22×C31—41no—
121C22×C31—41no—
122C22×C31C22×D411yescharacteristic
123C2×D31—42no—
124C2×D31—42no—
125C2×D31—42no—
126C4×C31—22no—
127C4×C31—21no—
128C4×C31—42no—
129C4×C31—41no—
130C4×C31#13512yescomplemented
131C4×C31C22×D411yescharacteristic
132C4×C31H(5,𝔽2)11yescharacteristic
133Dic31—22no—
134Dic31—41no—
135C2×#135—312no—
136#550—311no—
137#547D3111yescharacteristic, complemented
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—42no—
148C2×C31D4211yescharacteristic
149C2×C31C2×#13511yescharacteristic
150C2×C31#57711yescharacteristic
151D31—82no—
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⋊Q8C2×D3111yescharacteristic
162#127—311no—
163#127—621no—
164#123—22no—
165C22⋊Q8C2×D3112yes—
166#121—21no—
167#121—312no—
168#121—622no—
169C4×Q8—311no—
170C4×D4—621no—
171C4×D4C2×D3111yescharacteristic
172C4×D4C2×D3111yescharacteristic
173C2×C4⋊C4—21no—
174#106—311no—
175C4⋊C8—311no—
176#103—312no—
177#103—311no—
178#102—311no—
179#102—312no—
180C22⋊C8—312no—
181C4×C8—311no—
182C31#328411yescharacteristic, complemented, Sylow
183#41—24no—
184#41—42no—
185#41—624no—
186#41—1242no—
187C2×Q8—311no—
188C2×Q8C22×D3112yes—
189C2×D4—22no—
190C2×D4—312no—
191C2×D4—622no—
192C2×D4—622no—
193C2×D4—621no—
194C2×D4—621no—
195C2×D4C22⋊D3111yescharacteristic
196C2×D4C22×D3111yescharacteristic
197C22×C4—21no—
198C22×C4—22no—
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—22no—
212C4⋊C4—21no—
213C4⋊C4—311no—
214C4⋊C4—621no—
215C4⋊C4C22⋊D3111yescharacteristic
216C4⋊C4C22×D3111yescharacteristic
217C22⋊C4—22no—
218C22⋊C4—21no—
219C22⋊C4—622no—
220C22⋊C4C22⋊D3112yes—
221C42—621no—
222C42C22×D3111yescharacteristic
223C23—21no—
224C23—622no—
225C23C2×C22⋊D3112yes—
226Q8—22no—
227Q8—22no—
228Q8—312no—
229Q8—621no—
230D4—22no—
231D4—21no—
232D4—44no—
233D4—622no—
234D4—622no—
235D4—1244no—
236D4—1242no—
237D4—2481no—
238D4—2482no—
239D4D4×D3112yes—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—44no—
245C2×C4—42no—
246C2×C4—621no—
247C2×C4—621no—
248C2×C4—1244no—
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—22no—
260C22—41no—
261C22—41no—
262C22—1242no—
263C22—1242no—
264C22—1242no—
265C22C22×C22⋊D3111yescharacteristic
266C4—21no—
267C4—22no—
268C4—42no—
269C4—41no—
270C4—622no—
271C4—1241no—
272C4#1175852712yescomplemented
273C4C2×D4×D3111yescharacteristic
274C4#1175856111yescharacteristic
275C2—42no—
276C2—42no—
277C2—2482no—
278C2D4×C22⋊D3111yescharacteristic
279C2C2×#1175852711yescharacteristic
280C2#1452185711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#547C2
socleC22×C31C22×D4
hypercenter#547D31
hypocenterC31#3284
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#11758527C2semidirect
2. #14521720C2semidirect
3. #14521423C2semidirect
4. #14521280C2semidirect
5. #14521256C2semidirect
6. #14521244C2semidirect
7. #14521209C2semidirect
8. #14521194C2semidirect
9. C4×#92480C2semidirect
10. D4×C31⋊C8C2semidirect
11. C31×#547C2semidirect
72. #92480D4semidirect
130. C4×C31#135semidirect
137. #547D31semidirect
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
#145217202
#145214232
#145212802
#145212562
#145212442
#145212092
#145211942
C4×#924802
D4×C31⋊C82
C31×#5472
#328431