Finite Group Databasebeta

← #16855883

Subgroups — #16855883

Order 3968.

386 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 121 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#14521585C211yesmaximal, characteristic
2C2×Q8×Dic31C211yesmaximal, characteristic, complemented
3#14521382C211yesmaximal, characteristic, complemented
4#14521368C211yesmaximal, characteristic
5#14521354C212yesmaximal, complemented
6#14521273C212yesmaximal, complemented
7#14521188C211yesmaximal, characteristic
8#14521156C212yesmaximal, complemented
9#14521152C212yesmaximal, complemented
10#14521103C211yesmaximal, characteristic, complemented
11C31×#451C211yesmaximal, characteristic, complemented
12C22×Q8×C31C2211yescharacteristic
13C2×Q16×C31C2212yes—
14C31×C22⋊Q8C2211yescharacteristic
15#11758524C2211yescharacteristic
16Q8×Dic31—21no—
17Q8×Dic31—21no—
18Q8×Dic31C2212yes—
19Q8×Dic31C2212yes—
20Q8×Dic31C2211yescharacteristic
21C2×#92481C2212yes—
22#11758507C2211yescharacteristic
23C2×C4⋊Dic31—21no—
24C2×C4⋊Dic31C2211yescharacteristic
25C2×C4×Dic31—21no—
26C2×C4×Dic31C2211yescharacteristic
27#11758475—21no—
28#11758470C2211yescharacteristic
29#11758469—21no—
30#11758467C2211yescharacteristic
31#11758463—21no—
32(C22⋊D31)⋊C4—21no—
33C31×#103C2212yes—
34C31×C22⋊C8C2211yescharacteristic
35#11758423C2212yes—
36#11758423C2212yes—
37C22⋊(C31⋊C8)C2211yescharacteristic
38#11758409C2212yes—
39#11758405C2212yes—
40#11758403C2212yes—
41Dic31⋊C8C2212yes—
42#11758395C2212yes—
43C2×Q8×C31—24no—
44C2×Q8×C31—21no—
45C2×Q8×C31C2311yescharacteristic
46C2×Q8×C31C2312yes—
47C22×C4×C31—21no—
48C22×C4×C31C2311yescharacteristic
49C2×C22⋊D31—21no—
50C22×Dic31—21no—
51C22×Dic31D411yescharacteristic
52C22×Dic31D411yescharacteristic
53C2×D124C2311yescharacteristic
54C2×C4×D31—21no—
55Q16×C31—24no—
56C2×C8×C31C2312yes—
57C4⋊C4×C31—21no—
58C4⋊C4×C31C2311yescharacteristic
59C22⋊C4×C31—21no—
60#92481—24no—
61(C2×D31)⋊C4—21no—
62(C2×D31)⋊C4—21no—
63(C2×D31)⋊C4—21no—
64(C2×D31)⋊C4—21no—
65C4⋊Dic31—21no—
66C4⋊Dic31—21no—
67C4⋊Dic31—21no—
68C4⋊Dic31—22no—
69C4⋊Dic31—22no—
70C4⋊Dic31C2312yes—
71C4⋊Dic31C2311yescharacteristic
72Dic31⋊C4—21no—
73C4×Dic31—21no—
74C4×Dic31—22no—
75C4×Dic31—22no—
76C4×Dic31D411yescharacteristic
77C4×Dic31D411yescharacteristic
78C4×Dic31C2311yescharacteristic
79C4×Dic31C2311yescharacteristic
80C2×C31⋊C8C2312yes—
81C23×C31C2×D411yescharacteristic
82C22×D31—21no—
83Q8×C31—24no—
84Q8×C31—22no—
85Q8×C31—22no—
86Q8×C31#4114yes—
87C2×C4×C31—24no—
88C2×C4×C31—22no—
89C2×C4×C31—22no—
90C2×C4×C31—21no—
91C2×C4×C31—21no—
92C2×C4×C31C2×D411yescharacteristic
93C2×C4×C31C2411yescharacteristic
94C22⋊D31—42no—
95C2×Dic31—22no—
96C2×Dic31—22no—
97C2×Dic31—21no—
98C2×Dic31—41no—
99C2×Dic31—42no—
100C2×Dic31C2×D411yescharacteristic
101C2×Dic31C2×D411yescharacteristic
102C2×Dic31C2×D411yescharacteristic
103C2×Dic31C2×D411yescharacteristic
104D124—22no—
105C4×D31—42no—
106C8×C31—22no—
107C31⋊C8—22no—
108#3117—311nomaximal, Sylow
109C22×C31—21no—
110C22×C31—21no—
111C22×C31#13712yes—
112C22×C31C22×D411yescharacteristic
113C2×D31—41no—
114C2×D31—41no—
115C2×D31—41no—
116C4×C31—24no—
117C4×C31—21no—
118C4×C31—41no—
119C4×C31—41no—
120C4×C31C2×#4112yes—
121Dic31—21no—
122Dic31—21no—
123Dic31—21no—
124Dic31—21no—
125Dic31—41no—
126Dic31—42no—
127#529—311no—
128C2×C4×Q8—311no—
129#484—311no—
130#481—311no—
131#479—312no—
132#464—312no—
133#451D3111yescharacteristic, complemented
134#448—311no—
135#441—312no—
136#440—312no—
137#432—311no—
138C2×C31—22no—
139C2×C31#52511yescharacteristic
140C2×C31C2×#13711yescharacteristic
141C2×C31#57511yescharacteristic
142D31—81no—
143C22×Q8C2×D3111yescharacteristic
144C2×Q16C2×D3112yes—
145C2×SD16—312no—
146#126—621no—
147#125—311no—
148#124—311no—
149#123—621no—
150C22⋊Q8C2×D3111yescharacteristic
151#121—311no—
152C4×Q8—312no—
153C4×Q8—312no—
154C4×Q8—311no—
155C4×Q8—621no—
156C4×Q8—621no—
157C4×D4—311no—
158C4×D4—621no—
159#117—621no—
160C2×C4⋊C4—311no—
161C2×C4⋊C4—621no—
162C2×C42—311no—
163C2×C42—621no—
164#107—312no—
165#106—312no—
166C4⋊C8—312no—
167#103—312no—
168#103—312no—
169#103C2×D3112yes—
170#102—312no—
171C22⋊C8—311no—
172C22⋊C8C2×D3111yescharacteristic
173#97—312no—
174C31#311711yescharacteristic, complemented, Sylow
175C2×Q8—24no—
176C2×Q8—21no—
177C2×Q8C22×D3111yescharacteristic
178C2×Q8C22×D3112yes—
179C2×D4—311no—
180C2×D4—621no—
181C22×C4—21no—
182C22×C4—311no—
183C22×C4—311no—
184C22×C4—621no—
185C22×C4—621no—
186C22×C4C22×D3111yescharacteristic
187Q16—24no—
188SD16—624no—
189C2×C8—312no—
190C2×C8C22×D3112yes—
191C4⋊C4—21no—
192C4⋊C4—312no—
193C4⋊C4—311no—
194C4⋊C4—621no—
195C4⋊C4—622no—
196C4⋊C4—622no—
197C4⋊C4—621no—
198C4⋊C4—621no—
199C4⋊C4—621no—
200C4⋊C4C22×D3111yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—621no—
203C22⋊C4—621no—
204C22⋊C4—621no—
205C22⋊C4—621no—
206C42—311no—
207C42—311no—
208C42—311no—
209C42—311no—
210C42—622no—
211C42—622no—
212C42—621no—
213C23—621no—
214C23D4×D3111yescharacteristic
215Q8—24no—
216Q8—22no—
217Q8—22no—
218Q8#9249714yes—
219D4—622no—
220D4—1242no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—311no—
227C2×C4—311no—
228C2×C4—311no—
229C2×C4—311no—
230C2×C4—622no—
231C2×C4—622no—
232C2×C4—621no—
233C2×C4—1242no—
234C2×C4—1241no—
235C2×C4—1242no—
236C2×C4D4×D3111yescharacteristic
237C2×C4C23×D3111yescharacteristic
238C8—22no—
239C8—622no—
240C22—21no—
241C22—21no—
242C22—1241no—
243C22—1241no—
244C22—1241no—
245C22#1175849512yes—
246C22C2×D4×D3111yescharacteristic
247C4—24no—
248C4—21no—
249C4—41no—
250C4—41no—
251C4—621no—
252C4—621no—
253C4—621no—
254C4—621no—
255C4—1241no—
256C4—1242no—
257C4C2×#9249712yes—
258C2—22no—
259C2—2481no—
260C2#1452155511yescharacteristic
261C2C2×#1175849511yescharacteristic
262C2#1452184811yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#451C2
socleC22×C31C22×D4
prime radical#11758507C22
hypercenter#451D31
hypocenterC31#3117
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic31C2semidirect
3. #14521382C2semidirect
5. #14521354C2semidirect
6. #14521273C2semidirect
8. #14521156C2semidirect
9. #14521152C2semidirect
10. #14521103C2semidirect
11. C31×#451C2semidirect
133. #451D31semidirect
174. C31#3117semidirect

Maximal subgroups

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

subgroupindex
#145215852
C2×Q8×Dic312
#145213822
#145213682
#145213542
#145212732
#145211882
#145211562
#145211522
#145211032
C31×#4512
#311731