Finite Group Databasebeta

← #16845719

Subgroups — #16845719

Order 3968.

356 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 109 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×D4×Dic31C211yesmaximal, characteristic
2C2×Dic31⋊C8C211yesmaximal, characteristic, complemented
3C2×#11758420C211yesmaximal, characteristic
4#14520914C211yesmaximal, characteristic, complemented
5#14520812C211yesmaximal, characteristic, complemented
6#14520811C211yesmaximal, characteristic, complemented
7C2×C31×#102C211yesmaximal, characteristic
8C22×D4×C31C2211yescharacteristic
9C23×Dic31—21no—
10C22×C8×C31C2211yescharacteristic
11C2×C4⋊C4×C31C2211yescharacteristic
12C2×C22⋊Dic31—21no—
13D4×Dic31—22no—
14D4×Dic31C414yescomplemented
15C2×C4⋊Dic31C2211yescharacteristic
16C2×C4⋊Dic31C2211yescharacteristic
17C2×C4×Dic31C2211yescharacteristic
18C22×C31⋊C8C2211yescharacteristic
19C31×#102—22no—
20#11758420—22no—
21C31⋊#95—21no—
22C31⋊#95—21no—
23Dic31⋊C8—22no—
24C24×C31—21no—
25C2×D4×C31—21no—
26C2×D4×C31—21no—
27C2×D4×C31—21no—
28C2×D4×C31C2×C412yes—
29C2×D4×C31D412yes—
30C2×D4×C31Q812yes—
31C22×C4×C31—21no—
32C22×C4×C31C2311yescharacteristic
33C22×Dic31—21no—
34C22×Dic31—24no—
35C22×Dic31—21no—
36C22×Dic31—42no—
37C22×Dic31—42no—
38C22×Dic31D411yescharacteristic
39C22×Dic31D411yescharacteristic
40C2×C8×C31—21no—
41C2×C8×C31—21no—
42C2×C8×C31—21no—
43C4⋊C4×C31—21no—
44C4⋊C4×C31—21no—
45C22⋊Dic31—24no—
46C4⋊Dic31—21no—
47C4⋊Dic31—21no—
48C4⋊Dic31—21no—
49C4⋊Dic31C2×C412yes—
50C4×Dic31C2×C412yes—
51C4×Dic31D412yes—
52C2×C31⋊C8—21no—
53C2×C31⋊C8—21no—
54C2×C31⋊C8—21no—
55C23×C31—22no—
56C23×C31—22no—
57C23×C31—22no—
58C23×C31—41no—
59C23×C31—41no—
60C23×C31—41no—
61C23×C31—41no—
62C23×C31C2×D411yescharacteristic
63D4×C31—22no—
64D4×C31—22no—
65D4×C31—22no—
66D4×C31C4⋊C414yes—
67C2×C4×C31—41no—
68C2×C4×C31—41no—
69C2×C4×C31—41no—
70C2×C4×C31C22×C411yescharacteristic
71C2×C4×C31C2×D411yescharacteristic
72C2×C4×C31C2×D411yescharacteristic
73C2×C4×C31C2×Q811yescharacteristic
74C2×C4×C31#4111yescharacteristic
75C2×C4×C31#4111yescharacteristic
76C2×Dic31—21no—
77C2×Dic31—21no—
78C2×Dic31—21no—
79C2×Dic31—21no—
80C2×Dic31—22no—
81C2×Dic31—48no—
82C2×Dic31—41no—
83C2×Dic31—41no—
84C2×Dic31—41no—
85C2×Dic31—41no—
86C2×Dic31—41no—
87C2×Dic31C22⋊C412yes—
88C2×Dic31C22⋊C412yes—
89C8×C31—41no—
90C31⋊C8—41no—
91#1804—311nomaximal, Sylow
92C22×C31—24no—
93C22×C31—42no—
94C22×C31—42no—
95C22×C31—42no—
96C22×C31—42no—
97C22×C31—42no—
98C22×C31—42no—
99C22×C31C2×C22⋊C411yescharacteristic
100C22×C31#12111yescharacteristic
101C22×C31C22⋊Q811yescharacteristic
102C22×C31#13511yescharacteristic
103C22×C31#13511yescharacteristic
104C22×C31AGL(1,ℤ/8)11yescharacteristic
105C22×C31AGL(1,ℤ/8)11yescharacteristic
106C4×C31—81no—
107C4×C31C2×C4⋊C411yescharacteristic
108C4×C31#11711yescharacteristic
109C4×C31#12111yescharacteristic
110C4×C31C22⋊Q811yescharacteristic
111Dic31—22no—
112Dic31—22no—
113Dic31—42no—
114Dic31—81no—
115C2×C4×D4—311no—
116C2×C4⋊C8—311no—
117C2×#102—311no—
118C2×#102D3111yescharacteristic
119#382—311no—
120#340—311no—
121#340—311no—
122C2×C31—44no—
123C2×C31#38011yescharacteristic
124C2×C31#41611yescharacteristic
125C2×C31#41811yescharacteristic
126C2×C31#46311yescharacteristic
127C2×C31#46311yescharacteristic
128C2×C31#47811yescharacteristic
129C2×C31#47811yescharacteristic
130C22×D4C2×D3111yescharacteristic
131C23×C4—621no—
132C22×C8—311no—
133C22×C8C2×D3111yescharacteristic
134C4×D4—314no—
135C4×D4—622no—
136C2×C4⋊C4—311no—
137C2×C4⋊C4—311no—
138C2×C4⋊C4C2×D3111yescharacteristic
139C2×C22⋊C4—621no—
140C2×C42—311no—
141C4⋊C8—622no—
142#102—22no—
143#102—622no—
144#95—621no—
145#95—621no—
146C31#180411yescharacteristic, complemented, Sylow
147C24—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4Dic6212yes—
152C2×D4C4×D3112yes—
153C2×D4C22⋊D3112yes—
154C22×C4—21no—
155C22×C4—311no—
156C22×C4—311no—
157C22×C4—624no—
158C22×C4—621no—
159C22×C4—621no—
160C22×C4—1242no—
161C22×C4—1242no—
162C22×C4C22×D3111yescharacteristic
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—621no—
167C2×C8—621no—
168C2×C8—621no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—312no—
172C4⋊C4—621no—
173C4⋊C4—621no—
174C4⋊C4—621no—
175C22⋊C4—624no—
176C42—312no—
177C42—312no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23D4×D3111yescharacteristic
186D4—22no—
187D4—22no—
188D4—22no—
189D4Dic31⋊C414yes—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—312no—
194C2×C4—312no—
195C2×C4—621no—
196C2×C4—621no—
197C2×C4—621no—
198C2×C4—621no—
199C2×C4—622no—
200C2×C4—1248no—
201C2×C4—1241no—
202C2×C4—1241no—
203C2×C4—1241no—
204C2×C4—1241no—
205C2×C4—1241no—
206C2×C4C2×Dic6211yescharacteristic
207C2×C4C2×C4×D3111yescharacteristic
208C2×C4D4×D3111yescharacteristic
209C2×C4#9249711yescharacteristic
210C2×C4#9249711yescharacteristic
211C2×C4C2×C22⋊D3111yescharacteristic
212C8—41no—
213C8—1241no—
214C22—24no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—42no—
221C22#1175845511yescharacteristic
222C22C22⋊C4×D3111yescharacteristic
223C22#1175848811yescharacteristic
224C22#1175848911yescharacteristic
225C22#1175849111yescharacteristic
226C22#1175849311yescharacteristic
227C22#1175851611yescharacteristic
228C4—81no—
229C4—622no—
230C4—622no—
231C4—1242no—
232C4—2481no—
233C4#1175845411yescharacteristic
234C4#1175845511yescharacteristic
235C4C2×Dic31⋊C411yescharacteristic
236C4#1175851611yescharacteristic
237C2—44no—
238C2#1452090711yescharacteristic
239C2#1452104211yescharacteristic
240C2#1452105211yescharacteristic
241C2#1452126211yescharacteristic
242C2#1452126611yescharacteristic
243C2#1452134711yescharacteristic
244C2#1452135111yescharacteristic
subgroupG/H
centerC23D4×D31
commutatorC2×C4×C31C22×C4
FrattiniC22×C4C22×D31
FittingC2×C31×#102C2
socleC23×C31C2×D4
prime radicalC22×D4×C31C22
cosocleC22×C4×C31C23
hypercenterC2×#102D31
hypocenterC31#1804
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×Dic31⋊C8C2semidirect
4. #14520914C2semidirect
5. #14520812C2semidirect
6. #14520811C2semidirect
14. D4×Dic31C4semidirect
146. C31#1804semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic312
C2×Dic31⋊C82
C2×#117584202
#145209142
#145208122
#145208112
C2×C31×#1022
#180431