Finite Group Databasebeta

← #30664286

Subgroups — #30664286

Order 16768.

440 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 71 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×C22⋊C4×D131C211yesmaximal, characteristic
2#22553274C212yesmaximal, complemented
3#22553036C211yesmaximal, characteristic, complemented
4#22553035C212yesmaximal, complemented
5#323×C131C211yesmaximal, characteristic, complemented
6C24×D131C411yescharacteristic
7C22×C4×D131C411yescharacteristic
8C22×C4×D131C2211yescharacteristic
9C2×#14523496—22no—
10C2×C22⋊Dic131C411yescharacteristic
11C2×(C2×D131)⋊C4C411yescharacteristic
12C2×(C2×D131)⋊C4C2211yescharacteristic
13C22⋊C4×D131—22no—
14C22⋊C4×D131—22no—
15C22⋊(C131⋊C8)C2212yescomplemented
16(C2×D131)⋊C8—22no—
17C2×C22⋊C4×C131C2211yescharacteristic
18C22⋊C8×C131C2212yescomplemented
19C23×D131—21no—
20C23×D131—21no—
21C23×D131—22no—
22C23×D131—22no—
23C23×D131—42no—
24C23×D131—42no—
25C23×D131C2×C411yescharacteristic
26C23×D131C2×C411yescharacteristic
27C22×Dic131C2×C411yescharacteristic
28C22×Dic131C2×C411yescharacteristic
29C2×C4×D131—21no—
30C2×C4×D131—21no—
31C2×C4×D131—42no—
32C2×C4×D131—42no—
33C2×C4×D131D412yes—
34C2×C4×D131D412yes—
35#14523496—44no—
36C2×C131⋊C8—22no—
37C22⋊Dic131—22no—
38(C2×D131)⋊C4—22no—
39(C2×D131)⋊C4—22no—
40C24×C131C2×C411yescharacteristic
41C22×C4×C131C2×C411yescharacteristic
42C22×C4×C131C2311yescharacteristic
43C2×C8×C131—22no—
44C22⋊C4×C131—22no—
45C22×D131—21no—
46C22×D131—21no—
47C22×D131—22no—
48C22×D131—22no—
49C22×D131—24no—
50C22×D131—24no—
51C22×D131—21no—
52C22×D131—21no—
53C22×D131—48no—
54C22×D131—48no—
55C22×D131—42no—
56C22×D131—42no—
57C22×D131—42no—
58C22×D131—42no—
59C22×D131—41no—
60C22×D131—41no—
61C22×D131C22⋊C411yescharacteristic
62C22×D131C22⋊C411yescharacteristic
63C22×D131M1612yescomplemented
64C2×Dic131—21no—
65C2×Dic131—41no—
66C2×Dic131—41no—
67C2×Dic131C22⋊C412yes—
68C4×D131—24no—
69C4×D131—42no—
70C4×D131—42no—
71C4×D131—42no—
72C131⋊C8—42no—
73C23×C131—21no—
74C23×C131—21no—
75C23×C131—41no—
76C23×C131—41no—
77C23×C131M1612yescomplemented
78C23×C131C22×C411yescharacteristic
79C2×C4×C131—21no—
80C2×C4×C131—41no—
81C2×C4×C131—41no—
82C2×C4×C131C2×D412yes—
83C8×C131—42no—
84C22×C131—21no—
85C22×C131—21no—
86C22×C131—22no—
87C22×C131—42no—
88C22×C131—42no—
89C22×C131—41no—
90C22×C131—41no—
91C22×C131C2×C22⋊C411yescharacteristic
92C22×C131C2×M1611yescharacteristic
93C22×C131C2×M1611yescharacteristic
94C2×D131—24no—
95C2×D131—24no—
96C2×D131—21no—
97C2×D131—21no—
98C2×D131—21no—
99C2×D131—21no—
100C2×D131—22no—
101C2×D131—48no—
102C2×D131—48no—
103C2×D131—42no—
104C2×D131—41no—
105C2×D131—41no—
106C2×D131—42no—
107C2×D131#9912yes—
108C2×D131#10012yes—
109C4×C131—22no—
110C4×C131—41no—
111Dic131—22no—
112Dic131—41no—
113C2×C131—21no—
114C2×C131—21no—
115C2×C131—42no—
116C2×C131#40711yescharacteristic
117C2×C131C2×#9911yescharacteristic
118C2×C131C2×#10011yescharacteristic
119D131—22no—
120D131—22no—
121D131—42no—
122C131#146211yescharacteristic, complemented, Sylow
123#1462—1311nomaximal, Sylow
124C22×C22⋊C4—1311no—
125#407—1312no—
126#323—1311no—
127#323—1312no—
128#323D13111yescharacteristic, complemented
129C25—1311no—
130C23×C4—1311no—
131C23×C4—1311no—
132C2×M16—2622no—
133C2×C22⋊C4—2622no—
134C2×C22⋊C4—2622no—
135C2×C22⋊C4—1311no—
136C2×C22⋊C4—1311no—
137C2×C22⋊C4—1311no—
138C2×C22⋊C4C2×D13111yescharacteristic
139C22⋊C8—2622no—
140C22⋊C8—1312no—
141C22⋊C8C2×D13112yescomplemented
142C24—2621no—
143C24—2621no—
144C24—2622no—
145C24—2622no—
146C24—5242no—
147C24—5242no—
148C24—1311no—
149C24—1311no—
150C24C4×D13111yescharacteristic
151C22×C4—2621no—
152C22×C4—2621no—
153C22×C4—5242no—
154C22×C4—5242no—
155C22×C4—1312no—
156C22×C4—1311no—
157C22×C4—1311no—
158C22×C4—1312no—
159C22×C4C4×D13111yescharacteristic
160C22×C4C22×D13111yescharacteristic
161M16—5244no—
162C2×C8—22no—
163C2×C8—2622no—
164C22⋊C4—22no—
165C22⋊C4—2622no—
166C22⋊C4—2622no—
167C22⋊C4—2622no—
168C23—21no—
169C23—21no—
170C23—41no—
171C23—41no—
172C23—2622no—
173C23—2622no—
174C23—2624no—
175C23—2624no—
176C23—2621no—
177C23—2621no—
178C23—2621no—
179C23—2621no—
180C23—5241no—
181C23—5241no—
182C23—5248no—
183C23—5248no—
184C23—5242no—
185C23—5242no—
186C23—5242no—
187C23—5242no—
188C23—1311no—
189C23—1311no—
190C23—1312no—
191C23#1452349612yescomplemented
192C23C2×C4×D13111yescharacteristic
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—2624no—
197C2×C4—2621no—
198C2×C4—5242no—
199C2×C4—5242no—
200C2×C4—5242no—
201C2×C4—5241no—
202C2×C4—5241no—
203C2×C4—1312no—
204C2×C4D4×D13112yes—
205C8—42no—
206C8—5242no—
207C22—21no—
208C22—21no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22—41no—
213C22—41no—
214C22—2624no—
215C22—2624no—
216C22—2621no—
217C22—2621no—
218C22—2621no—
219C22—2621no—
220C22—2622no—
221C22—5242no—
222C22—5242no—
223C22—5248no—
224C22—5248no—
225C22—5241no—
226C22—5241no—
227C22—1312no—
228C22—1312no—
229C22C22⋊C4×D13111yescharacteristic
230C22M16×D13111yescharacteristic
231C22C2×#1452349611yescharacteristic
232C4—22no—
233C4—41no—
234C4—2622no—
235C4—5241no—
236C2—21no—
237C2—21no—
238C2—42no—
239C2—2622no—
240C2—2622no—
241C2—5242no—
242C2#2255327411yescharacteristic
243C2#99×D13111yescharacteristic
244C2#100×D13111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D131
commutatorC23×C131C22×C4
FrattiniC22×C4C22×D131
Fitting#323×C131C2
socleC22×C131C2×C22⋊C4
prime radicalC24×D131C4
cosocleC22×C4×C131C23
hypercenter#323D131
hypocenterC131#1462
LCSG ▷ C23×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #22553274C2semidirect
3. #22553036C2semidirect
4. #22553035C2semidirect
5. #323×C131C2semidirect
15. C22⋊(C131⋊C8)C22semidirect
18. C22⋊C8×C131C22semidirect
63. C22×D131M16semidirect
77. C23×C131M16semidirect
122. C131#1462semidirect
128. #323D131semidirect
141. C22⋊C8C2×D131semidirect
191. C23#14523496semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×D1312
#225532742
#225530362
#225530352
#323×C1312
#1462131