Finite Group Databasebeta

← #30670263

Subgroups — #30670263

Order 16768.

624 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 173 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×#16862229C212yesmaximal
2C2×C4×C22⋊D131C211yesmaximal, characteristic
3#22553243C212yesmaximal, complemented
4#22553238C212yesmaximal, complemented
5#22553208C212yesmaximal, complemented
6#22553175C212yesmaximal, complemented
7#22553172C211yesmaximal, characteristic, complemented
8#22553169C212yesmaximal, complemented
9C22×C4⋊C4×C131C211yesmaximal, characteristic, complemented
10C22×C22⋊D131C2211yescharacteristic
11C22×C4×D131C2211yescharacteristic
12#16862229—28no—
13C4×C22⋊D131—24no—
14C2×C4⋊Dic131C2212yes—
15C2×Dic131⋊C4C2214yes—
16C2×Dic131⋊C4C2211yescharacteristic
17C2×C22⋊Dic131C2212yes—
18C2×C22⋊Dic131C2211yescharacteristic
19C2×(C2×D131)⋊C4C2214yes—
20C2×(C2×D131)⋊C4C2211yescharacteristic
21C2×C4×Dic131C2211yescharacteristic
22C131⋊#95C2214yes—
23C131⋊#95C2212yes—
24C131⋊#95C2214yes—
25C23×C4×C131C2212yes—
26C23×C4×C131C2211yescharacteristic
27C2×C4⋊C4×C131—24no—
28C2×C4⋊C4×C131C2212yes—
29C2×C4⋊C4×C131C2212yes—
30C23×D131C2311yescharacteristic
31C2×C22⋊D131—22no—
32C2×C22⋊D131—22no—
33C2×C22⋊D131D414yes—
34C22×Dic131C2311yescharacteristic
35C22×Dic131C2311yescharacteristic
36C22×Dic131C2314yes—
37C22×Dic131C2311yescharacteristic
38C2×C4×D131—22no—
39C2×C4×D131—24no—
40C4⋊Dic131—24no—
41Dic131⋊C4—28no—
42Dic131⋊C4—22no—
43C22⋊Dic131—24no—
44C22⋊Dic131—22no—
45(C2×D131)⋊C4—28no—
46(C2×D131)⋊C4—22no—
47C4×Dic131—22no—
48C24×C131C2311yescharacteristic
49C22×C4×C131—24no—
50C22×C4×C131—24no—
51C22×C4×C131—24no—
52C22×C4×C131—24no—
53C22×C4×C131D414yes—
54C22×C4×C131C2314yes—
55C22×C4×C131C2311yescharacteristic
56C22×C4×C131C2311yescharacteristic
57C4⋊C4×C131—24no—
58C4⋊C4×C131—24no—
59C22×D131—21no—
60C22×D131—22no—
61C22×D131—21no—
62C22×D131—22no—
63C22×D131C2×D412yes—
64C22⋊D131—28no—
65C2×Dic131—21no—
66C2×Dic131—21no—
67C2×Dic131—21no—
68C2×Dic131—24no—
69C2×Dic131—21no—
70C2×Dic131—24no—
71C2×Dic131—22no—
72C2×Dic131—22no—
73C2×Dic131—24no—
74C2×Dic131C2×D412yes—
75C4×D131—44no—
76C23×C131—22no—
77C23×C131—22no—
78C23×C131C2×D412yes—
79C23×C131C2×D412yes—
80C23×C131#4112yes—
81C23×C131C2411yescharacteristic
82C2×C4×C131—22no—
83C2×C4×C131—28no—
84C2×C4×C131—24no—
85C2×C4×C131—24no—
86C2×C4×C131—24no—
87C2×C4×C131—48no—
88C2×C4×C131C2×D412yes—
89C2×C4×C131C2×D412yes—
90C2×C4×C131#4114yes—
91C22×C131—22no—
92C22×C131—22no—
93C22×C131—24no—
94C22×C131—24no—
95C22×C131#12114yes—
96C22×C131C22×D411yescharacteristic
97C22×C131C22×D411yescharacteristic
98C22×C131C2×#4111yescharacteristic
99C22×C131C2×#4112yes—
100C22×C1312−1+412yes—
101C2×D131—22no—
102C2×D131—24no—
103C2×D131—41no—
104C2×D131—41no—
105C2×D131—42no—
106C4×C131—22no—
107C4×C131—44no—
108C4×C131#12314yescomplemented
109Dic131—22no—
110Dic131—41no—
111Dic131—41no—
112Dic131—44no—
113C2×C131—24no—
114C2×C131C2×#12111yescharacteristic
115C2×C131C2×#12311yescharacteristic
116C2×C131#53611yescharacteristic
117C2×C131#54712yes—
118C2×C131#55212yes—
119D131—42no—
120C131#248611yescharacteristic, complemented, Sylow
121#2486—1311nomaximal, Sylow
122C2×#123—1312no—
123C2×C4×D4—1311no—
124C22×C4⋊C4D13111yescharacteristic, complemented
125#398—1312no—
126#397—1312no—
127#389—1312no—
128#381—1312no—
129#380—1311no—
130#380—1312no—
131C22×D4—1311no—
132C23×C4—1311no—
133C23×C4C2×D13112yes—
134C23×C4C2×D13111yescharacteristic
135#123—2628no—
136C4×D4—2624no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—1311no—
139C2×C4⋊C4—1314no—
140C2×C4⋊C4—1312no—
141C2×C4⋊C4C2×D13112yes—
142C2×C4⋊C4C2×D13112yes—
143C2×C22⋊C4—1311no—
144C2×C22⋊C4—1312no—
145C2×C22⋊C4—1311no—
146C2×C22⋊C4—1314no—
147C2×C42—1311no—
148#95—1314no—
149#95—1312no—
150#95—1314no—
151C24—1311no—
152C24C22×D13111yescharacteristic
153C2×D4—2622no—
154C2×D4—2622no—
155C2×D4—1314no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—2622no—
161C22×C4—2624no—
162C22×C4—1311no—
163C22×C4—1311no—
164C22×C4—1314no—
165C22×C4—1311no—
166C22×C4C22⋊D13114yes—
167C22×C4C22×D13114yes—
168C22×C4C22×D13111yescharacteristic
169C22×C4C22×D13111yescharacteristic
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—2628no—
173C4⋊C4—2622no—
174C4⋊C4—2624no—
175C22⋊C4—2628no—
176C22⋊C4—2624no—
177C22⋊C4—2622no—
178C22⋊C4—2622no—
179C42—2622no—
180C23—22no—
181C23—22no—
182C23—2621no—
183C23—2621no—
184C23—2622no—
185C23—2622no—
186C23—1312no—
187C23D4×D13112yes—
188C23C2×C22⋊D13112yes—
189C23#1452351212yes—
190C23C23×D13111yescharacteristic
191D4—2628no—
192C2×C4—22no—
193C2×C4—28no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—48no—
198C2×C4—2621no—
199C2×C4—2621no—
200C2×C4—2624no—
201C2×C4—2621no—
202C2×C4—2621no—
203C2×C4—2624no—
204C2×C4—2622no—
205C2×C4—2622no—
206C2×C4—2624no—
207C2×C4—5244no—
208C2×C4—1312no—
209C2×C4C2×C22⋊D13112yes—
210C2×C4C2×C22⋊D13112yes—
211C2×C4#1452351314yes—
212C22—22no—
213C22—24no—
214C22—24no—
215C22—22no—
216C22—2624no—
217C22—2622no—
218C22—5241no—
219C22—5242no—
220C22—5241no—
221C22#1686221714yes—
222C22C2×D4×D13111yescharacteristic
223C22C22×C22⋊D13111yescharacteristic
224C22C2×#1452351211yescharacteristic
225C22C2×#1452351312yes—
226C22#1686229412yes—
227C4—22no—
228C4—44no—
229C4—2622no—
230C4—5241no—
231C4—5244no—
232C4—5241no—
233C4#1686222914yescomplemented
234C2—24no—
235C2—5242no—
236C2C2×#1686221711yescharacteristic
237C2C2×#1686222911yescharacteristic
238C2#2255388611yescharacteristic
239C2#2255396312yes—
240C2#2255399712yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC22×C4⋊C4×C131C2
prime radicalC22×C22⋊D131C22
hypercenterC22×C4⋊C4D131
hypocenterC131#2486
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #22553243C2semidirect
4. #22553238C2semidirect
5. #22553208C2semidirect
6. #22553175C2semidirect
7. #22553172C2semidirect
8. #22553169C2semidirect
9. C22×C4⋊C4×C131C2semidirect
108. C4×C131#123semidirect
120. C131#2486semidirect
124. C22×C4⋊C4D131semidirect
233. C4#16862229semidirect

Maximal subgroups

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

subgroupindex
C2×#168622292
C2×C4×C22⋊D1312
#225532432
#225532382
#225532082
#225531752
#225531722
#225531692
C22×C4⋊C4×C1312
#2486131