Finite Group Databasebeta

← #30675461

Subgroups — #30675461

Order 16768.

526 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16862272C211yesmaximal, characteristic, complemented
2#22553879C211yesmaximal, characteristic, complemented
3#22553787C211yesmaximal, characteristic, complemented
4#22553525C214yesmaximal, complemented
5#22553505C214yesmaximal, complemented
6#22553338C211yesmaximal, characteristic, complemented
7#22553317C211yesmaximal, characteristic, complemented
8#22553309C211yesmaximal, characteristic, complemented
9C22×D8×C131C211yesmaximal, characteristic, complemented
10C2×#14523513C2211yescharacteristic
11#16862272—24no—
12C2×#14523501C2212yescomplemented
13C2×#14523498C2212yescomplemented
14C2×#14523497C2211yescharacteristic
15#16862242C2212yescomplemented
16#16862233C2212yescomplemented
17#16862217—22no—
18#16862215—22no—
19D4×Dic131—22no—
20D4×Dic131C2214yescomplemented
21#16862205C2211yescharacteristic
22#16862205C2211yescharacteristic
23C2×C22⋊Dic131—22no—
24Dic131⋊C8C2214yescomplemented
25#16862171C2212yescomplemented
26#16862168C2212yescomplemented
27#16862167C2214yescomplemented
28C22×D4×C131C2211yescharacteristic
29C22×D4×C131C2211yescharacteristic
30C2×D8×C131—24no—
31C2×D8×C131C2214yes—
32C22×C8×C131C2211yescharacteristic
33#14523513—22no—
34#14523513—22no—
35C2×Dic262C2311yescharacteristic
36C2×D524C2311yescharacteristic
37C2×C22⋊D131—24no—
38C2×C22⋊D131—21no—
39C22×Dic131—24no—
40C2×C4×D131—21no—
41#14523501—24no—
42#14523498—24no—
43#14523497—22no—
44C2×C131⋊C8C2312yes—
45C4⋊Dic131D412yes—
46C4⋊Dic131C2312yes—
47C22⋊Dic131—24no—
48C22⋊Dic131—24no—
49C22⋊Dic131—21no—
50C22⋊Dic131D412yes—
51C4×Dic131—21no—
52C4×Dic131C2312yes—
53C24×C131—21no—
54C24×C131—21no—
55C2×D4×C131—21no—
56C2×D4×C131—22no—
57C2×D4×C131—21no—
58C2×D4×C131—22no—
59C2×D4×C131—22no—
60C2×D4×C131D414yescomplemented
61C2×D4×C131C2312yes—
62C2×D4×C131C2312yes—
63C22×C4×C131C2311yescharacteristic
64D8×C131—28no—
65C2×C8×C131—22no—
66C2×C8×C131C2312yes—
67C22×D131—21no—
68Dic262—21no—
69Dic262—21no—
70D524—21no—
71D524—21no—
72C22⋊D131—42no—
73C22⋊D131—84no—
74C2×Dic131—22no—
75C2×Dic131—21no—
76C2×Dic131—84no—
77C2×Dic131C2×D414yes—
78C4×D131—42no—
79C131⋊C8—22no—
80C23×C131—22no—
81C23×C131—24no—
82C23×C131—22no—
83C23×C131—42no—
84C23×C131—82no—
85C23×C131—82no—
86C23×C131C2×D411yescharacteristic
87D4×C131—24no—
88D4×C131—22no—
89D4×C131—22no—
90D4×C131—22no—
91D4×C131—42no—
92D4×C131C2×D414yes—
93C2×C4×C131C2×D411yescharacteristic
94C2×C4×C131C2×D412yes—
95C2×C4×C131#4112yes—
96C2×C4×C131C2411yescharacteristic
97C8×C131—22no—
98C22×C131—22no—
99C22×C131—24no—
100C22×C131—44no—
101C22×C131—42no—
102C22×C131—44no—
103C22×C131—84no—
104C22×C131—84no—
105C22×C131#12112yes—
106C22×C131C22×D411yescharacteristic
107C2×D131—41no—
108C2×D131—42no—
109C4×C131#12112yes—
110C4×C131C22×D411yescharacteristic
111C4×C131C2×#4111yescharacteristic
112Dic131—24no—
113Dic131—41no—
114Dic131—42no—
115C2×C131—22no—
116C2×C131—44no—
117C2×C131—82no—
118C2×C131C2×#12111yescharacteristic
119C2×C131#57612yes—
120D131—81no—
121C131#304111yescharacteristic, complemented, Sylow
122#3041—1311nomaximal, Sylow
123C2×AGL(1,ℤ/8)—1311no—
124C22×D8D13111yescharacteristic, complemented
125#535—1311no—
126#518—1311no—
127#463—1314no—
128#459—1314no—
129#424—1311no—
130#418—1311no—
131#416—1311no—
132C2×#41—1311no—
133C22×D4C2×D13111yescharacteristic
134C22×D4C2×D13111yescharacteristic
135AGL(1,ℤ/8)—2624no—
136C2×SD16—1312no—
137C2×D8—24no—
138C2×D8—1312no—
139C2×D8C2×D13114yes—
140C2×M16—1311no—
141C22×C8C2×D13111yescharacteristic
142#127—1312no—
143#124—1312no—
144#121—2622no—
145#120—2622no—
146C4×D4—2622no—
147C4×D4—1314no—
148#117—1311no—
149#117—1311no—
150C2×C22⋊C4—2622no—
151C4⋊C8—1314no—
152#103—1312no—
153#102—1314no—
154#102—1312no—
155C24—21no—
156C24—21no—
157#41—2622no—
158#41—2622no—
159C2×Q8—1311no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—2621no—
166C2×D4—2624no—
167C2×D4—1311no—
168C2×D4C22⋊D13114yescomplemented
169C2×D4C22×D13112yes—
170C2×D4C22×D13112yes—
171C22×C4—2621no—
172C22×C4—2624no—
173C22×C4C22×D13111yescharacteristic
174SD16—2624no—
175D8—28no—
176D8—2624no—
177M16—2622no—
178C2×C8—22no—
179C2×C8—1312no—
180C2×C8C22×D13112yes—
181C4⋊C4—1312no—
182C4⋊C4—1312no—
183C22⋊C4—2621no—
184C22⋊C4—2624no—
185C22⋊C4—2624no—
186C22⋊C4—1312no—
187C42—2621no—
188C42—1312no—
189C23—22no—
190C23—24no—
191C23—22no—
192C23—42no—
193C23—2621no—
194C23—82no—
195C23—82no—
196C23D4×D13111yescharacteristic
197Q8—2621no—
198Q8—2621no—
199D4—24no—
200D4—22no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—2621no—
205D4—2621no—
206D4—5242no—
207D4—10484no—
208D4C2×C22⋊D13114yes—
209C2×C4—2622no—
210C2×C4—2621no—
211C2×C4—5242no—
212C2×C4—10484no—
213C2×C4—1314no—
214C2×C4D4×D13111yescharacteristic
215C2×C4C2×C22⋊D13112yes—
216C2×C4#1452351212yes—
217C2×C4C23×D13111yescharacteristic
218C8—22no—
219C8—2622no—
220C22—22no—
221C22—24no—
222C22—44no—
223C22—44no—
224C22—42no—
225C22—84no—
226C22—84no—
227C22—5242no—
228C22—5241no—
229C22#1686221812yes—
230C22C2×D4×D13111yescharacteristic
231C4—2624no—
232C4—5242no—
233C4—5241no—
234C4#1686221812yes—
235C4C22×C22⋊D13111yescharacteristic
236C4C2×#1452351211yescharacteristic
237C2—22no—
238C2—44no—
239C2—82no—
240C2—10481no—
241C2C2×#1686221811yescharacteristic
242C2#2255411312yes—
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC22×D8×C131C2
socleC22×C131C22×D4
hypercenterC22×D8D131
hypocenterC131#3041
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16862272C2semidirect
2. #22553879C2semidirect
3. #22553787C2semidirect
4. #22553525C2semidirect
5. #22553505C2semidirect
6. #22553338C2semidirect
7. #22553317C2semidirect
8. #22553309C2semidirect
9. C22×D8×C131C2semidirect
12. C2×#14523501C22semidirect
13. C2×#14523498C22semidirect
15. #16862242C22semidirect
16. #16862233C22semidirect
20. D4×Dic131C22semidirect
24. Dic131⋊C8C22semidirect
25. #16862171C22semidirect
26. #16862168C22semidirect
27. #16862167C22semidirect
60. C2×D4×C131D4semidirect
121. C131#3041semidirect
124. C22×D8D131semidirect
168. C2×D4C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622722
#225538792
#225537872
#225535252
#225535052
#225533382
#225533172
#225533092
C22×D8×C1312
#3041131