Finite Group Databasebeta

← #30675439

Subgroups — #30675439

Order 16768.

584 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 145 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×#16862271C211yesmaximal, characteristic, complemented
2C2×#16862242C211yesmaximal, characteristic, complemented
3#41×Dic131C211yesmaximal, characteristic, complemented
4#22553510C214yesmaximal, complemented
5#22553505C214yesmaximal, complemented
6#22553335C211yesmaximal, characteristic, complemented
7#22553318C211yesmaximal, characteristic, complemented
8#22553317C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C131C211yesmaximal, characteristic, complemented
10C22×D524C2211yescharacteristic
11C22×C22⋊D131—22no—
12#16862271—24no—
13C2×#14523500C2212yescomplemented
14C2×#14523498C2212yescomplemented
15C2×#14523497C2211yescharacteristic
16#16862242—21no—
17#16862242—21no—
18#16862242C2214yes—
19Q8×Dic131C2212yescomplemented
20D4×Dic131—22no—
21D4×Dic131C2212yescomplemented
22#16862205—21no—
23#16862205C2211yescharacteristic
24C2×C4×Dic131—21no—
25C2×C4×Dic131C2211yescharacteristic
26Dic131⋊C8C2214yescomplemented
27#16862168C2214yescomplemented
28#16862167C2214yescomplemented
29C2×#41×C131C2211yescharacteristic
30C22×D4×C131C2211yescharacteristic
31AGL(1,ℤ/8)×C131—24no—
32C2×SD16×C131C2212yescomplemented
33C2×D8×C131C2212yescomplemented
34C2×M16×C131C2211yescharacteristic
35C23×D131—21no—
36C2×D524—22no—
37C2×D524—21no—
38C2×D524—22no—
39C2×D524C2312yes—
40C2×C22⋊D131—24no—
41C2×C22⋊D131—24no—
42C2×C22⋊D131—48no—
43C22×Dic131—22no—
44C22×Dic131D412yes—
45#14523500—24no—
46#14523498—24no—
47#14523497—22no—
48C2×C131⋊C8C2312yes—
49C4⋊Dic131—22no—
50C4⋊Dic131C2312yes—
51C22⋊Dic131—22no—
52C22⋊Dic131—21no—
53C4×Dic131—22no—
54C4×Dic131—21no—
55C4×Dic131D411yescharacteristic
56C4×Dic131D411yescharacteristic
57C4×Dic131C2312yes—
58C24×C131—21no—
59#41×C131—22no—
60#41×C131D414yescomplemented
61C2×Q8×C131C2311yescharacteristic
62C2×D4×C131—21no—
63C2×D4×C131—22no—
64C2×D4×C131—21no—
65C2×D4×C131—22no—
66C2×D4×C131C2312yes—
67C2×D4×C131C2311yescharacteristic
68C22×C4×C131—21no—
69C22×C4×C131C2311yescharacteristic
70SD16×C131—24no—
71D8×C131—24no—
72M16×C131—22no—
73C2×C8×C131C2312yes—
74C22×D131—22no—
75C22×D131—44no—
76C22×D131—42no—
77D524—22no—
78D524—22no—
79D524—42no—
80C22⋊D131—48no—
81C22⋊D131—48no—
82C2×Dic131—24no—
83C2×Dic131—22no—
84C2×Dic131—44no—
85C2×Dic131C2×D412yes—
86C2×Dic131C2×D412yes—
87C131⋊C8—22no—
88C23×C131—21no—
89C23×C131—22no—
90C23×C131—44no—
91C23×C131—42no—
92C23×C131C2×D411yescharacteristic
93Q8×C131—21no—
94Q8×C131C2×D412yescomplemented
95D4×C131—24no—
96D4×C131—22no—
97D4×C131—21no—
98D4×C131—22no—
99D4×C131—42no—
100D4×C131C2×D412yescomplemented
101C2×C4×C131—21no—
102C2×C4×C131—24no—
103C2×C4×C131C2×D411yescharacteristic
104C2×C4×C131C2×D412yes—
105C2×C4×C131#4112yes—
106C2×C4×C131C2411yescharacteristic
107C8×C131—22no—
108C22×C131—22no—
109C22×C131—22no—
110C22×C131—42no—
111C22×C131—42no—
112C22×C131—44no—
113C22×C131—84no—
114C22×C131#12112yes—
115C22×C131C22×D411yescharacteristic
116C2×D131—42no—
117C2×D131—44no—
118C2×D131—84no—
119C4×C131—22no—
120C4×C131#12112yes—
121C4×C131C22×D411yescharacteristic
122C4×C131C2×#4111yescharacteristic
123Dic131—22no—
124Dic131—42no—
125Dic131AGL(1,ℤ/8)14yescomplemented
126C2×C131—22no—
127C2×C131—42no—
128C2×C131—82no—
129C2×C131C2×#12111yescharacteristic
130C2×C131C2×AGL(1,ℤ/8)12yes—
131D131—82no—
132C131#303811yescharacteristic, complemented, Sylow
133#3038—1311nomaximal, Sylow
134C2×AGL(1,ℤ/8)—1311no—
135C2×AGL(1,ℤ/8)D13111yescharacteristic, complemented
136C2×#127—1311no—
137C4×#41—1311no—
138#460—1314no—
139#459—1314no—
140#423—1311no—
141#418—1311no—
142#418—1311no—
143C2×#41C2×D13111yescharacteristic
144C22×D4—2622no—
145C22×D4—1311no—
146C22×D4C2×D13111yescharacteristic
147AGL(1,ℤ/8)—24no—
148AGL(1,ℤ/8)—2624no—
149C2×SD16—1312no—
150C2×SD16C2×D13112yescomplemented
151C2×D8—1312no—
152C2×D8C2×D13112yescomplemented
153C2×M16—1311no—
154C2×M16C2×D13111yescharacteristic
155#127—2621no—
156#127—2621no—
157#127—1314no—
158C4×Q8—1312no—
159C4×D4—2622no—
160C4×D4—1312no—
161#117—2621no—
162#117—1311no—
163C2×C42—2621no—
164C2×C42—1311no—
165C4⋊C8—1314no—
166#102—1314no—
167#102—1314no—
168C24—21no—
169C24—2621no—
170#41—22no—
171#41C22⋊D13114yescomplemented
172C2×Q8C22×D13111yescharacteristic
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—2624no—
178C2×D4—2624no—
179C2×D4—2621no—
180C2×D4—2622no—
181C2×D4—2622no—
182C2×D4—5248no—
183C2×D4—1312no—
184C2×D4C22×D13112yes—
185C2×D4C22×D13111yescharacteristic
186C22×C4—21no—
187C22×C4—2622no—
188C22×C4—1312no—
189C22×C4C22×D13111yescharacteristic
190SD16—24no—
191SD16—2624no—
192D8—24no—
193D8—2624no—
194M16—22no—
195M16—2622no—
196C2×C8—1312no—
197C2×C8C22×D13112yes—
198C4⋊C4—2622no—
199C4⋊C4—1312no—
200C22⋊C4—2621no—
201C22⋊C4—2622no—
202C42—2622no—
203C42—2621no—
204C42—1311no—
205C42—1311no—
206C42—1312no—
207C23—21no—
208C23—22no—
209C23—44no—
210C23—42no—
211C23—2622no—
212C23—5244no—
213C23—5242no—
214C23D4×D13111yescharacteristic
215Q8—21no—
216Q8C2×C22⋊D13112yescomplemented
217D4—24no—
218D4—22no—
219D4—21no—
220D4—22no—
221D4—42no—
222D4—2622no—
223D4—2622no—
224D4—5248no—
225D4—5248no—
226D4—5242no—
227D4C2×C22⋊D13112yescomplemented
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—2624no—
231C2×C4—2622no—
232C2×C4—5244no—
233C2×C4—1312no—
234C2×C4—1312no—
235C2×C4D4×D13111yescharacteristic
236C2×C4C2×C22⋊D13112yes—
237C2×C4#1452351212yes—
238C2×C4C23×D13111yescharacteristic
239C8—22no—
240C8—2622no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—42no—
245C22—44no—
246C22—84no—
247C22—5242no—
248C22—5244no—
249C22—10484no—
250C22#1686221812yes—
251C22C2×D4×D13111yescharacteristic
252C4—22no—
253C4—2622no—
254C4—5242no—
255C4—1314no—
256C4#1686221812yes—
257C4C22×C22⋊D13111yescharacteristic
258C4C2×#1452351211yescharacteristic
259C2—22no—
260C2—42no—
261C2—82no—
262C2—10482no—
263C2C2×#1686221811yescharacteristic
264C2AGL(1,ℤ/8)×D13112yes—
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC2×AGL(1,ℤ/8)×C131C2
socleC22×C131C22×D4
hypercenterC2×AGL(1,ℤ/8)D131
hypocenterC131#3038
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16862271C2semidirect
2. C2×#16862242C2semidirect
3. #41×Dic131C2semidirect
4. #22553510C2semidirect
5. #22553505C2semidirect
6. #22553335C2semidirect
7. #22553318C2semidirect
8. #22553317C2semidirect
9. C2×AGL(1,ℤ/8)×C131C2semidirect
13. C2×#14523500C22semidirect
14. C2×#14523498C22semidirect
19. Q8×Dic131C22semidirect
21. D4×Dic131C22semidirect
26. Dic131⋊C8C22semidirect
27. #16862168C22semidirect
28. #16862167C22semidirect
32. C2×SD16×C131C22semidirect
33. C2×D8×C131C22semidirect
60. #41×C131D4semidirect
94. Q8×C131C2×D4semidirect
100. D4×C131C2×D4semidirect
125. Dic131AGL(1,ℤ/8)semidirect
132. C131#3038semidirect
135. C2×AGL(1,ℤ/8)D131semidirect
150. C2×SD16C2×D131semidirect
152. C2×D8C2×D131semidirect
171. #41C22⋊D131semidirect
216. Q8C2×C22⋊D131semidirect
227. D4C2×C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622712
C2×#168622422
#41×Dic1312
#225535102
#225535052
#225533352
#225533182
#225533172
C2×AGL(1,ℤ/8)×C1312
#3038131