Finite Group Databasebeta

← #30675321

Subgroups — #30675321

Order 16768.

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

normal subgroups only

This group has 762 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×#16862291C211yesmaximal, characteristic, complemented
2#22554112C212yesmaximal, complemented
3#22553879C211yesmaximal, characteristic, complemented
4#22553492C214yesmaximal, complemented
5#22553482C214yesmaximal, complemented
6#22553328C212yesmaximal, complemented
7C2×#100×C131C211yesmaximal, characteristic, complemented
8#16862291—24no—
9#16862291—22no—
10#16862291C2214yes—
11C2×#14523513C2212yes—
12C2×#14523513C2211yescharacteristic, complemented
13C2×#14523512—22no—
14C22×C22⋊D131—22no—
15C2×D4×D131—22no—
16#16862279—22no—
17#16862279C2214yescomplemented
18#16862274—22no—
19#16862274C2214yescomplemented
20#16862268—24no—
21#16862242C2212yescomplemented
22#16862233C2212yescomplemented
23#16862217—22no—
24#16862215—22no—
25#16862205C2211yescharacteristic, complemented
26#16862172C2218yescomplemented
27C22×D4×C131C2211yescharacteristic
28C2×M16×C131C2212yescomplemented
29#100×C131C2214yescomplemented
30C23×D131—22no—
31#14523513—22no—
32#14523513—22no—
33#14523513—22no—
34#14523513—22no—
35#14523513D414yescomplemented
36#14523513C2314yes—
37#14523512—28no—
38#14523512—28no—
39C2×Dic262D412yescomplemented
40C2×Dic262C2311yescharacteristic, complemented
41C2×D524D412yescomplemented
42C2×D524C2311yescharacteristic, complemented
43C2×C22⋊D131—24no—
44C2×C22⋊D131—28no—
45C2×C22⋊D131—22no—
46C2×C22⋊D131—28no—
47C2×C22⋊D131—21no—
48C2×C22⋊D131—24no—
49D4×D131—28no—
50D4×D131—28no—
51C22×Dic131—22no—
52C2×C4×D131—22no—
53C2×C4×D131—21no—
54Dic524—24no—
55#14523502—24no—
56#14523502—24no—
57D1048—24no—
58C4⋊Dic131—21no—
59C22⋊Dic131—21no—
60C22⋊Dic131—24no—
61C4×Dic131C2312yescomplemented
62C24×C131D412yes—
63C2×D4×C131—22no—
64C2×D4×C131—22no—
65C2×D4×C131D412yes—
66C2×D4×C131C2312yes—
67C22×C4×C131C2311yescharacteristic
68M16×C131—22no—
69M16×C131C2314yescomplemented
70C2×C8×C131—22no—
71C22×D131—22no—
72C22×D131—28no—
73C22×D131—21no—
74C22×D131—84no—
75Dic262—22no—
76Dic262—22no—
77Dic262C2×D414yes—
78D524—22no—
79D524—22no—
80D524C2×D414yes—
81C22⋊D131—216no—
82C22⋊D131—24no—
83C22⋊D131—24no—
84C22⋊D131—41no—
85C22⋊D131—41no—
86C22⋊D131—84no—
87C2×Dic131—28no—
88C2×Dic131—22no—
89C2×Dic131—22no—
90C2×Dic131—21no—
91C4×D131—24no—
92C4×D131—24no—
93C4×D131—42no—
94C23×C131—22no—
95C23×C131—22no—
96C23×C131—82no—
97C23×C131C2×D414yes—
98C23×C131C2×D411yescharacteristic
99D4×C131—24no—
100D4×C131—24no—
101C2×C4×C131C2×D412yes—
102C2×C4×C131C2×D412yes—
103C2×C4×C131C2×D411yescharacteristic
104C2×C4×C131C2411yescharacteristic
105C8×C131—24no—
106C22×C131—24no—
107C22×C131—41no—
108C22×C131—44no—
109C22×C131—44no—
110C22×C131—82no—
111C22×C131#12011yescharacteristic
112C22×C131#12011yescharacteristic
113C22×C131C22×D411yescharacteristic
114C2×D131—24no—
115C2×D131—44no—
116C2×D131—42no—
117C2×D131—41no—
118C2×D131—88no—
119C4×C131#12012yes—
120C4×C131C22×D412yes—
121Dic131—24no—
122Dic131—42no—
123Dic131—41no—
124C2×C131—21no—
125C2×C131—21no—
126C2×C131—21no—
127C2×C131—44no—
128C2×C131C2×#12011yescharacteristic
129D131—44no—
130D131—81no—
131C131#301911yescharacteristic, complemented, Sylow
132#3019—1311nomaximal, Sylow
133C2×H(5,𝔽2)—1311no—
134#575—1312no—
135#535—1311no—
136#455—1314no—
137#453—1314no—
138#421—1312no—
139C2×#100D13111yescharacteristic, complemented
140H(5,𝔽2)—2624no—
141H(5,𝔽2)—2622no—
142H(5,𝔽2)—1314no—
143C2×#41—2622no—
144C2×#41—1311no—
145C2×#41—1312no—
146C22×D4—2622no—
147C22×D4—2622no—
148C22×D4C2×D13111yescharacteristic
149#137—2622no—
150#137—1314no—
151AGL(1,ℤ/8)—2622no—
152AGL(1,ℤ/8)—1314no—
153#135—2624no—
154C2×M16C2×D13112yescomplemented
155#127—1312no—
156#124—1312no—
157#121—2622no—
158#120—2622no—
159#117—1311no—
160#104—1318no—
161#100C2×D13114yescomplemented
162C24—2622no—
163C24D52412yes—
164#41—2628no—
165#41—2628no—
166#41—2622no—
167#41—2622no—
168#41—2622no—
169#41—2622no—
170#41—1314no—
171#41—1314no—
172C2×Q8—1312no—
173C2×Q8—1311no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—2628no—
177C2×D4—2628no—
178C2×D4—2624no—
179C2×D4—2628no—
180C2×D4—2622no—
181C2×D4—2628no—
182C2×D4—2621no—
183C2×D4—2624no—
184C2×D4—1312no—
185C2×D4—1311no—
186C2×D4D52412yes—
187C2×D4C22×D13112yes—
188C22×C4—2622no—
189C22×C4—2622no—
190C22×C4—2621no—
191C22×C4C22×D13111yescharacteristic
192Q16—2624no—
193SD16—2624no—
194SD16—2624no—
195D8—2624no—
196M16—22no—
197M16C22×D13114yescomplemented
198C2×C8—22no—
199C4⋊C4—2621no—
200C22⋊C4—2621no—
201C22⋊C4—2624no—
202C42—1312no—
203C23—22no—
204C23—22no—
205C23—2628no—
206C23—2622no—
207C23—2621no—
208C23—82no—
209C23—10484no—
210C23C2×D52414yes—
211C23C2×D52411yescharacteristic
212Q8—2622no—
213Q8—2622no—
214Q8—1314no—
215D4—24no—
216D4—24no—
217D4—26216no—
218D4—2624no—
219D4—2624no—
220D4—2622no—
221D4—2622no—
222D4—5241no—
223D4—5241no—
224D4—10484no—
225D4—1314no—
226C2×C4—2628no—
227C2×C4—2622no—
228C2×C4—2624no—
229C2×C4—2624no—
230C2×C4—2622no—
231C2×C4—2621no—
232C2×C4—5242no—
233C2×C4D4×D13112yes—
234C2×C4D4×D13112yes—
235C2×C4C2×D52411yescharacteristic
236C2×C4C23×D13111yescharacteristic
237C8—24no—
238C22—24no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—2624no—
243C22—82no—
244C22—5244no—
245C22—5242no—
246C22—5241no—
247C22—10488no—
248C22#1686221611yescharacteristic
249C22#1686221611yescharacteristic
250C22C22×D52411yescharacteristic
251C4—2624no—
252C4—5242no—
253C4—5241no—
254C4#1686221612yes—
255C4C2×D4×D13112yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—5244no—
261C2—10481no—
262C2C2×#1686221611yescharacteristic
subgroupG/H
centerC2C2×#16862216
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC2×#100×C131C2
socleC2×C131C2×#120
hypercenterC2×#100D131
hypocenterC131#3019
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
1. C2×#16862291C2semidirect
2. #22554112C2semidirect
3. #22553879C2semidirect
4. #22553492C2semidirect
5. #22553482C2semidirect
6. #22553328C2semidirect
7. C2×#100×C131C2semidirect
12. C2×#14523513C22semidirect
17. #16862279C22semidirect
19. #16862274C22semidirect
21. #16862242C22semidirect
22. #16862233C22semidirect
25. #16862205C22semidirect
26. #16862172C22semidirect
28. C2×M16×C131C22semidirect
29. #100×C131C22semidirect
35. #14523513D4semidirect
39. C2×Dic262D4semidirect
40. C2×Dic262C23semidirect
41. C2×D524D4semidirect
42. C2×D524C23semidirect
61. C4×Dic131C23semidirect
69. M16×C131C23semidirect
131. C131#3019semidirect
139. C2×#100D131semidirect
154. C2×M16C2×D131semidirect
161. #100C2×D131semidirect
197. M16C22×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622912
#225541122
#225538792
#225534922
#225534822
#225533282
C2×#100×C1312
#3019131