Finite Group Databasebeta

← #30674245

Subgroups — #30674245

Order 16768.

1068 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 201 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
1C23×C4×D131C211yesmaximal, characteristic
2C2×#16862222C212yesmaximal, complemented
3#22553184C212yesmaximal, complemented
4#22553183C214yesmaximal, complemented
5#22553172C211yesmaximal, characteristic, complemented
6#22553169C212yesmaximal, complemented
7#22553167C212yesmaximal
8C2×C22⋊Q8×C131C211yesmaximal, characteristic, complemented
9C24×D131C2211yescharacteristic
10C23×Dic131C2211yescharacteristic
11C22×C4×D131—28no—
12C22×C4×D131—24no—
13C22×C4×D131C2212yes—
14C22×C4×D131C2212yes—
15#16862222—28no—
16C2×C4⋊Dic131C2212yescomplemented
17C2×Dic131⋊C4C2214yescomplemented
18C2×C22⋊Dic131C2212yes—
19C2×(C2×D131)⋊C4C2214yes—
20C2×(C2×D131)⋊C4C2214yes—
21C131⋊#95C2214yescomplemented
22C131⋊#95C2212yescomplemented
23C22×Q8×C131C2211yescharacteristic, complemented
24C23×C4×C131C2211yescharacteristic
25C22⋊Q8×C131—24no—
26C2×C4⋊C4×C131C2211yescharacteristic, complemented
27C2×C4⋊C4×C131C2212yescomplemented
28C2×C22⋊C4×C131C2212yes—
29C23×D131—24no—
30C23×D131—24no—
31C23×D131—22no—
32C23×D131—22no—
33C23×D131Q814yescomplemented
34C23×D131C2312yes—
35C22×Dic131—24no—
36C22×Dic131—22no—
37C22×Dic131C2314yes—
38C22×Dic131C2312yes—
39C2×C4×D131—232no—
40C2×C4×D131—24no—
41C2×C4×D131—24no—
42C2×C4×D131—24no—
43C2×C4×D131—24no—
44C2×C4×D131—24no—
45C2×C4×D131D418yes—
46C4⋊Dic131—24no—
47Dic131⋊C4—28no—
48C22⋊Dic131—24no—
49(C2×D131)⋊C4—28no—
50(C2×D131)⋊C4—28no—
51C24×C131C2311yescharacteristic
52C2×Q8×C131—24no—
53C2×Q8×C131—22no—
54C22×C4×C131—24no—
55C22×C4×C131D414yes—
56C22×C4×C131C2312yes—
57C22×C4×C131C2312yes—
58C22×C4×C131C2311yescharacteristic
59C22×C4×C131C2311yescharacteristic
60C4⋊C4×C131—22no—
61C4⋊C4×C131—24no—
62C22⋊C4×C131—24no—
63C22×D131—28no—
64C22×D131—216no—
65C22×D131—28no—
66C22×D131—28no—
67C22×D131—22no—
68C22×D131—22no—
69C22×D131—22no—
70C22×D131—22no—
71C22×D131—216no—
72C22×D131C2×D414yes—
73C22×D131C2×Q814yes—
74C22×D131#4114yes—
75C2×Dic131—28no—
76C2×Dic131—22no—
77C2×Dic131—22no—
78C2×Dic131—24no—
79C2×Dic131—24no—
80C2×Dic131—24no—
81C2×Dic131C2×D414yes—
82C4×D131—216no—
83C4×D131—216no—
84C23×C131—22no—
85C23×C131—22no—
86C23×C131C2×D412yes—
87C23×C131C2×Q812yes—
88C23×C131#4112yes—
89C23×C131C2411yescharacteristic
90Q8×C131—44no—
91C2×C4×C131—22no—
92C2×C4×C131—28no—
93C2×C4×C131—22no—
94C2×C4×C131—22no—
95C2×C4×C131—24no—
96C2×C4×C131—24no—
97C2×C4×C131C2×D414yes—
98C2×C4×C131C2×D412yes—
99C2×C4×C131C2×D412yes—
100C22×C131—22no—
101C22×C131—24no—
102C22×C131—22no—
103C22×C131—24no—
104C22×C131C22⋊Q814yescomplemented
105C22×C131C22×D412yes—
106C22×C131C22×D411yescharacteristic
107C22×C131C22×Q811yescharacteristic
108C22×C131C2×#4112yes—
109C22×C131C2×#4111yescharacteristic
110C2×D131—232no—
111C2×D131—24no—
112C2×D131—24no—
113C2×D131—24no—
114C2×D131—24no—
115C2×D131—24no—
116C2×D131—24no—
117C2×D131C22⋊Q818yescomplemented
118C4×C131—22no—
119C4×C131—42no—
120C4×C131—42no—
121C4×C131#12014yes—
122Dic131—24no—
123Dic131—44no—
124C2×C131—24no—
125C2×C131C2×#12011yescharacteristic
126C2×C131C2×C22⋊Q812yes—
127C2×C131C2×C22⋊Q811yescharacteristic
128C2×C131#52511yescharacteristic
129C2×C131#52512yes—
130D131—28no—
131C131#284811yescharacteristic, complemented, Sylow
132#2848—1311nomaximal, Sylow
133C24×C4—1311no—
134C2×C22⋊Q8—1312no—
135C2×C22⋊Q8D13111yescharacteristic, complemented
136#385—1314no—
137#385—1312no—
138#380—1311no—
139#380—1312no—
140#379—1312no—
141C25—1311no—
142C22×Q8C2×D13111yescharacteristic, complemented
143C23×C4—2628no—
144C23×C4—2624no—
145C23×C4—1311no—
146C23×C4—1312no—
147C23×C4—1312no—
148C23×C4C2×D13111yescharacteristic
149C22⋊Q8—24no—
150C22⋊Q8—2628no—
151C2×C4⋊C4—1312no—
152C2×C4⋊C4—1314no—
153C2×C4⋊C4C2×D13111yescharacteristic, complemented
154C2×C4⋊C4C2×D13112yescomplemented
155C2×C22⋊C4—1312no—
156C2×C22⋊C4—1314no—
157C2×C22⋊C4—1314no—
158C2×C22⋊C4C2×D13112yes—
159#95—1312no—
160#95—1314no—
161C24—2624no—
162C24—2624no—
163C24—2622no—
164C24—2622no—
165C24—1314no—
166C24—1312no—
167C24C22×D13111yescharacteristic
168C2×Q8—24no—
169C2×Q8—22no—
170C22×C4—24no—
171C22×C4—26232no—
172C22×C4—2624no—
173C22×C4—2624no—
174C22×C4—2622no—
175C22×C4—2624no—
176C22×C4—2624no—
177C22×C4—2624no—
178C22×C4—2624no—
179C22×C4—1314no—
180C22×C4—1312no—
181C22×C4—1318no—
182C22×C4C22⋊D13114yes—
183C22×C4C22×D13112yes—
184C22×C4C22×D13112yes—
185C22×C4C22×D13111yescharacteristic
186C22×C4C22×D13111yescharacteristic
187C4⋊C4—22no—
188C4⋊C4—24no—
189C4⋊C4—2628no—
190C4⋊C4—2624no—
191C22⋊C4—24no—
192C22⋊C4—2628no—
193C22⋊C4—2628no—
194C22⋊C4—2624no—
195C23—22no—
196C23—22no—
197C23—2628no—
198C23—2628no—
199C23—26216no—
200C23—26216no—
201C23—2628no—
202C23—2622no—
203C23—2622no—
204C23—2622no—
205C23—2622no—
206C23—1314no—
207C23—1314no—
208C23—1314no—
209C23C2×C22⋊D13112yes—
210C23Q8×D13112yes—
211C23#1452351412yes—
212C23C23×D13111yescharacteristic
213Q8—44no—
214C2×C4—22no—
215C2×C4—28no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—26216no—
221C2×C4—2628no—
222C2×C4—2622no—
223C2×C4—2622no—
224C2×C4—26216no—
225C2×C4—2624no—
226C2×C4—2624no—
227C2×C4—2624no—
228C2×C4—1314no—
229C2×C4D4×D13114yes—
230C2×C4C2×C22⋊D13112yes—
231C2×C4C2×C22⋊D13112yes—
232C22—22no—
233C22—24no—
234C22—22no—
235C22—24no—
236C22—26232no—
237C22—2624no—
238C22—2624no—
239C22—2624no—
240C22—2624no—
241C22—2624no—
242C22—2624no—
243C22—1318no—
244C22#1686222214yescomplemented
245C22C2×D4×D13112yes—
246C22C22×C22⋊D13111yescharacteristic
247C22C2×Q8×D13111yescharacteristic
248C22#41×D13112yes—
249C22C2×#1452351411yescharacteristic
250C4—22no—
251C4—42no—
252C4—42no—
253C4—2624no—
254C4—5244no—
255C4#1686221414yes—
256C2—24no—
257C2—2628no—
258C2C2×#1686221411yescharacteristic
259C2C22⋊Q8×D13112yes—
260C2C2×#1686222211yescharacteristic
261C2#2255381911yescharacteristic
262C2#2255382512yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC2×C22⋊Q8×C131C2
prime radicalC24×D131C22
hypercenterC2×C22⋊Q8D131
hypocenterC131#2848
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16862222C2semidirect
3. #22553184C2semidirect
4. #22553183C2semidirect
5. #22553172C2semidirect
6. #22553169C2semidirect
8. C2×C22⋊Q8×C131C2semidirect
16. C2×C4⋊Dic131C22semidirect
17. C2×Dic131⋊C4C22semidirect
21. C131⋊#95C22semidirect
22. C131⋊#95C22semidirect
23. C22×Q8×C131C22semidirect
26. C2×C4⋊C4×C131C22semidirect
27. C2×C4⋊C4×C131C22semidirect
33. C23×D131Q8semidirect
104. C22×C131C22⋊Q8semidirect
117. C2×D131C22⋊Q8semidirect
131. C131#2848semidirect
135. C2×C22⋊Q8D131semidirect
142. C22×Q8C2×D131semidirect
153. C2×C4⋊C4C2×D131semidirect
154. C2×C4⋊C4C2×D131semidirect
244. C22#16862222semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D1312
C2×#168622222
#225531842
#225531832
#225531722
#225531692
#225531672
C2×C22⋊Q8×C1312
#2848131