Finite Group Databasebeta

← #30664464

Subgroups — #30664464

Order 16768.

334 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 65 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
1#22553825C211yesmaximal, characteristic
2#22553274C211yesmaximal, characteristic, complemented
3#95×D131C211yesmaximal, characteristic, complemented
4#22553049C211yesmaximal, characteristic, complemented
5#22553047C211yesmaximal, characteristic, complemented
6#22553045C211yesmaximal, characteristic, complemented
7#328×C131C211yesmaximal, characteristic, complemented
8C2×#14523514C411yescharacteristic, complemented
9C22×C4×D131—21no—
10C22×C4×D131C2211yescharacteristic
11C2×#14523496—21no—
12#16862231—21no—
13#16862222C411yescharacteristic, complemented
14#16862221C411yescharacteristic, complemented
15#16862220C2211yescharacteristic
16#16862216—21no—
17#16862208—21no—
18(C22⋊D131)⋊C4—21no—
19#16862204C411yescharacteristic, complemented
20C22⋊(C131⋊C8)C2211yescharacteristic
21(C2×D131)⋊C8—21no—
22C131⋊#95—21no—
23C131⋊#95C2211yescharacteristic
24C22⋊Q8×C131C2211yescharacteristic
25C22⋊C8×C131C2211yescharacteristic
26#95×C131C2211yescharacteristic
27C23×D131D411yescharacteristic
28#14523514—42no—
29C2×D524—21no—
30C2×D524—21no—
31C2×D524C2×C411yescharacteristic, complemented
32C2×C22⋊D131—21no—
33C22×Dic131—21no—
34C22×Dic131D411yescharacteristic
35C2×C4×D131—24no—
36C2×C4×D131—21no—
37C2×C4×D131—42no—
38C2×C4×D131—42no—
39C2×C4×D131—42no—
40C2×C4×D131C2×C411yescharacteristic
41C2×C4×D131C2×C411yescharacteristic
42C2×C4×D131D411yescharacteristic
43C2×C4×D131D411yescharacteristic
44#14523496—42no—
45C2×C131⋊C8—21no—
46C4⋊Dic131C2×C411yescharacteristic, complemented
47Dic131⋊C4—21no—
48(C2×D131)⋊C4—21no—
49(C2×D131)⋊C4—21no—
50(C2×D131)⋊C4—21no—
51(C2×D131)⋊C4—21no—
52C4×Dic131—21no—
53C2×Q8×C131C2×C411yescharacteristic, complemented
54C22×C4×C131—21no—
55C22×C4×C131C2311yescharacteristic
56C2×C8×C131—21no—
57C4⋊C4×C131—21no—
58C4⋊C4×C131C2×C411yescharacteristic, complemented
59C22⋊C4×C131—21no—
60C22×D131—21no—
61C22×D131—21no—
62C22×D131—22no—
63C22×D131—22no—
64C22×D131—21no—
65C22×D131C22⋊C411yescharacteristic
66C22×D131C22⋊C411yescharacteristic
67D524—41no—
68D524—42no—
69D524—42no—
70C22⋊D131—41no—
71C22⋊D131—41no—
72C2×Dic131—22no—
73C2×Dic131—21no—
74C2×Dic131—41no—
75C2×Dic131—41no—
76C2×Dic131—41no—
77C2×Dic131C22⋊C411yescharacteristic
78C2×Dic131C22⋊C411yescharacteristic
79C4×D131—22no—
80C4×D131—22no—
81C4×D131—41no—
82C4×D131—41no—
83C4×D131—42no—
84C4×D131—48no—
85C131⋊C8—41no—
86C23×C131C2×D411yescharacteristic
87Q8×C131—41no—
88C2×C4×C131—22no—
89C2×C4×C131—21no—
90C2×C4×C131—21no—
91C2×C4×C131—41no—
92C2×C4×C131—41no—
93C2×C4×C131—41no—
94C2×C4×C131C22×C411yescharacteristic
95C2×C4×C131C2×D411yescharacteristic
96C8×C131—41no—
97C22×C131—21no—
98C22×C131—21no—
99C22×C131C2×C22⋊C411yescharacteristic
100C22×C131AGL(1,ℤ/8)11yescharacteristic
101C22×C131#13711yescharacteristic
102C2×D131—24no—
103C2×D131—24no—
104C2×D131—21no—
105C2×D131—21no—
106C2×D131—21no—
107C2×D131—21no—
108C2×D131—41no—
109C2×D131—41no—
110C2×D131—41no—
111C2×D131#9912yes—
112C2×D131#10412yes—
113C4×C131—21no—
114C4×C131—21no—
115C4×C131—42no—
116C4×C131—41no—
117C4×C131—41no—
118Dic131—21no—
119Dic131—21no—
120Dic131—42no—
121Dic131—41no—
122C2×C131—21no—
123C2×C131—21no—
124C2×C131C2×#9911yescharacteristic
125C2×C131#41711yescharacteristic
126C2×C131C2×#10411yescharacteristic
127D131—22no—
128D131—22no—
129D131—81no—
130C131#150411yescharacteristic, complemented, Sylow
131#1504—1311nomaximal, Sylow
132#525—1311no—
133#407—1311no—
134C2×#95—1311no—
135#328—1311no—
136#328D13111yescharacteristic, complemented
137#327—1311no—
138#327—1311no—
139C2×#41—1311no—
140C23×C4—2621no—
141C23×C4—1311no—
142C2×M16—2621no—
143#123—2621no—
144C22⋊Q8—1311no—
145C22⋊Q8C2×D13111yescharacteristic
146#121—1311no—
147#121—1311no—
148#120—2621no—
149C4×D4—2621no—
150C4×D4—2621no—
151#117—1311no—
152C22⋊C8—2621no—
153C22⋊C8—1311no—
154C22⋊C8C2×D13111yescharacteristic
155#95—2621no—
156#95—1311no—
157#95C2×D13111yescharacteristic
158C24—1311no—
159#41—5242no—
160C2×Q8C4×D13111yescharacteristic, complemented
161C2×D4—2621no—
162C2×D4—2621no—
163C2×D4—2621no—
164C2×D4—1311no—
165C22×C4—21no—
166C22×C4—2624no—
167C22×C4—2621no—
168C22×C4—2621no—
169C22×C4—5242no—
170C22×C4—5242no—
171C22×C4—5242no—
172C22×C4—1311no—
173C22×C4—1311no—
174C22×C4—1311no—
175C22×C4—1311no—
176C22×C4—1311no—
177C22×C4C22×D13111yescharacteristic
178M16—5242no—
179C2×C8—21no—
180C2×C8—2621no—
181C4⋊C4—21no—
182C4⋊C4—2621no—
183C4⋊C4—1311no—
184C4⋊C4C4×D13111yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—2621no—
187C22⋊C4—2621no—
188C22⋊C4—2621no—
189C22⋊C4—2621no—
190C42—2621no—
191C23—2622no—
192C23—2622no—
193C23—2621no—
194C23—2621no—
195C23—2621no—
196C23—1311no—
197C23—1311no—
198C23D4×D13111yescharacteristic
199Q8—41no—
200D4—5241no—
201D4—5242no—
202D4—5242no—
203D4—5241no—
204D4—5241no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—2622no—
212C2×C4—2622no—
213C2×C4—2622no—
214C2×C4—2621no—
215C2×C4—5241no—
216C2×C4—5241no—
217C2×C4—5242no—
218C2×C4—5248no—
219C2×C4—5241no—
220C2×C4—5241no—
221C2×C4—5241no—
222C2×C4—1311no—
223C2×C4—1311no—
224C2×C4C2×C4×D13111yescharacteristic
225C2×C4D4×D13111yescharacteristic
226C8—41no—
227C8—5241no—
228C22—21no—
229C22—21no—
230C22—2624no—
231C22—2624no—
232C22—2621no—
233C22—2621no—
234C22—2621no—
235C22—2621no—
236C22—5241no—
237C22—5241no—
238C22—5241no—
239C22—1312no—
240C22—1312no—
241C22C22⋊C4×D13111yescharacteristic
242C22#1686227311yescharacteristic
243C22#1686227811yescharacteristic
244C4—21no—
245C4—21no—
246C4—42no—
247C4—41no—
248C4—41no—
249C4—2621no—
250C4—2621no—
251C4—5242no—
252C4—5241no—
253C2—21no—
254C2—21no—
255C2—2622no—
256C2—2622no—
257C2—10481no—
258C2#99×D13111yescharacteristic
259C2#2255331211yescharacteristic
260C2#104×D13111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D131
commutatorC2×C4×C131C22×C4
FrattiniC22×C4C22×D131
Fitting#328×C131C2
socleC22×C131C2×C22⋊C4
prime radical#22553825C2
cosocleC22×C4×C131C23
hypercenter#328D131
hypocenterC131#1504
LCSG ▷ C2×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #22553274C2semidirect
3. #95×D131C2semidirect
4. #22553049C2semidirect
5. #22553047C2semidirect
6. #22553045C2semidirect
7. #328×C131C2semidirect
8. C2×#14523514C4semidirect
13. #16862222C4semidirect
14. #16862221C4semidirect
19. #16862204C4semidirect
31. C2×D524C2×C4semidirect
46. C4⋊Dic131C2×C4semidirect
53. C2×Q8×C131C2×C4semidirect
58. C4⋊C4×C131C2×C4semidirect
130. C131#1504semidirect
136. #328D131semidirect
160. C2×Q8C4×D131semidirect
184. C4⋊C4C4×D131semidirect

Maximal subgroups

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

subgroupindex
#225538252
#225532742
#95×D1312
#225530492
#225530472
#225530452
#328×C1312
#1504131