Finite Group Databasebeta

← #30671136

Subgroups — #30671136

Order 16768.

584 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 151 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×#16862226C211yesmaximal, characteristic, complemented
2C2×(C22⋊D131)⋊C4C211yesmaximal, characteristic
3#22553250C211yesmaximal, characteristic, complemented
4#22553233C211yesmaximal, characteristic, complemented
5#22553224C211yesmaximal, characteristic, complemented
6#22553198C212yesmaximal, complemented
7#22553188C212yesmaximal
8#22553187C212yesmaximal, complemented
9#22553169C212yesmaximal, complemented
10#95×D131C211yesmaximal, characteristic, complemented
11#399×C131C211yesmaximal, characteristic, complemented
12C22×Dic262C2211yescharacteristic
13C22×C22⋊D131C2211yescharacteristic
14C23×Dic131C2211yescharacteristic
15C22×C4×D131C2212yes—
16C22×C4×D131C2211yescharacteristic
17#16862226—24no—
18(C22⋊D131)⋊C4—24no—
19C2×C4⋊Dic131C2212yes—
20C2×C4⋊Dic131C2212yes—
21C2×Dic131⋊C4C2211yescharacteristic
22C2×C22⋊Dic131C2212yes—
23C2×(C2×D131)⋊C4C2211yescharacteristic
24C2×(C2×D131)⋊C4C2212yes—
25C2×C4×Dic131C2211yescharacteristic
26C131⋊#95C2212yes—
27C131⋊#95C2211yescharacteristic
28C131⋊#95C2212yes—
29C131⋊#95C2212yes—
30C131⋊#95C2212yes—
31C131⋊#95C2212yes—
32C2×C4⋊C4×C131C2212yes—
33C2×C4⋊C4×C131C2211yescharacteristic
34C2×C22⋊C4×C131C2212yes—
35C2×C22⋊C4×C131C2211yescharacteristic
36#95×C131C2211yescharacteristic
37C23×D131C2311yescharacteristic
38C2×Dic262—22no—
39C2×Dic262—42no—
40C2×C22⋊D131—22no—
41C2×C22⋊D131—22no—
42C2×C22⋊D131Q814yescomplemented
43C22×Dic131—22no—
44C22×Dic131—42no—
45C22×Dic131C2312yes—
46C22×Dic131C2311yescharacteristic
47C22×Dic131C2312yes—
48C22×Dic131C2311yescharacteristic
49C22×Dic131C2311yescharacteristic
50C2×C4×D131—24no—
51C2×C4×D131—24no—
52C2×C4×D131—42no—
53C2×C4×D131—42no—
54C2×C4×D131D414yes—
55C4⋊Dic131—24no—
56C4⋊Dic131—42no—
57Dic131⋊C4—22no—
58C22⋊Dic131—42no—
59(C2×D131)⋊C4—24no—
60(C2×D131)⋊C4—22no—
61C4×Dic131—22no—
62C24×C131C2311yescharacteristic
63C22×C4×C131C2311yescharacteristic
64C22×C4×C131C2312yes—
65C22×C4×C131C2311yescharacteristic
66C22×C4×C131C2312yes—
67C4⋊C4×C131—22no—
68C4⋊C4×C131—24no—
69C22⋊C4×C131—22no—
70C22⋊C4×C131—24no—
71C22×D131—22no—
72C22×D131—22no—
73C22×D131C2×D412yes—
74C22×D131C2×Q812yes—
75C22×D131#4112yes—
76Dic262—44no—
77C22⋊D131—28no—
78C2×Dic131—21no—
79C2×Dic131—21no—
80C2×Dic131—22no—
81C2×Dic131—21no—
82C2×Dic131—22no—
83C2×Dic131—22no—
84C2×Dic131—22no—
85C2×Dic131—22no—
86C2×Dic131—22no—
87C2×Dic131—22no—
88C2×Dic131—22no—
89C2×Dic131—44no—
90C2×Dic131C2×D412yes—
91C2×Dic131C2×Q812yes—
92C4×D131—44no—
93C4×D131—48no—
94C23×C131—21no—
95C23×C131—22no—
96C23×C131—21no—
97C23×C131—22no—
98C23×C131C2×Q812yes—
99C23×C131C2411yescharacteristic
100C2×C4×C131—22no—
101C2×C4×C131—22no—
102C2×C4×C131—22no—
103C2×C4×C131—21no—
104C2×C4×C131—22no—
105C2×C4×C131—22no—
106C2×C4×C131—22no—
107C2×C4×C131—22no—
108C2×C4×C131C2×D412yes—
109C2×C4×C131#4114yes—
110C22×C131—22no—
111C22×C131—24no—
112C22×C131—41no—
113C22×C131—41no—
114C22×C131—42no—
115C22×C131C22×D411yescharacteristic
116C22×C131C22×Q811yescharacteristic
117C22×C131C2×#4111yescharacteristic
118C22×C131C2×#4112yes—
119C22×C131H(5,𝔽2)12yes—
120C2×D131—22no—
121C2×D131—22no—
122C2×D131—24no—
123C2×D131—24no—
124C2×D131C22⋊Q814yes—
125C4×C131—22no—
126C4×C131—24no—
127C4×C131—41no—
128C4×C131—42no—
129Dic131—22no—
130Dic131—22no—
131Dic131—41no—
132Dic131—42no—
133Dic131—42no—
134C2×C131—42no—
135C2×C131C2×C22⋊Q811yescharacteristic
136C2×C131#53211yescharacteristic
137C2×C131#53411yescharacteristic
138C2×C131#55112yes—
139C2×C131#55412yes—
140D131—24no—
141C131#256411yescharacteristic, complemented, Sylow
142#2564—1311nomaximal, Sylow
143C2×C22⋊Q8—1311no—
144C2×C4×D4—1311no—
145#400—1311no—
146#399D13111yescharacteristic, complemented
147#395—1311no—
148#393—1311no—
149#387—1312no—
150#386—1312no—
151#385—1312no—
152#380—1312no—
153C2×#95—1311no—
154C22×Q8—1311no—
155C22×D4—1311no—
156C23×C4—1312no—
157C23×C4—1311no—
158C23×C4—1311no—
159C22⋊Q8—2624no—
160C4×D4—2624no—
161C2×C4⋊C4—1311no—
162C2×C4⋊C4—1312no—
163C2×C4⋊C4—1312no—
164C2×C4⋊C4C2×D13111yescharacteristic
165C2×C4⋊C4C2×D13112yes—
166C2×C22⋊C4—1311no—
167C2×C22⋊C4—1312no—
168C2×C22⋊C4—1312no—
169C2×C22⋊C4C2×D13111yescharacteristic
170C2×C22⋊C4C2×D13112yes—
171C2×C42—1311no—
172#95—1312no—
173#95—1311no—
174#95—1312no—
175#95—1312no—
176#95—1312no—
177#95—1312no—
178#95C2×D13111yescharacteristic
179C24—1311no—
180C24C22×D13111yescharacteristic
181C2×Q8—2622no—
182C2×Q8—5242no—
183C2×D4—2622no—
184C2×D4—2622no—
185C2×D4—1314no—
186C22×C4—2624no—
187C22×C4—2622no—
188C22×C4—2624no—
189C22×C4—5242no—
190C22×C4—5242no—
191C22×C4—5242no—
192C22×C4—1312no—
193C22×C4—1312no—
194C22×C4—1311no—
195C22×C4—1311no—
196C22×C4—1311no—
197C22×C4—1314no—
198C22×C4C22×D13111yescharacteristic
199C22×C4C22×D13112yes—
200C22×C4C22×D13111yescharacteristic
201C22×C4C22×D13112yes—
202C4⋊C4—22no—
203C4⋊C4—24no—
204C4⋊C4—2624no—
205C4⋊C4—2622no—
206C4⋊C4—5242no—
207C22⋊C4—22no—
208C22⋊C4—24no—
209C22⋊C4—2622no—
210C22⋊C4—2624no—
211C22⋊C4—5242no—
212C42—2622no—
213C23—21no—
214C23—21no—
215C23—22no—
216C23—22no—
217C23—2622no—
218C23—2622no—
219C23—1312no—
220C23—1312no—
221C23—1312no—
222C23Q8×D13112yes—
223C23C23×D13111yescharacteristic
224Q8—5244no—
225D4—2628no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—2621no—
235C2×C4—2621no—
236C2×C4—2622no—
237C2×C4—2621no—
238C2×C4—2622no—
239C2×C4—2622no—
240C2×C4—2622no—
241C2×C4—2622no—
242C2×C4—2622no—
243C2×C4—2622no—
244C2×C4—2622no—
245C2×C4—5244no—
246C2×C4—5248no—
247C2×C4—5244no—
248C2×C4—1312no—
249C2×C4—1312no—
250C2×C4D4×D13112yes—
251C2×C4#1452351214yes—
252C22—24no—
253C22—22no—
254C22—41no—
255C22—41no—
256C22—42no—
257C22—2622no—
258C22—2622no—
259C22—2624no—
260C22—2624no—
261C22—1314no—
262C22C2×D4×D13111yescharacteristic
263C22C2×Q8×D13111yescharacteristic
264C22#41×D13111yescharacteristic
265C22C2×#1452351212yes—
266C22#1686229212yes—
267C4—22no—
268C4—24no—
269C4—41no—
270C4—42no—
271C4—2622no—
272C4—2622no—
273C4—5241no—
274C4—5242no—
275C4—5242no—
276C2—42no—
277C2—2624no—
278C2C22⋊Q8×D13111yescharacteristic
279C2#2255386011yescharacteristic
280C2#2255387311yescharacteristic
281C2#2255398512yes—
282C2#2255402012yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#399×C131C2
prime radicalC22×C22⋊D131C22
hypercenter#399D131
hypocenterC131#2564
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#16862226C2semidirect
3. #22553250C2semidirect
4. #22553233C2semidirect
5. #22553224C2semidirect
6. #22553198C2semidirect
8. #22553187C2semidirect
9. #22553169C2semidirect
10. #95×D131C2semidirect
11. #399×C131C2semidirect
42. C2×C22⋊D131Q8semidirect
141. C131#2564semidirect
146. #399D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622262
C2×(C22⋊D131)⋊C42
#225532502
#225532332
#225532242
#225531982
#225531882
#225531872
#225531692
#95×D1312
#399×C1312
#2564131