Finite Group Databasebeta

← #30670429

Subgroups — #30670429

Order 16768.

708 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 169 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×#16862231C214yesmaximal, complemented
2C2×#16862221C211yesmaximal, characteristic
3C2×C4⋊C4×D131C211yesmaximal, characteristic, complemented
4#22553247C214yesmaximal, complemented
5#22553214C211yesmaximal, characteristic, complemented
6#22553213C211yesmaximal, characteristic, complemented
7#22553181C211yesmaximal, characteristic, complemented
8C4×(C2×D131)⋊C4C211yesmaximal, characteristic, complemented
9#384×C131C211yesmaximal, characteristic, complemented
10C22×D524C2211yescharacteristic
11C22×D524C2212yes—
12C22×C4×D131C2212yes—
13C22×C4×D131C2211yescharacteristic
14#16862231—216no—
15#16862221—24no—
16C2×C4⋊Dic131C2211yescharacteristic
17C2×Dic131⋊C4C2214yes—
18C2×Dic131⋊C4C2212yes—
19C4⋊C4×D131—24no—
20C2×(C2×D131)⋊C4C2214yes—
21C2×(C2×D131)⋊C4C2214yes—
22C2×(C2×D131)⋊C4C2212yes—
23C2×(C2×D131)⋊C4C2212yes—
24C2×C4×Dic131C2211yescharacteristic
25C131⋊#95C2212yes—
26C2×C4⋊C4×C131C2211yescharacteristic
27C2×C4⋊C4×C131C2211yescharacteristic
28C2×C4⋊C4×C131C2214yes—
29C2×C42×C131C2211yescharacteristic
30C23×D131C2312yes—
31C23×D131C2311yescharacteristic
32C2×D524—24no—
33C2×D524—28no—
34C2×D524—22no—
35C2×D524—22no—
36C2×D524—22no—
37C22×Dic131C2312yes—
38C22×Dic131C2311yescharacteristic
39C22×Dic131C2312yes—
40C2×C4×D131—28no—
41C2×C4×D131—24no—
42C2×C4×D131—22no—
43C2×C4×D131—22no—
44C2×C4×D131D414yes—
45C4⋊Dic131—22no—
46Dic131⋊C4—28no—
47Dic131⋊C4—24no—
48(C2×D131)⋊C4—24no—
49(C2×D131)⋊C4—28no—
50(C2×D131)⋊C4—28no—
51(C2×D131)⋊C4D418yescomplemented
52C4×Dic131—22no—
53C22×C4×C131C2312yes—
54C22×C4×C131C2312yes—
55C22×C4×C131C2312yes—
56C22×C4×C131C2311yescharacteristic
57C4⋊C4×C131—22no—
58C4⋊C4×C131—28no—
59C4⋊C4×C131—22no—
60C42×C131—22no—
61C22×D131—21no—
62C22×D131—21no—
63C22×D131—22no—
64C22×D131—24no—
65C22×D131—22no—
66C22×D131—22no—
67C22×D131—22no—
68C22×D131—24no—
69C22×D131C2×D414yes—
70C22×D131C2×D412yes—
71D524—48no—
72D524—44no—
73C2×Dic131—22no—
74C2×Dic131—21no—
75C2×Dic131—21no—
76C2×Dic131—22no—
77C2×Dic131—24no—
78C2×Dic131—22no—
79C2×Dic131C2×D414yes—
80C2×Dic131C2×D412yes—
81C4×D131—28no—
82C4×D131—48no—
83C23×C131C2411yescharacteristic
84C2×C4×C131—22no—
85C2×C4×C131—22no—
86C2×C4×C131—24no—
87C2×C4×C131—22no—
88C2×C4×C131—22no—
89C2×C4×C131—24no—
90C2×C4×C131C2×D414yes—
91C2×C4×C131C2×D412yes—
92C2×C4×C131#4112yes—
93C2×C4×C131#4112yes—
94C22×C131C22×D411yescharacteristic
95C22×C131C22×D412yes—
96C22×C131C2×#4111yescharacteristic
97C22×C131C2×#4111yescharacteristic
98C22×C1312−1+412yes—
99C2×D131—22no—
100C2×D131—22no—
101C2×D131—24no—
102C2×D131—24no—
103C2×D131—42no—
104C2×D131—42no—
105C2×D131—42no—
106C2×D131—44no—
107C2×D131—44no—
108C2×D131#12714yes—
109C4×C131—24no—
110C4×C131—42no—
111C4×C131—42no—
112C4×C131#12314yescomplemented
113Dic131—22no—
114Dic131—24no—
115Dic131—42no—
116C2×C131C2×#12311yescharacteristic
117C2×C131C2×#12711yescharacteristic
118C2×C131#53611yescharacteristic
119C2×C131#54712yes—
120C2×C131#54712yes—
121D131—24no—
122D131—82no—
123C131#250111yescharacteristic, complemented, Sylow
124#2501—1311nomaximal, Sylow
125C2×#123—1314no—
126C2×#121—1311no—
127C22×C4⋊C4—1311no—
128#399—1314no—
129#390—1311no—
130#390—1311no—
131#384—1311no—
132#384D13111yescharacteristic, complemented
133C4×C22⋊C4—1311no—
134C22×D4—1312no—
135C22×D4—1311no—
136C23×C4—1312no—
137C23×C4—1311no—
138#123—26216no—
139#121—2624no—
140C2×C4⋊C4—2624no—
141C2×C4⋊C4—1311no—
142C2×C4⋊C4—1312no—
143C2×C4⋊C4—1314no—
144C2×C4⋊C4C2×D13111yescharacteristic
145C2×C4⋊C4C2×D13111yescharacteristic
146C2×C4⋊C4C2×D13114yes—
147C2×C22⋊C4—1312no—
148C2×C22⋊C4—1314no—
149C2×C22⋊C4—1314no—
150C2×C22⋊C4—1312no—
151C2×C42—1311no—
152C2×C42C2×D13111yescharacteristic
153#95—1312no—
154C24—1312no—
155C24—1311no—
156C2×D4—2624no—
157C2×D4—2628no—
158C2×D4—2622no—
159C2×D4—2622no—
160C2×D4—2622no—
161C22×C4—2624no—
162C22×C4—2628no—
163C22×C4—2622no—
164C22×C4—2622no—
165C22×C4—1312no—
166C22×C4—1311no—
167C22×C4—1312no—
168C22×C4—1314no—
169C22×C4C22×D13112yes—
170C22×C4C22×D13112yes—
171C22×C4C22×D13112yes—
172C22×C4C22×D13111yescharacteristic
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—28no—
176C4⋊C4—2624no—
177C4⋊C4—2628no—
178C4⋊C4—2622no—
179C22⋊C4—2624no—
180C22⋊C4—2628no—
181C22⋊C4—2628no—
182C22⋊C4—1318no—
183C42—22no—
184C42—2622no—
185C23—2621no—
186C23—2621no—
187C23—2622no—
188C23—2624no—
189C23—2622no—
190C23—2622no—
191C23—2622no—
192C23—2624no—
193C23—1314no—
194C23—1312no—
195C23C23×D13111yescharacteristic
196D4—5248no—
197D4—5244no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—24no—
204C2×C4—2621no—
205C2×C4—2622no—
206C2×C4—2621no—
207C2×C4—2622no—
208C2×C4—2628no—
209C2×C4—2624no—
210C2×C4—2622no—
211C2×C4—5248no—
212C2×C4—1314no—
213C2×C4—1312no—
214C2×C4D4×D13114yes—
215C2×C4D4×D13112yes—
216C2×C4#1452351312yes—
217C2×C4#1452351412yes—
218C22—2622no—
219C22—2624no—
220C22—2624no—
221C22—2622no—
222C22—5242no—
223C22—5244no—
224C22—5244no—
225C22—5242no—
226C22—5242no—
227C22—1314no—
228C22C2×D4×D13111yescharacteristic
229C22C2×D4×D13112yes—
230C22C2×#1452351311yescharacteristic
231C22C2×#1452351411yescharacteristic
232C22#1686229412yes—
233C4—24no—
234C4—42no—
235C4—42no—
236C4—2624no—
237C4—2622no—
238C4—5242no—
239C4#1686223114yescomplemented
240C2—2624no—
241C2—10482no—
242C2C2×#1686223111yescharacteristic
243C2#127×D13111yescharacteristic
244C2#2255388711yescharacteristic
245C2#2255396312yes—
246C2#2255396512yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#384×C131C2
prime radicalC2×#16862221C2
hypercenter#384D131
hypocenterC131#2501
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
1. C2×#16862231C2semidirect
3. C2×C4⋊C4×D131C2semidirect
4. #22553247C2semidirect
5. #22553214C2semidirect
6. #22553213C2semidirect
7. #22553181C2semidirect
8. C4×(C2×D131)⋊C4C2semidirect
9. #384×C131C2semidirect
51. (C2×D131)⋊C4D4semidirect
112. C4×C131#123semidirect
123. C131#2501semidirect
132. #384D131semidirect
239. C4#16862231semidirect

Maximal subgroups

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

subgroupindex
C2×#168622312
C2×#168622212
C2×C4⋊C4×D1312
#225532472
#225532142
#225532132
#225531812
C4×(C2×D131)⋊C42
#384×C1312
#2501131