Finite Group Databasebeta

← #30670912

Subgroups — #30670912

Order 16768.

676 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 185 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×#16862245C211yesmaximal, characteristic, complemented
2C2×#16862223C214yesmaximal, complemented
3C2×D4×Dic131C211yesmaximal, characteristic
4#22553248C214yesmaximal, complemented
5#22553215C211yesmaximal, characteristic
6#22553182C211yesmaximal, characteristic, complemented
7#22553181C211yesmaximal, characteristic, complemented
8C4×Dic131⋊C4C211yesmaximal, characteristic, complemented
9#390×C131C211yesmaximal, characteristic
10C22×Dic262C2212yes—
11C23×Dic131C2212yes—
12#16862245—24no—
13#16862223—216no—
14D4×Dic131—24no—
15C2×C4⋊Dic131C2211yescharacteristic
16C2×C4⋊Dic131C2211yescharacteristic
17C2×C4⋊Dic131C2214yes—
18C2×Dic131⋊C4C2214yes—
19C2×Dic131⋊C4C2212yes—
20C2×Dic131⋊C4C2212yes—
21C2×C22⋊Dic131C2214yes—
22C2×C22⋊Dic131C2212yes—
23C2×C4×Dic131C2211yescharacteristic
24C2×C4×Dic131C2211yescharacteristic
25C131⋊#95C2212yes—
26C22×D4×C131C2211yescharacteristic
27C2×C4⋊C4×C131C2211yescharacteristic
28C2×C22⋊C4×C131C2214yes—
29C2×C42×C131C2211yescharacteristic
30C2×Dic262—28no—
31C2×Dic262—24no—
32C22×Dic131—24no—
33C22×Dic131—28no—
34C22×Dic131C2312yes—
35C22×Dic131C2312yes—
36C22×Dic131C2311yescharacteristic
37C22×Dic131C2312yes—
38C22×Dic131C2311yescharacteristic
39C4⋊Dic131—22no—
40C4⋊Dic131—28no—
41C4⋊Dic131—22no—
42Dic131⋊C4—28no—
43Dic131⋊C4—24no—
44Dic131⋊C4D418yescomplemented
45C22⋊Dic131—24no—
46C22⋊Dic131—28no—
47C4×Dic131—22no—
48C4×Dic131D414yes—
49C24×C131C2312yes—
50C2×D4×C131—22no—
51C2×D4×C131—22no—
52C2×D4×C131Q814yescomplemented
53C22×C4×C131C2312yes—
54C22×C4×C131C2312yes—
55C22×C4×C131C2311yescharacteristic
56C4⋊C4×C131—22no—
57C22⋊C4×C131—28no—
58C42×C131—22no—
59Dic262—48no—
60C2×Dic131—22no—
61C2×Dic131—22no—
62C2×Dic131—22no—
63C2×Dic131—22no—
64C2×Dic131—24no—
65C2×Dic131—24no—
66C2×Dic131—22no—
67C2×Dic131—48no—
68C2×Dic131C2×D412yes—
69C2×Dic131C2×D414yes—
70C2×Dic131C2×D412yes—
71C2×Dic131C2×D414yes—
72C23×C131—24no—
73C23×C131—22no—
74C23×C131—22no—
75C23×C131—24no—
76C23×C131C2×Q814yes—
77C23×C131C2411yescharacteristic
78D4×C131—28no—
79C2×C4×C131—22no—
80C2×C4×C131—22no—
81C2×C4×C131—24no—
82C2×C4×C131—22no—
83C2×C4×C131C2×D414yes—
84C2×C4×C131C2×D412yes—
85C2×C4×C131C2×Q812yes—
86C2×C4×C131#4112yes—
87C22×C131—24no—
88C22×C131—28no—
89C22×C131—42no—
90C22×C131—42no—
91C22×C131—44no—
92C22×C131C22×D411yescharacteristic
93C22×C131C22×D412yes—
94C22×C131C22×Q811yescharacteristic
95C22×C131C2×#4111yescharacteristic
96C22×C1312−1+412yes—
97C4×C131—24no—
98C4×C131—42no—
99C4×C131C22⋊Q814yescomplemented
100Dic131—22no—
101Dic131—24no—
102Dic131—42no—
103Dic131—42no—
104Dic131#12714yes—
105C2×C131—44no—
106C2×C131C2×C22⋊Q811yescharacteristic
107C2×C131C2×#12711yescharacteristic
108C2×C131#53611yescharacteristic
109C2×C131#54712yes—
110C2×C131D4×Q812yes—
111C131#254211yescharacteristic, complemented, Sylow
112#2542—1311nomaximal, Sylow
113C2×C4⋊Q8—1311no—
114C2×C22⋊Q8—1314no—
115C2×C4×D4—1311no—
116#399—1314no—
117#390—1311no—
118#390D13111yescharacteristic
119#384—1311no—
120#384—1311no—
121C4×C4⋊C4—1311no—
122C22×Q8—1312no—
123C22×D4C2×D13111yescharacteristic
124C23×C4—1312no—
125C4⋊Q8—2624no—
126C22⋊Q8—26216no—
127C4×D4—2624no—
128C2×C4⋊C4—1311no—
129C2×C4⋊C4—1314no—
130C2×C4⋊C4—1314no—
131C2×C4⋊C4—1311no—
132C2×C4⋊C4—1312no—
133C2×C4⋊C4—1312no—
134C2×C4⋊C4C2×D13111yescharacteristic
135C2×C22⋊C4—1312no—
136C2×C22⋊C4—1314no—
137C2×C22⋊C4C2×D13114yes—
138C2×C42—1311no—
139C2×C42—1311no—
140C2×C42C2×D13111yescharacteristic
141#95—1312no—
142C24C22×D13112yes—
143C2×Q8—2628no—
144C2×Q8—2624no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4Dic26214yescomplemented
148C22×C4—2624no—
149C22×C4—2628no—
150C22×C4—1312no—
151C22×C4—1312no—
152C22×C4—1311no—
153C22×C4—1312no—
154C22×C4—1311no—
155C22×C4C22×D13112yes—
156C22×C4C22×D13112yes—
157C22×C4C22×D13111yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—2628no—
160C4⋊C4—2624no—
161C4⋊C4—2622no—
162C4⋊C4—2628no—
163C4⋊C4—2622no—
164C4⋊C4—1318no—
165C22⋊C4—28no—
166C22⋊C4—2624no—
167C22⋊C4—2628no—
168C42—22no—
169C42—2622no—
170C42—1314no—
171C23—24no—
172C23—22no—
173C23—22no—
174C23—24no—
175C23C2×Dic26214yes—
176C23C23×D13111yescharacteristic
177Q8—5248no—
178D4—28no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—22no—
183C2×C4—2622no—
184C2×C4—2622no—
185C2×C4—2622no—
186C2×C4—2622no—
187C2×C4—2624no—
188C2×C4—2624no—
189C2×C4—2622no—
190C2×C4—5248no—
191C2×C4—1312no—
192C2×C4—1314no—
193C2×C4—1312no—
194C2×C4—1314no—
195C2×C4D4×D13114yes—
196C2×C4D4×D13112yes—
197C2×C4C2×Dic26212yes—
198C2×C4#1452351212yes—
199C22—28no—
200C22—24no—
201C22—42no—
202C22—44no—
203C22—42no—
204C22C2×D4×D13111yescharacteristic
205C22C2×D4×D13112yes—
206C22C22×Dic26211yescharacteristic
207C22C2×#1452351211yescharacteristic
208C22#1686229312yes—
209C4—24no—
210C4—42no—
211C4—2622no—
212C4—2624no—
213C4—5242no—
214C4—5242no—
215C4—1314no—
216C4#1686222314yescomplemented
217C2—44no—
218C2C2×#1686222311yescharacteristic
219C2#127×D13111yescharacteristic
220C2#2255388211yescharacteristic
221C2#2255396212yes—
222C2D4×Dic26212yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#390×C131C2
prime radicalC22×D4×C131C22
hypercenter#390D131
hypocenterC131#2542
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#16862245C2semidirect
2. C2×#16862223C2semidirect
4. #22553248C2semidirect
6. #22553182C2semidirect
7. #22553181C2semidirect
8. C4×Dic131⋊C4C2semidirect
44. Dic131⋊C4D4semidirect
52. C2×D4×C131Q8semidirect
99. C4×C131C22⋊Q8semidirect
111. C131#2542semidirect
147. C2×D4Dic262semidirect
216. C4#16862223semidirect

Maximal subgroups

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

subgroupindex
C2×#168622452
C2×#168622232
C2×D4×Dic1312
#225532482
#225532152
#225531822
#225531812
C4×Dic131⋊C42
#390×C1312
#2542131