Finite Group Databasebeta

← #30670986

Subgroups — #30670986

Order 16768.

556 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 181 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×#16862245C212yesmaximal
2C2×#16862244C212yesmaximal
3#22553233C212yesmaximal
4#22553232C211yesmaximal, characteristic
5#22553209C212yesmaximal
6#22553207C212yesmaximal
7#22553181C211yesmaximal, characteristic
8#22553163C212yesmaximal
9#395×C131C211yesmaximal, characteristic
10C22×Dic262C2211yescharacteristic
11C22×Dic262C2212yes—
12#16862245—28no—
13#16862244—28no—
14C2×C4⋊Dic131C2212yes—
15C2×C4⋊Dic131C2211yescharacteristic
16C2×Dic131⋊C4C2212yes—
17C2×Dic131⋊C4C2212yes—
18C2×Dic131⋊C4C2212yes—
19C2×Dic131⋊C4C2212yes—
20C2×Dic131⋊C4C2212yes—
21C2×Dic131⋊C4C2212yes—
22C2×C4×Dic131C2212yes—
23C2×C4×Dic131C2212yes—
24C2×C4×Dic131C2211yescharacteristic
25C131⋊#95C2212yes—
26C131⋊#95C2211yescharacteristic
27C131⋊#95C2212yes—
28C22×Q8×C131C2211yescharacteristic
29C2×C4⋊C4×C131C2212yes—
30C2×C4⋊C4×C131C2211yescharacteristic
31#95×C131C2212yes—
32#95×C131C2211yescharacteristic
33C2×Dic262—24no—
34C2×Dic262—24no—
35C2×Dic262—24no—
36C2×Dic262—42no—
37C2×Dic262—41no—
38C22×Dic131C2312yes—
39C22×Dic131C2312yes—
40C22×Dic131C2312yes—
41C22×Dic131C2312yes—
42C4⋊Dic131—24no—
43C4⋊Dic131—41no—
44Dic131⋊C4—24no—
45Dic131⋊C4—24no—
46Dic131⋊C4—24no—
47Dic131⋊C4—24no—
48Dic131⋊C4D418yes—
49Dic131⋊C4Q818yescomplemented
50C4×Dic131—24no—
51C4×Dic131—24no—
52C4×Dic131Q814yescomplemented
53C2×Q8×C131—24no—
54C2×Q8×C131—41no—
55C22×C4×C131C2311yescharacteristic
56C22×C4×C131C2311yescharacteristic
57C22×C4×C131C2312yes—
58C22×C4×C131C2311yescharacteristic
59C22×C4×C131C2312yes—
60C4⋊C4×C131—24no—
61C4⋊C4×C131—41no—
62Dic262—44no—
63Dic262—48no—
64C2×Dic131—22no—
65C2×Dic131—22no—
66C2×Dic131—22no—
67C2×Dic131—22no—
68C2×Dic131—22no—
69C2×Dic131—22no—
70C2×Dic131—22no—
71C2×Dic131C2×D414yes—
72C2×Dic131C2×D414yes—
73C2×Dic131C2×Q814yes—
74C2×Dic131C2×Q814yes—
75C2×Dic131C2×Q814yes—
76C23×C131C2411yescharacteristic
77Q8×C131—44no—
78C2×C4×C131—22no—
79C2×C4×C131—22no—
80C2×C4×C131—21no—
81C2×C4×C131—22no—
82C2×C4×C131—21no—
83C2×C4×C131—22no—
84C2×C4×C131—22no—
85C2×C4×C131—22no—
86C2×C4×C131—21no—
87C2×C4×C131—21no—
88C2×C4×C131C2×D414yes—
89C2×C4×C131C2×Q812yes—
90C2×C4×C131C2×Q814yes—
91C22×C131C22×D412yes—
92C22×C131C22×Q811yescharacteristic
93C22×C131C22×Q812yes—
94C22×C131H(5,𝔽2)11yescharacteristic
95C22×C1312−1+411yescharacteristic
96C4×C131—22no—
97C4×C131—24no—
98C4×C131—24no—
99C4×C131—41no—
100C4×C131—41no—
101Dic131—24no—
102Dic131—24no—
103Dic131—42no—
104Dic131C4⋊Q818yescomplemented
105C2×C131C2×C4⋊Q812yes—
106C2×C131#54411yescharacteristic
107C2×C131D4211yescharacteristic
108C2×C131D4×Q812yes—
109C2×C131Q8211yescharacteristic
110C131#255111yescharacteristic, complemented, Sylow
111#2551—1311nomaximal, Sylow
112C2×C4⋊Q8—1312no—
113C2×C4⋊Q8—1312no—
114#395—1311no—
115#395—1312no—
116#395D13111yescharacteristic
117#389—1312no—
118#389—1312no—
119#384—1311no—
120C4×C4⋊C4—1312no—
121C22×Q8—1312no—
122C22×Q8—1311no—
123C22×Q8C2×D13111yescharacteristic
124C4⋊Q8—2628no—
125C4⋊Q8—2628no—
126C2×C4⋊C4—1312no—
127C2×C4⋊C4—1312no—
128C2×C4⋊C4—1312no—
129C2×C4⋊C4—1312no—
130C2×C4⋊C4—1312no—
131C2×C4⋊C4—1311no—
132C2×C4⋊C4—1312no—
133C2×C4⋊C4—1312no—
134C2×C4⋊C4C2×D13112yes—
135C2×C4⋊C4C2×D13111yescharacteristic
136C2×C42—1312no—
137C2×C42—1312no—
138C2×C42—1311no—
139#95—1312no—
140#95—1312no—
141#95—1311no—
142#95C2×D13112yes—
143#95C2×D13111yescharacteristic
144C2×Q8—24no—
145C2×Q8—41no—
146C2×Q8—2624no—
147C2×Q8—2624no—
148C2×Q8—2624no—
149C2×Q8—5242no—
150C2×Q8—5241no—
151C22×C4—1312no—
152C22×C4—1312no—
153C22×C4—1312no—
154C22×C4—1312no—
155C22×C4C22×D13111yescharacteristic
156C22×C4C22×D13111yescharacteristic
157C22×C4C22×D13112yes—
158C22×C4C22×D13111yescharacteristic
159C22×C4C22×D13112yes—
160C4⋊C4—24no—
161C4⋊C4—41no—
162C4⋊C4—2624no—
163C4⋊C4—2624no—
164C4⋊C4—2624no—
165C4⋊C4—2624no—
166C4⋊C4—2624no—
167C4⋊C4—5241no—
168C4⋊C4—1318no—
169C4⋊C4—1318no—
170C42—2624no—
171C42—2624no—
172C42—1314no—
173C23C23×D13111yescharacteristic
174Q8—44no—
175Q8—5248no—
176Q8—5244no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—2622no—
188C2×C4—2622no—
189C2×C4—2622no—
190C2×C4—2622no—
191C2×C4—2622no—
192C2×C4—2622no—
193C2×C4—2622no—
194C2×C4—1314no—
195C2×C4—1314no—
196C2×C4—1314no—
197C2×C4—1314no—
198C2×C4—1314no—
199C2×C4D4×D13114yes—
200C2×C4Q8×D13112yes—
201C2×C4Q8×D13114yes—
202C22C2×D4×D13112yes—
203C22C2×Q8×D13111yescharacteristic
204C22C2×Q8×D13112yes—
205C22#1686229111yescharacteristic
206C22#1686229311yescharacteristic
207C4—22no—
208C4—24no—
209C4—24no—
210C4—41no—
211C4—41no—
212C4—2624no—
213C4—2624no—
214C4—5242no—
215C4—1318no—
216C2C4⋊Q8×D13112yes—
217C2#2255394111yescharacteristic
218C2#2255394411yescharacteristic
219C2#2255397512yes—
220C2#2255404211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, prime radical, cosocleC23×C131C24
Fitting#395×C131C2
hypercenter#395D131
hypocenterC131#2551
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
49. Dic131⋊C4Q8semidirect
52. C4×Dic131Q8semidirect
104. Dic131C4⋊Q8semidirect
110. C131#2551semidirect

Maximal subgroups

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

subgroupindex
C2×#168622452
C2×#168622442
#225532332
#225532322
#225532092
#225532072
#225531812
#225531632
#395×C1312
#2551131