Finite Group Databasebeta

← #30670947

Subgroups — #30670947

Order 16768.

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

normal subgroups only

This group has 540 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×#16862244C211yesmaximal, characteristic
2C2×#16862244C211yesmaximal, characteristic
3C2×Q8×Dic131C211yesmaximal, characteristic
4#22553245C214yesmaximal
5#22553219C211yesmaximal, characteristic
6#22553218C211yesmaximal, characteristic
7#22553207C214yesmaximal
8C4×Dic131⋊C4C211yesmaximal, characteristic
9C2×C4⋊Q8×C131C211yesmaximal, characteristic
10C22×Dic262C2211yescharacteristic
11#16862244—24no—
12#16862244—24no—
13Q8×Dic131—24no—
14C2×C4⋊Dic131C2211yescharacteristic
15C2×C4⋊Dic131C2212yes—
16C2×Dic131⋊C4C2212yes—
17C2×Dic131⋊C4C2214yes—
18C2×Dic131⋊C4C2212yes—
19C2×Dic131⋊C4C2212yes—
20C2×C4×Dic131C2211yescharacteristic
21C2×C4×Dic131C2212yes—
22C2×C4×Dic131C2211yescharacteristic
23C131⋊#95C2212yes—
24C131⋊#95C2214yes—
25C131⋊#95C2214yes—
26C22×Q8×C131C2211yescharacteristic
27C22×Q8×C131C2211yescharacteristic
28C4⋊Q8×C131—24no—
29C2×C4⋊C4×C131C2214yes—
30C2×C42×C131C2211yescharacteristic
31C2×Dic262—22no—
32C2×Dic262—22no—
33C2×Dic262—22no—
34C22×Dic131C2312yes—
35C22×Dic131C2312yes—
36C22×Dic131C2311yescharacteristic
37C22×Dic131C2312yes—
38C22×Dic131C2311yescharacteristic
39C4⋊Dic131—24no—
40C4⋊Dic131—22no—
41Dic131⋊C4—24no—
42Dic131⋊C4—24no—
43Dic131⋊C4—28no—
44Dic131⋊C4Q818yescomplemented
45C4×Dic131—24no—
46C4×Dic131—22no—
47C4×Dic131D414yes—
48C2×Q8×C131—22no—
49C2×Q8×C131—22no—
50C2×Q8×C131—22no—
51C2×Q8×C131—22no—
52C2×Q8×C131—22no—
53C2×Q8×C131D414yes—
54C22×C4×C131C2312yes—
55C22×C4×C131C2312yes—
56C22×C4×C131C2312yes—
57C22×C4×C131C2311yescharacteristic
58C4⋊C4×C131—28no—
59C42×C131—22no—
60Dic262—44no—
61C2×Dic131—22no—
62C2×Dic131—22no—
63C2×Dic131—22no—
64C2×Dic131—22no—
65C2×Dic131—22no—
66C2×Dic131—22no—
67C2×Dic131—22no—
68C2×Dic131—24no—
69C2×Dic131C2×D412yes—
70C2×Dic131C2×D412yes—
71C2×Dic131C2×Q814yes—
72C2×Dic131C2×Q814yes—
73C23×C131C2411yescharacteristic
74Q8×C131—28no—
75Q8×C131—44no—
76C2×C4×C131—22no—
77C2×C4×C131—22no—
78C2×C4×C131—22no—
79C2×C4×C131—22no—
80C2×C4×C131—22no—
81C2×C4×C131—22no—
82C2×C4×C131—22no—
83C2×C4×C131C2×D412yes—
84C2×C4×C131C2×D414yes—
85C2×C4×C131C2×D412yes—
86C2×C4×C131C2×Q814yes—
87C2×C4×C131#4112yes—
88C22×C131C22×D411yescharacteristic
89C22×C131C22×D411yescharacteristic
90C22×C131C22×Q812yes—
91C22×C131C2×#4111yescharacteristic
92C22×C1312−1+412yes—
93C4×C131—24no—
94C4×C131—24no—
95C4×C131—42no—
96C4×C131#12114yes—
97Dic131—22no—
98Dic131—24no—
99Dic131—42no—
100Dic131—42no—
101Dic131C4⋊Q814yescomplemented
102C2×C131C2×#12111yescharacteristic
103C2×C131C2×C4⋊Q811yescharacteristic
104C2×C131#53611yescharacteristic
105C2×C131D4×Q812yes—
106C2×C131#55712yes—
107C131#254611yescharacteristic, complemented, Sylow
108#2546—1311nomaximal, Sylow
109C2×C4⋊Q8—1311no—
110C2×C4⋊Q8—1311no—
111C2×C4⋊Q8D13111yescharacteristic
112C2×C4×Q8—1311no—
113#398—1314no—
114#391—1311no—
115#391—1311no—
116#389—1314no—
117C4×C4⋊C4—1311no—
118C22×Q8—1311no—
119C22×Q8C2×D13111yescharacteristic
120C22×Q8C2×D13111yescharacteristic
121C4⋊Q8—24no—
122C4⋊Q8—2624no—
123C4⋊Q8—2624no—
124C4×Q8—2624no—
125C2×C4⋊C4—1311no—
126C2×C4⋊C4—1312no—
127C2×C4⋊C4—1312no—
128C2×C4⋊C4—1314no—
129C2×C4⋊C4—1312no—
130C2×C4⋊C4—1312no—
131C2×C4⋊C4C2×D13114yes—
132C2×C42—1312no—
133C2×C42—1311no—
134C2×C42—1311no—
135C2×C42C2×D13111yescharacteristic
136#95—1312no—
137#95—1314no—
138#95—1314no—
139C2×Q8—22no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—22no—
143C2×Q8—22no—
144C2×Q8—2622no—
145C2×Q8—2622no—
146C2×Q8—2622no—
147C2×Q8C22⋊D13114yes—
148C22×C4—1312no—
149C22×C4—1312no—
150C22×C4—1312no—
151C22×C4—1311no—
152C22×C4—1311no—
153C22×C4C22×D13112yes—
154C22×C4C22×D13112yes—
155C22×C4C22×D13112yes—
156C22×C4C22×D13111yescharacteristic
157C4⋊C4—28no—
158C4⋊C4—2624no—
159C4⋊C4—2624no—
160C4⋊C4—2624no—
161C4⋊C4—2622no—
162C4⋊C4—2628no—
163C4⋊C4—1318no—
164C42—22no—
165C42—2624no—
166C42—2622no—
167C42—1314no—
168C23C23×D13111yescharacteristic
169Q8—28no—
170Q8—44no—
171Q8—5244no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—2622no—
180C2×C4—2622no—
181C2×C4—2622no—
182C2×C4—2622no—
183C2×C4—2622no—
184C2×C4—2622no—
185C2×C4—2622no—
186C2×C4—2624no—
187C2×C4—1312no—
188C2×C4—1314no—
189C2×C4—1312no—
190C2×C4—1314no—
191C2×C4D4×D13112yes—
192C2×C4C2×C22⋊D13114yes—
193C2×C4C2×C22⋊D13112yes—
194C2×C4Q8×D13114yes—
195C2×C4#1452351212yes—
196C22C2×D4×D13111yescharacteristic
197C22C22×C22⋊D13111yescharacteristic
198C22C2×Q8×D13112yes—
199C22C2×#1452351211yescharacteristic
200C22#1686229412yes—
201C4—24no—
202C4—24no—
203C4—42no—
204C4—2622no—
205C4—2624no—
206C4—5242no—
207C4—5242no—
208C4—1314no—
209C4#1686221814yes—
210C2C2×#1686221811yescharacteristic
211C2C4⋊Q8×D13111yescharacteristic
212C2#2255388711yescharacteristic
213C2Q8×C22⋊D13112yes—
214C2#2255403712yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, prime radical, cosocleC23×C131C24
FittingC2×C4⋊Q8×C131C2
hypercenterC2×C4⋊Q8D131
hypocenterC131#2546
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
44. Dic131⋊C4Q8semidirect
101. Dic131C4⋊Q8semidirect
107. C131#2546semidirect

Maximal subgroups

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

subgroupindex
C2×#168622442
C2×Q8×Dic1312
#225532452
#225532192
#225532182
#225532072
C4×Dic131⋊C42
C2×C4⋊Q8×C1312
#2546131