Finite Group Databasebeta

← #30670247

Subgroups — #30670247

Order 16768.

512 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16862202C211yesmaximal, characteristic
2C22×C4⋊Dic131C211yesmaximal, characteristic
3#22553254C212yesmaximal, complemented
4#22553245C212yesmaximal, complemented
5#22553199C212yesmaximal, complemented
6#22553198C212yesmaximal, complemented
7#22553185C211yesmaximal, characteristic
8#22553173C211yesmaximal, characteristic
9#22553173C211yesmaximal, characteristic
10#22553173C211yesmaximal, characteristic
11#385×C131C211yesmaximal, characteristic
12C23×Dic131C2211yescharacteristic
13C23×Dic131C2211yescharacteristic
14#16862202—24no—
15C2×C4⋊Dic131—24no—
16C2×C4⋊Dic131C2212yes—
17C2×C4⋊Dic131C2212yes—
18C2×Dic131⋊C4C2212yes—
19C2×Dic131⋊C4C2211yescharacteristic
20C2×Dic131⋊C4C2211yescharacteristic
21C2×C22⋊Dic131C2211yescharacteristic
22C2×C22⋊Dic131C2211yescharacteristic
23C2×C4×Dic131C2211yescharacteristic
24C2×C4×Dic131C2211yescharacteristic
25C131⋊#95C2212yes—
26C131⋊#95C2212yes—
27C131⋊#95C2212yes—
28C131⋊#95C2212yes—
29C131⋊#95C2212yes—
30C131⋊#95C2212yes—
31C131⋊#95C2212yes—
32C23×C4×C131C2211yescharacteristic
33C2×C4⋊C4×C131C2212yes—
34C2×C22⋊C4×C131C2211yescharacteristic
35C2×C22⋊C4×C131C2211yescharacteristic
36#95×C131C2212yes—
37C22×Dic131—22no—
38C22×Dic131—22no—
39C22×Dic131—22no—
40C22×Dic131—22no—
41C22×Dic131—22no—
42C22×Dic131D414yes—
43C22×Dic131C2312yes—
44C22×Dic131C2312yes—
45C22×Dic131C2311yescharacteristic
46C22×Dic131C2311yescharacteristic
47C22×Dic131C2311yescharacteristic
48C22×Dic131C2311yescharacteristic
49C4⋊Dic131—24no—
50C4⋊Dic131—24no—
51Dic131⋊C4—22no—
52Dic131⋊C4—22no—
53Dic131⋊C4—24no—
54C22⋊Dic131—22no—
55C22⋊Dic131—22no—
56C4×Dic131—22no—
57C4×Dic131—22no—
58C24×C131C2311yescharacteristic
59C22×C4×C131—22no—
60C22×C4×C131—24no—
61C22×C4×C131C2311yescharacteristic
62C22×C4×C131C2312yes—
63C22×C4×C131C2311yescharacteristic
64C22×C4×C131C2312yes—
65C4⋊C4×C131—24no—
66C22⋊C4×C131—22no—
67C22⋊C4×C131—22no—
68C2×Dic131—28no—
69C2×Dic131—21no—
70C2×Dic131—21no—
71C2×Dic131—21no—
72C2×Dic131—21no—
73C2×Dic131—21no—
74C2×Dic131—21no—
75C2×Dic131—22no—
76C2×Dic131—22no—
77C2×Dic131—22no—
78C2×Dic131—22no—
79C2×Dic131—22no—
80C2×Dic131—22no—
81C2×Dic131—22no—
82C2×Dic131—22no—
83C2×Dic131—44no—
84C2×Dic131C2×D412yes—
85C2×Dic131C2×D412yes—
86C23×C131—22no—
87C23×C131—22no—
88C23×C131C2×D412yes—
89C23×C131#4112yes—
90C23×C131#4112yes—
91C23×C131C2411yescharacteristic
92C2×C4×C131—28no—
93C2×C4×C131—22no—
94C2×C4×C131—22no—
95C2×C4×C131—21no—
96C2×C4×C131—21no—
97C2×C4×C131—22no—
98C2×C4×C131—22no—
99C2×C4×C131—22no—
100C2×C4×C131—22no—
101C2×C4×C131—22no—
102C2×C4×C131#4114yes—
103C22×C131—22no—
104C22×C131—22no—
105C22×C131—24no—
106C22×C131—24no—
107C22×C131#12314yes—
108C22×C131C22×D411yescharacteristic
109C22×C131C2×#4112yes—
110C22×C131C2×#4111yescharacteristic
111C22×C131C2×#4111yescharacteristic
112C22×C1312−1+412yes—
113C4×C131—24no—
114C4×C131—41no—
115C4×C131—41no—
116C4×C131—42no—
117Dic131—22no—
118Dic131—22no—
119Dic131—42no—
120Dic131—41no—
121Dic131—41no—
122Dic131—42no—
123C2×C131—24no—
124C2×C131C2×#12311yescharacteristic
125C2×C131#52511yescharacteristic
126C2×C131#53611yescharacteristic
127C2×C131#55212yes—
128C2×C131#55212yes—
129C131#248411yescharacteristic, complemented, Sylow
130#2484—1311nomaximal, Sylow
131C2×#117—1311no—
132C22×C4⋊C4—1311no—
133#400—1312no—
134#398—1312no—
135#387—1312no—
136#387—1312no—
137#385—1311no—
138#385D13111yescharacteristic
139#381—1311no—
140#381—1311no—
141#381—1311no—
142C23×C4—1311no—
143C23×C4—1311no—
144C23×C4C2×D13111yescharacteristic
145#117—2624no—
146C2×C4⋊C4—2624no—
147C2×C4⋊C4—1312no—
148C2×C4⋊C4—1312no—
149C2×C4⋊C4—1311no—
150C2×C4⋊C4—1311no—
151C2×C4⋊C4—1312no—
152C2×C4⋊C4C2×D13112yes—
153C2×C22⋊C4—1311no—
154C2×C22⋊C4—1311no—
155C2×C22⋊C4C2×D13111yescharacteristic
156C2×C22⋊C4C2×D13111yescharacteristic
157C2×C42—1311no—
158C2×C42—1311no—
159#95—1312no—
160#95—1312no—
161#95—1312no—
162#95—1312no—
163#95—1312no—
164#95—1312no—
165#95—1312no—
166#95C2×D13112yes—
167C24C22×D13111yescharacteristic
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—2622no—
171C22×C4—2622no—
172C22×C4—2622no—
173C22×C4—2622no—
174C22×C4—2622no—
175C22×C4—1312no—
176C22×C4—1311no—
177C22×C4—1312no—
178C22×C4—1311no—
179C22×C4—1314no—
180C22×C4—1311no—
181C22×C4—1311no—
182C22×C4C22×D13112yes—
183C22×C4C22×D13111yescharacteristic
184C22×C4C22×D13111yescharacteristic
185C22×C4C22×D13112yes—
186C4⋊C4—24no—
187C4⋊C4—2622no—
188C4⋊C4—2622no—
189C4⋊C4—2624no—
190C4⋊C4—2624no—
191C4⋊C4—2624no—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—2622no—
195C22⋊C4—2622no—
196C42—2622no—
197C42—2622no—
198C23—22no—
199C23—22no—
200C23D4×D13112yes—
201C23#1452351312yes—
202C23#1452351412yes—
203C23C23×D13111yescharacteristic
204C2×C4—28no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—2628no—
215C2×C4—2621no—
216C2×C4—2621no—
217C2×C4—2621no—
218C2×C4—2621no—
219C2×C4—2621no—
220C2×C4—2621no—
221C2×C4—2622no—
222C2×C4—2622no—
223C2×C4—2622no—
224C2×C4—2622no—
225C2×C4—2622no—
226C2×C4—2622no—
227C2×C4—2622no—
228C2×C4—2622no—
229C2×C4—5244no—
230C2×C4—1312no—
231C2×C4—1312no—
232C2×C4#1452351214yes—
233C22—22no—
234C22—24no—
235C22—24no—
236C22—22no—
237C22#1686223114yes—
238C22C2×D4×D13111yescharacteristic
239C22C2×#1452351212yes—
240C22C2×#1452351311yescharacteristic
241C22C2×#1452351411yescharacteristic
242C22#1686229312yes—
243C4—24no—
244C4—41no—
245C4—42no—
246C4—41no—
247C4—2622no—
248C4—2622no—
249C4—5241no—
250C4—5242no—
251C4—5242no—
252C4—5241no—
253C2—24no—
254C2C2×#1686223111yescharacteristic
255C2#2255381811yescharacteristic
256C2#2255388411yescharacteristic
257C2#2255400212yes—
258C2#2255400412yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#385×C131C2
prime radicalC24×C131C23
hypercenter#385D131
hypocenterC131#2484
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. #22553254C2semidirect
4. #22553245C2semidirect
5. #22553199C2semidirect
6. #22553198C2semidirect
129. C131#2484semidirect

Maximal subgroups

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

subgroupindex
C2×#168622022
C22×C4⋊Dic1312
#225532542
#225532452
#225531992
#225531982
#225531852
#225531732
#385×C1312
#2484131