Finite Group Databasebeta

← #30674298

Subgroups — #30674298

Order 16768.

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

normal subgroups only

This group has 564 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×#16862223C211yesmaximal, characteristic
2C22×C4×Dic131C211yesmaximal, characteristic
3#22553216C212yesmaximal, complemented
4#22553209C212yesmaximal, complemented
5#22553199C212yesmaximal, complemented
6#22553195C212yesmaximal, complemented
7#22553187C211yesmaximal, characteristic
8#22553184C211yesmaximal, characteristic
9#22553175C211yesmaximal, characteristic
10#22553170C211yesmaximal, characteristic
11#381×C131C211yesmaximal, characteristic
12C22×Dic262C2211yescharacteristic
13C23×Dic131C2211yescharacteristic
14C23×Dic131C2211yescharacteristic
15#16862223—24no—
16C2×C4⋊Dic131C2212yes—
17C2×C4⋊Dic131C2211yescharacteristic
18C2×Dic131⋊C4C2212yes—
19C2×Dic131⋊C4C2212yes—
20C2×Dic131⋊C4C2211yescharacteristic
21C2×Dic131⋊C4C2211yescharacteristic
22C2×C22⋊Dic131C2211yescharacteristic
23C2×C22⋊Dic131C2211yescharacteristic
24C2×C4×Dic131—24no—
25C2×C4×Dic131C2212yes—
26C2×C4×Dic131C2212yes—
27C131⋊#95C2212yes—
28C131⋊#95C2212yes—
29C131⋊#95C2212yes—
30C131⋊#95C2212yes—
31C131⋊#95C2212yes—
32C23×C4×C131C2211yescharacteristic
33C2×C22⋊C4×C131C2211yescharacteristic
34C2×C22⋊C4×C131C2211yescharacteristic
35#95×C131C2212yes—
36#95×C131C2212yes—
37C2×Dic262—22no—
38C2×Dic262—24no—
39C22×Dic131—24no—
40C22×Dic131—24no—
41C22×Dic131—22no—
42C22×Dic131D414yes—
43C22×Dic131C2311yescharacteristic
44C22×Dic131C2312yes—
45C22×Dic131C2311yescharacteristic
46C22×Dic131C2311yescharacteristic
47C22×Dic131C2312yes—
48C22×Dic131C2311yescharacteristic
49C4⋊Dic131—22no—
50C4⋊Dic131—24no—
51Dic131⋊C4—22no—
52Dic131⋊C4—22no—
53Dic131⋊C4—24no—
54Dic131⋊C4—24no—
55C22⋊Dic131—22no—
56C22⋊Dic131—22no—
57C4×Dic131—24no—
58C4×Dic131—24no—
59C24×C131C2311yescharacteristic
60C22×C4×C131—22no—
61C22×C4×C131—22no—
62C22×C4×C131Q814yescomplemented
63C22×C4×C131C2311yescharacteristic
64C22×C4×C131C2312yes—
65C22×C4×C131C2311yescharacteristic
66C22×C4×C131C2312yes—
67C22⋊C4×C131—22no—
68C22⋊C4×C131—22no—
69Dic262—44no—
70C2×Dic131—28no—
71C2×Dic131—22no—
72C2×Dic131—22no—
73C2×Dic131—22no—
74C2×Dic131—28no—
75C2×Dic131—21no—
76C2×Dic131—21no—
77C2×Dic131—22no—
78C2×Dic131—22no—
79C2×Dic131—22no—
80C2×Dic131—22no—
81C2×Dic131—22no—
82C2×Dic131C2×D412yes—
83C2×Dic131C2×D412yes—
84C2×Dic131#4114yes—
85C2×Dic131#4114yes—
86C23×C131—22no—
87C23×C131—22no—
88C23×C131C2×D412yes—
89C23×C131C2×Q812yes—
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×C131C2×Q814yes—
103C22×C131—22no—
104C22×C131—22no—
105C22×C131—24no—
106C22×C131—24no—
107C22×C131C22⋊Q814yes—
108C22×C131C22×D411yescharacteristic
109C22×C131C22×Q811yescharacteristic
110C22×C131C2×#4112yes—
111C22×C131C2×#4112yes—
112C22×C131C2×#4111yescharacteristic
113C4×C131—24no—
114C4×C131—41no—
115C4×C131—41no—
116C4×C131—42no—
117Dic131—22no—
118Dic131—24no—
119Dic131—41no—
120Dic131—41no—
121Dic131—42no—
122Dic131#12314yes—
123C2×C131—24no—
124C2×C131C2×C22⋊Q811yescharacteristic
125C2×C131C2×#12311yescharacteristic
126C2×C131#52511yescharacteristic
127C2×C131#52912yes—
128C2×C131#53312yes—
129C131#285511yescharacteristic, complemented, Sylow
130#2855—1311nomaximal, Sylow
131C2×C22⋊Q8—1311no—
132C22×C42—1311no—
133#391—1312no—
134#389—1312no—
135#387—1312no—
136#387—1312no—
137#385—1311no—
138#385—1311no—
139#381—1311no—
140#381D13111yescharacteristic
141#380—1311no—
142C22×Q8—1311no—
143C23×C4—1311no—
144C23×C4—1311no—
145C23×C4C2×D13111yescharacteristic
146C22⋊Q8—2624no—
147C2×C4⋊C4—1311no—
148C2×C4⋊C4—1311no—
149C2×C4⋊C4—1311no—
150C2×C4⋊C4—1312no—
151C2×C4⋊C4—1312no—
152C2×C4⋊C4—1312no—
153C2×C22⋊C4—1311no—
154C2×C22⋊C4—1311no—
155C2×C22⋊C4C2×D13111yescharacteristic
156C2×C22⋊C4C2×D13111yescharacteristic
157C2×C42—2624no—
158C2×C42—1312no—
159C2×C42—1312no—
160#95—1312no—
161#95—1312no—
162#95—1312no—
163#95—1312no—
164#95—1312no—
165#95C2×D13112yes—
166#95C2×D13112yes—
167C24C22×D13111yescharacteristic
168C2×Q8—2622no—
169C2×Q8—2624no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—2622no—
173C22×C4—2624no—
174C22×C4—2624no—
175C22×C4—1312no—
176C22×C4—1311no—
177C22×C4—1311no—
178C22×C4—1314no—
179C22×C4—1312no—
180C22×C4—1311no—
181C22×C4—1311no—
182C22×C4Dic26214yescomplemented
183C22×C4C22×D13112yes—
184C22×C4C22×D13111yescharacteristic
185C22×C4C22×D13111yescharacteristic
186C22×C4C22×D13112yes—
187C4⋊C4—2624no—
188C4⋊C4—2624no—
189C4⋊C4—2622no—
190C4⋊C4—2622no—
191C4⋊C4—2624no—
192C4⋊C4—2622no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—2622no—
196C22⋊C4—2622no—
197C42—2624no—
198C42—2624no—
199C23—22no—
200C23—22no—
201C23D4×D13112yes—
202C23C2×Dic26212yes—
203C23#1452351212yes—
204C23C23×D13111yescharacteristic
205Q8—5244no—
206C2×C4—28no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—2628no—
217C2×C4—2622no—
218C2×C4—2622no—
219C2×C4—2622no—
220C2×C4—2628no—
221C2×C4—2621no—
222C2×C4—2621no—
223C2×C4—2622no—
224C2×C4—2622no—
225C2×C4—2622no—
226C2×C4—2622no—
227C2×C4—2622no—
228C2×C4—1312no—
229C2×C4—1314no—
230C2×C4—1312no—
231C2×C4—1314no—
232C2×C4C2×Dic26214yes—
233C22—24no—
234C22—22no—
235C22—22no—
236C22—24no—
237C22#1686222314yes—
238C22C2×D4×D13111yescharacteristic
239C22C22×Dic26211yescharacteristic
240C22#41×D13112yes—
241C22#41×D13112yes—
242C22C2×#1452351211yescharacteristic
243C4—24no—
244C4—41no—
245C4—41no—
246C4—42no—
247C4—2622no—
248C4—2624no—
249C4—5241no—
250C4—5241no—
251C4—5242no—
252C4—1314no—
253C2—24no—
254C2C2×#1686222311yescharacteristic
255C2#123×D13111yescharacteristic
256C2#2255382011yescharacteristic
257C2#2255383912yes—
258C2#2255386512yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#381×C131C2
prime radicalC24×C131C23
hypercenter#381D131
hypocenterC131#2855
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
3. #22553216C2semidirect
4. #22553209C2semidirect
5. #22553199C2semidirect
6. #22553195C2semidirect
62. C22×C4×C131Q8semidirect
129. C131#2855semidirect
182. C22×C4Dic262semidirect

Maximal subgroups

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

subgroupindex
C2×#168622232
C22×C4×Dic1312
#225532162
#225532092
#225531992
#225531952
#225531872
#225531842
#225531752
#225531702
#381×C1312
#2855131