Finite Group Databasebeta

← #30670411

Subgroups — #30670411

Order 16768.

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

normal subgroups only

This group has 484 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×#16862226C211yesmaximal, characteristic
2C2×#16862204C211yesmaximal, characteristic
3#22553238C214yesmaximal
4#22553218C211yesmaximal, characteristic, complemented
5#22553217C211yesmaximal, characteristic, complemented
6#22553198C214yesmaximal, complemented
7#22553177C211yesmaximal, characteristic, complemented
8C4×(C2×D131)⋊C4C211yesmaximal, characteristic
9#384×C131C211yesmaximal, characteristic, complemented
10C22×Dic262C2211yescharacteristic
11C22×C4×D131C2211yescharacteristic
12#16862226—24no—
13#16862204—24no—
14C2×C4⋊Dic131C2211yescharacteristic
15C2×C4⋊Dic131C2212yes—
16C2×(C2×D131)⋊C4C2212yes—
17C2×(C2×D131)⋊C4C2212yes—
18C2×(C2×D131)⋊C4C2212yes—
19C2×C4×Dic131C2211yescharacteristic
20C2×C4×Dic131C2212yes—
21C131⋊#95C2212yes—
22C131⋊#95C2214yes—
23C131⋊#95C2214yes—
24C131⋊#95C2214yes—
25C2×C4⋊C4×C131C2211yescharacteristic
26C2×C4⋊C4×C131C2214yes—
27C2×C4⋊C4×C131C2211yescharacteristic
28C2×C42×C131C2211yescharacteristic
29C23×D131C2311yescharacteristic
30C2×Dic262—22no—
31C2×Dic262—22no—
32C2×Dic262—22no—
33C22×Dic131C2312yes—
34C22×Dic131C2312yes—
35C22×Dic131C2312yes—
36C22×Dic131C2311yescharacteristic
37C2×C4×D131—22no—
38C2×C4×D131—22no—
39C2×C4×D131D414yes—
40C4⋊Dic131—24no—
41C4⋊Dic131—22no—
42(C2×D131)⋊C4—24no—
43(C2×D131)⋊C4—24no—
44(C2×D131)⋊C4—24no—
45C4×Dic131—24no—
46C4×Dic131—22no—
47C22×C4×C131C2312yes—
48C22×C4×C131C2312yes—
49C22×C4×C131C2312yes—
50C22×C4×C131C2311yescharacteristic
51C4⋊C4×C131—22no—
52C4⋊C4×C131—22no—
53C4⋊C4×C131—28no—
54C42×C131—22no—
55C22×D131—22no—
56C22×D131—21no—
57C22×D131—21no—
58C22×D131—22no—
59C22×D131C2×D412yes—
60Dic262—44no—
61C2×Dic131—21no—
62C2×Dic131—21no—
63C2×Dic131—22no—
64C2×Dic131—22no—
65C2×Dic131—22no—
66C2×Dic131—22no—
67C2×Dic131—22no—
68C2×Dic131—22no—
69C2×Dic131—24no—
70C2×Dic131—22no—
71C2×Dic131C2×D412yes—
72C4×D131—28no—
73C23×C131C2411yescharacteristic
74C2×C4×C131—22no—
75C2×C4×C131—22no—
76C2×C4×C131—22no—
77C2×C4×C131—22no—
78C2×C4×C131—24no—
79C2×C4×C131—24no—
80C2×C4×C131C2×D412yes—
81C2×C4×C131#4114yes—
82C2×C4×C131#4112yes—
83C2×C4×C131#4112yes—
84C22×C131C22×D411yescharacteristic
85C22×C131C2×#4112yes—
86C22×C131C2×#4111yescharacteristic
87C22×C131C2×#4111yescharacteristic
88C22×C1312−1+412yes—
89C2×D131—22no—
90C2×D131—22no—
91C2×D131—22no—
92C2×D131—42no—
93C2×D131—42no—
94C4×C131—24no—
95C4×C131—42no—
96C4×C131—42no—
97C4×C131#12314yes—
98Dic131—22no—
99Dic131—42no—
100Dic131—42no—
101Dic131—42no—
102C2×C131C2×#12311yescharacteristic
103C2×C131#53211yescharacteristic
104C2×C131#53611yescharacteristic
105C2×C131#55212yes—
106C2×C131#55612yes—
107D131—42no—
108C131#249911yescharacteristic, complemented, Sylow
109#2499—1311nomaximal, Sylow
110C2×C22⋊Q8—1311no—
111C2×#117—1311no—
112#397—1314no—
113#391—1311no—
114#391—1311no—
115#387—1314no—
116#384D13111yescharacteristic, complemented
117#382—1311no—
118C4×C22⋊C4—1311no—
119C22×Q8—1311no—
120C23×C4—1311no—
121C22⋊Q8—2624no—
122#117—2624no—
123C2×C4⋊C4—1311no—
124C2×C4⋊C4—1312no—
125C2×C4⋊C4C2×D13111yescharacteristic
126C2×C4⋊C4C2×D13111yescharacteristic
127C2×C4⋊C4C2×D13114yes—
128C2×C22⋊C4—1312no—
129C2×C22⋊C4—1312no—
130C2×C22⋊C4—1312no—
131C2×C42—1312no—
132C2×C42—1311no—
133C2×C42C2×D13111yescharacteristic
134#95—1312no—
135#95—1314no—
136#95—1314no—
137#95—1314no—
138C24—1311no—
139C2×Q8—2622no—
140C2×Q8—2622no—
141C2×Q8—2622no—
142C22×C4—2622no—
143C22×C4—2622no—
144C22×C4—1312no—
145C22×C4—1312no—
146C22×C4—1312no—
147C22×C4—1311no—
148C22×C4—1314no—
149C22×C4C22×D13112yes—
150C22×C4C22×D13112yes—
151C22×C4C22×D13112yes—
152C22×C4C22×D13111yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—28no—
156C4⋊C4—2624no—
157C4⋊C4—2622no—
158C22⋊C4—2624no—
159C22⋊C4—2624no—
160C22⋊C4—2624no—
161C42—22no—
162C42—2624no—
163C42—2622no—
164C23—2622no—
165C23—2621no—
166C23—2621no—
167C23—2622no—
168C23—1312no—
169C23C23×D13111yescharacteristic
170Q8—5244no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—2621no—
178C2×C4—2621no—
179C2×C4—2628no—
180C2×C4—2622no—
181C2×C4—2622no—
182C2×C4—2622no—
183C2×C4—2622no—
184C2×C4—2622no—
185C2×C4—2622no—
186C2×C4—2624no—
187C2×C4—2622no—
188C2×C4—1312no—
189C2×C4D4×D13112yes—
190C2×C4#1452351214yes—
191C2×C4#1452351312yes—
192C2×C4#1452351412yes—
193C22—2622no—
194C22—2622no—
195C22—2622no—
196C22—5242no—
197C22—5242no—
198C22C2×D4×D13111yescharacteristic
199C22C2×#1452351212yes—
200C22C2×#1452351311yescharacteristic
201C22C2×#1452351411yescharacteristic
202C22#1686229412yes—
203C4—24no—
204C4—42no—
205C4—42no—
206C4—2622no—
207C4—5242no—
208C4—5242no—
209C4—5242no—
210C4#1686223114yes—
211C2—5242no—
212C2C2×#1686223111yescharacteristic
213C2#2255386011yescharacteristic
214C2#2255388711yescharacteristic
215C2#2255399712yes—
216C2#2255403612yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#384×C131C2
prime radicalC23×D131C23
hypercenter#384D131
hypocenterC131#2499
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
4. #22553218C2semidirect
5. #22553217C2semidirect
6. #22553198C2semidirect
7. #22553177C2semidirect
9. #384×C131C2semidirect
108. C131#2499semidirect
116. #384D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622262
C2×#168622042
#225532382
#225532182
#225532172
#225531982
#225531772
C4×(C2×D131)⋊C42
#384×C1312
#2499131