Finite Group Databasebeta

← #30670943

Subgroups — #30670943

Order 16768.

576 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 157 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×#16862246C211yesmaximal, characteristic, complemented
2C2×#16862233C211yesmaximal, characteristic
3C2×#16862233C211yesmaximal, characteristic
4C2×#16862211C211yesmaximal, characteristic, complemented
5#22553235C214yesmaximal, complemented
6#22553217C211yesmaximal, characteristic, complemented
7#22553205C214yesmaximal, complemented
8C4×Dic131⋊C4C211yesmaximal, characteristic, complemented
9#390×C131C211yesmaximal, characteristic
10C22×Dic262C2211yescharacteristic
11C22×Dic262C2211yescharacteristic
12#16862246—24no—
13#16862233—24no—
14#16862233—24no—
15#16862211—24no—
16C2×C4⋊Dic131C2214yes—
17C2×Dic131⋊C4C2212yes—
18C2×Dic131⋊C4C2212yes—
19C2×C22⋊Dic131C2212yes—
20C2×C22⋊Dic131C2214yes—
21C2×C22⋊Dic131C2212yes—
22C2×C4×Dic131C2211yescharacteristic
23C2×C4×Dic131C2212yes—
24C2×C4×Dic131C2211yescharacteristic
25C131⋊#95C2212yes—
26C131⋊#95C2214yes—
27C22×D4×C131C2211yescharacteristic
28C2×C4⋊C4×C131C2211yescharacteristic
29C2×C22⋊C4×C131C2214yes—
30C2×C42×C131C2211yescharacteristic
31C2×Dic262—22no—
32C2×Dic262—22no—
33C2×Dic262—22no—
34C2×Dic262—22no—
35C2×Dic262—22no—
36C2×Dic262D414yes—
37C22×Dic131C2312yes—
38C22×Dic131C2312yes—
39C22×Dic131C2312yes—
40C22×Dic131C2311yescharacteristic
41C22×Dic131C2311yescharacteristic
42C4⋊Dic131—28no—
43Dic131⋊C4—24no—
44Dic131⋊C4—24no—
45C22⋊Dic131—28no—
46C22⋊Dic131—24no—
47C22⋊Dic131—24no—
48C4×Dic131—24no—
49C4×Dic131—22no—
50C4×Dic131D414yes—
51C24×C131C2312yes—
52C2×D4×C131—22no—
53C2×D4×C131—22no—
54C2×D4×C131—22no—
55C22×C4×C131C2312yes—
56C22×C4×C131C2312yes—
57C22×C4×C131C2311yescharacteristic
58C4⋊C4×C131—22no—
59C22⋊C4×C131—28no—
60C42×C131—22no—
61Dic262—28no—
62Dic262—44no—
63C2×Dic131—22no—
64C2×Dic131—22no—
65C2×Dic131—22no—
66C2×Dic131—22no—
67C2×Dic131—22no—
68C2×Dic131—22no—
69C2×Dic131—22no—
70C2×Dic131—22no—
71C2×Dic131—22no—
72C2×Dic131—22no—
73C2×Dic131C2×D412yes—
74C2×Dic131C2×D412yes—
75C2×Dic131C2×D414yes—
76C23×C131—22no—
77C23×C131—22no—
78C23×C131—24no—
79C23×C131—22no—
80C23×C131—24no—
81C23×C131C2411yescharacteristic
82D4×C131—44no—
83C2×C4×C131—22no—
84C2×C4×C131—22no—
85C2×C4×C131—22no—
86C2×C4×C131—24no—
87C2×C4×C131C2×D412yes—
88C2×C4×C131C2×D412yes—
89C2×C4×C131#4114yes—
90C2×C4×C131#4112yes—
91C22×C131—42no—
92C22×C131—42no—
93C22×C131—42no—
94C22×C131—44no—
95C22×C131—44no—
96C22×C131C22×D411yescharacteristic
97C22×C131C22×D411yescharacteristic
98C22×C131C2×#4112yes—
99C22×C131C2×#4111yescharacteristic
100C22×C1312−1+412yes—
101C4×C131—24no—
102C4×C131—42no—
103C4×C131#12114yes—
104Dic131—24no—
105Dic131—22no—
106Dic131—22no—
107Dic131—42no—
108Dic131—42no—
109C2×C131—82no—
110C2×C131C2×#12111yescharacteristic
111C2×C131#53211yescharacteristic
112C2×C131#53611yescharacteristic
113C2×C131#54812yes—
114C2×C131#55612yes—
115C131#254511yescharacteristic, complemented, Sylow
116#2545—1311nomaximal, Sylow
117C2×C4⋊Q8—1311no—
118C2×#124—1311no—
119C2×#124—1311no—
120C2×C4×Q8—1311no—
121#396—1314no—
122#391—1311no—
123#390D13111yescharacteristic
124#388—1314no—
125C4×C4⋊C4—1311no—
126C22×Q8—1311no—
127C22×Q8—1311no—
128C22×D4C2×D13111yescharacteristic
129C4⋊Q8—2624no—
130#124—2624no—
131#124—2624no—
132C4×Q8—2624no—
133C2×C4⋊C4—1312no—
134C2×C4⋊C4—1314no—
135C2×C4⋊C4—1312no—
136C2×C4⋊C4C2×D13111yescharacteristic
137C2×C22⋊C4—1312no—
138C2×C22⋊C4—1314no—
139C2×C22⋊C4—1312no—
140C2×C22⋊C4C2×D13114yes—
141C2×C42—1312no—
142C2×C42—1311no—
143C2×C42—1311no—
144C2×C42C2×D13111yescharacteristic
145#95—1312no—
146#95—1314no—
147C24C22×D13112yes—
148C2×Q8—2622no—
149C2×Q8—2622no—
150C2×Q8—2622no—
151C2×Q8—2622no—
152C2×Q8—2622no—
153C2×Q8—1314no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—22no—
157C22×C4—1312no—
158C22×C4—1312no—
159C22×C4—1311no—
160C22×C4—1311no—
161C22×C4—1312no—
162C22×C4C22×D13112yes—
163C22×C4C22×D13112yes—
164C22×C4C22×D13111yescharacteristic
165C4⋊C4—22no—
166C4⋊C4—2624no—
167C4⋊C4—2628no—
168C4⋊C4—2624no—
169C22⋊C4—28no—
170C22⋊C4—2628no—
171C22⋊C4—2624no—
172C22⋊C4—2624no—
173C42—22no—
174C42—2624no—
175C42—2622no—
176C42—1314no—
177C23—22no—
178C23—22no—
179C23—24no—
180C23—22no—
181C23—24no—
182C23C23×D13111yescharacteristic
183Q8—2628no—
184Q8—5244no—
185D4—44no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—2622no—
191C2×C4—2622no—
192C2×C4—2622no—
193C2×C4—2622no—
194C2×C4—2622no—
195C2×C4—2622no—
196C2×C4—2622no—
197C2×C4—2622no—
198C2×C4—2622no—
199C2×C4—2622no—
200C2×C4—1312no—
201C2×C4—1312no—
202C2×C4—1314no—
203C2×C4D4×D13112yes—
204C2×C4D4×D13112yes—
205C2×C4#1452351214yes—
206C2×C4#1452351312yes—
207C22—42no—
208C22—42no—
209C22—44no—
210C22—44no—
211C22—42no—
212C22C2×D4×D13111yescharacteristic
213C22C2×D4×D13111yescharacteristic
214C22C2×#1452351212yes—
215C22C2×#1452351311yescharacteristic
216C22#1686229312yes—
217C4—24no—
218C4—42no—
219C4—2622no—
220C4—2622no—
221C4—2624no—
222C4—5242no—
223C4—5242no—
224C4#1686221914yes—
225C2—82no—
226C2C2×#1686221911yescharacteristic
227C2#2255386011yescharacteristic
228C2#2255388211yescharacteristic
229C2#2255396812yes—
230C2#2255403212yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#390×C131C2
prime radicalC22×D4×C131C22
hypercenter#390D131
hypocenterC131#2545
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#16862246C2semidirect
4. C2×#16862211C2semidirect
5. #22553235C2semidirect
6. #22553217C2semidirect
7. #22553205C2semidirect
8. C4×Dic131⋊C4C2semidirect
115. C131#2545semidirect

Maximal subgroups

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

subgroupindex
C2×#168622462
C2×#168622332
C2×#168622112
#225532352
#225532172
#225532052
C4×Dic131⋊C42
#390×C1312
#2545131