Finite Group Databasebeta

← #30671122

Subgroups — #30671122

Order 16768.

532 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 153 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×#16862238C211yesmaximal, characteristic, complemented
2C2×C4×D524C211yesmaximal, characteristic
3#22553252C212yesmaximal, complemented
4#22553230C212yesmaximal
5#22553203C214yesmaximal, complemented
6#22553169C212yesmaximal, complemented
7#22553169C212yesmaximal, complemented
8C4×C4⋊C4×C131C211yesmaximal, characteristic, complemented
9C22×D524C2211yescharacteristic
10C22×C4×D131C2212yes—
11#16862238—24no—
12C4×D524—24no—
13C2×C4⋊Dic131C2211yescharacteristic
14C2×C4⋊Dic131C2211yescharacteristic
15C2×C4⋊Dic131C2211yescharacteristic
16C2×Dic131⋊C4C2212yes—
17C2×Dic131⋊C4C2212yes—
18C2×(C2×D131)⋊C4C2214yes—
19C2×(C2×D131)⋊C4C2212yes—
20C2×(C2×D131)⋊C4C2214yes—
21C131⋊#95C2214yes—
22C131⋊#95C2214yes—
23C2×C4⋊C4×C131C2212yes—
24C2×C42×C131C2212yes—
25C2×C42×C131C2211yescharacteristic
26#95×C131C2212yes—
27C23×D131C2312yes—
28C2×D524—24no—
29C2×D524—22no—
30C22×Dic131C2312yes—
31C22×Dic131C2312yes—
32C22×Dic131C2312yes—
33C2×C4×D131—24no—
34C2×C4×D131—24no—
35C2×C4×D131—24no—
36C4⋊Dic131—22no—
37C4⋊Dic131—22no—
38C4⋊Dic131—22no—
39Dic131⋊C4—24no—
40Dic131⋊C4—24no—
41(C2×D131)⋊C4—28no—
42(C2×D131)⋊C4—24no—
43(C2×D131)⋊C4—44no—
44C22×C4×C131C2314yes—
45C22×C4×C131C2311yescharacteristic
46C22×C4×C131C2311yescharacteristic
47C22×C4×C131C2311yescharacteristic
48C4⋊C4×C131—24no—
49C42×C131—42no—
50C42×C131D414yes—
51C22×D131—22no—
52C22×D131—22no—
53C22×D131—22no—
54C22×D131—22no—
55C22×D131—22no—
56C22×D131—22no—
57C22×D131#4114yes—
58D524—28no—
59C2×Dic131—22no—
60C2×Dic131—22no—
61C2×Dic131—22no—
62C2×Dic131—24no—
63C2×Dic131—22no—
64C2×Dic131—22no—
65C2×Dic131—22no—
66C2×Dic131—22no—
67C4×D131—48no—
68C23×C131C2411yescharacteristic
69C2×C4×C131—22no—
70C2×C4×C131—21no—
71C2×C4×C131—21no—
72C2×C4×C131—24no—
73C2×C4×C131—24no—
74C2×C4×C131—24no—
75C2×C4×C131C2×D412yes—
76C2×C4×C131C2×D412yes—
77C2×C4×C131C2×D412yes—
78C2×C4×C131#4114yes—
79C22×C131C22×D411yescharacteristic
80C22×C131C2×#4112yes—
81C22×C131C2×#4112yes—
82C22×C131H(5,𝔽2)11yescharacteristic
83C22×C1312−1+411yescharacteristic
84C2×D131—24no—
85C2×D131—24no—
86C2×D131—24no—
87C2×D131—42no—
88C2×D131—42no—
89C2×D131—42no—
90C2×D131—42no—
91C4×C131—22no—
92C4×C131—22no—
93C4×C131—44no—
94C4×C131#12414yes—
95Dic131—42no—
96Dic131—42no—
97Dic131—42no—
98C2×C131#52511yescharacteristic
99C2×C131C2×#12411yescharacteristic
100C2×C131#53711yescharacteristic
101C2×C131#55212yes—
102C2×C131#55312yes—
103D131—44no—
104C131#256211yescharacteristic, complemented, Sylow
105#2562—1311nomaximal, Sylow
106C2×#125—1311no—
107C2×C4×D4—1311no—
108#400—1312no—
109#394—1312no—
110#388—1314no—
111#380—1312no—
112#380—1312no—
113C4×C4⋊C4D13111yescharacteristic, complemented
114C22×D4—1311no—
115C23×C4—1312no—
116#125—2624no—
117C4×D4—2624no—
118C2×C4⋊C4—1311no—
119C2×C4⋊C4—1311no—
120C2×C4⋊C4—1311no—
121C2×C4⋊C4—1312no—
122C2×C4⋊C4—1312no—
123C2×C4⋊C4C2×D13112yes—
124C2×C22⋊C4—1312no—
125C2×C22⋊C4—1314no—
126C2×C22⋊C4—1314no—
127C2×C42C2×D13112yes—
128C2×C42C2×D13111yescharacteristic
129#95—1314no—
130#95—1314no—
131#95C2×D13112yes—
132C24—1312no—
133C2×D4—2624no—
134C2×D4—2622no—
135C22×C4—2624no—
136C22×C4—2624no—
137C22×C4—2624no—
138C22×C4—1312no—
139C22×C4—1312no—
140C22×C4—1312no—
141C22×C4C22×D13114yes—
142C22×C4C22×D13111yescharacteristic
143C22×C4C22×D13111yescharacteristic
144C22×C4C22×D13111yescharacteristic
145C4⋊C4—24no—
146C4⋊C4—2624no—
147C4⋊C4—2624no—
148C4⋊C4—2622no—
149C4⋊C4—2622no—
150C4⋊C4—2622no—
151C22⋊C4—2628no—
152C22⋊C4—2624no—
153C22⋊C4—5244no—
154C42—42no—
155C42D52414yes—
156C23—2622no—
157C23—2622no—
158C23—2622no—
159C23—2622no—
160C23—2622no—
161C23—2622no—
162C23—1314no—
163C23C23×D13111yescharacteristic
164D4—2628no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—24no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—2622no—
172C2×C4—2622no—
173C2×C4—2622no—
174C2×C4—2624no—
175C2×C4—2622no—
176C2×C4—2622no—
177C2×C4—2622no—
178C2×C4—2622no—
179C2×C4—5248no—
180C2×C4C2×D52412yes—
181C2×C4C2×D52412yes—
182C2×C4C2×D52412yes—
183C2×C4#1452351314yes—
184C22—2624no—
185C22—2624no—
186C22—2624no—
187C22—5242no—
188C22—5242no—
189C22—5242no—
190C22—5242no—
191C22C22×D52411yescharacteristic
192C22#41×D13112yes—
193C22C2×#1452351312yes—
194C22#1686229111yescharacteristic
195C22#1686229411yescharacteristic
196C4—22no—
197C4—22no—
198C4—44no—
199C4—5242no—
200C4—5242no—
201C4—5242no—
202C4#1686223414yes—
203C2—5244no—
204C2#2255382411yescharacteristic
205C2C2×#1686223411yescharacteristic
206C2#2255389311yescharacteristic
207C2#2255400312yes—
208C2#2255401212yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC4×C4⋊C4×C131C2
prime radicalC22×D524C22
hypercenterC4×C4⋊C4D131
hypocenterC131#2562
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16862238C2semidirect
3. #22553252C2semidirect
5. #22553203C2semidirect
6. #22553169C2semidirect
7. #22553169C2semidirect
8. C4×C4⋊C4×C131C2semidirect
104. C131#2562semidirect
113. C4×C4⋊C4D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622382
C2×C4×D5242
#225532522
#225532302
#225532032
#225531692
C4×C4⋊C4×C1312
#2562131