Finite Group Databasebeta

← #30637664

Subgroups — #30637664

Order 16256.

310 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 53 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
1#22550457C211yesmaximal, characteristic, complemented
2#22550010C212yesmaximal, complemented
3#22549609C212yesmaximal, complemented
4#22549602C211yesmaximal, characteristic, complemented
5#394×C127C211yesmaximal, characteristic, complemented
6C2×#14523248—21no—
7C2×#14523233—22no—
8#16861189—22no—
9#16861184—22no—
10#16861183—21no—
11#16861183C2212yes—
12#16861130—22no—
13C22⋊(C127⋊C8)C2212yes—
14C22×D4×C127C2211yescharacteristic
15C2×C4⋊C4×C127C2211yescharacteristic
16C2×C22⋊C4×C127—21no—
17C2×C22⋊C4×C127—21no—
18#95×C127C2211yescharacteristic
19#14523248—42no—
20#14523248—42no—
21C2×Dic254—21no—
22C2×D508—21no—
23C2×D508—21no—
24C2×D508D412yes—
25C2×C22⋊D127—22no—
26C2×C4×D127—22no—
27#14523233—44no—
28C2×C127⋊C8—22no—
29C4⋊Dic127—21no—
30C4⋊Dic127D412yes—
31(C2×D127)⋊C4—22no—
32(C2×D127)⋊C4—22no—
33C24×C127—21no—
34C2×D4×C127—22no—
35C2×D4×C127—42no—
36C22×C4×C127—21no—
37C22×C4×C127D411yescharacteristic
38C22×C4×C127D411yescharacteristic
39C22×C4×C127C2311yescharacteristic
40C4⋊C4×C127—22no—
41C22⋊C4×C127—42no—
42C22⋊C4×C127—42no—
43C22×D127—22no—
44Dic254—42no—
45D508—42no—
46D508—42no—
47D508—81no—
48C22⋊D127—42no—
49C22⋊D127—42no—
50C2×Dic127—22no—
51C4×D127—44no—
52C127⋊C8—42no—
53C23×C127—22no—
54C23×C127—41no—
55C23×C127—41no—
56C23×C127—41no—
57C23×C127—41no—
58C23×C127—41no—
59C23×C127—41no—
60C23×C127C2×D411yescharacteristic
61D4×C127—24no—
62D4×C127—42no—
63C2×C4×C127—21no—
64C2×C4×C127—21no—
65C2×C4×C127—21no—
66C2×C4×C127—41no—
67C2×C4×C127—41no—
68C2×C4×C127—41no—
69C2×C4×C127—41no—
70C2×C4×C127—41no—
71C2×C4×C127D812yescomplemented
72C2×C4×C127C2×D412yes—
73C22×C127—21no—
74C22×C127—21no—
75C22×C127—42no—
76C22×C127—42no—
77C22×C127—42no—
78C22×C127—81no—
79C22×C127—81no—
80C22×C127—81no—
81C22×C127—81no—
82C22×C127#12011yescharacteristic
83C22×C127C2×D811yescharacteristic
84C22×C127AGL(1,ℤ/8)11yescharacteristic
85C2×D127—42no—
86C2×D127—42no—
87C2×D127—42no—
88C4×C127—22no—
89C4×C127—22no—
90C4×C127—41no—
91C4×C127—81no—
92Dic127—42no—
93C2×C127—21no—
94C2×C127—21no—
95C2×C127—82no—
96C2×C127#44711yescharacteristic
97C2×C127#45411yescharacteristic
98C2×C127B(4,𝔽2)11yescharacteristic
99D127—82no—
100#1598—1271nomaximal, Sylow
101C127#159811yescharacteristic, complemented, Sylow
102#537—1271no—
103#447—1272no—
104#394D12711yescharacteristic, complemented
105#327—1272no—
106#324—1271no—
107C2×#41—2541no—
108C22×D4C2×D12711yescharacteristic
109C2×D8—2542no—
110C22⋊Q8—2542no—
111#121—1272no—
112#121—2542no—
113#121—2541no—
114C2×C4⋊C4C2×D12711yescharacteristic
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117#102—2542no—
118C22⋊C8—1272no—
119#95C2×D12711yescharacteristic
120C24—21no—
121#41—5082no—
122#41—5082no—
123C2×Q8—2541no—
124C2×D4—22no—
125C2×D4—42no—
126C2×D4—1272no—
127C2×D4—2541no—
128C2×D4—2541no—
129C2×D4—2542no—
130C22×C4—21no—
131C22×C4—2542no—
132C22×C4C22⋊D12711yescharacteristic
133C22×C4C22⋊D12711yescharacteristic
134C22×C4C22×D12711yescharacteristic
135D8—5084no—
136C2×C8—2542no—
137C4⋊C4—22no—
138C4⋊C4—1272no—
139C4⋊C4—2541no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C22⋊C4—2542no—
143C22⋊C4—2542no—
144C23—22no—
145C23—41no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—2542no—
152C23C2×C22⋊D12711yescharacteristic
153Q8—5082no—
154D4—24no—
155D4—42no—
156D4—10161no—
157D4—5082no—
158D4—5082no—
159D4—5082no—
160D4—5082no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—5084no—
170C2×C4—2542no—
171C2×C4#1452323312yescomplemented
172C2×C4D4×D12712yes—
173C8—5082no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—81no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22—5082no—
184C22—5082no—
185C22—5082no—
186C22#1686117711yescharacteristic
187C22C2×#1452323311yescharacteristic
188C22#1686123411yescharacteristic
189C4—22no—
190C4—22no—
191C4—41no—
192C4—81no—
193C4—5082no—
194C2—21no—
195C2—21no—
196C2—82no—
197C2—10162no—
198C2#2255000911yescharacteristic
199C2#2255004811yescharacteristic
200C2#2255006211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#394×C127C2
socleC22×C127#120
hypercenter#394D127
hypocenterC127#1598
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #22550457C2semidirect
2. #22550010C2semidirect
3. #22549609C2semidirect
4. #22549602C2semidirect
5. #394×C127C2semidirect
71. C2×C4×C127D8semidirect
101. C127#1598semidirect
104. #394D127semidirect
171. C2×C4#14523233semidirect

Maximal subgroups

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

subgroupindex
#225504572
#225500102
#225496092
#225496022
#394×C1272
#1598127