Finite Group Databasebeta

← #30639774

Subgroups — #30639774

Order 16256.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 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×#16861237C212yesmaximal, complemented
2#22550457C211yesmaximal, characteristic, complemented
3C2×#16861135C212yesmaximal, complemented
4#22549777C211yesmaximal, characteristic, complemented
5#343×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C2212yes—
7C22×D508C2211yescharacteristic
8#16861237—24no—
9#16861237—24no—
10C2×#14523237—22no—
11C2×D1016—22no—
12#16861189—22no—
13#16861184—22no—
14#16861183—22no—
15C2×(C2×D127)⋊C4—22no—
16C2×C4×Dic127C2211yescharacteristic
17#16861135—24no—
18C2×M16×C127C2212yes—
19C2×C4⋊C4×C127C2211yescharacteristic
20C23×D127—21no—
21#14523248—22no—
22#14523248—22no—
23#14523248—22no—
24#14523248—22no—
25C2×Dic254D412yes—
26C2×D508—22no—
27C2×D508—22no—
28C2×D508—22no—
29C2×D508—21no—
30C2×D508D412yes—
31C2×D508D412yescomplemented
32C2×C22⋊D127—22no—
33C22×Dic127—21no—
34C2×C4×D127—22no—
35#14523237—28no—
36D1016—28no—
37C4⋊Dic127—22no—
38(C2×D127)⋊C4—24no—
39(C2×D127)⋊C4—44no—
40C4×Dic127—22no—
41C22×C4×C127D412yes—
42C22×C4×C127C2311yescharacteristic
43M16×C127—22no—
44M16×C127D414yescomplemented
45C2×C8×C127—22no—
46C4⋊C4×C127—22no—
47C22×D127—22no—
48C22×D127—22no—
49C22×D127—42no—
50C22×D127—42no—
51C22×D127—41no—
52C22×D127—41no—
53Dic254—22no—
54Dic254—22no—
55D508—22no—
56D508—24no—
57D508—22no—
58D508—42no—
59D508—82no—
60C22⋊D127—42no—
61C22⋊D127—42no—
62C2×Dic127—22no—
63C2×Dic127—22no—
64C2×Dic127—41no—
65C2×Dic127—41no—
66C4×D127—42no—
67C4×D127—42no—
68C23×C127C2×D411yescharacteristic
69C2×C4×C127—22no—
70C2×C4×C127—42no—
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127C2×D412yes—
73C2×C4×C127C2×D412yes—
74C2×C4×C127#4111yescharacteristic
75C8×C127—24no—
76C22×C127—21no—
77C22×C127—21no—
78C22×C127#12011yescharacteristic
79C22×C127#12111yescharacteristic
80C22×C127#12711yescharacteristic
81C2×D127—44no—
82C2×D127—42no—
83C2×D127—42no—
84C2×D127—42no—
85C2×D127—42no—
86C2×D127—82no—
87C2×D127—81no—
88C2×D127—81no—
89C4×C127—42no—
90C4×C127#12012yes—
91C4×C127#12112yes—
92Dic127—42no—
93Dic127—42no—
94C2×C127—21no—
95C2×C127—21no—
96C2×C127#39211yescharacteristic
97C2×C127#45311yescharacteristic
98C2×C127#45411yescharacteristic
99D127—82no—
100D127—82no—
101#2006—1271nomaximal, Sylow
102C127#200611yescharacteristic, complemented, Sylow
103C2×AGL(1,ℤ/8)—1272no—
104#537—1271no—
105C2×#104—1272no—
106#390—1271no—
107#343D12711yescharacteristic, complemented
108C2×#41—1272no—
109C22×D4—1271no—
110AGL(1,ℤ/8)—2544no—
111AGL(1,ℤ/8)—2544no—
112C2×SD16—2542no—
113C2×D8—2542no—
114C2×M16C2×D12712yes—
115C22⋊Q8—2542no—
116#121—2542no—
117#121—2542no—
118C2×C4⋊C4C2×D12711yescharacteristic
119C2×C22⋊C4—2542no—
120C2×C42—1271no—
121#104—2544no—
122C24—2541no—
123#41—2542no—
124#41—2542no—
125#41—2542no—
126#41—2542no—
127C2×Q8—1272no—
128C2×D4—1272no—
129C2×D4—1272no—
130C2×D4—2542no—
131C2×D4—2542no—
132C2×D4—2542no—
133C2×D4—2542no—
134C2×D4—2541no—
135C22×C4—2541no—
136C22×C4—2542no—
137C22×C4D50812yes—
138C22×C4C22×D12711yescharacteristic
139SD16—2548no—
140D8—2548no—
141M16—22no—
142M16D50814yescomplemented
143C2×C8—22no—
144C4⋊C4—22no—
145C4⋊C4—2542no—
146C22⋊C4—5084no—
147C22⋊C4—2544no—
148C42—2542no—
149C23—5082no—
150C23—5082no—
151C23—5081no—
152C23—5081no—
153C23—2542no—
154C23—2542no—
155C23C2×D50811yescharacteristic
156Q8—2542no—
157Q8—2542no—
158D4—10162no—
159D4—5082no—
160D4—5082no—
161D4—5082no—
162D4—2542no—
163D4—2542no—
164D4—2544no—
165C2×C4—22no—
166C2×C4—42no—
167C2×C4—5081no—
168C2×C4—5081no—
169C2×C4—5082no—
170C2×C4—5082no—
171C2×C4—2542no—
172C2×C4—2542no—
173C2×C4D4×D12712yes—
174C2×C4D4×D12711yescharacteristic
175C2×C4C2×D50812yes—
176C2×C4#1452324911yescharacteristic
177C8—24no—
178C22—21no—
179C22—21no—
180C22—10162no—
181C22—10161no—
182C22—10161no—
183C22—5084no—
184C22—5082no—
185C22—5082no—
186C22—5082no—
187C22—5082no—
188C22#1686117911yescharacteristic
189C22#1686118411yescharacteristic
190C22#1686120611yescharacteristic
191C4—42no—
192C4—5082no—
193C4—5082no—
194C4#1686117912yes—
195C4#1686118412yes—
196C2—21no—
197C2—21no—
198C2—10162no—
199C2—10162no—
200C2#2254978611yescharacteristic
201C2#2255004611yescharacteristic
202C2#2255005111yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#343×C127C2
socleC22×C127#120
hypercenter#343D127
hypocenterC127#2006
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. C2×#16861237C2semidirect
2. #22550457C2semidirect
3. C2×#16861135C2semidirect
4. #22549777C2semidirect
5. #343×C127C2semidirect
31. C2×D508D4semidirect
44. M16×C127D4semidirect
102. C127#2006semidirect
107. #343D127semidirect
142. M16D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168612372
#225504572
C2×#168611352
#225497772
#343×C1272
#2006127