Finite Group Databasebeta

← #30639771

Subgroups — #30639771

Order 16256.

406 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 73 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×#16861235C212yesmaximal, complemented
2#22550457C211yesmaximal, characteristic, complemented
3C2×#16861137C212yesmaximal, complemented
4#22549645C211yesmaximal, characteristic, complemented
5#390×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C2212yes—
7#16861235—24no—
8#16861235—24no—
9C2×#14523236—22no—
10C2×#14523233—22no—
11C2×#14523232C2212yes—
12#16861189—22no—
13#16861184—22no—
14#16861183—22no—
15#16861137—24no—
16C22×D4×C127C2211yescharacteristic
17C2×C4⋊C4×C127C2211yescharacteristic
18C2×C22⋊C4×C127—22no—
19C2×C42×C127C2211yescharacteristic
20#14523248—22no—
21#14523248—22no—
22#14523248—22no—
23#14523248—22no—
24C2×Dic254D412yes—
25C2×D508—22no—
26C2×D508D412yes—
27C2×C22⋊D127—22no—
28C2×C4×D127—22no—
29#14523236—28no—
30#14523233—28no—
31#14523232—22no—
32#14523232D414yescomplemented
33C2×C127⋊C8—22no—
34C4⋊Dic127—22no—
35(C2×D127)⋊C4—24no—
36C24×C127—21no—
37C2×D4×C127—22no—
38C2×D4×C127—22no—
39C2×D4×C127—21no—
40C2×D4×C127D412yescomplemented
41C22×C4×C127—21no—
42C22×C4×C127D412yes—
43C22×C4×C127C2311yescharacteristic
44C4⋊C4×C127—22no—
45C22⋊C4×C127—44no—
46C42×C127—22no—
47C22×D127—22no—
48Dic254—22no—
49Dic254—22no—
50D508—22no—
51D508—22no—
52D508—82no—
53C22⋊D127—42no—
54C22⋊D127—42no—
55C2×Dic127—22no—
56C4×D127—42no—
57C4×D127—42no—
58C127⋊C8—24no—
59C23×C127—22no—
60C23×C127—42no—
61C23×C127—42no—
62C23×C127—41no—
63C23×C127—41no—
64C23×C127C2×D411yescharacteristic
65D4×C127—24no—
66D4×C127—42no—
67C2×C4×C127—22no—
68C2×C4×C127—22no—
69C2×C4×C127—41no—
70C2×C4×C127—41no—
71C2×C4×C127—42no—
72C2×C4×C127C2×D412yes—
73C2×C4×C127C2×D412yes—
74C2×C4×C127C2×D411yescharacteristic
75C2×C4×C127#4111yescharacteristic
76C22×C127—21no—
77C22×C127—21no—
78C22×C127—44no—
79C22×C127—42no—
80C22×C127—82no—
81C22×C127—81no—
82C22×C127—81no—
83C22×C127#12011yescharacteristic
84C22×C127#12111yescharacteristic
85C22×C127#12711yescharacteristic
86C2×D127—42no—
87C2×D127—42no—
88C2×D127—42no—
89C4×C127—42no—
90C4×C127—42no—
91C4×C127#12012yes—
92C4×C127#12112yes—
93Dic127—42no—
94C2×C127—21no—
95C2×C127—21no—
96C2×C127—82no—
97C2×C127#39211yescharacteristic
98C2×C127#45311yescharacteristic
99C2×C127#45411yescharacteristic
100D127—82no—
101#2006—1271nomaximal, Sylow
102C127#200611yescharacteristic, complemented, Sylow
103C2×AGL(1,ℤ/8)—1272no—
104#537—1271no—
105C2×#104—1272no—
106#390D12711yescharacteristic, complemented
107#343—1271no—
108C2×#41—1272no—
109C22×D4C2×D12711yescharacteristic
110AGL(1,ℤ/8)—2544no—
111AGL(1,ℤ/8)—2544no—
112C2×SD16—2542no—
113C2×D8—2542no—
114C2×M16—1272no—
115C22⋊Q8—2542no—
116#121—2542no—
117#121—2542no—
118C2×C4⋊C4C2×D12711yescharacteristic
119C2×C22⋊C4—22no—
120C2×C42C2×D12711yescharacteristic
121#104—2544no—
122C24—21no—
123#41—2542no—
124#41—2542no—
125#41—2542no—
126#41—2542no—
127C2×Q8—1272no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—1272no—
132C2×D4—2542no—
133C2×D4—2542no—
134C2×D4D50812yescomplemented
135C22×C4—21no—
136C22×C4—2542no—
137C22×C4C22⋊D12712yes—
138C22×C4C22×D12711yescharacteristic
139SD16—2548no—
140D8—2548no—
141M16—1274no—
142M16—2542no—
143C2×C8—2542no—
144C4⋊C4—22no—
145C4⋊C4—2542no—
146C22⋊C4—44no—
147C22⋊C4—2544no—
148C42—22no—
149C23—22no—
150C23—42no—
151C23—42no—
152C23—41no—
153C23—41no—
154C23—2542no—
155C23C2×C22⋊D12711yescharacteristic
156Q8—2542no—
157Q8—2542no—
158D4—24no—
159D4—42no—
160D4—10162no—
161D4—5082no—
162D4—5082no—
163D4—2542no—
164D4—2542no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—5082no—
171C2×C4—5082no—
172C2×C4—2542no—
173C2×C4D4×D12712yes—
174C2×C4D4×D12712yes—
175C2×C4C2×D50811yescharacteristic
176C2×C4#1452324811yescharacteristic
177C8—2544no—
178C22—21no—
179C22—21no—
180C22—44no—
181C22—42no—
182C22—82no—
183C22—81no—
184C22—81no—
185C22—5082no—
186C22—5082no—
187C22—5082no—
188C22#1686117711yescharacteristic
189C22#1686118311yescharacteristic
190C22#1686120511yescharacteristic
191C4—42no—
192C4—42no—
193C4—5082no—
194C4#1686117912yes—
195C4#1686118212yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—10162no—
200C2#2254978511yescharacteristic
201C2#2255004311yescharacteristic
202C2#2255004811yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#390×C127C2
socleC22×C127#120
hypercenter#390D127
hypocenterC127#2006
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#16861235C2semidirect
2. #22550457C2semidirect
3. C2×#16861137C2semidirect
4. #22549645C2semidirect
5. #390×C127C2semidirect
32. #14523232D4semidirect
40. C2×D4×C127D4semidirect
102. C127#2006semidirect
106. #390D127semidirect
134. C2×D4D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
#225504572
C2×#168611372
#225496452
#390×C1272
#2006127