Finite Group Databasebeta

← #30639779

Subgroups — #30639779

Order 16256.

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

normal subgroups only

This group has 370 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×#16861242C212yesmaximal, complemented
2#22550457C211yesmaximal, characteristic
3C2×#16861135C212yesmaximal, complemented
4#22549781C211yesmaximal, characteristic, complemented
5#343×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C2212yes—
7C22×Dic254C2211yescharacteristic
8#16861242—24no—
9#16861242—24no—
10C2×Dic508—22no—
11C2×#14523237—22no—
12#16861189—22no—
13#16861184—22no—
14#16861183—22no—
15C2×C4×Dic127C2211yescharacteristic
16#16861135—24no—
17C127⋊#95—22no—
18C2×M16×C127C2212yes—
19C2×C4⋊C4×C127C2211yescharacteristic
20#14523248—22no—
21#14523248—22no—
22#14523248—22no—
23#14523248—22no—
24C2×Dic254—22no—
25C2×Dic254—22no—
26C2×Dic254—21no—
27C2×Dic254D412yes—
28C2×Dic254D412yescomplemented
29C2×D508—22no—
30C2×D508D412yes—
31C2×C22⋊D127—22no—
32C22×Dic127—21no—
33C22×Dic127—21no—
34C2×C4×D127—22no—
35Dic508—28no—
36#14523237—28no—
37C4⋊Dic127—22no—
38(C2×D127)⋊C4—24no—
39C4×Dic127—22no—
40C22×C4×C127D412yes—
41C22×C4×C127C2311yescharacteristic
42M16×C127—22no—
43M16×C127D414yescomplemented
44C2×C8×C127—22no—
45C4⋊C4×C127—22no—
46C22×D127—22no—
47Dic254—24no—
48Dic254—22no—
49Dic254—22no—
50Dic254—42no—
51D508—22no—
52D508—22no—
53D508—82no—
54C22⋊D127—42no—
55C22⋊D127—42no—
56C2×Dic127—22no—
57C2×Dic127—22no—
58C2×Dic127—22no—
59C2×Dic127—41no—
60C2×Dic127—41no—
61C2×Dic127—41no—
62C2×Dic127—41no—
63C4×D127—42no—
64C4×D127—42no—
65C23×C127C2×D411yescharacteristic
66C2×C4×C127—22no—
67C2×C4×C127—42no—
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×D412yes—
70C2×C4×C127C2×D412yes—
71C2×C4×C127#4111yescharacteristic
72C8×C127—24no—
73C22×C127—21no—
74C22×C127—21no—
75C22×C127#12011yescharacteristic
76C22×C127#12111yescharacteristic
77C22×C127#12711yescharacteristic
78C2×D127—42no—
79C2×D127—42no—
80C2×D127—42no—
81C4×C127—42no—
82C4×C127#12012yes—
83C4×C127#12112yes—
84Dic127—42no—
85Dic127—42no—
86Dic127—42no—
87C2×C127—21no—
88C2×C127—21no—
89C2×C127#39211yescharacteristic
90C2×C127#45511yescharacteristic
91C2×C127#45611yescharacteristic
92D127—82no—
93#2007—1271nomaximal, Sylow
94C127#200711yescharacteristic, complemented, Sylow
95C2×#137—1272no—
96#537—1271no—
97C2×#104—1272no—
98#391—1271no—
99#343D12711yescharacteristic, complemented
100C2×#41—1272no—
101C22×Q8—1271no—
102#137—2544no—
103#137—2544no—
104C2×Q16—2542no—
105C2×SD16—2542no—
106C2×M16C2×D12712yes—
107C22⋊Q8—2542no—
108#121—2542no—
109#121—2542no—
110C2×C4⋊C4C2×D12711yescharacteristic
111C2×C42—1271no—
112#104—2544no—
113#95—2542no—
114#41—2542no—
115#41—2542no—
116#41—2542no—
117#41—2542no—
118C2×Q8—1272no—
119C2×Q8—1272no—
120C2×Q8—2542no—
121C2×Q8—2542no—
122C2×Q8—2541no—
123C2×D4—1272no—
124C2×D4—2542no—
125C2×D4—2542no—
126C22×C4—2541no—
127C22×C4—2541no—
128C22×C4—2542no—
129C22×C4D50812yes—
130C22×C4C22×D12711yescharacteristic
131Q16—2548no—
132SD16—2548no—
133M16—22no—
134M16D50814yescomplemented
135C2×C8—22no—
136C4⋊C4—22no—
137C4⋊C4—2542no—
138C22⋊C4—2544no—
139C42—2542no—
140C23—2542no—
141C23C2×D50811yescharacteristic
142Q8—5082no—
143Q8—2542no—
144Q8—2544no—
145Q8—2542no—
146D4—10162no—
147D4—5082no—
148D4—5082no—
149D4—2542no—
150D4—2542no—
151C2×C4—22no—
152C2×C4—42no—
153C2×C4—5081no—
154C2×C4—5081no—
155C2×C4—5081no—
156C2×C4—5081no—
157C2×C4—5082no—
158C2×C4—5082no—
159C2×C4—2542no—
160C2×C4—2542no—
161C2×C4—2542no—
162C2×C4D4×D12712yes—
163C2×C4D4×D12711yescharacteristic
164C2×C4C2×D50812yes—
165C2×C4#1452324911yescharacteristic
166C8—24no—
167C22—21no—
168C22—21no—
169C22—5082no—
170C22—5082no—
171C22—5082no—
172C22#1686117911yescharacteristic
173C22#1686118411yescharacteristic
174C22#1686120611yescharacteristic
175C4—42no—
176C4—5082no—
177C4—5082no—
178C4—5082no—
179C4#1686117912yes—
180C4#1686118412yes—
181C2—21no—
182C2—21no—
183C2—10162no—
184C2#2254978611yescharacteristic
185C2#2255005611yescharacteristic
186C2#2255006111yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#343×C127C2
socleC22×C127#120
prime radical#22550457C2
hypercenter#343D127
hypocenterC127#2007
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×#16861242C2semidirect
3. C2×#16861135C2semidirect
4. #22549781C2semidirect
5. #343×C127C2semidirect
28. C2×Dic254D4semidirect
43. M16×C127D4semidirect
94. C127#2007semidirect
99. #343D127semidirect
134. M16D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168612422
#225504572
C2×#168611352
#225497812
#343×C1272
#2007127