Finite Group Databasebeta

← #30649022

Subgroups — #30649022

Order 16256.

382 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 121 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#22550481C211yesmaximal, characteristic
2#22550353C211yesmaximal, characteristic, complemented
3#22550201C212yesmaximal, complemented
4#22550184C212yesmaximal, complemented
5#22550077C212yesmaximal, complemented
6#22550017C212yesmaximal, complemented
7#22549978C212yesmaximal, complemented
8C4×#14523237C212yesmaximal, complemented
9#433×C127C211yesmaximal, characteristic, complemented
10C22×Dic254C2211yescharacteristic
11C2×#14523237C2212yes—
12C2×#14523237C2212yes—
13#16861206C2211yescharacteristic
14#16861201C2211yescharacteristic
15#16861194—22no—
16#16861183C2212yes—
17#16861182—22no—
18C4×Dic254C2212yes—
19C4×Dic254C2212yes—
20#16861174—22no—
21C4×D508C2212yes—
22#16861168C2211yescharacteristic
23#16861165—22no—
24#16861143C2212yes—
25#16861141C2212yes—
26#16861134C2212yes—
27#16861134C2212yes—
28#16861131C2212yes—
29#16861131C2212yes—
30#117×C127C2211yescharacteristic
31C4⋊C8×C127C2212yes—
32C22⋊C8×C127C2212yes—
33#97×C127C2211yescharacteristic
34C4×C8×C127C2211yescharacteristic
35C2×Dic254—24no—
36C2×Dic254—21no—
37C2×Dic254C2312yes—
38C2×D508—22no—
39C2×D508C2312yes—
40C2×C22⋊D127—22no—
41C22×Dic127—21no—
42C2×C4×D127—22no—
43#14523237—24no—
44#14523237—24no—
45C4⋊Dic127C2312yes—
46C4⋊Dic127C2312yes—
47Dic127⋊C4—22no—
48Dic127⋊C4—22no—
49Dic127⋊C4—22no—
50C22⋊Dic127—21no—
51(C2×D127)⋊C4—22no—
52(C2×D127)⋊C4—22no—
53C4×Dic127—22no—
54C4×Dic127—22no—
55C4×Dic127—21no—
56C22×C4×C127C2311yescharacteristic
57C2×C8×C127C2312yes—
58C2×C8×C127C2312yes—
59C4⋊C4×C127D412yes—
60C22⋊C4×C127D412yes—
61C42×C127C2311yescharacteristic
62C42×C127C2311yescharacteristic
63C22×D127—22no—
64Dic254—24no—
65Dic254—22no—
66Dic254#4114yes—
67D508—22no—
68D508—22no—
69D508—82no—
70C22⋊D127—82no—
71C2×Dic127—24no—
72C2×Dic127—22no—
73C2×Dic127—22no—
74C2×Dic127—22no—
75C4×D127—82no—
76C23×C127C2×D411yescharacteristic
77C2×C4×C127—22no—
78C2×C4×C127C2×D412yes—
79C2×C4×C127C2×D412yes—
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2411yescharacteristic
82C8×C127—22no—
83C8×C127—22no—
84C22×C127—21no—
85C22×C127—22no—
86C22×C127C22×D411yescharacteristic
87C2×D127—42no—
88C2×D127—42no—
89C2×D127—42no—
90C4×C127—21no—
91C4×C127—22no—
92C4×C127—22no—
93C4×C127C2×#4112yes—
94Dic127—24no—
95Dic127—42no—
96Dic127—42no—
97C2×C127—41no—
98C2×C127#52511yescharacteristic
99C2×C127#57712yes—
100D127—82no—
101#3126—1271nomaximal, Sylow
102C127#312611yescharacteristic, complemented, Sylow
103#540—1271no—
104#519—1271no—
105#482—1272no—
106#479—1272no—
107#460—1272no—
108#448—1272no—
109#440—1272no—
110C4×SD16—1272no—
111#433D12711yescharacteristic, complemented
112C22×Q8—1271no—
113C2×SD16—1272no—
114C2×SD16—1272no—
115#127—1271no—
116#125—1271no—
117#123—2542no—
118#121—1272no—
119#121—2542no—
120C4×Q8—1272no—
121C4×Q8—1272no—
122C4×Q8—2542no—
123C4×D4—1272no—
124#117—1271no—
125#117—2542no—
126#117C2×D12711yescharacteristic
127#107—1272no—
128#106—1272no—
129C4⋊C8C2×D12712yes—
130#103—1272no—
131#103—1272no—
132#102—1272no—
133#102—1272no—
134C22⋊C8C2×D12712yes—
135#97C2×D12711yescharacteristic
136C4×C8C2×D12711yescharacteristic
137C2×Q8—1272no—
138C2×Q8—2544no—
139C2×Q8—2541no—
140C2×D4—1272no—
141C2×D4—2542no—
142C2×D4—2542no—
143C22×C4—2542no—
144C22×C4—2541no—
145C22×C4C22×D12711yescharacteristic
146SD16—2544no—
147SD16—2544no—
148C2×C8C22×D12712yes—
149C2×C8C22×D12712yes—
150C4⋊C4—1272no—
151C4⋊C4—1272no—
152C4⋊C4—2542no—
153C4⋊C4—2542no—
154C4⋊C4—2542no—
155C4⋊C4D50812yes—
156C22⋊C4—2541no—
157C22⋊C4—2542no—
158C22⋊C4—2542no—
159C22⋊C4D50812yes—
160C42—2541no—
161C42—2542no—
162C42—2542no—
163C42C22×D12711yescharacteristic
164C42C22×D12711yescharacteristic
165C23—2542no—
166C23C2×D50811yescharacteristic
167Q8—1274no—
168Q8—2544no—
169Q8—2542no—
170D4—10162no—
171D4—10162no—
172D4—2542no—
173D4—2542no—
174C2×C4—22no—
175C2×C4—10162no—
176C2×C4—2542no—
177C2×C4—2542no—
178C2×C4—2544no—
179C2×C4—2542no—
180C2×C4C2×D50812yes—
181C2×C4C2×D50812yes—
182C2×C4C2×D50811yescharacteristic
183C2×C4C23×D12711yescharacteristic
184C8—22no—
185C8—22no—
186C22—21no—
187C22—22no—
188C22—5082no—
189C22—5082no—
190C22—5082no—
191C22C22×D50811yescharacteristic
192C4—21no—
193C4—22no—
194C4—22no—
195C4—5082no—
196C4—5082no—
197C4—2544no—
198C4#41×D12712yes—
199C2—41no—
200C2—10162no—
201C2#2255038811yescharacteristic
202C2#2255068812yes—
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#433×C127C2
socleC22×C127C22×D4
prime radical#22550481C2
hypercenter#433D127
hypocenterC127#3126
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
2. #22550353C2semidirect
3. #22550201C2semidirect
4. #22550184C2semidirect
5. #22550077C2semidirect
6. #22550017C2semidirect
7. #22549978C2semidirect
8. C4×#14523237C2semidirect
9. #433×C127C2semidirect
102. C127#3126semidirect
111. #433D127semidirect

Maximal subgroups

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

subgroupindex
#225504812
#225503532
#225502012
#225501842
#225500772
#225500172
#225499782
C4×#145232372
#433×C1272
#3126127