Finite Group Databasebeta

← #30637854

Subgroups — #30637854

Order 16256.

256 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 55 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#22550546C211yesmaximal, characteristic
2#22550168C211yesmaximal, characteristic, complemented
3#22550163C211yesmaximal, characteristic, complemented
4#22549620C211yesmaximal, characteristic, complemented
5#22549608C211yesmaximal, characteristic, complemented
6#22549605C211yesmaximal, characteristic, complemented
7#492×C127C211yesmaximal, characteristic, complemented
8C2×#14523247—21no—
9#16861205—21no—
10#16861181—22no—
11#16861180—21no—
12C4×Dic254C2211yescharacteristic
13C4×D508C2211yescharacteristic
14D4×Dic127—21no—
15#16861142—21no—
16#16861140—21no—
17C4⋊(C127⋊C8)C2211yescharacteristic
18C4⋊(C127⋊C8)C2211yescharacteristic
19#16861132—21no—
20#16861129—21no—
21C2×SD16×C127—22no—
22C4⋊Q8×C127C2211yescharacteristic
23#127×C127C2211yescharacteristic
24C4×C8×C127C2211yescharacteristic
25#14523247—42no—
26#14523247—42no—
27C2×Dic254D411yescharacteristic
28C2×D508D411yescharacteristic
29C2×C22⋊D127—22no—
30C2×C22⋊D127—22no—
31C22×Dic127—22no—
32C2×C4×D127—21no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127D411yescharacteristic
36C4⋊Dic127D411yescharacteristic
37Dic127⋊C4—21no—
38C22⋊Dic127—22no—
39(C2×D127)⋊C4—21no—
40C4×Dic127—21no—
41C2×Q8×C127—21no—
42C2×D4×C127—21no—
43C2×D4×C127—21no—
44C2×D4×C127D412yes—
45SD16×C127—44no—
46C2×C8×C127—21no—
47C4⋊C4×C127—21no—
48C4⋊C4×C127—21no—
49C42×C127C2311yescharacteristic
50C22×D127—21no—
51Dic254—21no—
52Dic254SD1612yescomplemented
53D508—21no—
54D508—21no—
55C22⋊D127—44no—
56C22⋊D127—82no—
57C2×Dic127—21no—
58C2×Dic127—21no—
59C2×Dic127—21no—
60C2×Dic127—44no—
61C4×D127—41no—
62C4×D127—41no—
63C127⋊C8—41no—
64C127⋊C8—41no—
65C23×C127—22no—
66Q8×C127—42no—
67D4×C127—42no—
68D4×C127—41no—
69D4×C127—41no—
70D4×C127—41no—
71D4×C127—41no—
72C2×C4×C127—41no—
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127C2×D411yescharacteristic
75C2×C4×C127C2×D411yescharacteristic
76C8×C127—22no—
77C22×C127—42no—
78C22×C127—42no—
79C22×C127—42no—
80C22×C127#12011yescharacteristic
81C2×D127—41no—
82C2×D127—41no—
83C2×D127—41no—
84C4×C127—21no—
85C4×C127—81no—
86C4×C127C2×SD1611yescharacteristic
87C4×C127#13511yescharacteristic
88C4×C127AGL(1,ℤ/8)11yescharacteristic
89C4×C127#13711yescharacteristic
90Dic127—22no—
91Dic127—41no—
92Dic127—41no—
93C2×C127—82no—
94C2×C127#45011yescharacteristic
95C2×C127#45211yescharacteristic
96C2×C127#45311yescharacteristic
97D127—81no—
98#1631—1271nomaximal, Sylow
99C127#163111yescharacteristic, complemented, Sylow
100#550—1271no—
101#492D12711yescharacteristic, complemented
102#477—1271no—
103#476—1271no—
104#331—1271no—
105Q8⋊C8—1271no—
106D4⋊C8—1271no—
107C2×#41—2541no—
108C2×SD16—22no—
109C4⋊Q8C2×D12711yescharacteristic
110#127—2541no—
111#127C2×D12711yescharacteristic
112#121—2541no—
113#121—2542no—
114C4×Q8—1271no—
115C4×D4—1271no—
116C4×D4—2541no—
117#107—2541no—
118#106—2541no—
119C4⋊C8—1271no—
120C4⋊C8—1271no—
121#103—2541no—
122#102—2541no—
123C4×C8C2×D12711yescharacteristic
124#41—5082no—
125#41—5082no—
126C2×Q8—21no—
127C2×Q8—1271no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—1271no—
131C2×D4—2542no—
132C2×D4—2542no—
133C2×D4C22⋊D12712yes—
134C22×C4—2542no—
135C22×C4—2541no—
136SD16—44no—
137C2×C8—21no—
138C2×C8—2541no—
139C2×C8—2541no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—1271no—
143C4⋊C4—1271no—
144C4⋊C4—2541no—
145C22⋊C4—2542no—
146C22⋊C4—2541no—
147C42—2541no—
148C42C22×D12711yescharacteristic
149C23—22no—
150C23—2541no—
151Q8—42no—
152Q8—1272no—
153Q8—2541no—
154D4—42no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—41no—
159D4—10162no—
160D4—5084no—
161D4—2541no—
162D4—2541no—
163C2×C4—41no—
164C2×C4—5084no—
165C2×C4—5081no—
166C2×C4—5081no—
167C2×C4—2541no—
168C2×C4—2541no—
169C2×C4—2541no—
170C2×C4D4×D12711yescharacteristic
171C2×C4D4×D12711yescharacteristic
172C2×C4C2×C22⋊D12711yescharacteristic
173C8—22no—
174C8—5081no—
175C8—5081no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—5081no—
180C22—5081no—
181C22—5081no—
182C22#1686117711yescharacteristic
183C4—21no—
184C4—81no—
185C4—5081no—
186C4—5081no—
187C4—2542no—
188C4SD16×D12711yescharacteristic
189C4#1686123011yescharacteristic
190C4#1686123611yescharacteristic
191C4#1686123811yescharacteristic
192C2—82no—
193C2—10161no—
194C2#2255002511yescharacteristic
195C2#2255003511yescharacteristic
196C2#2255004311yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#492×C127C2
socleC22×C127#120
prime radical#22550546C2
hypercenter#492D127
hypocenterC127#1631
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
2. #22550168C2semidirect
3. #22550163C2semidirect
4. #22549620C2semidirect
5. #22549608C2semidirect
6. #22549605C2semidirect
7. #492×C127C2semidirect
52. Dic254SD16semidirect
99. C127#1631semidirect
101. #492D127semidirect

Maximal subgroups

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

subgroupindex
#225505462
#225501682
#225501632
#225496202
#225496082
#225496052
#492×C1272
#1631127