Finite Group Databasebeta

← #30637990

Subgroups — #30637990

Order 16256.

234 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550541C211yesmaximal, characteristic
2#22550158C211yesmaximal, characteristic, complemented
3#22550078C211yesmaximal, characteristic
4#22549622C211yesmaximal, characteristic, complemented
5#22549608C211yesmaximal, characteristic, complemented
6#22549605C211yesmaximal, characteristic
7#498×C127C211yesmaximal, characteristic, complemented
8C2×Q8×D127—21no—
9C2×#14523235—21no—
10#16861208—21no—
11#16861187—22no—
12#16861185—21no—
13C4×Dic254C2211yescharacteristic
14C4×D508C2211yescharacteristic
15#16861171—21no—
16#16861140—21no—
17C4⋊(C127⋊C8)C2211yescharacteristic
18C4⋊(C127⋊C8)C2211yescharacteristic
19#16861133—21no—
20#16861132—21no—
21C4⋊Q8×C127C2211yescharacteristic
22C4⋊Q8×C127C2211yescharacteristic
23#106×C127—22no—
24C4×C8×C127C2211yescharacteristic
25C2×Dic254—22no—
26C2×Dic254D411yescharacteristic
27Q8×D127—42no—
28Q8×D127—42no—
29C2×D508D411yescharacteristic
30C2×C4×D127—22no—
31C2×C4×D127—21no—
32#14523235—42no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127D411yescharacteristic
36C4⋊Dic127D411yescharacteristic
37Dic127⋊C4—22no—
38Dic127⋊C4—21no—
39(C2×D127)⋊C4—22no—
40(C2×D127)⋊C4—21no—
41C4×Dic127—21no—
42C2×Q8×C127—21no—
43C2×Q8×C127—21no—
44C2×C8×C127—21no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127—21no—
47C4⋊C4×C127—21no—
48C4⋊C4×C127D412yes—
49C42×C127C2311yescharacteristic
50C22×D127—21no—
51Dic254—21no—
52Dic254—44no—
53Dic254SD1612yescomplemented
54D508—21no—
55D508SD1612yes—
56C2×Dic127—21no—
57C2×Dic127—21no—
58C2×Dic127—21no—
59C4×D127—44no—
60C4×D127—41no—
61C4×D127—41no—
62C127⋊C8—41no—
63C127⋊C8—41no—
64Q8×C127—41no—
65Q8×C127—41no—
66Q8×C127—41no—
67Q8×C127—41no—
68C2×C4×C127—22no—
69C2×C4×C127—41no—
70C2×C4×C127C2×D411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127C2×D411yescharacteristic
73C8×C127—22no—
74C22×C127#12011yescharacteristic
75C2×D127—22no—
76C2×D127—41no—
77C2×D127—41no—
78C4×C127—21no—
79C4×C127—42no—
80C4×C127—81no—
81C4×C127C2×SD1611yescharacteristic
82C4×C127C2×SD1611yescharacteristic
83C4×C127AGL(1,ℤ/8)11yescharacteristic
84C4×C127#13711yescharacteristic
85Dic127—22no—
86Dic127—41no—
87Dic127—41no—
88C2×C127#44811yescharacteristic
89C2×C127#45011yescharacteristic
90C2×C127#45611yescharacteristic
91D127—42no—
92#1651—1271nomaximal, Sylow
93C127#165111yescharacteristic, complemented, Sylow
94D4×Q8—1271no—
95#498D12711yescharacteristic, complemented
96#475—1271no—
97#461—1271no—
98#332—1271no—
99Q8⋊C8—1271no—
100D4⋊C8—1271no—
101C22×Q8—2541no—
102C2×SD16—2541no—
103C4⋊Q8—2541no—
104C4⋊Q8C2×D12711yescharacteristic
105C4⋊Q8C2×D12711yescharacteristic
106C22⋊Q8—2541no—
107C22⋊Q8—2542no—
108C4×Q8—1271no—
109C4×D4—1271no—
110C4×D4—2541no—
111#106—22no—
112#106—2541no—
113C4⋊C8—1271no—
114C4⋊C8—1271no—
115#103—2541no—
116#103—2541no—
117C4×C8C2×D12711yescharacteristic
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—1271no—
121C2×Q8—5082no—
122C2×Q8—5082no—
123C2×Q8—2542no—
124C2×D4—1271no—
125C22×C4—2542no—
126C22×C4—2541no—
127SD16—5082no—
128C2×C8—21no—
129C2×C8—2541no—
130C2×C8—2541no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—1271no—
135C4⋊C4—1271no—
136C4⋊C4—2542no—
137C4⋊C4—2541no—
138C4⋊C4C22⋊D12712yes—
139C22⋊C4—2542no—
140C22⋊C4—2541no—
141C42—2541no—
142C42C22×D12711yescharacteristic
143C23—2541no—
144Q8—41no—
145Q8—41no—
146Q8—41no—
147Q8—41no—
148Q8—1272no—
149Q8—5084no—
150Q8—2541no—
151D4—1272no—
152D4—2541no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—5084no—
156C2×C4—5081no—
157C2×C4—5081no—
158C2×C4—2541no—
159C2×C4—2541no—
160C2×C4—2541no—
161C2×C4D4×D12711yescharacteristic
162C2×C4D4×D12711yescharacteristic
163C2×C4C2×C22⋊D12711yescharacteristic
164C8—22no—
165C8—5081no—
166C8—5081no—
167C22—5081no—
168C22—5081no—
169C22—2542no—
170C22#1686117711yescharacteristic
171C4—21no—
172C4—42no—
173C4—81no—
174C4—5081no—
175C4—5081no—
176C4—2542no—
177C4SD16×D12711yescharacteristic
178C4SD16×D12711yescharacteristic
179C4#1686123611yescharacteristic
180C4#1686123811yescharacteristic
181C2—5082no—
182C2#2255001311yescharacteristic
183C2#2255002511yescharacteristic
184C2#2255005811yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#498×C127C2
socleC22×C127#120
prime radicalC2×D508D4
hypercenter#498D127
hypocenterC127#1651
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
2. #22550158C2semidirect
4. #22549622C2semidirect
5. #22549608C2semidirect
7. #498×C127C2semidirect
53. Dic254SD16semidirect
93. C127#1651semidirect
95. #498D127semidirect

Maximal subgroups

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

subgroupindex
#225505412
#225501582
#225500782
#225496222
#225496082
#225496052
#498×C1272
#1651127