Finite Group Databasebeta

← #30640073

Subgroups — #30640073

Order 16256.

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

normal subgroups only

This group has 246 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
1#22550504C211yesmaximal, characteristic
2#22549916C211yesmaximal, characteristic
3#22549745C211yesmaximal, characteristic
4#22549644C211yesmaximal, characteristic
5#22549637C211yesmaximal, characteristic
6#22549635C211yesmaximal, characteristic
7#428×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9#16861209—21no—
10#16861208—21no—
11#16861201—21no—
12#16861200—21no—
13#16861186—22no—
14#16861168C2211yescharacteristic
15C2×C4⋊Dic127C2211yescharacteristic
16C2×Dic127⋊C4—22no—
17C2×C4×Dic127C2211yescharacteristic
18#16861142—21no—
19#16861140—21no—
20C2×M16×C127C2211yescharacteristic
21#117×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23#107×C127—21no—
24#106×C127—21no—
25C2×Dic254—22no—
26C22×Dic127—21no—
27C22×Dic127D412yes—
28#14523232—21no—
29#14523232Q812yescomplemented
30C2×C127⋊C8—21no—
31C4⋊Dic127—22no—
32C4⋊Dic127—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127D411yescharacteristic
35C4⋊Dic127D411yescharacteristic
36Dic127⋊C4—22no—
37Dic127⋊C4—22no—
38Dic127⋊C4—44no—
39C22⋊Dic127—21no—
40C4×Dic127—21no—
41C4×Dic127—21no—
42C4×Dic127Q812yescomplemented
43C22×C4×C127—21no—
44C22×C4×C127C2311yescharacteristic
45M16×C127—21no—
46M16×C127Q812yescomplemented
47C2×C8×C127—21no—
48C4⋊C4×C127—21no—
49C4⋊C4×C127—21no—
50C4⋊C4×C127D411yescharacteristic
51C4⋊C4×C127D411yescharacteristic
52C22⋊C4×C127—21no—
53C42×C127—21no—
54Dic254—82no—
55C2×Dic127—22no—
56C2×Dic127—22no—
57C2×Dic127—22no—
58C2×Dic127—21no—
59C2×Dic127—21no—
60C2×Dic127—22no—
61C2×Dic127—42no—
62C127⋊C8—22no—
63C23×C127C2×D411yescharacteristic
64C2×C4×C127—21no—
65C2×C4×C127—21no—
66C2×C4×C127—42no—
67C2×C4×C127—41no—
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×Q811yescharacteristic
71C2×C4×C127C2×Q811yescharacteristic
72C2×C4×C127C2×Q811yescharacteristic
73C2×C4×C127#4111yescharacteristic
74C8×C127—22no—
75C22×C127—21no—
76C22×C127—21no—
77C22×C127#12011yescharacteristic
78C22×C127C22⋊Q811yescharacteristic
79C22×C127C4⋊Q811yescharacteristic
80C4×C127—41no—
81C4×C127—41no—
82C4×C127—81no—
83C4×C127C22⋊Q811yescharacteristic
84C4×C127C22⋊Q811yescharacteristic
85C4×C127C4⋊Q811yescharacteristic
86C4×C127C4⋊Q811yescharacteristic
87Dic127—24no—
88Dic127—42no—
89Dic127—41no—
90Dic127—41no—
91C2×C127—21no—
92C2×C127—21no—
93C2×C127#39511yescharacteristic
94C2×C127#45311yescharacteristic
95C2×C127#45611yescharacteristic
96#2060—1271nomaximal, Sylow
97C127#206011yescharacteristic, complemented, Sylow
98#544—1271no—
99#428—1271no—
100#428D12711yescharacteristic
101#384—1271no—
102#343—1271no—
103#339—1271no—
104#339—1271no—
105C2×M16—1271no—
106C2×M16C2×D12711yescharacteristic
107C4⋊Q8—2541no—
108C4⋊Q8—2541no—
109#125—2541no—
110#125—2541no—
111C22⋊Q8—2542no—
112#117—1271no—
113#117C2×D12711yescharacteristic
114C2×C4⋊C4—1271no—
115C2×C4⋊C4—2542no—
116C2×C4⋊C4C2×D12711yescharacteristic
117C2×C42—1271no—
118#107—21no—
119#107—2541no—
120#106—21no—
121#106—2541no—
122C2×Q8—2542no—
123C22×C4—21no—
124C22×C4—1272no—
125C22×C4—2541no—
126C22×C4C22×D12711yescharacteristic
127M16—21no—
128M16—1272no—
129M16—2541no—
130M16Dic25412yescomplemented
131C2×C8—21no—
132C2×C8—2541no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—1271no—
136C4⋊C4—1271no—
137C4⋊C4—5084no—
138C4⋊C4—2542no—
139C4⋊C4—2542no—
140C4⋊C4—2542no—
141C4⋊C4—2541no—
142C4⋊C4—2541no—
143C4⋊C4C22⋊D12711yescharacteristic
144C4⋊C4C22⋊D12711yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—2541no—
147C42—21no—
148C42—1272no—
149C42—2541no—
150C42—2541no—
151C23D4×D12711yescharacteristic
152Q8—10162no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—5082no—
158C2×C4—2541no—
159C2×C4—2541no—
160C2×C4—2542no—
161C2×C4—2542no—
162C2×C4—2542no—
163C2×C4—2542no—
164C2×C4D4×D12711yescharacteristic
165C2×C4C2×C22⋊D12711yescharacteristic
166C2×C4Q8×D12711yescharacteristic
167C2×C4Q8×D12711yescharacteristic
168C2×C4C2×Dic25411yescharacteristic
169C2×C4#1452324811yescharacteristic
170C8—22no—
171C8—2542no—
172C22—21no—
173C22—21no—
174C22#1686117711yescharacteristic
175C22#1686118711yescharacteristic
176C22#1686120811yescharacteristic
177C4—41no—
178C4—41no—
179C4—81no—
180C4—5082no—
181C4—5081no—
182C4—5081no—
183C4—2544no—
184C4#1686118711yescharacteristic
185C4#1686118811yescharacteristic
186C4#1686120711yescharacteristic
187C4#1686120811yescharacteristic
188C2—21no—
189C2—21no—
190C2#2254979611yescharacteristic
191C2#2255004211yescharacteristic
192C2#2255005711yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#428×C127C2
socleC22×C127#120
prime radicalC23×C127C2×D4
hypercenter#428D127
hypocenterC127#2060
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
29. #14523232Q8semidirect
42. C4×Dic127Q8semidirect
46. M16×C127Q8semidirect
97. C127#2060semidirect
130. M16Dic254semidirect

Maximal subgroups

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

subgroupindex
#225505042
#225499162
#225497452
#225496442
#225496372
#225496352
#428×C1272
#2060127