Finite Group Databasebeta

← #30651304

Subgroups — #30651304

Order 16256.

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

normal subgroups only

This group has 336 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#22550566C212yesmaximal
2#22550278C211yesmaximal, characteristic, complemented
3#22550218C212yesmaximal, complemented
4#22550212C212yesmaximal, complemented
5#22550184C212yesmaximal, complemented
6#22550134C212yesmaximal, complemented
7C4×Dic508C211yesmaximal, characteristic, complemented
8#22549918C212yesmaximal, complemented
9#436×C127C211yesmaximal, characteristic
10C2×Dic508C2211yescharacteristic
11#16861202—22no—
12#16861201C2212yes—
13#16861199—22no—
14#16861190—22no—
15#16861188C2212yes—
16#16861188C2212yes—
17C4×Dic254C2212yes—
18#16861168C2212yes—
19#16861165—22no—
20C2×C4⋊Dic127C2212yes—
21#16861143C2211yescharacteristic
22#16861143C2211yescharacteristic
23#16861143C2211yescharacteristic
24#16861143C2212yes—
25#16861141C2212yes—
26#16861141C2212yes—
27#16861134C2212yes—
28#16861134C2212yes—
29#16861134C2212yes—
30C2×M16×C127C2212yes—
31C4×D4×C127C2211yescharacteristic
32C4⋊C8×C127C2211yescharacteristic
33C22⋊C8×C127C2212yes—
34C4×C8×C127C2211yescharacteristic
35C2×Dic254C2312yes—
36C22×Dic127—22no—
37Dic508—22no—
38C4⋊Dic127—22no—
39C4⋊Dic127C2312yes—
40C4⋊Dic127C2312yes—
41C4⋊Dic127C2312yes—
42Dic127⋊C4—22no—
43Dic127⋊C4—22no—
44Dic127⋊C4—22no—
45Dic127⋊C4—22no—
46Dic127⋊C4—22no—
47C22⋊Dic127—22no—
48C22⋊Dic127—22no—
49C22⋊Dic127—22no—
50C4×Dic127—22no—
51C4×Dic127—22no—
52C2×D4×C127D411yescharacteristic
53C22×C4×C127C2312yes—
54M16×C127—24no—
55C2×C8×C127C2311yescharacteristic
56C2×C8×C127C2312yes—
57C2×C8×C127C2311yescharacteristic
58C4⋊C4×C127D411yescharacteristic
59C22⋊C4×C127D412yes—
60C42×C127C2311yescharacteristic
61Dic254—22no—
62Dic254—22no—
63C2×Dic127—22no—
64C2×Dic127—22no—
65C2×Dic127—22no—
66C2×Dic127—22no—
67C2×Dic127—44no—
68C23×C127C2×D412yes—
69D4×C127—41no—
70C2×C4×C127—22no—
71C2×C4×C127—22no—
72C2×C4×C127C2×D411yescharacteristic
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127C2×D412yes—
75C2×C4×C127C2411yescharacteristic
76C8×C127—22no—
77C8×C127—21no—
78C8×C127#4112yes—
79C22×C127—22no—
80C22×C127—22no—
81C22×C127—22no—
82C22×C127C22×D411yescharacteristic
83C4×C127—22no—
84C4×C127—21no—
85C4×C127—21no—
86C4×C127C2×#4111yescharacteristic
87C4×C1272−1+411yescharacteristic
88Dic127—42no—
89Dic127—42no—
90Dic127—42no—
91Dic127—42no—
92C2×C127—42no—
93C2×C127#54711yescharacteristic
94C2×C127#57511yescharacteristic
95C2×C127#57811yescharacteristic
96#3352—1271nomaximal, Sylow
97C127#335211yescharacteristic, complemented, Sylow
98#552—1272no—
99#499—1271no—
100#485—1272no—
101#484—1272no—
102#479—1272no—
103#470—1272no—
104C4×Q16—1271no—
105#436D12711yescharacteristic
106#428—1272no—
107C2×Q16—1271no—
108C2×M16C2×D12712yes—
109#126—2542no—
110#125—1272no—
111#125—2542no—
112#123—2542no—
113C22⋊Q8—1272no—
114C22⋊Q8—1272no—
115C4×Q8—1272no—
116C4×D4C2×D12711yescharacteristic
117#117—1272no—
118#117—2542no—
119C2×C4⋊C4—1272no—
120#107—1271no—
121#107—1272no—
122#107—1271no—
123#107—1271no—
124#106—1272no—
125#106—1272no—
126C4⋊C8C2×D12711yescharacteristic
127#103—1272no—
128#103—1272no—
129#103—1272no—
130C22⋊C8C2×D12712yes—
131C4×C8C2×D12711yescharacteristic
132C2×Q8—1272no—
133C2×D4D50811yescharacteristic
134C22×C4—2542no—
135C22×C4C22×D12712yes—
136Q16—2542no—
137M16—24no—
138C2×C8C22×D12711yescharacteristic
139C2×C8C22×D12712yes—
140C2×C8C22×D12711yescharacteristic
141C4⋊C4—1272no—
142C4⋊C4—1272no—
143C4⋊C4—1272no—
144C4⋊C4—2542no—
145C4⋊C4—2542no—
146C4⋊C4—2542no—
147C4⋊C4—2542no—
148C4⋊C4—2542no—
149C4⋊C4—2542no—
150C4⋊C4D50811yescharacteristic
151C22⋊C4—2542no—
152C22⋊C4—2542no—
153C22⋊C4—2542no—
154C22⋊C4D50812yes—
155C42—2542no—
156C42—2542no—
157C42C22×D12711yescharacteristic
158C23C2×D50812yes—
159Q8—2542no—
160Q8—2542no—
161D4—41no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—5084no—
165C2×C4—2542no—
166C2×C4—2542no—
167C2×C4—2542no—
168C2×C4—2542no—
169C2×C4C2×D50811yescharacteristic
170C2×C4C2×D50812yes—
171C2×C4C2×D50811yescharacteristic
172C2×C4C23×D12711yescharacteristic
173C8—22no—
174C8—21no—
175C8#1452324712yes—
176C22—22no—
177C22—22no—
178C22—22no—
179C22C22×D50811yescharacteristic
180C4—22no—
181C4—21no—
182C4—21no—
183C4—5082no—
184C4—5082no—
185C4—5082no—
186C4—5082no—
187C4C2×#1452324711yescharacteristic
188C4#1686125611yescharacteristic
189C2—42no—
190C2#2255053011yescharacteristic
191C2#2255067611yescharacteristic
192C2#2255069311yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#436×C127C2
socleC22×C127C22×D4
prime radicalC2×D4×C127D4
hypercenter#436D127
hypocenterC127#3352
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
2. #22550278C2semidirect
3. #22550218C2semidirect
4. #22550212C2semidirect
5. #22550184C2semidirect
6. #22550134C2semidirect
7. C4×Dic508C2semidirect
8. #22549918C2semidirect
97. C127#3352semidirect

Maximal subgroups

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

subgroupindex
#225505662
#225502782
#225502182
#225502122
#225501842
#225501342
C4×Dic5082
#225499182
#436×C1272
#3352127