Finite Group Databasebeta

← #30652870

Subgroups — #30652870

Order 16256.

1058 conjugacy classes of subgroups; only the 451 normal subgroups are shown.

normal subgroups only

This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 451 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#22550626C213yesmaximal, complemented
2#22550620C216yesmaximal, complemented
3#22550615C216yesmaximal, complemented
4#22550425C213yesmaximal, complemented
5#22550405C216yesmaximal, complemented
6#22550382C216yesmaximal, complemented
7H(3,𝔽4)×C127C211yesmaximal, characteristic, complemented
8C2×#14523249C2211yescharacteristic, complemented
9C2×#14523247C2213yes—
10C23×Dic127C2212yescomplemented
11#16861207C2213yescomplemented
12#16861205C2213yes—
13#16861202C22112yescomplemented
14#16861201C2216yescomplemented
15#16861197C2213yescomplemented
16#16861193C2216yescomplemented
17#16861191C22112yescomplemented
18#16861190C2216yes—
19#16861186C22112yescomplemented
20#16861185C2216yescomplemented
21#16861182C22112yescomplemented
22#16861181C22112yes—
23#16861180C2216yes—
24#16861178C2216yescomplemented
25Q8×Dic127C2212yescomplemented
26D4×Dic127C2216yes—
27(C22⋊D127)⋊C4C22112yescomplemented
28#16861165C2216yescomplemented
29C42×D127C2213yescomplemented
30#124×C127C2216yes—
31#124×C127C2213yes—
32#120×C127C2216yes—
33#14523249—42no—
34#14523247—46no—
35C2×Dic254C2313yes—
36C2×D508C2313yes—
37C2×C22⋊D127C23112yes—
38C2×C22⋊D127C2316yes—
39C22×Dic127—44no—
40C22×Dic127C2316yes—
41C22×Dic127C2316yes—
42C2×C4×D127C2313yes—
43C2×C4×D127C2316yes—
44C4⋊Dic127C2316yes—
45C4⋊Dic127C2316yes—
46Dic127⋊C4C23112yes—
47Dic127⋊C4C23112yes—
48C22⋊Dic127C23112yes—
49C22⋊Dic127C2316yes—
50(C2×D127)⋊C4C23112yes—
51C4×Dic127C2313yes—
52C4×Dic127C2316yes—
53C24×C127C2312yes—
54C2×Q8×C127C2312yes—
55C2×Q8×C127C2311yescharacteristic
56C2×D4×C127C2313yes—
57C2×D4×C127C2316yes—
58C22⋊C4×C127C23112yes—
59C22⋊C4×C127C2316yes—
60C42×C127C2313yes—
61C22×D127C2413yes—
62Dic254—43no—
63D508—43no—
64C22⋊D127—412no—
65C22⋊D127—46no—
66C2×Dic127—212no—
67C2×Dic127—46no—
68C2×Dic127C2416yes—
69C2×Dic127C2416yes—
70C2×Dic127C2411yescharacteristic
71C4×D127—26no—
72C4×D127—46no—
73C23×C127—44no—
74C23×C127—42no—
75C23×C127C2416yes—
76Q8×C127—42no—
77Q8×C127—41no—
78D4×C127—46no—
79D4×C127—43no—
80C2×C4×C127C2416yes—
81C2×C4×C127C2413yes—
82C22×C127—26no—
83C22×C127—212no—
84C22×C127—48no—
85C22×C127C2511yescharacteristic
86C2×D127—23no—
87C2×D127—26no—
88C4×C127—23no—
89C4×C127—26no—
90Dic127—26no—
91Dic127—26no—
92Dic127H(5,𝔽2)12yes—
93C2×C127—46no—
94C2×C127C2×H(5,𝔽2)11yescharacteristic
95C2×C127#58512yes—
96D127—43no—
97#3529—1271nomaximal, Sylow
98C127#352911yescharacteristic, complemented, Sylow
99#563—1273no—
100#562—1276no—
101H(3,𝔽4)D12711yescharacteristic, complemented
102#560—1276no—
103#532—1273no—
104#529—1276no—
105#525—1276no—
106C2×#41—1271no—
107C2×#41—1273no—
108C23×C4—1272no—
109C4⋊Q8—1273no—
110#127—1273no—
111#126—12712no—
112#125—1276no—
113#124—1273no—
114#124C2×D12716yes—
115#124C2×D12713yes—
116#123—1276no—
117#123—12712no—
118#123—1276no—
119C22⋊Q8—1276no—
120C22⋊Q8—12712no—
121#121—12712no—
122#121—1276no—
123#121—12712no—
124#120—1276no—
125#120C2×D12716yes—
126C4×Q8—1272no—
127C4×D4—1276no—
128C4×D4—12712no—
129#117—1276no—
130C2×C42—1273no—
131C24C22×D12712yes—
132#41—5086no—
133#41—5082no—
134C2×Q8—1273no—
135C2×Q8C22×D12712yes—
136C2×Q8C22×D12711yescharacteristic
137C2×D4—12712no—
138C2×D4—1276no—
139C2×D4—1273no—
140C2×D4C22×D12716yes—
141C2×D4C22×D12713yes—
142C22×C4—1276no—
143C22×C4—1276no—
144C22×C4—1276no—
145C22×C4—1273no—
146C22×C4—5084no—
147C4⋊C4—12712no—
148C4⋊C4—12712no—
149C4⋊C4—1276no—
150C4⋊C4—1276no—
151C22⋊C4—1276no—
152C22⋊C4—12712no—
153C22⋊C4—12712no—
154C22⋊C4C22×D12716yes—
155C22⋊C4C22×D127112yes—
156C42—1276no—
157C42—1273no—
158C42C22×D12713yes—
159C23—42no—
160C23—44no—
161C23—1273no—
162C23C23×D12716yes—
163Q8—41no—
164Q8—42no—
165Q8—5083no—
166D4—43no—
167D4—46no—
168D4—5083no—
169D4—5086no—
170D4—50812no—
171C2×C4—1271no—
172C2×C4—1276no—
173C2×C4—1276no—
174C2×C4—5086no—
175C2×C4—5086no—
176C2×C4—2546no—
177C2×C4—25412no—
178C2×C4C23×D12716yes—
179C2×C4C23×D12713yes—
180C22—26no—
181C22—212no—
182C22—48no—
183C22—2543no—
184C22—2546no—
185C22C24×D12711yescharacteristic
186C4—23no—
187C4—26no—
188C4—1272no—
189C4—2546no—
190C4—2546no—
191C2—46no—
192C2—5083no—
193C2H(5,𝔽2)×D12711yescharacteristic
194C2#2255071112yes—
subgroupG/H
center, FrattiniC22C24×D127
commutator, socle, cosocleC22×C127C25
FittingH(3,𝔽4)×C127C2
hypercenterH(3,𝔽4)D127
hypocenterC127#3529
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #22550626C2semidirect
2. #22550620C2semidirect
3. #22550615C2semidirect
4. #22550425C2semidirect
5. #22550405C2semidirect
6. #22550382C2semidirect
7. H(3,𝔽4)×C127C2semidirect
8. C2×#14523249C22semidirect
10. C23×Dic127C22semidirect
11. #16861207C22semidirect
13. #16861202C22semidirect
14. #16861201C22semidirect
15. #16861197C22semidirect
16. #16861193C22semidirect
17. #16861191C22semidirect
19. #16861186C22semidirect
20. #16861185C22semidirect
21. #16861182C22semidirect
24. #16861178C22semidirect
25. Q8×Dic127C22semidirect
27. (C22⋊D127)⋊C4C22semidirect
28. #16861165C22semidirect
29. C42×D127C22semidirect
98. C127#3529semidirect
101. H(3,𝔽4)D127semidirect

Maximal subgroups

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

subgroupindex
#225506262
#225506202
#225506152
#225504252
#225504052
#225503822
H(3,𝔽4)×C1272
#3529127