Finite Group Databasebeta

← #44565059

Subgroups — #44565059

Order 32128.

958 conjugacy classes of subgroups; only the 449 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 449 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#30632437C213yesmaximal, complemented
2#30632406C216yesmaximal, complemented
3#30632368C216yesmaximal, complemented
4#30632301C216yesmaximal, complemented
5#30632272C216yesmaximal, complemented
6#30632216C213yesmaximal, complemented
7H(3,𝔽4)×C251C211yesmaximal, characteristic, complemented
8#22548713C2216yescomplemented
9#22548709C2212yescomplemented
10#22548708C2216yescomplemented
11#22548708C2216yescomplemented
12#22548708C2216yescomplemented
13#22548708C2216yescomplemented
14#22548707C2213yescomplemented
15#22548702C2216yes—
16#22548701C2216yescomplemented
17#22548697C2216yescomplemented
18#22548696C2216yes—
19#22548696C2216yes—
20#22548692C2216yescomplemented
21#22548692C2216yescomplemented
22#22548692C2216yescomplemented
23#22548691C2213yescomplemented
24#22548687C2216yes—
25#22548686C2213yes—
26#22548684C2212yescomplemented
27Q8×Dic251C2213yescomplemented
28C4×Dic502C2216yescomplemented
29D4×Dic251C2213yes—
30D4×Dic251C2216yes—
31(C22⋊D251)⋊C4C2216yescomplemented
32#22548671C2216yescomplemented
33#22548671C2216yescomplemented
34C2×C22⋊Dic251C2216yescomplemented
35C42×D251C2211yescharacteristic, complemented
36#124×C251C2213yes—
37#124×C251C2216yes—
38#120×C251C2216yes—
39C2×Dic502C2316yes—
40C2×C22⋊D251C2316yes—
41C22×Dic251C2316yes—
42C22×Dic251C2316yes—
43C2×C4×D251C2313yes—
44C4⋊Dic251C2316yes—
45C4⋊Dic251C2313yes—
46C4⋊Dic251C2313yes—
47Dic251⋊C4C2316yes—
48Dic251⋊C4C2316yes—
49Dic251⋊C4C2316yes—
50Dic251⋊C4C2316yes—
51Dic251⋊C4C2316yes—
52C22⋊Dic251C2316yes—
53C22⋊Dic251C2316yes—
54C22⋊Dic251C2316yes—
55C22⋊Dic251C2316yes—
56C22⋊Dic251C2316yes—
57(C2×D251)⋊C4C2316yes—
58C4×Dic251C2313yes—
59C4×Dic251C2316yes—
60C4×Dic251C2316yes—
61C24×C251C2312yes—
62C2×Q8×C251C2313yes—
63C2×D4×C251C2313yes—
64C2×D4×C251C2316yes—
65C22⋊C4×C251C2316yes—
66C22⋊C4×C251C2316yes—
67C22⋊C4×C251C2316yes—
68C42×C251C2311yescharacteristic
69C42×C251C2312yes—
70C22×D251C2411yescharacteristic
71Dic502—46no—
72C22⋊D251—46no—
73C2×Dic251—46no—
74C2×Dic251—46no—
75C2×Dic251C2413yes—
76C2×Dic251C2416yes—
77C2×Dic251C2416yes—
78C4×D251—43no—
79C23×C251—46no—
80C23×C251C2416yes—
81Q8×C251—43no—
82D4×C251—43no—
83D4×C251—46no—
84C2×C4×C251C2416yes—
85C2×C4×C251C2413yes—
86C22×C251—26no—
87C22×C251—26no—
88C22×C251—26no—
89C22×C251—84no—
90C22×C251C2511yescharacteristic
91C2×D251—23no—
92C4×C251—23no—
93C4×C251—26no—
94Dic251—23no—
95Dic251—26no—
96Dic251—26no—
97C2×C251—46no—
98C2×C251#58513yes—
99D251—41no—
100C251#356811yescharacteristic, complemented, Sylow
101#3568—2511nomaximal, Sylow
102#563—2513no—
103H(3,𝔽4)D25111yescharacteristic, complemented
104#556—2516no—
105#551—2516no—
106#542—2516no—
107#538—2516no—
108#529—2513no—
109C4⋊Q8—2516no—
110#126—2516no—
111#126—2516no—
112#126—2516no—
113#126—2516no—
114#126—2512no—
115#125—2513no—
116#124—2516no—
117#124—2516no—
118#124C2×D25113yes—
119#124C2×D25116yes—
120#123—2516no—
121#123—2516no—
122#123—2516no—
123C22⋊Q8—2513no—
124C22⋊Q8—2516no—
125C22⋊Q8—2516no—
126C22⋊Q8—2516no—
127#121—2513no—
128#121—2516no—
129#120—2512no—
130#120C2×D25116yes—
131C4×Q8—2513no—
132C4×Q8—2516no—
133C4×D4—2513no—
134C4×D4—2516no—
135C4×D4—2516no—
136#117—2516no—
137#117—2516no—
138C2×C22⋊C4—2516no—
139C2×C42—2511no—
140C24C22×D25112yes—
141C2×Q8—2516no—
142C2×Q8C22×D25113yes—
143C2×D4—2516no—
144C2×D4C22×D25113yes—
145C2×D4C22×D25116yes—
146C22×C4—2516no—
147C22×C4—2516no—
148C22×C4—2513no—
149C4⋊C4—2516no—
150C4⋊C4—2516no—
151C4⋊C4—2516no—
152C4⋊C4—2516no—
153C4⋊C4—2516no—
154C4⋊C4—2516no—
155C4⋊C4—2513no—
156C4⋊C4—2513no—
157C22⋊C4—2516no—
158C22⋊C4—2516no—
159C22⋊C4—2516no—
160C22⋊C4—2516no—
161C22⋊C4—2516no—
162C22⋊C4—2516no—
163C22⋊C4C22×D25116yes—
164C22⋊C4C22×D25116yes—
165C22⋊C4C22×D25116yes—
166C42—2513no—
167C42—2516no—
168C42—2516no—
169C42C22×D25111yescharacteristic
170C42C22×D25112yes—
171C23—46no—
172C23—2511no—
173C23C23×D25116yes—
174Q8—43no—
175Q8—10046no—
176D4—43no—
177D4—46no—
178D4—10046no—
179C2×C4—10043no—
180C2×C4—10046no—
181C2×C4—10046no—
182C2×C4—2513no—
183C2×C4—2516no—
184C2×C4—2516no—
185C2×C4C23×D25116yes—
186C2×C4C23×D25113yes—
187C22—26no—
188C22—26no—
189C22—26no—
190C22—84no—
191C22—5023no—
192C22C24×D25111yescharacteristic
193C4—23no—
194C4—26no—
195C4—5023no—
196C4—5026no—
197C4—5026no—
198C2—46no—
199C2—10041no—
200C2#3063252213yes—
subgroupG/H
center, FrattiniC22C24×D251
commutator, socle, cosocleC22×C251C25
FittingH(3,𝔽4)×C251C2
hypercenterH(3,𝔽4)D251
hypocenterC251#3568
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #30632437C2semidirect
2. #30632406C2semidirect
3. #30632368C2semidirect
4. #30632301C2semidirect
5. #30632272C2semidirect
6. #30632216C2semidirect
7. H(3,𝔽4)×C251C2semidirect
8. #22548713C22semidirect
9. #22548709C22semidirect
10. #22548708C22semidirect
11. #22548708C22semidirect
12. #22548708C22semidirect
13. #22548708C22semidirect
14. #22548707C22semidirect
16. #22548701C22semidirect
17. #22548697C22semidirect
20. #22548692C22semidirect
21. #22548692C22semidirect
22. #22548692C22semidirect
23. #22548691C22semidirect
26. #22548684C22semidirect
27. Q8×Dic251C22semidirect
28. C4×Dic502C22semidirect
31. (C22⋊D251)⋊C4C22semidirect
32. #22548671C22semidirect
33. #22548671C22semidirect
34. C2×C22⋊Dic251C22semidirect
35. C42×D251C22semidirect
100. C251#3568semidirect
103. H(3,𝔽4)D251semidirect

Maximal subgroups

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

subgroupindex
#306324372
#306324062
#306323682
#306323012
#306322722
#306322162
H(3,𝔽4)×C2512
#3568251