Finite Group Databasebeta

← #30680365

Subgroups — #30680365

Order 16768.

1258 conjugacy classes of subgroups; only the 455 normal subgroups are shown.

normal subgroups only

This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 455 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#22554054C213yesmaximal, complemented
2#22554024C216yesmaximal, complemented
3#22553944C216yesmaximal, complemented
4#22553880C216yesmaximal, complemented
5#22553877C216yesmaximal, complemented
6#22553861C213yesmaximal, complemented
7H(3,𝔽4)×C131C211yesmaximal, characteristic, complemented
8C2×#14523514C2213yescomplemented
9C2×#14523512C2213yescomplemented
10C22×C22⋊D131C2216yescomplemented
11C2×D4×D131C2216yescomplemented
12#16862244C2213yescomplemented
13#16862243C2212yescomplemented
14#16862242C2213yes—
15#16862242C2216yescomplemented
16#16862235C2216yescomplemented
17#16862234C2213yescomplemented
18#16862232C22112yescomplemented
19#16862219C22112yescomplemented
20#16862219C22112yescomplemented
21#16862218C22112yescomplemented
22#16862216C2216yescomplemented
23#16862216C22112yescomplemented
24#16862215C2212yescomplemented
25#16862214C2216yescomplemented
26(C22⋊D131)⋊C4C22112yescomplemented
27#16862203C2216yescomplemented
28#16862202C2216yescomplemented
29C42×D131C2211yescharacteristic, complemented
30#124×C131C2213yescomplemented
31#124×C131C2216yescomplemented
32#120×C131C2216yescomplemented
33C23×D131C2316yes—
34#14523514—46no—
35#14523512—46no—
36C2×Dic262C2313yes—
37C2×D524C2313yes—
38C2×D524C2316yes—
39C2×D524C2316yes—
40C2×C22⋊D131—412no—
41C2×C22⋊D131C23112yes—
42C2×C22⋊D131C23112yes—
43C2×C22⋊D131C2316yes—
44D4×D131—412no—
45C22×Dic131C2316yes—
46C2×C4×D131C2313yes—
47C2×C4×D131C2316yes—
48Dic131⋊C4C23112yes—
49C22⋊Dic131C2316yes—
50(C2×D131)⋊C4C23112yes—
51(C2×D131)⋊C4C23112yes—
52C4×Dic131C2316yes—
53C4×Dic131C2313yes—
54C24×C131C2312yes—
55C2×Q8×C131C2313yes—
56C2×D4×C131C2313yes—
57C2×D4×C131C2316yes—
58C22⋊C4×C131C2316yes—
59C22⋊C4×C131C23112yes—
60C42×C131C2311yescharacteristic
61C42×C131C2312yes—
62C22×D131—412no—
63C22×D131—46no—
64C22×D131C2416yes—
65C22×D131C2411yescharacteristic
66Dic262—43no—
67D524—46no—
68D524—46no—
69D524—43no—
70C22⋊D131—224no—
71C22⋊D131—412no—
72C22⋊D131—46no—
73C2×Dic131—212no—
74C2×Dic131C2416yes—
75C2×Dic131C2413yes—
76C4×D131—212no—
77C4×D131—26no—
78C23×C131—46no—
79C23×C131C2416yes—
80Q8×C131—43no—
81D4×C131—46no—
82D4×C131—43no—
83C2×C4×C131C2413yes—
84C2×C4×C131C2416yes—
85C22×C131—26no—
86C22×C131—212no—
87C22×C131—84no—
88C22×C131C2511yescharacteristic
89C2×D131—23no—
90C2×D131—26no—
91C2×D131—212no—
92C2×D131—812no—
93C4×C131—26no—
94C4×C131—23no—
95Dic131—26no—
96Dic131H(5,𝔽2)16yes—
97C2×C131—46no—
98C2×C131C2×H(5,𝔽2)13yes—
99D131—41no—
100D131—46no—
101C131#354911yescharacteristic, complemented, Sylow
102#3549—1311nomaximal, Sylow
103H(3,𝔽4)—1313no—
104H(3,𝔽4)D13111yescharacteristic, complemented
105#555—1316no—
106D42—1316no—
107#535—1316no—
108#535—1316no—
109#532—1313no—
110C2×#41—1313no—
111C2×#41—1313no—
112C22×D4—1316no—
113C22×D4—1316no—
114C4⋊Q8—1313no—
115#127—1313no—
116#127—1316no—
117#127—1312no—
118#124—1316no—
119#124—13112no—
120#124—1313no—
121#124C2×D13113yescomplemented
122#124C2×D13116yescomplemented
123#121—13112no—
124#121—13112no—
125#121—13112no—
126#120—1312no—
127#120—1316no—
128#120—1316no—
129#120—13112no—
130#120C2×D13116yescomplemented
131C4×D4—13112no—
132#117—1316no—
133#117—1316no—
134C2×C42—1311no—
135C24—1316no—
136C24C22×D13112yes—
137#41—5246no—
138#41—5246no—
139C2×Q8—1313no—
140C2×Q8C22×D13113yes—
141C2×D4—52412no—
142C2×D4—52412no—
143C2×D4—13112no—
144C2×D4—13112no—
145C2×D4—1316no—
146C2×D4—1313no—
147C2×D4—1316no—
148C2×D4—1316no—
149C2×D4C22×D13113yes—
150C2×D4C22×D13116yes—
151C22×C4—1316no—
152C22×C4—1313no—
153C22×C4—1316no—
154C4⋊C4—13112no—
155C22⋊C4—1316no—
156C22⋊C4—13112no—
157C22⋊C4—13112no—
158C22⋊C4C22×D13116yes—
159C22⋊C4C22×D131112yes—
160C42—1313no—
161C42—1316no—
162C42C22×D13111yescharacteristic
163C42C22×D13112yes—
164C23—46no—
165C23—5246no—
166C23—52412no—
167C23—1316no—
168C23—1311no—
169C23C23×D13116yes—
170Q8—43no—
171Q8—5243no—
172D4—46no—
173D4—43no—
174D4—26224no—
175D4—5246no—
176D4—5246no—
177D4—5243no—
178D4—52412no—
179D4—5246no—
180C2×C4—26212no—
181C2×C4—2626no—
182C2×C4—26212no—
183C2×C4—1316no—
184C2×C4—1313no—
185C2×C4C23×D13113yes—
186C2×C4C23×D13116yes—
187C22—26no—
188C22—212no—
189C22—2623no—
190C22—2626no—
191C22—26212no—
192C22—84no—
193C22—104812no—
194C22C24×D13111yescharacteristic
195C4—26no—
196C4—23no—
197C4—2626no—
198C4—1316no—
199C2—46no—
200C2—5246no—
201C2—5241no—
202C2H(5,𝔽2)×D13113yes—
subgroupG/H
center, FrattiniC22C24×D131
commutator, socle, cosocleC22×C131C25
FittingH(3,𝔽4)×C131C2
hypercenterH(3,𝔽4)D131
hypocenterC131#3549
LCSG ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #22554054C2semidirect
2. #22554024C2semidirect
3. #22553944C2semidirect
4. #22553880C2semidirect
5. #22553877C2semidirect
6. #22553861C2semidirect
7. H(3,𝔽4)×C131C2semidirect
8. C2×#14523514C22semidirect
9. C2×#14523512C22semidirect
10. C22×C22⋊D131C22semidirect
11. C2×D4×D131C22semidirect
12. #16862244C22semidirect
13. #16862243C22semidirect
15. #16862242C22semidirect
16. #16862235C22semidirect
17. #16862234C22semidirect
18. #16862232C22semidirect
19. #16862219C22semidirect
20. #16862219C22semidirect
21. #16862218C22semidirect
22. #16862216C22semidirect
23. #16862216C22semidirect
24. #16862215C22semidirect
25. #16862214C22semidirect
26. (C22⋊D131)⋊C4C22semidirect
27. #16862203C22semidirect
28. #16862202C22semidirect
29. C42×D131C22semidirect
30. #124×C131C22semidirect
31. #124×C131C22semidirect
32. #120×C131C22semidirect
101. C131#3549semidirect
104. H(3,𝔽4)D131semidirect
121. #124C2×D131semidirect
122. #124C2×D131semidirect
130. #120C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225540542
#225540242
#225539442
#225538802
#225538772
#225538612
H(3,𝔽4)×C1312
#3549131