Finite Group Databasebeta

← #30648527

Subgroups — #30648527

Order 16256.

440 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 153 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#22550433C211yesmaximal, characteristic, complemented
2#22550176C214yesmaximal, complemented
3#22550166C212yesmaximal, complemented
4#22550156C212yesmaximal, complemented
5#22549916C212yesmaximal, complemented
6#22549900C211yesmaximal, characteristic, complemented
7#22549881C212yesmaximal
8C2×C4×D4×C127C211yesmaximal, characteristic
9C2×#14523232C2212yes—
10#16861209C2211yescharacteristic, complemented
11#16861209C2211yescharacteristic, complemented
12#16861201C2212yescomplemented
13#16861188—21no—
14#16861188—21no—
15C4×Dic254—21no—
16C4×Dic254—21no—
17#16861168C2212yes—
18#16861142C2214yescomplemented
19#16861140C2214yescomplemented
20C4⋊(C127⋊C8)C2212yescomplemented
21C4⋊(C127⋊C8)C2212yescomplemented
22#16861130C2214yes—
23#16861130C2214yes—
24C22×D4×C127C2211yescharacteristic
25C23×C4×C127—21no—
26C4×D4×C127—21no—
27C4×D4×C127—21no—
28C4×D4×C127C2212yes—
29C4×D4×C127C2212yes—
30C2×C4⋊C4×C127C2211yescharacteristic
31C2×C22⋊C4×C127—21no—
32C2×C42×C127C2211yescharacteristic
33C2×Dic254—21no—
34C2×Dic254—21no—
35#14523232—24no—
36C2×C127⋊C8C2314yes—
37C4⋊Dic127—21no—
38C4⋊Dic127—21no—
39C4⋊Dic127C2312yes—
40C4⋊Dic127C2312yes—
41Dic127⋊C4—22no—
42Dic127⋊C4—22no—
43C22⋊Dic127—22no—
44C4×Dic127—22no—
45C24×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127—22no—
48C2×D4×C127Q814yescomplemented
49C2×D4×C127C2312yes—
50C22×C4×C127—21no—
51C22×C4×C127—22no—
52C22×C4×C127—22no—
53C22×C4×C127—44no—
54C22×C4×C127D411yescharacteristic
55C22×C4×C127D411yescharacteristic
56C22×C4×C127C2311yescharacteristic
57C4⋊C4×C127—21no—
58C4⋊C4×C127C2312yes—
59C22⋊C4×C127—22no—
60C22⋊C4×C127—22no—
61C42×C127D411yescharacteristic
62C42×C127D411yescharacteristic
63C42×C127C2311yescharacteristic
64C42×C127C2311yescharacteristic
65Dic254—42no—
66Dic254—42no—
67C2×Dic127—22no—
68C2×Dic127—22no—
69C127⋊C8—24no—
70C23×C127—24no—
71C23×C127—22no—
72C23×C127—42no—
73C23×C127—42no—
74C23×C127C2×D411yescharacteristic
75D4×C127—24no—
76D4×C127—22no—
77D4×C127C2×Q814yes—
78C2×C4×C127—22no—
79C2×C4×C127—22no—
80C2×C4×C127—41no—
81C2×C4×C127—44no—
82C2×C4×C127—42no—
83C2×C4×C127—44no—
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2×D411yescharacteristic
89C2×C4×C127C2×Q812yes—
90C2×C4×C127#4112yes—
91C2×C4×C127C2411yescharacteristic
92C22×C127—21no—
93C22×C127—21no—
94C22×C127—24no—
95C22×C127—48no—
96C22×C127—42no—
97C22×C127—42no—
98C22×C127C22⋊Q812yes—
99C22×C127C22×D411yescharacteristic
100C4×C127—21no—
101C4×C127—21no—
102C4×C127—21no—
103C4×C127—42no—
104C4×C127C22⋊Q812yes—
105C4×C127AGL(1,ℤ/8)12yes—
106C4×C127C22×Q811yescharacteristic
107C4×C127C2×#4111yescharacteristic
108Dic127—42no—
109Dic127—42no—
110C2×C127—22no—
111C2×C127—44no—
112C2×C127C2×C22⋊Q811yescharacteristic
113C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
114C2×C127#57511yescharacteristic
115#3077—1271nomaximal, Sylow
116C127#307711yescharacteristic, complemented, Sylow
117#533—1271no—
118C2×C4×D4D12711yescharacteristic
119#478—1274no—
120#476—1272no—
121#474—1272no—
122#428—1272no—
123#423—1271no—
124#418—1272no—
125C22×D4C2×D12711yescharacteristic
126C23×C4—21no—
127C2×M16—1272no—
128C4⋊Q8—1271no—
129C4⋊Q8—1271no—
130#125—1272no—
131C22⋊Q8—2541no—
132C22⋊Q8—2541no—
133C4×Q8—2541no—
134C4×Q8—2541no—
135C4×D4—21no—
136C4×D4—21no—
137C4×D4C2×D12712yes—
138C4×D4C2×D12712yes—
139#117—1272no—
140C2×C4⋊C4C2×D12711yescharacteristic
141C2×C22⋊C4—21no—
142C2×C42C2×D12711yescharacteristic
143#107—1274no—
144#106—1274no—
145C4⋊C8—1272no—
146C4⋊C8—1272no—
147#102—1274no—
148#102—1274no—
149C24—21no—
150C2×Q8—2541no—
151C2×Q8—2541no—
152C2×D4—21no—
153C2×D4—22no—
154C2×D4Dic25414yescomplemented
155C2×D4C22×D12712yes—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—44no—
160C22×C4C22⋊D12711yescharacteristic
161C22×C4C22⋊D12711yescharacteristic
162C22×C4C22×D12711yescharacteristic
163M16—2544no—
164C2×C8—1274no—
165C4⋊C4—21no—
166C4⋊C4—1272no—
167C4⋊C4—1272no—
168C4⋊C4—2542no—
169C4⋊C4—2542no—
170C4⋊C4—2541no—
171C4⋊C4—2541no—
172C4⋊C4C22×D12712yes—
173C22⋊C4—22no—
174C22⋊C4—22no—
175C22⋊C4—2542no—
176C42—2542no—
177C42C22⋊D12711yescharacteristic
178C42C22⋊D12711yescharacteristic
179C42C22×D12711yescharacteristic
180C42C22×D12711yescharacteristic
181C23—24no—
182C23—22no—
183C23—42no—
184C23—42no—
185C23C2×C22⋊D12711yescharacteristic
186Q8—5082no—
187Q8—5082no—
188D4—24no—
189D4—22no—
190D4C2×Dic25414yes—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—41no—
194C2×C4—44no—
195C2×C4—42no—
196C2×C4—44no—
197C2×C4—2542no—
198C2×C4—2542no—
199C2×C4C2×C22⋊D12711yescharacteristic
200C2×C4C2×C22⋊D12711yescharacteristic
201C2×C4C2×C22⋊D12711yescharacteristic
202C2×C4C2×C22⋊D12711yescharacteristic
203C2×C4C2×C22⋊D12711yescharacteristic
204C2×C4C2×Dic25412yes—
205C2×C4#1452324812yes—
206C2×C4C23×D12711yescharacteristic
207C8—2544no—
208C22—21no—
209C22—21no—
210C22—24no—
211C22—48no—
212C22—42no—
213C22—42no—
214C22#1686118812yes—
215C22C22×C22⋊D12711yescharacteristic
216C4—21no—
217C4—21no—
218C4—21no—
219C4—42no—
220C4—5082no—
221C4—5082no—
222C4#1686118812yes—
223C4#1686123512yes—
224C4C22×Dic25411yescharacteristic
225C4C2×#1452324811yescharacteristic
226C2—22no—
227C2—44no—
228C2C2×#1686118811yescharacteristic
229C2C2×#1686123511yescharacteristic
230C2#2255067411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×C4×D4×C127C2
socleC22×C127C22×D4
prime radicalC22×D4×C127C22
hypercenterC2×C4×D4D127
hypocenterC127#3077
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #22550433C2semidirect
2. #22550176C2semidirect
3. #22550166C2semidirect
4. #22550156C2semidirect
5. #22549916C2semidirect
6. #22549900C2semidirect
10. #16861209C22semidirect
11. #16861209C22semidirect
12. #16861201C22semidirect
18. #16861142C22semidirect
19. #16861140C22semidirect
20. C4⋊(C127⋊C8)C22semidirect
21. C4⋊(C127⋊C8)C22semidirect
48. C2×D4×C127Q8semidirect
116. C127#3077semidirect
154. C2×D4Dic254semidirect

Maximal subgroups

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

subgroupindex
#225504332
#225501762
#225501662
#225501562
#225499162
#225499002
#225498812
C2×C4×D4×C1272
#3077127