Finite Group Databasebeta

← #30650118

Subgroups — #30650118

Order 16256.

386 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550426C211yesmaximal, characteristic
3#22550172C212yesmaximal, complemented
4#22550157C212yesmaximal, complemented
5#22550122C214yesmaximal, complemented
6#22550082C212yesmaximal, complemented
7#22550077C212yesmaximal, complemented
8#423×C127C211yesmaximal, characteristic, complemented
9C2×#14523248—21no—
10C2×#14523237C2214yes—
11#16861209C2211yescharacteristic
12#16861209C2211yescharacteristic
13#16861209C2211yescharacteristic
14#16861206C2211yescharacteristic
15#16861201—21no—
16#16861197—21no—
17#16861188—21no—
18#16861188C2212yes—
19#16861183—21no—
20#16861183C2212yes—
21C4×Dic254—21no—
22C4×Dic254C2212yes—
23C4×D508—21no—
24C4×D508C2212yes—
25#16861168—21no—
26#16861143C2214yes—
27#16861134C2214yes—
28#16861131C2214yes—
29C2×M16×C127C2212yes—
30C2×C42×C127C2211yescharacteristic
31C4⋊C8×C127C2212yes—
32C4⋊C8×C127C2212yes—
33#14523248—44no—
34C2×Dic254—21no—
35C2×Dic254—21no—
36C2×Dic254C2312yes—
37C2×D508—21no—
38C2×D508—21no—
39C2×D508C2312yes—
40C2×C22⋊D127—22no—
41C2×C4×D127—22no—
42#14523237—44no—
43C4⋊Dic127—21no—
44C4⋊Dic127—21no—
45C4⋊Dic127—21no—
46C4⋊Dic127—21no—
47C4⋊Dic127C2312yes—
48C4⋊Dic127C2312yes—
49Dic127⋊C4—22no—
50Dic127⋊C4—22no—
51C22⋊Dic127—22no—
52(C2×D127)⋊C4—22no—
53(C2×D127)⋊C4—22no—
54C4×Dic127—22no—
55C22×C4×C127D411yescharacteristic
56C22×C4×C127D411yescharacteristic
57C22×C4×C127C2311yescharacteristic
58M16×C127—42no—
59C2×C8×C127C2314yes—
60C42×C127D411yescharacteristic
61C42×C127D411yescharacteristic
62C42×C127C2311yescharacteristic
63C42×C127C2311yescharacteristic
64C22×D127—22no—
65Dic254—22no—
66Dic254—22no—
67Dic254—42no—
68Dic254—42no—
69D508—22no—
70D508—22no—
71D508—42no—
72D508—42no—
73C22⋊D127—44no—
74C2×Dic127—22no—
75C2×Dic127—22no—
76C2×Dic127—22no—
77C4×D127—44no—
78C23×C127C2×D411yescharacteristic
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—21no—
82C2×C4×C127—21no—
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2411yescharacteristic
89C8×C127—24no—
90C22×C127—21no—
91C22×C127—21no—
92C22×C127—21no—
93C22×C127C22×D411yescharacteristic
94C2×D127—42no—
95C2×D127—42no—
96C2×D127—42no—
97C4×C127—21no—
98C4×C127—21no—
99C4×C127—21no—
100C4×C127AGL(1,ℤ/8)12yes—
101C4×C127#13712yes—
102C4×C127H(5,𝔽2)11yescharacteristic
103C4×C1272−1+411yescharacteristic
104Dic127—42no—
105Dic127—42no—
106Dic127—42no—
107C2×C127—41no—
108C2×C127#53711yescharacteristic
109C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
110C2×C127C2×#13711yescharacteristic
111D127—82no—
112#3254—1271nomaximal, Sylow
113C127#325411yescharacteristic, complemented, Sylow
114#533—1271no—
115#532—1271no—
116#477—1272no—
117#474—1272no—
118#468—1274no—
119#461—1272no—
120#460—1272no—
121#423D12711yescharacteristic, complemented
122C2×#41—2541no—
123C2×SD16—1274no—
124C2×M16C2×D12712yes—
125C4⋊Q8—1271no—
126C4⋊Q8—1271no—
127C4⋊Q8—1271no—
128#127—1271no—
129#125—2541no—
130#124—2541no—
131C22⋊Q8—1272no—
132C22⋊Q8—2541no—
133#121—1272no—
134#121—2541no—
135C4×Q8—1272no—
136C4×Q8—2541no—
137C4×D4—1272no—
138C4×D4—2541no—
139#117—2541no—
140C2×C42C2×D12711yescharacteristic
141#107—1274no—
142C4⋊C8C2×D12712yes—
143C4⋊C8C2×D12712yes—
144#103—1274no—
145#102—1274no—
146#41—5084no—
147C2×Q8—1272no—
148C2×Q8—2541no—
149C2×Q8—2541no—
150C2×D4—1272no—
151C2×D4—2542no—
152C2×D4—2541no—
153C2×D4—2541no—
154C22×C4—2542no—
155C22×C4D50811yescharacteristic
156C22×C4D50811yescharacteristic
157C22×C4C22×D12711yescharacteristic
158SD16—5084no—
159M16—42no—
160C2×C8C22×D12714yes—
161C4⋊C4—1272no—
162C4⋊C4—1272no—
163C4⋊C4—2542no—
164C4⋊C4—2542no—
165C4⋊C4—2541no—
166C4⋊C4—2541no—
167C4⋊C4—2541no—
168C4⋊C4—2541no—
169C22⋊C4—2542no—
170C22⋊C4—2542no—
171C22⋊C4—2542no—
172C42—2542no—
173C42D50811yescharacteristic
174C42D50811yescharacteristic
175C42C22×D12711yescharacteristic
176C42C22×D12711yescharacteristic
177C23—2542no—
178C23C2×D50811yescharacteristic
179Q8—5082no—
180Q8—5082no—
181Q8—2542no—
182Q8—2542no—
183D4—5082no—
184D4—5084no—
185D4—5082no—
186D4—2542no—
187D4—2542no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—5084no—
193C2×C4—2542no—
194C2×C4—2542no—
195C2×C4—2542no—
196C2×C4C2×D50811yescharacteristic
197C2×C4C2×D50811yescharacteristic
198C2×C4C2×D50811yescharacteristic
199C2×C4C2×D50811yescharacteristic
200C2×C4C2×D50811yescharacteristic
201C2×C4C23×D12711yescharacteristic
202C8—24no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—5082no—
207C22—5082no—
208C22—5082no—
209C22C22×D50811yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—5082no—
214C4—5082no—
215C4—5082no—
216C4#1686123712yes—
217C4#1686124212yes—
218C4#1686125411yescharacteristic
219C4#1686125711yescharacteristic
220C2—41no—
221C2—10162no—
222C2#2255045711yescharacteristic
223C2C2×#1686123711yescharacteristic
224C2C2×#1686124211yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#423×C127C2
socleC22×C127C22×D4
prime radical#22550426C2
hypercenter#423D127
hypocenterC127#3254
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
1. #22550433C2semidirect
3. #22550172C2semidirect
4. #22550157C2semidirect
5. #22550122C2semidirect
6. #22550082C2semidirect
7. #22550077C2semidirect
8. #423×C127C2semidirect
113. C127#3254semidirect
121. #423D127semidirect

Maximal subgroups

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

subgroupindex
#225504332
#225504262
#225501722
#225501572
#225501222
#225500822
#225500772
#423×C1272
#3254127