Finite Group Databasebeta

← #30650113

Subgroups — #30650113

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#22550426C211yesmaximal, characteristic
2#22550171C212yesmaximal, complemented
3#22550153C212yesmaximal, complemented
4#22550124C214yesmaximal, complemented
5#22550078C212yesmaximal, complemented
6#22550076C212yesmaximal, complemented
7#22549900C211yesmaximal, characteristic, complemented
8#533×C127C211yesmaximal, characteristic, complemented
9C2×#14523248—21no—
10C2×#14523235C2214yes—
11C2×#14523232C2212yes—
12#16861209C2211yescharacteristic
13#16861206C2211yescharacteristic
14#16861197—21no—
15#16861183—21no—
16#16861183C2212yes—
17C4×D508—21no—
18C4×D508C2212yes—
19#16861142C2214yes—
20C4⋊(C127⋊C8)C2212yes—
21C4⋊(C127⋊C8)C2212yes—
22#16861133C2214yes—
23#16861129C2214yes—
24C4⋊Q8×C127C2211yescharacteristic
25C4⋊Q8×C127C2211yescharacteristic
26#125×C127—21no—
27C22⋊Q8×C127—21no—
28C22⋊Q8×C127C2212yes—
29C4×Q8×C127—21no—
30C4×Q8×C127C2212yes—
31#117×C127—21no—
32C2×C42×C127C2211yescharacteristic
33#14523248—44no—
34C2×Dic254—21no—
35C2×D508—21no—
36C2×D508—21no—
37C2×D508C2312yes—
38C2×C22⋊D127—22no—
39C2×C4×D127—22no—
40#14523235—44no—
41#14523232—42no—
42C2×C127⋊C8C2314yes—
43C4⋊Dic127—21no—
44C4⋊Dic127C2312yes—
45(C2×D127)⋊C4—22no—
46(C2×D127)⋊C4—22no—
47C2×Q8×C127—21no—
48C2×Q8×C127C2312yes—
49C22×C4×C127D411yescharacteristic
50C22×C4×C127D411yescharacteristic
51C22×C4×C127C2311yescharacteristic
52C4⋊C4×C127—21no—
53C4⋊C4×C127—21no—
54C4⋊C4×C127—21no—
55C4⋊C4×C127—22no—
56C4⋊C4×C127—22no—
57C4⋊C4×C127C2312yes—
58C22⋊C4×C127—22no—
59C42×C127—22no—
60C42×C127D411yescharacteristic
61C42×C127D411yescharacteristic
62C42×C127C2311yescharacteristic
63C42×C127C2311yescharacteristic
64C22×D127—22no—
65Dic254—42no—
66D508—22no—
67D508—22no—
68D508—42no—
69D508—42no—
70C22⋊D127—44no—
71C2×Dic127—22no—
72C4×D127—44no—
73C127⋊C8—24no—
74C23×C127C2×D411yescharacteristic
75Q8×C127—22no—
76Q8×C127—22no—
77Q8×C127—42no—
78C2×C4×C127—22no—
79C2×C4×C127—22no—
80C2×C4×C127—21no—
81C2×C4×C127—21no—
82C2×C4×C127—22no—
83C2×C4×C127—22no—
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2×D411yescharacteristic
89C2×C4×C127C2411yescharacteristic
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×C127—42no—
101C4×C127—42no—
102C4×C127AGL(1,ℤ/8)12yes—
103C4×C127#13712yes—
104C4×C127H(5,𝔽2)11yescharacteristic
105C4×C1272−1+411yescharacteristic
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#533D12711yescharacteristic, complemented
115#532—1271no—
116#477—1272no—
117#474—1272no—
118#468—1274no—
119#461—1272no—
120#460—1272no—
121#423—1271no—
122C2×#41—2541no—
123C2×SD16—1274no—
124C2×M16—1272no—
125C4⋊Q8—1271no—
126C4⋊Q8C2×D12711yescharacteristic
127C4⋊Q8C2×D12711yescharacteristic
128#127—1271no—
129#125—21no—
130#124—2541no—
131C22⋊Q8—21no—
132C22⋊Q8C2×D12712yes—
133#121—1272no—
134#121—2541no—
135C4×Q8—21no—
136C4×Q8C2×D12712yes—
137C4×D4—1272no—
138C4×D4—2541no—
139#117—21no—
140C2×C42C2×D12711yescharacteristic
141#107—1274no—
142C4⋊C8—1272no—
143C4⋊C8—1272no—
144#103—1274no—
145#102—1274no—
146#41—5084no—
147C2×Q8—21no—
148C2×Q8—2541no—
149C2×Q8C22×D12712yes—
150C2×D4—1272no—
151C2×D4—2542no—
152C2×D4—2541no—
153C2×D4—2541no—
154C22×C4—2542no—
155C22×C4C22⋊D12711yescharacteristic
156C22×C4C22⋊D12711yescharacteristic
157C22×C4C22×D12711yescharacteristic
158SD16—5084no—
159M16—5082no—
160C2×C8—1274no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—1272no—
167C4⋊C4—2541no—
168C4⋊C4C22×D12712yes—
169C22⋊C4—22no—
170C22⋊C4—2542no—
171C22⋊C4—2542no—
172C42—22no—
173C42C22⋊D12711yescharacteristic
174C42C22⋊D12711yescharacteristic
175C42C22×D12711yescharacteristic
176C42C22×D12711yescharacteristic
177C23—2542no—
178C23C2×C22⋊D12711yescharacteristic
179Q8—22no—
180Q8—22no—
181Q8—42no—
182Q8—5082no—
183D4—5082no—
184D4—5084no—
185D4—5082no—
186D4—2542no—
187D4—2542no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—5084no—
195C2×C4—2542no—
196C2×C4C2×C22⋊D12711yescharacteristic
197C2×C4C2×C22⋊D12711yescharacteristic
198C2×C4C2×C22⋊D12711yescharacteristic
199C2×C4C2×C22⋊D12711yescharacteristic
200C2×C4C2×C22⋊D12711yescharacteristic
201C2×C4C23×D12711yescharacteristic
202C8—2544no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—5082no—
207C22—5082no—
208C22—5082no—
209C22C22×C22⋊D12711yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—42no—
214C4—42no—
215C4—5082no—
216C4#1686123412yes—
217C4#1686124012yes—
218C4#1686125511yescharacteristic
219C4#1686125611yescharacteristic
220C2—41no—
221C2—10162no—
222C2#2255045411yescharacteristic
223C2C2×#1686123411yescharacteristic
224C2C2×#1686124011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#533×C127C2
socleC22×C127C22×D4
prime radical#22550426C2
hypercenter#533D127
hypocenterC127#3254
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #22550171C2semidirect
3. #22550153C2semidirect
4. #22550124C2semidirect
5. #22550078C2semidirect
6. #22550076C2semidirect
7. #22549900C2semidirect
8. #533×C127C2semidirect
113. C127#3254semidirect
114. #533D127semidirect

Maximal subgroups

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

subgroupindex
#225504262
#225501712
#225501532
#225501242
#225500782
#225500762
#225499002
#533×C1272
#3254127