Finite Group Databasebeta

← #30637685

Subgroups — #30637685

Order 16256.

256 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 51 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#22550501C211yesmaximal, characteristic
2#22550032C211yesmaximal, characteristic, complemented
3#22549787C211yesmaximal, characteristic
4#22549614C211yesmaximal, characteristic, complemented
5#22549613C211yesmaximal, characteristic, complemented
6#22549599C211yesmaximal, characteristic
7#451×C127C211yesmaximal, characteristic, complemented
8C2×#14523238—21no—
9#16861189—21no—
10#16861188C2211yescharacteristic
11#16861187—21no—
12#16861187—21no—
13#16861186—21no—
14#16861185—21no—
15C2×(C2×D127)⋊C4—21no—
16C2×(C2×D127)⋊C4C2211yescharacteristic
17C22⋊C4×D127—21no—
18#16861133—21no—
19C22⋊(C127⋊C8)C2211yescharacteristic
20C127⋊#95—21no—
21C127⋊#95C2211yescharacteristic
22C22×Q8×C127C2211yescharacteristic
23C2×Q16×C127—21no—
24C22⋊Q8×C127C2211yescharacteristic
25#103×C127—21no—
26C22⋊C8×C127C2211yescharacteristic
27C23×D127D411yescharacteristic
28C2×Dic254—21no—
29C2×Dic254D411yescharacteristic
30C22×Dic127—21no—
31C22×Dic127D411yescharacteristic
32C2×C4×D127—21no—
33C2×C4×D127—21no—
34#14523238—42no—
35C2×C127⋊C8—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127D411yescharacteristic
38Dic127⋊C4—21no—
39Dic127⋊C4—21no—
40Dic127⋊C4—21no—
41C22⋊Dic127—21no—
42(C2×D127)⋊C4—21no—
43(C2×D127)⋊C4—21no—
44(C2×D127)⋊C4—21no—
45(C2×D127)⋊C4—21no—
46(C2×D127)⋊C4—42no—
47C2×Q8×C127—21no—
48C2×Q8×C127—21no—
49C2×Q8×C127—42no—
50C2×Q8×C127D411yescharacteristic
51C22×C4×C127—21no—
52C22×C4×C127C2311yescharacteristic
53Q16×C127—42no—
54C2×C8×C127—21no—
55C4⋊C4×C127—21no—
56C4⋊C4×C127D411yescharacteristic
57C22⋊C4×C127—21no—
58C22×D127—21no—
59C22×D127—21no—
60C22×D127—41no—
61C22×D127—41no—
62C22×D127Q1612yescomplemented
63Dic254—41no—
64Dic254—81no—
65C2×Dic127—21no—
66C2×Dic127—21no—
67C2×Dic127—21no—
68C2×Dic127—41no—
69C2×Dic127—41no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C4×D127—81no—
73C4×D127—81no—
74C127⋊C8—41no—
75C23×C127C2×D411yescharacteristic
76Q8×C127—22no—
77Q8×C127—22no—
78Q8×C127—41no—
79Q8×C127—41no—
80Q8×C127—41no—
81C2×C4×C127—22no—
82C2×C4×C127—21no—
83C2×C4×C127—21no—
84C2×C4×C127—41no—
85C2×C4×C127—41no—
86C2×C4×C127—41no—
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2×D411yescharacteristic
89C8×C127—41no—
90C22×C127—21no—
91C22×C127—21no—
92C22×C127#12011yescharacteristic
93C22×C127C2×Q1611yescharacteristic
94C22×C127#13711yescharacteristic
95C2×D127—22no—
96C2×D127—41no—
97C2×D127—41no—
98C2×D127—42no—
99C2×D127—81no—
100C4×C127—21no—
101C4×C127—21no—
102C4×C127—41no—
103C4×C127—42no—
104C4×C127—41no—
105Dic127—41no—
106Dic127—41no—
107Dic127—41no—
108Dic127—81no—
109C2×C127—21no—
110C2×C127—21no—
111C2×C127#45111yescharacteristic
112C2×C127#45511yescharacteristic
113C2×C127B(4,𝔽2)11yescharacteristic
114D127—42no—
115#1602—1271nomaximal, Sylow
116C127#160211yescharacteristic, complemented, Sylow
117#543—1271no—
118#451—1271no—
119#451D12711yescharacteristic, complemented
120#393—1271no—
121#328—1271no—
122#328—1271no—
123#323—1271no—
124C22×Q8C2×D12711yescharacteristic
125C2×Q16—21no—
126C2×Q16—2541no—
127C22⋊Q8—1271no—
128C22⋊Q8—2541no—
129C22⋊Q8—2541no—
130C22⋊Q8—2541no—
131C22⋊Q8—2541no—
132C22⋊Q8—2541no—
133C22⋊Q8C2×D12711yescharacteristic
134C2×C22⋊C4—1271no—
135C2×C22⋊C4—2541no—
136C2×C22⋊C4—2541no—
137#103—21no—
138#103—2541no—
139C22⋊C8—1271no—
140C22⋊C8C2×D12711yescharacteristic
141#95—1271no—
142#95—2541no—
143C24—1271no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—42no—
147C2×Q8—1271no—
148C2×Q8—2541no—
149C2×Q8C22⋊D12711yescharacteristic
150C22×C4—21no—
151C22×C4—1271no—
152C22×C4—2541no—
153C22×C4—2541no—
154C22×C4—2541no—
155C22×C4C22×D12711yescharacteristic
156Q16—42no—
157Q16—5082no—
158C2×C8—21no—
159C2×C8—2541no—
160C4⋊C4—21no—
161C4⋊C4—1271no—
162C4⋊C4—2541no—
163C4⋊C4—2541no—
164C4⋊C4—2541no—
165C4⋊C4—2541no—
166C4⋊C4C22⋊D12711yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—5082no—
169C22⋊C4—2541no—
170C22⋊C4—2541no—
171C22⋊C4—2541no—
172C22⋊C4—2541no—
173C22⋊C4—2541no—
174C23—1272no—
175C23—5081no—
176C23—5081no—
177C23—2541no—
178C23—2541no—
179C23D4×D12711yescharacteristic
180Q8—22no—
181Q8—22no—
182Q8—41no—
183Q8—41no—
184Q8—41no—
185Q8—10161no—
186Q8—5081no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—10161no—
194C2×C4—10161no—
195C2×C4—5081no—
196C2×C4—5081no—
197C2×C4—5081no—
198C2×C4—5081no—
199C2×C4—2541no—
200C2×C4—2541no—
201C2×C4—2541no—
202C2×C4D4×D12711yescharacteristic
203C2×C4C2×C22⋊D12711yescharacteristic
204C8—41no—
205C8—5081no—
206C22—21no—
207C22—21no—
208C22—10161no—
209C22—5082no—
210C22—5081no—
211C22—5081no—
212C22—2542no—
213C22#1686117711yescharacteristic
214C22Q16×D12711yescharacteristic
215C22#1686124111yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—42no—
220C4—41no—
221C4—10161no—
222C4—5081no—
223C4—5081no—
224C4—5081no—
225C2—21no—
226C2—21no—
227C2—5082no—
228C2#2255003011yescharacteristic
229C2#2255005211yescharacteristic
230C2#2255006211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#451×C127C2
socleC22×C127#120
prime radicalC23×D127D4
hypercenter#451D127
hypocenterC127#1602
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. #22550032C2semidirect
4. #22549614C2semidirect
5. #22549613C2semidirect
7. #451×C127C2semidirect
62. C22×D127Q16semidirect
116. C127#1602semidirect
119. #451D127semidirect

Maximal subgroups

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

subgroupindex
#225505012
#225500322
#225497872
#225496142
#225496132
#225495992
#451×C1272
#1602127