Finite Group Databasebeta

← #30637779

Subgroups — #30637779

Order 16256.

240 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550463C211yesmaximal, characteristic
2#22550212C211yesmaximal, characteristic, complemented
3#22550192C211yesmaximal, characteristic, complemented
4#22549813C211yesmaximal, characteristic, complemented
5#22549612C211yesmaximal, characteristic, complemented
6#22549599C211yesmaximal, characteristic, complemented
7#327×C127C211yesmaximal, characteristic, complemented
8#16861204—21no—
9#16861195—21no—
10#16861193—21no—
11#16861188C2211yescharacteristic
12#16861181—21no—
13#16861177—21no—
14D4×Dic127—21no—
15C2×C4⋊Dic127—21no—
16C2×C4⋊Dic127C2211yescharacteristic
17C2×(C2×D127)⋊C4—21no—
18C2×(C2×D127)⋊C4C2211yescharacteristic
19#16861143—21no—
20#16861140—21no—
21#16861134—21no—
22#16861130—21no—
23C22⋊(C127⋊C8)C2211yescharacteristic
24#121×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27C23×D127D411yescharacteristic
28C2×Dic254D411yescharacteristic
29C2×C22⋊D127—21no—
30C2×C22⋊D127—21no—
31C22×Dic127—21no—
32C22×Dic127—21no—
33C22×Dic127D411yescharacteristic
34C2×C127⋊C8—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127—21no—
38C4⋊Dic127—42no—
39C4⋊Dic127D411yescharacteristic
40Dic127⋊C4—21no—
41C22⋊Dic127—21no—
42C22⋊Dic127—21no—
43(C2×D127)⋊C4—21no—
44(C2×D127)⋊C4—21no—
45(C2×D127)⋊C4—21no—
46(C2×D127)⋊C4—21no—
47(C2×D127)⋊C4—42no—
48C4×Dic127—21no—
49C2×D4×C127—21no—
50C2×D4×C127D411yescharacteristic
51C22×C4×C127—21no—
52C22×C4×C127C2311yescharacteristic
53C2×C8×C127—21no—
54C4⋊C4×C127D411yescharacteristic
55C22⋊C4×C127—21no—
56C22×D127—21no—
57C22×D127—21no—
58C22×D127—21no—
59C22×D127—41no—
60C22×D127—41no—
61Dic254—41no—
62C22⋊D127—81no—
63C22⋊D127—81no—
64C2×Dic127—21no—
65C2×Dic127—21no—
66C2×Dic127—21no—
67C2×Dic127—41no—
68C2×Dic127—41no—
69C2×Dic127—41no—
70C2×Dic127—41no—
71C2×Dic127—81no—
72C127⋊C8—41no—
73C23×C127—21no—
74C23×C127C2×D411yescharacteristic
75D4×C127—41no—
76D4×C127—41no—
77D4×C127—41no—
78C2×C4×C127—22no—
79C2×C4×C127—21no—
80C2×C4×C127—41no—
81C2×C4×C127—41no—
82C2×C4×C127—41no—
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C8×C127—41no—
86C22×C127—21no—
87C22×C127—21no—
88C22×C127—41no—
89C22×C127—41no—
90C22×C127—41no—
91C22×C127#12011yescharacteristic
92C22×C127#13511yescharacteristic
93C22×C127#13711yescharacteristic
94C2×D127—42no—
95C2×D127—41no—
96C2×D127—41no—
97C2×D127—41no—
98C2×D127—81no—
99C4×C127—21no—
100C4×C127—21no—
101C4×C127—42no—
102C4×C127—41no—
103Dic127—41no—
104Dic127—41no—
105Dic127—41no—
106Dic127—81no—
107C2×C127—21no—
108C2×C127—21no—
109C2×C127—81no—
110C2×C127#45211yescharacteristic
111C2×C127#45311yescharacteristic
112C2×C127#45811yescharacteristic
113D127—81no—
114#1618—1271nomaximal, Sylow
115C127#161811yescharacteristic, complemented, Sylow
116#538—1271no—
117#484—1271no—
118#481—1271no—
119#399—1271no—
120#328—1271no—
121#327D12711yescharacteristic, complemented
122#323—1271no—
123#126—2541no—
124#124—2541no—
125#123—2541no—
126C22⋊Q8—1271no—
127#121—2541no—
128#121C2×D12711yescharacteristic
129#120—2541no—
130C4×D4—2541no—
131C2×C4⋊C4—1271no—
132C2×C4⋊C4—2541no—
133C2×C22⋊C4—1271no—
134C2×C22⋊C4—2541no—
135#107—2541no—
136#106—2541no—
137#103—2541no—
138#102—2541no—
139C22⋊C8—1271no—
140C22⋊C8C2×D12711yescharacteristic
141#95C2×D12711yescharacteristic
142C24—1271no—
143C2×Q8—1271no—
144C2×D4—21no—
145C2×D4—2541no—
146C2×D4—2541no—
147C2×D4D50811yescharacteristic
148C22×C4—21no—
149C22×C4—1271no—
150C22×C4—2541no—
151C22×C4—2541no—
152C22×C4C22×D12711yescharacteristic
153C2×C8—21no—
154C2×C8—2541no—
155C4⋊C4—1271no—
156C4⋊C4—5082no—
157C4⋊C4—2541no—
158C4⋊C4—2541no—
159C4⋊C4—2541no—
160C4⋊C4—2541no—
161C4⋊C4D50811yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—5082no—
164C22⋊C4—2541no—
165C22⋊C4—2541no—
166C22⋊C4—2541no—
167C22⋊C4—2541no—
168C22⋊C4—2541no—
169C22⋊C4—2541no—
170C42—2541no—
171C23—21no—
172C23—5081no—
173C23—5081no—
174C23—2541no—
175C23—2541no—
176C23—2541no—
177C23D4×D12711yescharacteristic
178Q8—5081no—
179D4—41no—
180D4—41no—
181D4—41no—
182D4—10161no—
183D4—10161no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—10161no—
190C2×C4—5081no—
191C2×C4—5081no—
192C2×C4—5081no—
193C2×C4—5081no—
194C2×C4—2541no—
195C2×C4—2541no—
196C2×C4—2541no—
197C2×C4D4×D12711yescharacteristic
198C2×C4C2×D50811yescharacteristic
199C8—41no—
200C8—5081no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—10161no—
207C22—5082no—
208C22—5081no—
209C22—5081no—
210C22—5081no—
211C22#1686117911yescharacteristic
212C22#1686122811yescharacteristic
213C22#1686123811yescharacteristic
214C4—21no—
215C4—21no—
216C4—42no—
217C4—41no—
218C4—10161no—
219C4—5081no—
220C4—5081no—
221C4—5081no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—10161no—
226C2#2255003911yescharacteristic
227C2#2255004511yescharacteristic
228C2#2255006711yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#327×C127C2
socleC22×C127#120
prime radical#22550463C2
hypercenter#327D127
hypocenterC127#1618
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #22550212C2semidirect
3. #22550192C2semidirect
4. #22549813C2semidirect
5. #22549612C2semidirect
6. #22549599C2semidirect
7. #327×C127C2semidirect
115. C127#1618semidirect
121. #327D127semidirect

Maximal subgroups

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

subgroupindex
#225504632
#225502122
#225501922
#225498132
#225496122
#225495992
#327×C1272
#1618127