Finite Group Databasebeta

← #30637695

Subgroups — #30637695

Order 16256.

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

normal subgroups only

This group has 244 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#22550473C211yesmaximal, characteristic
2#22550034C211yesmaximal, characteristic, complemented
3#22550017C211yesmaximal, characteristic
4#22549796C211yesmaximal, characteristic, complemented
5#22549614C211yesmaximal, characteristic, complemented
6#22549611C211yesmaximal, characteristic
7#324×C127C211yesmaximal, characteristic, complemented
8C22×Dic254C2211yescharacteristic
9C2×Dic508—21no—
10C2×#14523237—21no—
11#16861204—21no—
12#16861199—21no—
13#16861194—21no—
14#16861192—21no—
15#16861188C2211yescharacteristic
16#16861187—21no—
17#16861183C2211yescharacteristic
18C4×C22⋊D127—21no—
19C2×Dic127⋊C4—21no—
20#16861134—21no—
21#16861134—21no—
22C127⋊#95—21no—
23C127⋊#95C2211yescharacteristic
24C2×C4⋊C4×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26C22⋊C8×C127C2211yescharacteristic
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—42no—
30C2×Dic254D411yescharacteristic
31C2×D508D411yescharacteristic
32C2×C22⋊D127—21no—
33C22×Dic127—21no—
34C22×Dic127—21no—
35C2×C4×D127—21no—
36Dic508—42no—
37#14523237—42no—
38C4⋊Dic127D411yescharacteristic
39C4⋊Dic127D411yescharacteristic
40Dic127⋊C4—21no—
41Dic127⋊C4—21no—
42Dic127⋊C4—21no—
43Dic127⋊C4—21no—
44Dic127⋊C4—42no—
45C22⋊Dic127—21no—
46(C2×D127)⋊C4—21no—
47(C2×D127)⋊C4—21no—
48(C2×D127)⋊C4—21no—
49(C2×D127)⋊C4—21no—
50C4×Dic127—21no—
51C22×C4×C127D411yescharacteristic
52C22×C4×C127D411yescharacteristic
53C22×C4×C127C2311yescharacteristic
54C2×C8×C127—21no—
55C2×C8×C127—21no—
56C4⋊C4×C127—21no—
57C4⋊C4×C127—21no—
58C22×D127—21no—
59Dic254—22no—
60Dic254—22no—
61Dic254—41no—
62Dic254—41no—
63Dic254—41no—
64D508—41no—
65C22⋊D127—41no—
66C22⋊D127—41no—
67C2×Dic127—22no—
68C2×Dic127—22no—
69C2×Dic127—21no—
70C2×Dic127—21no—
71C2×Dic127—21no—
72C2×Dic127—41no—
73C2×Dic127—41no—
74C2×Dic127—41no—
75C2×Dic127—41no—
76C4×D127—81no—
77C23×C127C2×D411yescharacteristic
78C2×C4×C127—21no—
79C2×C4×C127—21no—
80C2×C4×C127—21no—
81C2×C4×C127—21no—
82C2×C4×C127—41no—
83C2×C4×C127—41no—
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C8×C127—41no—
87C8×C127—41no—
88C22×C127—21no—
89C22×C127—21no—
90C22×C127#12011yescharacteristic
91C22×C127#13511yescharacteristic
92C22×C127#13711yescharacteristic
93C2×D127—41no—
94C2×D127—41no—
95C2×D127—41no—
96C4×C127—21no—
97C4×C127—21no—
98C4×C127—41no—
99C4×C127—41no—
100Dic127—42no—
101Dic127—41no—
102Dic127—41no—
103Dic127—41no—
104Dic127—42no—
105C2×C127—21no—
106C2×C127—21no—
107C2×C127#45211yescharacteristic
108C2×C127#45611yescharacteristic
109C2×C127B(4,𝔽2)11yescharacteristic
110D127—81no—
111#1603—1271nomaximal, Sylow
112C127#160311yescharacteristic, complemented, Sylow
113#539—1271no—
114#451—1271no—
115#448—1271no—
116#395—1271no—
117#328—1271no—
118#327—1271no—
119#324D12711yescharacteristic, complemented
120C22×Q8—1271no—
121C2×Q16—2541no—
122C2×SD16—2541no—
123#126—2541no—
124#125—2541no—
125#123—2541no—
126#123—2541no—
127C22⋊Q8—1271no—
128C22⋊Q8—2541no—
129#121—1271no—
130C4×D4—2541no—
131C2×C4⋊C4—2541no—
132C2×C4⋊C4C2×D12711yescharacteristic
133#103—2541no—
134#103—2541no—
135C22⋊C8C2×D12711yescharacteristic
136C22⋊C8C2×D12711yescharacteristic
137#95—1271no—
138#95—2541no—
139C2×Q8—1271no—
140C2×Q8—5082no—
141C2×Q8—2541no—
142C2×Q8—2541no—
143C2×D4—1271no—
144C2×D4—2541no—
145C22×C4—2541no—
146C22×C4—2541no—
147C22×C4—2541no—
148C22×C4D50811yescharacteristic
149C22×C4D50811yescharacteristic
150C22×C4C22×D12711yescharacteristic
151Q16—5082no—
152SD16—5082no—
153C2×C8—21no—
154C2×C8—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—1271no—
158C4⋊C4—1271no—
159C4⋊C4—5082no—
160C4⋊C4—2541no—
161C4⋊C4—2541no—
162C4⋊C4—2541no—
163C4⋊C4—2541no—
164C22⋊C4—2541no—
165C22⋊C4—2541no—
166C22⋊C4—2541no—
167C22⋊C4—2541no—
168C22⋊C4—2541no—
169C42—2541no—
170C23—2541no—
171C23C2×D50811yescharacteristic
172Q8—5081no—
173Q8—5081no—
174Q8—5081no—
175Q8—2542no—
176Q8—2542no—
177D4—5081no—
178D4—5081no—
179D4—5081no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—10161no—
187C2×C4—5081no—
188C2×C4—5081no—
189C2×C4—5081no—
190C2×C4—5081no—
191C2×C4—2542no—
192C2×C4—2542no—
193C2×C4—2541no—
194C2×C4—2541no—
195C2×C4—2541no—
196C2×C4D4×D12711yescharacteristic
197C2×C4D4×D12711yescharacteristic
198C8—41no—
199C8—41no—
200C22—21no—
201C22—21no—
202C22—5081no—
203C22—5081no—
204C22—5081no—
205C22#1686117911yescharacteristic
206C22#1686123111yescharacteristic
207C22#1686124211yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—41no—
212C4—5082no—
213C4—5081no—
214C4—5081no—
215C4—5081no—
216C4—5082no—
217C2—21no—
218C2—21no—
219C2—10161no—
220C2#2255004011yescharacteristic
221C2#2255006111yescharacteristic
222C2#2255006411yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#324×C127C2
socleC22×C127#120
prime radical#16861183C22
hypercenter#324D127
hypocenterC127#1603
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #22550034C2semidirect
4. #22549796C2semidirect
5. #22549614C2semidirect
7. #324×C127C2semidirect
112. C127#1603semidirect
119. #324D127semidirect

Maximal subgroups

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

subgroupindex
#225504732
#225500342
#225500172
#225497962
#225496142
#225496112
#324×C1272
#1603127