Finite Group Databasebeta

← #30637780

Subgroups — #30637780

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#22550464C211yesmaximal, characteristic
2#22550212C211yesmaximal, characteristic, complemented
3#22550195C211yesmaximal, characteristic, complemented
4#22549812C211yesmaximal, characteristic, complemented
5#22549614C211yesmaximal, characteristic, complemented
6#22549611C211yesmaximal, characteristic, complemented
7#323×C127C211yesmaximal, characteristic, complemented
8#16861202—21no—
9#16861195—21no—
10#16861191—21no—
11#16861188C2211yescharacteristic
12#16861183C2211yescharacteristic
13#16861182—21no—
14#16861178—21no—
15C4×C22⋊D127—21no—
16C2×C4⋊Dic127C2211yescharacteristic
17C2×Dic127⋊C4—21no—
18C2×C22⋊Dic127—21no—
19#16861143—21no—
20#16861141—21no—
21#16861134—21no—
22#16861131—21no—
23C127⋊#95C2211yescharacteristic
24C2×C22⋊C4×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26C22⋊C8×C127C2211yescharacteristic
27C2×Dic254D411yescharacteristic
28C2×D508D411yescharacteristic
29C2×C22⋊D127—21no—
30C2×C22⋊D127—21no—
31C2×C22⋊D127—21no—
32C22×Dic127—21no—
33C22×Dic127—21no—
34C2×C4×D127—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127D411yescharacteristic
38C4⋊Dic127D411yescharacteristic
39Dic127⋊C4—21no—
40Dic127⋊C4—21no—
41Dic127⋊C4—42no—
42C22⋊Dic127—21no—
43C22⋊Dic127—21no—
44C22⋊Dic127—21no—
45C22⋊Dic127—42no—
46(C2×D127)⋊C4—21no—
47(C2×D127)⋊C4—21no—
48C4×Dic127—21no—
49C24×C127D411yescharacteristic
50C22×C4×C127D411yescharacteristic
51C22×C4×C127C2311yescharacteristic
52C2×C8×C127—21no—
53C2×C8×C127—21no—
54C22⋊C4×C127—21no—
55C22⋊C4×C127—21no—
56C22×D127—21no—
57Dic254—41no—
58D508—41no—
59C22⋊D127—41no—
60C22⋊D127—41no—
61C22⋊D127—81no—
62C22⋊D127—81no—
63C2×Dic127—22no—
64C2×Dic127—21no—
65C2×Dic127—21no—
66C2×Dic127—21no—
67C2×Dic127—41no—
68C2×Dic127—41no—
69C2×Dic127—41no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C4×D127—81no—
73C23×C127—21no—
74C23×C127—21no—
75C23×C127—21no—
76C23×C127—41no—
77C23×C127—41no—
78C23×C127C2×D411yescharacteristic
79C2×C4×C127—21no—
80C2×C4×C127—41no—
81C2×C4×C127—41no—
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C8×C127—41no—
85C8×C127—41no—
86C22×C127—21no—
87C22×C127—21no—
88C22×C127—42no—
89C22×C127—41no—
90C22×C127—41no—
91C22×C127—41no—
92C22×C127—81no—
93C22×C127#12011yescharacteristic
94C22×C127#13511yescharacteristic
95C22×C127#13711yescharacteristic
96C2×D127—41no—
97C2×D127—41no—
98C2×D127—41no—
99C4×C127—21no—
100C4×C127—21no—
101C4×C127—41no—
102Dic127—42no—
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#327—1271no—
122#323D12711yescharacteristic, complemented
123#126—2541no—
124#124—2541no—
125#123—2541no—
126C22⋊Q8—1271no—
127#121—1271no—
128#121—2541no—
129#120—2541no—
130C4×D4—2541no—
131C2×C4⋊C4—1271no—
132C2×C4⋊C4—2541no—
133C2×C22⋊C4—2541no—
134C2×C22⋊C4C2×D12711yescharacteristic
135#107—2541no—
136#106—2541no—
137#103—2541no—
138#102—2541no—
139C22⋊C8C2×D12711yescharacteristic
140C22⋊C8C2×D12711yescharacteristic
141#95—1271no—
142C24D50811yescharacteristic
143C2×Q8—1271no—
144C2×D4—1271no—
145C2×D4—2541no—
146C2×D4—2541no—
147C2×D4—2541no—
148C22×C4—2541no—
149C22×C4—2541no—
150C22×C4—2541no—
151C22×C4D50811yescharacteristic
152C22×C4C22×D12711yescharacteristic
153C2×C8—21no—
154C2×C8—21no—
155C4⋊C4—1271no—
156C4⋊C4—1271no—
157C4⋊C4—5082no—
158C4⋊C4—2541no—
159C4⋊C4—2541no—
160C4⋊C4—2541no—
161C4⋊C4—2541no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—5082no—
165C22⋊C4—2541no—
166C22⋊C4—2541no—
167C22⋊C4—2541no—
168C22⋊C4—2541no—
169C22⋊C4—2541no—
170C42—2541no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—41no—
175C23—41no—
176C23—2541no—
177C23C2×D50811yescharacteristic
178Q8—5081no—
179D4—10161no—
180D4—10161no—
181D4—5081no—
182D4—5081no—
183D4—5081no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—10161no—
188C2×C4—5081no—
189C2×C4—5081no—
190C2×C4—5081no—
191C2×C4—5081no—
192C2×C4—5081no—
193C2×C4—2542no—
194C2×C4—2541no—
195C2×C4—2541no—
196C2×C4—2541no—
197C2×C4D4×D12711yescharacteristic
198C2×C4D4×D12711yescharacteristic
199C8—41no—
200C8—41no—
201C22—21no—
202C22—21no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—81no—
208C22—5081no—
209C22—5081no—
210C22—5081no—
211C22#1686117911yescharacteristic
212C22#1686123111yescharacteristic
213C22#1686124211yescharacteristic
214C4—21no—
215C4—21no—
216C4—41no—
217C4—10161no—
218C4—5082no—
219C4—5081no—
220C4—5081no—
221C4—5081no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—10161no—
226C2#2255004011yescharacteristic
227C2#2255004611yescharacteristic
228C2#2255006711yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#323×C127C2
socleC22×C127#120
prime radical#22550464C2
hypercenter#323D127
hypocenterC127#1618
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #22550212C2semidirect
3. #22550195C2semidirect
4. #22549812C2semidirect
5. #22549614C2semidirect
6. #22549611C2semidirect
7. #323×C127C2semidirect
115. C127#1618semidirect
122. #323D127semidirect

Maximal subgroups

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

subgroupindex
#225504642
#225502122
#225501952
#225498122
#225496142
#225496112
#323×C1272
#1618127