Finite Group Databasebeta

← #30637734

Subgroups — #30637734

Order 16256.

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

normal subgroups only

This group has 254 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#22550453C211yesmaximal, characteristic
2#22550190C211yesmaximal, characteristic, complemented
3#22550187C211yesmaximal, characteristic, complemented
4#22549808C211yesmaximal, characteristic, complemented
5#22549609C211yesmaximal, characteristic, complemented
6#22549601C211yesmaximal, characteristic, complemented
7#327×C127C211yesmaximal, characteristic, complemented
8C2×#14523247—21no—
9#16861187—21no—
10#16861186—21no—
11#16861183C2211yescharacteristic
12#16861182—21no—
13#16861181—21no—
14#16861181—21no—
15C2×C4⋊Dic127—21no—
16C2×C4⋊Dic127C2211yescharacteristic
17C2×Dic127⋊C4C2211yescharacteristic
18#16861143—21no—
19#16861142—21no—
20#16861131—21no—
21#16861130—21no—
22C22⋊(C127⋊C8)C2211yescharacteristic
23C127⋊#95—21no—
24#121×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27#14523247—42no—
28#14523247—42no—
29C2×Dic254—21no—
30C2×D508D411yescharacteristic
31C2×C22⋊D127—21no—
32C2×C22⋊D127—21no—
33C2×C22⋊D127—21no—
34C22×Dic127—21no—
35C22×Dic127—21no—
36C22×Dic127D411yescharacteristic
37C22×Dic127D411yescharacteristic
38C2×C4×D127—21no—
39C2×C127⋊C8—21no—
40C4⋊Dic127—21no—
41C4⋊Dic127—21no—
42C4⋊Dic127—42no—
43C4⋊Dic127D411yescharacteristic
44Dic127⋊C4—21no—
45Dic127⋊C4—21no—
46Dic127⋊C4—21no—
47C22⋊Dic127—21no—
48(C2×D127)⋊C4—21no—
49(C2×D127)⋊C4—21no—
50C2×D4×C127—21no—
51C2×D4×C127D411yescharacteristic
52C22×C4×C127—21no—
53C22×C4×C127C2311yescharacteristic
54C2×C8×C127—21no—
55C4⋊C4×C127D411yescharacteristic
56C22⋊C4×C127—21no—
57C22×D127—21no—
58Dic254—42no—
59D508—41no—
60C22⋊D127—42no—
61C22⋊D127—41no—
62C22⋊D127—41no—
63C22⋊D127—81no—
64C2×Dic127—21no—
65C2×Dic127—21no—
66C2×Dic127—21no—
67C2×Dic127—21no—
68C2×Dic127—42no—
69C2×Dic127—41no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C2×Dic127—41no—
73C2×Dic127D812yes—
74C4×D127—42no—
75C127⋊C8—41no—
76C23×C127—21no—
77C23×C127C2×D411yescharacteristic
78D4×C127—41no—
79D4×C127—41no—
80D4×C127—41no—
81C2×C4×C127—22no—
82C2×C4×C127—21no—
83C2×C4×C127—41no—
84C2×C4×C127—41no—
85C2×C4×C127—41no—
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C8×C127—41no—
89C22×C127—21no—
90C22×C127—21no—
91C22×C127—41no—
92C22×C127—41no—
93C22×C127—41no—
94C22×C127#12011yescharacteristic
95C22×C127C2×D811yescharacteristic
96C22×C127AGL(1,ℤ/8)11yescharacteristic
97C2×D127—41no—
98C2×D127—41no—
99C2×D127—41no—
100C4×C127—21no—
101C4×C127—21no—
102C4×C127—42no—
103C4×C127—41no—
104Dic127—22no—
105Dic127—41no—
106Dic127—41no—
107Dic127—81no—
108C2×C127—21no—
109C2×C127—21no—
110C2×C127—81no—
111C2×C127#44711yescharacteristic
112C2×C127#45611yescharacteristic
113C2×C127#45811yescharacteristic
114D127—81no—
115#1610—1271nomaximal, Sylow
116C127#161011yescharacteristic, complemented, Sylow
117#537—1271no—
118#480—1271no—
119#480—1271no—
120#398—1271no—
121#327—1271no—
122#327D12711yescharacteristic, complemented
123#324—1271no—
124C2×#41—2541no—
125C22⋊Q8—2541no—
126C22⋊Q8—2541no—
127#121—1271no—
128#121—2541no—
129#121—2541no—
130#121—2541no—
131#121C2×D12711yescharacteristic
132C2×C4⋊C4—1271no—
133C2×C4⋊C4—1271no—
134C2×C4⋊C4—2541no—
135#107—2541no—
136#107—2541no—
137#102—2541no—
138#102—2541no—
139C22⋊C8—1271no—
140C22⋊C8C2×D12711yescharacteristic
141#95—2541no—
142#95C2×D12711yescharacteristic
143#41—5082no—
144#41—5082no—
145C2×Q8—2541no—
146C2×D4—21no—
147C2×D4—1271no—
148C2×D4—2541no—
149C2×D4—2541no—
150C2×D4—2541no—
151C2×D4D50811yescharacteristic
152C22×C4—21no—
153C22×C4—1271no—
154C22×C4—1271no—
155C22×C4—2541no—
156C22×C4—2541no—
157C22×C4—2541no—
158C22×C4C22×D12711yescharacteristic
159C2×C8—21no—
160C2×C8—2541no—
161C4⋊C4—1271no—
162C4⋊C4—5082no—
163C4⋊C4—2541no—
164C4⋊C4—2541no—
165C4⋊C4—2541no—
166C4⋊C4—2541no—
167C4⋊C4—2541no—
168C4⋊C4D50811yescharacteristic
169C22⋊C4—21no—
170C22⋊C4—2541no—
171C22⋊C4—2541no—
172C22⋊C4—2541no—
173C23—21no—
174C23—2541no—
175C23D4×D12711yescharacteristic
176Q8—5082no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—10161no—
181D4—5081no—
182D4—5081no—
183D4—5081no—
184D4—5082no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—1272no—
191C2×C4—5082no—
192C2×C4—5081no—
193C2×C4—5081no—
194C2×C4—5081no—
195C2×C4—5082no—
196C2×C4—5081no—
197C2×C4—2541no—
198C2×C4—2541no—
199C2×C4—2541no—
200C2×C4—2541no—
201C2×C4D4×D12711yescharacteristic
202C2×C4C2×D50811yescharacteristic
203C8—41no—
204C8—5081no—
205C22—21no—
206C22—21no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—5081no—
211C22—5081no—
212C22—5081no—
213C22#1686117911yescharacteristic
214C22D8×D12711yescharacteristic
215C22#1686123611yescharacteristic
216C4—21no—
217C4—21no—
218C4—42no—
219C4—41no—
220C4—10161no—
221C4—5081no—
222C4—5081no—
223C4—2542no—
224C2—21no—
225C2—21no—
226C2—81no—
227C2—10161no—
228C2#2255001111yescharacteristic
229C2#2255006011yescharacteristic
230C2#2255006711yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#327×C127C2
socleC22×C127#120
prime radical#22550453C2
hypercenter#327D127
hypocenterC127#1610
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. #22550190C2semidirect
3. #22550187C2semidirect
4. #22549808C2semidirect
5. #22549609C2semidirect
6. #22549601C2semidirect
7. #327×C127C2semidirect
116. C127#1610semidirect
122. #327D127semidirect

Maximal subgroups

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

subgroupindex
#225504532
#225501902
#225501872
#225498082
#225496092
#225496012
#327×C1272
#1610127