Finite Group Databasebeta

← #30637768

Subgroups — #30637768

Order 16256.

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

normal subgroups only

This group has 246 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#22550195C211yesmaximal, characteristic, complemented
3#22550192C211yesmaximal, characteristic, complemented
4#22549816C211yesmaximal, 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⋊Dic127C2211yescharacteristic
16C2×Dic127⋊C4C2211yescharacteristic
17#16861141—21no—
18#16861140—21no—
19#16861131—21no—
20#16861130—21no—
21C22⋊(C127⋊C8)C2211yescharacteristic
22C127⋊#95—21no—
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⋊Dic127D411yescharacteristic
43Dic127⋊C4—21no—
44Dic127⋊C4—21no—
45Dic127⋊C4—21no—
46C22⋊Dic127—21no—
47(C2×D127)⋊C4—21no—
48(C2×D127)⋊C4—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—
57Dic254—42no—
58D508—41no—
59C22⋊D127—42no—
60C22⋊D127—41no—
61C22⋊D127—41no—
62C22⋊D127—81no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—21no—
66C2×Dic127—21no—
67C2×Dic127—42no—
68C2×Dic127—41no—
69C2×Dic127—42no—
70C2×Dic127—41no—
71C2×Dic127SD1612yes—
72C4×D127—42no—
73C127⋊C8—41no—
74C23×C127—21no—
75C23×C127C2×D411yescharacteristic
76D4×C127—41no—
77D4×C127—41no—
78D4×C127—41no—
79C2×C4×C127—21no—
80C2×C4×C127—41no—
81C2×C4×C127—41no—
82C2×C4×C127—41no—
83C2×C4×C127—41no—
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C8×C127—41no—
87C22×C127—21no—
88C22×C127—21no—
89C22×C127—41no—
90C22×C127—41no—
91C22×C127—41no—
92C22×C127#12011yescharacteristic
93C22×C127C2×SD1611yescharacteristic
94C22×C127AGL(1,ℤ/8)11yescharacteristic
95C2×D127—41no—
96C2×D127—41no—
97C2×D127—41no—
98C4×C127—21no—
99C4×C127—21no—
100C4×C127—41no—
101C4×C127—81no—
102Dic127—22no—
103Dic127—41no—
104Dic127—41no—
105Dic127—81no—
106C2×C127—21no—
107C2×C127—21no—
108C2×C127—81no—
109C2×C127#45011yescharacteristic
110C2×C127#45411yescharacteristic
111C2×C127#45811yescharacteristic
112D127—81no—
113#1616—1271nomaximal, Sylow
114C127#161611yescharacteristic, complemented, Sylow
115#537—1271no—
116#481—1271no—
117#481—1271no—
118#400—1271no—
119#327—1271no—
120#327D12711yescharacteristic, complemented
121#324—1271no—
122C2×#41—2541no—
123C22⋊Q8—2541no—
124C22⋊Q8—2541no—
125#121—1271no—
126#121—2541no—
127#121—2541no—
128#121—2541no—
129#121C2×D12711yescharacteristic
130C2×C4⋊C4—1271no—
131C2×C4⋊C4—1271no—
132#106—2541no—
133#106—2541no—
134#102—2541no—
135#102—2541no—
136C22⋊C8—1271no—
137C22⋊C8C2×D12711yescharacteristic
138#95—2541no—
139#95—2541no—
140#95C2×D12711yescharacteristic
141#41—5082no—
142#41—5082no—
143C2×Q8—2541no—
144C2×D4—21no—
145C2×D4—1271no—
146C2×D4—2541no—
147C2×D4—2541no—
148C2×D4—2541no—
149C2×D4D50811yescharacteristic
150C22×C4—21no—
151C22×C4—1271no—
152C22×C4—1271no—
153C22×C4—2541no—
154C22×C4—2541no—
155C22×C4—2541no—
156C22×C4C22×D12711yescharacteristic
157C2×C8—21no—
158C2×C8—2541no—
159C4⋊C4—1271no—
160C4⋊C4—2541no—
161C4⋊C4—2541no—
162C4⋊C4—2541no—
163C4⋊C4—2541no—
164C4⋊C4—2541no—
165C4⋊C4D50811yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—2541no—
168C22⋊C4—2541no—
169C22⋊C4—2541no—
170C23—21no—
171C23—2541no—
172C23D4×D12711yescharacteristic
173Q8—5082no—
174D4—41no—
175D4—41no—
176D4—41no—
177D4—10161no—
178D4—5081no—
179D4—5081no—
180D4—5081no—
181D4—5082no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—1272no—
188C2×C4—5082no—
189C2×C4—5082no—
190C2×C4—5081no—
191C2×C4—5082no—
192C2×C4—5081no—
193C2×C4—2541no—
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—5081no—
207C22—5081no—
208C22—5081no—
209C22#1686117911yescharacteristic
210C22SD16×D12711yescharacteristic
211C22#1686123311yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—81no—
216C4—10161no—
217C4—5081no—
218C4—5081no—
219C4—2542no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—10161no—
224C2#2255002811yescharacteristic
225C2#2255005011yescharacteristic
226C2#2255006711yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#327×C127C2
socleC22×C127#120
prime radical#22550453C2
hypercenter#327D127
hypocenterC127#1616
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. #22550195C2semidirect
3. #22550192C2semidirect
4. #22549816C2semidirect
5. #22549609C2semidirect
6. #22549601C2semidirect
7. #327×C127C2semidirect
114. C127#1616semidirect
120. #327D127semidirect

Maximal subgroups

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

subgroupindex
#225504532
#225501952
#225501922
#225498162
#225496092
#225496012
#327×C1272
#1616127