Finite Group Databasebeta

← #30637098

Subgroups — #30637098

Order 16256.

214 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 61 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
1C2×#16861204C211yesmaximal, characteristic
2#22549930C211yesmaximal, characteristic, complemented
3#22549926C211yesmaximal, characteristic, complemented
4#22549602C211yesmaximal, characteristic, complemented
5#22549599C211yesmaximal, characteristic
6#22549599C211yesmaximal, characteristic
7#324×C127C211yesmaximal, characteristic, complemented
8#16861204—21no—
9#16861204—21no—
10#16861204—21no—
11#16861204—21no—
12C2×C4⋊Dic127C411yescharacteristic
13C2×Dic127⋊C4C411yescharacteristic
14C2×(C2×D127)⋊C4C411yescharacteristic
15C2×(C2×D127)⋊C4C411yescharacteristic
16C2×(C2×D127)⋊C4C2211yescharacteristic
17C2×C4×Dic127C2211yescharacteristic
18Dic127⋊C8—21no—
19Dic127⋊C8—21no—
20C22⋊(C127⋊C8)C2211yescharacteristic
21C22⋊(C127⋊C8)C2211yescharacteristic
22#16861119—21no—
23C8×Dic127—21no—
24C2×C4⋊C4×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26C22⋊C8×C127C2211yescharacteristic
27C23×D127C2×C411yescharacteristic
28C22×Dic127C2×C411yescharacteristic
29C22×Dic127C2×C411yescharacteristic
30C22×Dic127C2×C411yescharacteristic
31C2×C127⋊C8—21no—
32C2×C127⋊C8—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127—21no—
35Dic127⋊C4—21no—
36Dic127⋊C4—21no—
37(C2×D127)⋊C4—21no—
38(C2×D127)⋊C4—21no—
39(C2×D127)⋊C4—21no—
40(C2×D127)⋊C4—21no—
41(C2×D127)⋊C4—21no—
42(C2×D127)⋊C4—21no—
43C4×Dic127D411yescharacteristic
44C4×Dic127D411yescharacteristic
45C4×Dic127D411yescharacteristic
46C4×Dic127D411yescharacteristic
47C22×C4×C127C2×C411yescharacteristic
48C22×C4×C127C2×C411yescharacteristic
49C22×C4×C127C2311yescharacteristic
50C2×C8×C127—21no—
51C2×C8×C127—21no—
52C4⋊C4×C127—21no—
53C4⋊C4×C127—21no—
54C22×D127—21no—
55C22×D127—21no—
56C22×D127—21no—
57C22×D127—41no—
58C22×D127—41no—
59C2×Dic127—21no—
60C2×Dic127—21no—
61C2×Dic127—21no—
62C2×Dic127—41no—
63C2×Dic127—41no—
64C2×Dic127C22⋊C411yescharacteristic
65C2×Dic127C22⋊C411yescharacteristic
66C2×Dic127C22⋊C411yescharacteristic
67C2×Dic127C22⋊C411yescharacteristic
68C127⋊C8—41no—
69C127⋊C8—41no—
70C23×C127C22×C411yescharacteristic
71C2×C4×C127—21no—
72C2×C4×C127—21no—
73C2×C4×C127—21no—
74C2×C4×C127—21no—
75C2×C4×C127—41no—
76C2×C4×C127—41no—
77C2×C4×C127C2×D411yescharacteristic
78C2×C4×C127C2×D411yescharacteristic
79C8×C127—41no—
80C8×C127—41no—
81C22×C127—21no—
82C22×C127—21no—
83C22×C127C2×C22⋊C411yescharacteristic
84C22×C127#13111yescharacteristic
85C22×C127#13111yescharacteristic
86C2×D127—41no—
87C2×D127—41no—
88C2×D127—41no—
89C2×D127—81no—
90C2×D127—81no—
91C4×C127—21no—
92C4×C127—21no—
93C4×C127—41no—
94C4×C127—41no—
95Dic127—21no—
96Dic127—21no—
97Dic127—21no—
98Dic127—21no—
99Dic127—41no—
100C2×C127—21no—
101C2×C127—21no—
102C2×C127#40811yescharacteristic
103C2×C127#41011yescharacteristic
104C2×C127#41311yescharacteristic
105D127—81no—
106#1473—1271nomaximal, Sylow
107C127#147311yescharacteristic, complemented, Sylow
108C2×#126—1271no—
109#432—1271no—
110#431—1271no—
111#324—1271no—
112#324D12711yescharacteristic, complemented
113#323—1271no—
114#323—1271no—
115#126—2541no—
116#126—2541no—
117#126—2541no—
118#126—2541no—
119C2×C4⋊C4—1271no—
120C2×C4⋊C4—1271no—
121C2×C4⋊C4C2×D12711yescharacteristic
122C2×C22⋊C4—1271no—
123C2×C22⋊C4—1271no—
124C2×C22⋊C4—1271no—
125C2×C42—1271no—
126C4⋊C8—2541no—
127C4⋊C8—2541no—
128C22⋊C8—1271no—
129C22⋊C8—1271no—
130C22⋊C8C2×D12711yescharacteristic
131C22⋊C8C2×D12711yescharacteristic
132#97—2541no—
133C4×C8—2541no—
134C24—1271no—
135C22×C4—1271no—
136C22×C4—1271no—
137C22×C4—1271no—
138C22×C4C4×D12711yescharacteristic
139C22×C4C4×D12711yescharacteristic
140C22×C4C22×D12711yescharacteristic
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—2541no—
144C2×C8—2541no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—2541no—
148C4⋊C4—2541no—
149C4⋊C4—2541no—
150C4⋊C4—2541no—
151C22⋊C4—2541no—
152C22⋊C4—2541no—
153C22⋊C4—2541no—
154C22⋊C4—2541no—
155C22⋊C4—2541no—
156C22⋊C4—2541no—
157C42—1271no—
158C42—1271no—
159C42—1271no—
160C42—1271no—
161C23—5081no—
162C23—5081no—
163C23—2541no—
164C23—2541no—
165C23—2541no—
166C23C2×C4×D12711yescharacteristic
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—1271no—
174C2×C4—1271no—
175C2×C4—1271no—
176C2×C4—1271no—
177C2×C4—5081no—
178C2×C4—5081no—
179C2×C4—2541no—
180C2×C4—2541no—
181C2×C4—2541no—
182C2×C4D4×D12711yescharacteristic
183C2×C4D4×D12711yescharacteristic
184C8—41no—
185C8—41no—
186C8—5081no—
187C8—5081no—
188C22—21no—
189C22—21no—
190C22—10161no—
191C22—10161no—
192C22—5081no—
193C22—5081no—
194C22—5081no—
195C22C22⋊C4×D12711yescharacteristic
196C22#1686121511yescharacteristic
197C22#1686121711yescharacteristic
198C4—21no—
199C4—21no—
200C4—41no—
201C4—41no—
202C4—5081no—
203C4—2541no—
204C4—2541no—
205C4—2541no—
206C4—2541no—
207C2—21no—
208C2—21no—
209C2—10161no—
210C2#2254984111yescharacteristic
211C2#2254984811yescharacteristic
212C2#2254985911yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#324×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC23×D127C2×C4
cosocleC22×C4×C127C23
hypercenter#324D127
hypocenterC127#1473
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #22549930C2semidirect
3. #22549926C2semidirect
4. #22549602C2semidirect
7. #324×C127C2semidirect
107. C127#1473semidirect
112. #324D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612042
#225499302
#225499262
#225496022
#225495992
#324×C1272
#1473127