Finite Group Databasebeta

← #30637668

Subgroups — #30637668

Order 16256.

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

normal subgroups only

This group has 266 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#22550505C211yesmaximal, characteristic, complemented
2#22550027C211yesmaximal, characteristic
3#22549790C211yesmaximal, characteristic
4#22549614C211yesmaximal, characteristic, complemented
5#22549613C211yesmaximal, characteristic, complemented
6#22549601C211yesmaximal, characteristic, complemented
7#450×C127C211yesmaximal, characteristic
8C2×#14523236—21no—
9#16861208—21no—
10#16861207—21no—
11#16861199—21no—
12#16861199—21no—
13#16861188C2211yescharacteristic
14#16861186—21no—
15#16861165—21no—
16C2×Dic127⋊C4C2211yescharacteristic
17C2×C22⋊Dic127—21no—
18C2×C22⋊Dic127—21no—
19#16861130—21no—
20C22⋊(C127⋊C8)C2211yescharacteristic
21C127⋊#95C2211yescharacteristic
22C22×D4×C127C2211yescharacteristic
23C2×SD16×C127—21no—
24C22⋊Q8×C127C2211yescharacteristic
25#102×C127—21no—
26C22⋊C8×C127C2211yescharacteristic
27C2×Dic254—21no—
28C2×Dic254D411yescharacteristic
29C22×Dic127—21no—
30C22×Dic127D411yescharacteristic
31C22×Dic127D411yescharacteristic
32#14523236—42no—
33C2×C127⋊C8—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127D411yescharacteristic
36Dic127⋊C4—21no—
37Dic127⋊C4—21no—
38Dic127⋊C4—21no—
39Dic127⋊C4—21no—
40Dic127⋊C4—21no—
41Dic127⋊C4—21no—
42Dic127⋊C4—21no—
43C22⋊Dic127—21no—
44C22⋊Dic127—42no—
45C22⋊Dic127—42no—
46C4×Dic127—21no—
47C4×Dic127—21no—
48C24×C127—21no—
49C2×Q8×C127D411yescharacteristic
50C2×D4×C127—21no—
51C2×D4×C127—21no—
52C2×D4×C127—42no—
53C22×C4×C127C2311yescharacteristic
54SD16×C127—42no—
55C2×C8×C127—21no—
56C4⋊C4×C127—21no—
57C4⋊C4×C127D411yescharacteristic
58C22⋊C4×C127—21no—
59Dic254—41no—
60Dic254—81no—
61C2×Dic127—21no—
62C2×Dic127—21no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—21no—
66C2×Dic127—41no—
67C2×Dic127—41no—
68C2×Dic127—41no—
69C2×Dic127—41no—
70C2×Dic127SD1612yescomplemented
71C127⋊C8—41no—
72C23×C127—22no—
73C23×C127—41no—
74C23×C127—41no—
75C23×C127—41no—
76C23×C127—41no—
77C23×C127—41no—
78C23×C127—41no—
79C23×C127C2×D411yescharacteristic
80Q8×C127—41no—
81D4×C127—22no—
82D4×C127—22no—
83D4×C127—41no—
84D4×C127—41no—
85C2×C4×C127—21no—
86C2×C4×C127—21no—
87C2×C4×C127—41no—
88C2×C4×C127C2×D411yescharacteristic
89C2×C4×C127C2×D411yescharacteristic
90C8×C127—41no—
91C22×C127—21no—
92C22×C127—21no—
93C22×C127—42no—
94C22×C127—42no—
95C22×C127—42no—
96C22×C127—81no—
97C22×C127—81no—
98C22×C127—81no—
99C22×C127—81no—
100C22×C127#12011yescharacteristic
101C22×C127C2×SD1611yescharacteristic
102C22×C127#13711yescharacteristic
103C4×C127—21no—
104C4×C127—21no—
105C4×C127—41no—
106C4×C127—41no—
107Dic127—22no—
108Dic127—41no—
109Dic127—41no—
110Dic127—41no—
111Dic127—81no—
112C2×C127—21no—
113C2×C127—21no—
114C2×C127—82no—
115C2×C127#44811yescharacteristic
116C2×C127#45411yescharacteristic
117C2×C127B(4,𝔽2)11yescharacteristic
118#1599—1271nomaximal, Sylow
119C127#159911yescharacteristic, complemented, Sylow
120#544—1271no—
121#450—1271no—
122#450D12711yescharacteristic
123#394—1271no—
124#328—1271no—
125#328—1271no—
126#324—1271no—
127C22×D4C2×D12711yescharacteristic
128C2×SD16—21no—
129C2×SD16—2541no—
130C4⋊Q8—2541no—
131C4⋊Q8—2541no—
132#125—2541no—
133#125—2541no—
134C22⋊Q8—1271no—
135C22⋊Q8—2541no—
136C22⋊Q8C2×D12711yescharacteristic
137#117—2541no—
138C2×C4⋊C4—1271no—
139C2×C22⋊C4—2541no—
140C2×C22⋊C4—2541no—
141#102—21no—
142#102—2541no—
143C22⋊C8—1271no—
144C22⋊C8C2×D12711yescharacteristic
145#95—1271no—
146C24—21no—
147C2×Q8—1271no—
148C2×Q8—2541no—
149C2×Q8C22⋊D12711yescharacteristic
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—42no—
153C22×C4—1271no—
154C22×C4—1271no—
155C22×C4—2541no—
156C22×C4C22×D12711yescharacteristic
157SD16—42no—
158SD16—5082no—
159C2×C8—21no—
160C2×C8—2541no—
161C4⋊C4—21no—
162C4⋊C4—1271no—
163C4⋊C4—2541no—
164C4⋊C4—2541no—
165C4⋊C4—2541no—
166C4⋊C4—2541no—
167C4⋊C4—2541no—
168C4⋊C4—2541no—
169C4⋊C4—2541no—
170C4⋊C4—2541no—
171C4⋊C4C22⋊D12711yescharacteristic
172C22⋊C4—21no—
173C22⋊C4—5082no—
174C22⋊C4—5082no—
175C22⋊C4—2541no—
176C42—2541no—
177C42—2541no—
178C23—22no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23D4×D12711yescharacteristic
186Q8—41no—
187Q8—10161no—
188Q8—5081no—
189D4—22no—
190D4—22no—
191D4—41no—
192D4—41no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—1272no—
197C2×C4—5081no—
198C2×C4—5081no—
199C2×C4—5081no—
200C2×C4—5081no—
201C2×C4—2541no—
202C2×C4—2541no—
203C2×C4—2541no—
204C2×C4—2541no—
205C2×C4—2541no—
206C2×C4D4×D12711yescharacteristic
207C2×C4C2×C22⋊D12711yescharacteristic
208C8—41no—
209C8—5081no—
210C22—21no—
211C22—21no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—81no—
216C22—81no—
217C22—81no—
218C22—81no—
219C22#1686117711yescharacteristic
220C22SD16×D12711yescharacteristic
221C22#1686123811yescharacteristic
222C4—21no—
223C4—21no—
224C4—41no—
225C4—41no—
226C4—10161no—
227C4—5081no—
228C4—5081no—
229C4—5081no—
230C4—2542no—
231C2—21no—
232C2—21no—
233C2—82no—
234C2#2255001311yescharacteristic
235C2#2255004711yescharacteristic
236C2#2255006211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#450×C127C2
socleC22×C127#120
prime radicalC22×D4×C127C22
hypercenter#450D127
hypocenterC127#1599
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #22550505C2semidirect
4. #22549614C2semidirect
5. #22549613C2semidirect
6. #22549601C2semidirect
70. C2×Dic127SD16semidirect
119. C127#1599semidirect

Maximal subgroups

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

subgroupindex
#225505052
#225500272
#225497902
#225496142
#225496132
#225496012
#450×C1272
#1599127