Finite Group Databasebeta

← #30637671

Subgroups — #30637671

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#22550029C211yesmaximal, characteristic
3#22550026C211yesmaximal, characteristic
4#22549794C211yesmaximal, characteristic
5#22549612C211yesmaximal, characteristic, complemented
6#22549601C211yesmaximal, characteristic, complemented
7#328×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C2×#14523237—21no—
10C2×#14523235—21no—
11#16861208—21no—
12#16861207—21no—
13#16861199—21no—
14#16861199—21no—
15#16861188C2211yescharacteristic
16#16861186—21no—
17#16861165—21no—
18C2×Dic127⋊C4C2211yescharacteristic
19C2×(C2×D127)⋊C4—21no—
20C2×(C2×D127)⋊C4—21no—
21#16861131—21no—
22#16861129—21no—
23C22⋊(C127⋊C8)C2211yescharacteristic
24C22⋊Q8×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27C23×D127—21no—
28C2×Dic254—21no—
29C2×Dic254D411yescharacteristic
30C2×D508—21no—
31C2×D508—21no—
32C2×D508—42no—
33C22×Dic127D411yescharacteristic
34C22×Dic127D411yescharacteristic
35#14523237—42no—
36#14523235—42no—
37C2×C127⋊C8—21no—
38C4⋊Dic127—21no—
39C4⋊Dic127D411yescharacteristic
40Dic127⋊C4—21no—
41Dic127⋊C4—21no—
42Dic127⋊C4—21no—
43Dic127⋊C4—21no—
44Dic127⋊C4—21no—
45Dic127⋊C4—21no—
46Dic127⋊C4—21no—
47C22⋊Dic127—21no—
48(C2×D127)⋊C4—42no—
49(C2×D127)⋊C4—42no—
50C4×Dic127—21no—
51C4×Dic127—21no—
52C2×Q8×C127D411yescharacteristic
53C22×C4×C127—21no—
54C22×C4×C127C2311yescharacteristic
55C2×C8×C127—21no—
56C4⋊C4×C127—21no—
57C4⋊C4×C127D411yescharacteristic
58C22⋊C4×C127—21no—
59C22×D127—22no—
60C22×D127—41no—
61C22×D127—41no—
62C22×D127—41no—
63C22×D127—41no—
64C22×D127—41no—
65C22×D127—41no—
66Dic254—41no—
67Dic254—81no—
68D508—22no—
69D508—22no—
70D508—41no—
71D508—41no—
72C2×Dic127—21no—
73C2×Dic127—21no—
74C2×Dic127—21no—
75C2×Dic127—21no—
76C2×Dic127—21no—
77C2×Dic127—41no—
78C2×Dic127SD1612yescomplemented
79C127⋊C8—41no—
80C23×C127C2×D411yescharacteristic
81Q8×C127—41no—
82C2×C4×C127—21no—
83C2×C4×C127—21no—
84C2×C4×C127—41no—
85C2×C4×C127—41no—
86C2×C4×C127—41no—
87C2×C4×C127—41no—
88C2×C4×C127C2×D411yescharacteristic
89C2×C4×C127C2×D411yescharacteristic
90C8×C127—41no—
91C22×C127—21no—
92C22×C127—21no—
93C22×C127#12011yescharacteristic
94C22×C127C2×SD1611yescharacteristic
95C22×C127#13711yescharacteristic
96C2×D127—42no—
97C2×D127—42no—
98C2×D127—42no—
99C2×D127—81no—
100C2×D127—81no—
101C2×D127—81no—
102C2×D127—81no—
103C4×C127—21no—
104C4×C127—21no—
105C4×C127—41no—
106C4×C127—41no—
107C4×C127—81no—
108Dic127—22no—
109Dic127—41no—
110Dic127—41no—
111Dic127—41no—
112C2×C127—21no—
113C2×C127—21no—
114C2×C127#44811yescharacteristic
115C2×C127#45411yescharacteristic
116C2×C127B(4,𝔽2)11yescharacteristic
117D127—82no—
118#1599—1271nomaximal, Sylow
119C127#159911yescharacteristic, complemented, Sylow
120#544—1271no—
121#450—1271no—
122#450—1271no—
123#394—1271no—
124#328—1271no—
125#328D12711yescharacteristic, complemented
126#324—1271no—
127C22×D4—1271no—
128C2×SD16—2541no—
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—2541no—
142#102—2541no—
143C22⋊C8—1271no—
144C22⋊C8C2×D12711yescharacteristic
145#95C2×D12711yescharacteristic
146C24—2541no—
147C2×Q8—1271no—
148C2×Q8—2541no—
149C2×Q8D50811yescharacteristic
150C2×D4—5082no—
151C2×D4—2541no—
152C2×D4—2541no—
153C22×C4—21no—
154C22×C4—1271no—
155C22×C4—1271no—
156C22×C4C22×D12711yescharacteristic
157SD16—5082no—
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⋊C4D50811yescharacteristic
172C22⋊C4—21no—
173C22⋊C4—5082no—
174C22⋊C4—5082no—
175C22⋊C4—2541no—
176C42—2541no—
177C42—2541no—
178C23—5081no—
179C23—5081no—
180C23—5081no—
181C23—5081no—
182C23—5081no—
183C23—5081no—
184C23—2542no—
185C23D4×D12711yescharacteristic
186Q8—41no—
187Q8—10161no—
188Q8—5081no—
189D4—5081no—
190D4—5081no—
191D4—2542no—
192D4—2542no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—1272no—
200C2×C4—5081no—
201C2×C4—2541no—
202C2×C4—2541no—
203C2×C4—2541no—
204C2×C4—2541no—
205C2×C4—2541no—
206C2×C4D4×D12711yescharacteristic
207C2×C4C2×D50811yescharacteristic
208C8—41no—
209C8—5081no—
210C22—21no—
211C22—21no—
212C22—10161no—
213C22—10161no—
214C22—10161no—
215C22—10161no—
216C22—5082no—
217C22—5082no—
218C22—5082no—
219C22#1686117911yescharacteristic
220C22SD16×D12711yescharacteristic
221C22#1686124111yescharacteristic
222C4—21no—
223C4—21no—
224C4—41no—
225C4—41no—
226C4—81no—
227C4—5081no—
228C4—5081no—
229C4—5081no—
230C4—2542no—
231C2—21no—
232C2—21no—
233C2—10162no—
234C2#2255001611yescharacteristic
235C2#2255005011yescharacteristic
236C2#2255006411yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#328×C127C2
socleC22×C127#120
prime radicalC22×D508C22
hypercenter#328D127
hypocenterC127#1599
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
1. #22550505C2semidirect
5. #22549612C2semidirect
6. #22549601C2semidirect
7. #328×C127C2semidirect
78. C2×Dic127SD16semidirect
119. C127#1599semidirect
125. #328D127semidirect

Maximal subgroups

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

subgroupindex
#225505052
#225500292
#225500262
#225497942
#225496122
#225496012
#328×C1272
#1599127