Finite Group Databasebeta

← #30637677

Subgroups — #30637677

Order 16256.

320 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550463C211yesmaximal, characteristic, complemented
2#22550029C211yesmaximal, characteristic, complemented
3#22550009C211yesmaximal, characteristic, complemented
4#22549786C211yesmaximal, characteristic, complemented
5#22549612C211yesmaximal, characteristic, complemented
6#22549599C211yesmaximal, characteristic, complemented
7#327×C127C211yesmaximal, characteristic, complemented
8C22×D508—21no—
9C22×D508C2211yescharacteristic
10C2×#14523237—21no—
11C2×#14523233—21no—
12#16861204—21no—
13#16861195—21no—
14#16861193—21no—
15#16861188C2211yescharacteristic
16#16861181—21no—
17#16861177—21no—
18D4×Dic127—21no—
19C2×(C2×D127)⋊C4—21no—
20C2×(C2×D127)⋊C4C2211yescharacteristic
21#16861131—21no—
22#16861129—21no—
23C22⋊(C127⋊C8)C2211yescharacteristic
24#121×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27C23×D127—21no—
28C23×D127D411yescharacteristic
29C2×Dic254D411yescharacteristic, complemented
30C2×D508—21no—
31C2×D508—21no—
32C2×D508—42no—
33C2×D508—42no—
34C2×D508—42no—
35C2×D508—42no—
36C2×C22⋊D127—21no—
37C2×C22⋊D127—21no—
38C22×Dic127—21no—
39C22×Dic127D411yescharacteristic
40#14523237—42no—
41#14523233—42no—
42C2×C127⋊C8—21no—
43C4⋊Dic127—21no—
44C4⋊Dic127D411yescharacteristic, complemented
45Dic127⋊C4—21no—
46C22⋊Dic127—21no—
47C22⋊Dic127—21no—
48(C2×D127)⋊C4—21no—
49(C2×D127)⋊C4—21no—
50(C2×D127)⋊C4—21no—
51(C2×D127)⋊C4—21no—
52(C2×D127)⋊C4—42no—
53C4×Dic127—21no—
54C2×D4×C127—21no—
55C2×D4×C127D411yescharacteristic, complemented
56C22×C4×C127—21no—
57C22×C4×C127C2311yescharacteristic
58C2×C8×C127—21no—
59C4⋊C4×C127D411yescharacteristic, complemented
60C22⋊C4×C127—21no—
61C22×D127—22no—
62C22×D127—21no—
63C22×D127—21no—
64C22×D127—21no—
65C22×D127—41no—
66C22×D127—41no—
67C22×D127—41no—
68C22×D127—41no—
69C22×D127—41no—
70C22×D127—41no—
71C22×D127—41no—
72C22×D127—41no—
73Dic254—41no—
74D508—22no—
75D508—22no—
76D508—41no—
77D508—41no—
78D508—48no—
79C22⋊D127—81no—
80C22⋊D127—81no—
81C2×Dic127—21no—
82C2×Dic127—21no—
83C2×Dic127—21no—
84C2×Dic127—41no—
85C2×Dic127—81no—
86C127⋊C8—41no—
87C23×C127—21no—
88C23×C127C2×D411yescharacteristic
89D4×C127—41no—
90D4×C127—41no—
91D4×C127—41no—
92C2×C4×C127—22no—
93C2×C4×C127—21no—
94C2×C4×C127—41no—
95C2×C4×C127—41no—
96C2×C4×C127—41no—
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C8×C127—41no—
100C22×C127—21no—
101C22×C127—21no—
102C22×C127—41no—
103C22×C127—41no—
104C22×C127—41no—
105C22×C127#12011yescharacteristic
106C22×C127#13511yescharacteristic
107C22×C127AGL(1,ℤ/8)11yescharacteristic
108C2×D127—42no—
109C2×D127—42no—
110C2×D127—42no—
111C2×D127—41no—
112C2×D127—41no—
113C2×D127—41no—
114C2×D127—42no—
115C2×D127—81no—
116C2×D127—81no—
117C2×D127—81no—
118C2×D127—81no—
119C2×D127—81no—
120C4×C127—21no—
121C4×C127—21no—
122C4×C127—42no—
123C4×C127—41no—
124Dic127—41no—
125Dic127—41no—
126Dic127—41no—
127C2×C127—21no—
128C2×C127—21no—
129C2×C127—81no—
130C2×C127#44911yescharacteristic
131C2×C127#45311yescharacteristic
132C2×C127B(4,𝔽2)11yescharacteristic
133D127—81no—
134D127—82no—
135#1600—1271nomaximal, Sylow
136C127#160011yescharacteristic, complemented, Sylow
137#538—1271no—
138#450—1271no—
139#447—1271no—
140#392—1271no—
141#328—1271no—
142#327D12711yescharacteristic, complemented
143#323—1271no—
144C22×D4—1271no—
145C22×D4—2541no—
146C2×SD16—2541no—
147C2×D8—2541no—
148#126—2541no—
149#124—2541no—
150#123—2541no—
151C22⋊Q8—1271no—
152#121—2541no—
153#121C2×D12711yescharacteristic
154#120—2541no—
155C4×D4—2541no—
156C2×C22⋊C4—1271no—
157C2×C22⋊C4—2541no—
158#102—2541no—
159#102—2541no—
160C22⋊C8—1271no—
161C22⋊C8C2×D12711yescharacteristic
162#95C2×D12711yescharacteristic
163C24—1271no—
164C24—2541no—
165C2×Q8—1271no—
166C2×D4—21no—
167C2×D4—5082no—
168C2×D4—5082no—
169C2×D4—5082no—
170C2×D4—5082no—
171C2×D4—2541no—
172C2×D4—2541no—
173C2×D4—2541no—
174C2×D4—2541no—
175C2×D4D50811yescharacteristic, complemented
176C22×C4—21no—
177C22×C4—1271no—
178C22×C4—2541no—
179C22×C4C22×D12711yescharacteristic
180SD16—5082no—
181D8—5082no—
182C2×C8—21no—
183C2×C8—2541no—
184C4⋊C4—1271no—
185C4⋊C4—2541no—
186C4⋊C4—2541no—
187C4⋊C4D50811yescharacteristic, complemented
188C22⋊C4—21no—
189C22⋊C4—5082no—
190C22⋊C4—2541no—
191C22⋊C4—2541no—
192C22⋊C4—2541no—
193C22⋊C4—2541no—
194C22⋊C4—2541no—
195C22⋊C4—2541no—
196C42—2541no—
197C23—21no—
198C23—5081no—
199C23—5081no—
200C23—5081no—
201C23—5081no—
202C23—5081no—
203C23—5081no—
204C23—5081no—
205C23—5081no—
206C23—2542no—
207C23—2541no—
208C23—2541no—
209C23—2541no—
210C23D4×D12711yescharacteristic
211Q8—5081no—
212D4—41no—
213D4—41no—
214D4—41no—
215D4—10161no—
216D4—10161no—
217D4—5081no—
218D4—5081no—
219D4—5088no—
220D4—2542no—
221D4—2542no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—10161no—
228C2×C4—5081no—
229C2×C4—2541no—
230C2×C4—2541no—
231C2×C4—2541no—
232C2×C4D4×D12711yescharacteristic
233C2×C4C2×D50811yescharacteristic
234C8—41no—
235C8—5081no—
236C22—21no—
237C22—21no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—10161no—
242C22—10161no—
243C22—10161no—
244C22—10161no—
245C22—10161no—
246C22—5082no—
247C22—5082no—
248C22—5082no—
249C22—5081no—
250C22—5081no—
251C22—5081no—
252C22—5082no—
253C22#1686117911yescharacteristic
254C22#1686123011yescharacteristic
255C22#1686123311yescharacteristic
256C4—21no—
257C4—21no—
258C4—42no—
259C4—41no—
260C4—5081no—
261C4—5081no—
262C4—5081no—
263C2—21no—
264C2—21no—
265C2—81no—
266C2—10162no—
267C2—10161no—
268C2#2255002211yescharacteristic
269C2#2255004511yescharacteristic
270C2#2255006411yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#327×C127C2
socleC22×C127#120
hypercenter#327D127
hypocenterC127#1600
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
1. #22550463C2semidirect
2. #22550029C2semidirect
3. #22550009C2semidirect
4. #22549786C2semidirect
5. #22549612C2semidirect
6. #22549599C2semidirect
7. #327×C127C2semidirect
29. C2×Dic254D4semidirect
44. C4⋊Dic127D4semidirect
55. C2×D4×C127D4semidirect
59. C4⋊C4×C127D4semidirect
136. C127#1600semidirect
142. #327D127semidirect
175. C2×D4D508semidirect
187. C4⋊C4D508semidirect

Maximal subgroups

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

subgroupindex
#225504632
#225500292
#225500092
#225497862
#225496122
#225495992
#327×C1272
#1600127