Finite Group Databasebeta

← #30637678

Subgroups — #30637678

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#22550464C211yesmaximal, characteristic, complemented
2#22550029C211yesmaximal, characteristic, complemented
3#22550012C211yesmaximal, characteristic, complemented
4#22549785C211yesmaximal, characteristic, complemented
5#22549614C211yesmaximal, characteristic, complemented
6#22549611C211yesmaximal, characteristic, complemented
7#323×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C22×C22⋊D127—21no—
10C2×#14523237—21no—
11C2×D1016—21no—
12#16861202—21no—
13#16861195—21no—
14#16861191—21no—
15#16861188C2211yescharacteristic
16#16861183C2211yescharacteristic
17#16861182—21no—
18#16861178—21no—
19C4×C22⋊D127—21no—
20C2×(C2×D127)⋊C4—21no—
21#16861131—21no—
22#16861131—21no—
23C127⋊#95C2211yescharacteristic
24C2×C22⋊C4×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26C22⋊C8×C127C2211yescharacteristic
27C23×D127—21no—
28C2×Dic254D411yescharacteristic, complemented
29C2×D508—21no—
30C2×D508—21no—
31C2×D508—42no—
32C2×D508D411yescharacteristic, complemented
33C2×C22⋊D127—21no—
34C2×C22⋊D127—21no—
35C2×C22⋊D127—21no—
36C2×C22⋊D127—42no—
37C2×C22⋊D127—42no—
38C2×C22⋊D127—42no—
39C22×Dic127—21no—
40C2×C4×D127—21no—
41#14523237—42no—
42D1016—42no—
43C4⋊Dic127D411yescharacteristic, complemented
44C4⋊Dic127D411yescharacteristic, complemented
45Dic127⋊C4—21no—
46Dic127⋊C4—21no—
47C22⋊Dic127—21no—
48C22⋊Dic127—21no—
49C22⋊Dic127—21no—
50(C2×D127)⋊C4—21no—
51(C2×D127)⋊C4—21no—
52(C2×D127)⋊C4—42no—
53C4×Dic127—21no—
54C24×C127D411yescharacteristic
55C22×C4×C127D411yescharacteristic
56C22×C4×C127C2311yescharacteristic
57C2×C8×C127—21no—
58C2×C8×C127—21no—
59C22⋊C4×C127—21no—
60C22⋊C4×C127—21no—
61C22×D127—22no—
62C22×D127—21no—
63C22×D127—41no—
64C22×D127—41no—
65C22×D127—41no—
66C22×D127—41no—
67C22×D127—41no—
68C22×D127—41no—
69Dic254—41no—
70D508—22no—
71D508—22no—
72D508—41no—
73D508—41no—
74D508—41no—
75C22⋊D127—48no—
76C22⋊D127—41no—
77C22⋊D127—41no—
78C22⋊D127—81no—
79C22⋊D127—81no—
80C2×Dic127—22no—
81C2×Dic127—21no—
82C2×Dic127—21no—
83C2×Dic127—21no—
84C2×Dic127—41no—
85C2×Dic127—41no—
86C4×D127—81no—
87C23×C127—21no—
88C23×C127—21no—
89C23×C127—21no—
90C23×C127—41no—
91C23×C127—41no—
92C23×C127C2×D411yescharacteristic
93C2×C4×C127—21no—
94C2×C4×C127—41no—
95C2×C4×C127—41no—
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C8×C127—41no—
99C8×C127—41no—
100C22×C127—21no—
101C22×C127—21no—
102C22×C127—42no—
103C22×C127—41no—
104C22×C127—41no—
105C22×C127—41no—
106C22×C127—81no—
107C22×C127#12011yescharacteristic
108C22×C127#13511yescharacteristic
109C22×C127AGL(1,ℤ/8)11yescharacteristic
110C2×D127—42no—
111C2×D127—42no—
112C2×D127—41no—
113C2×D127—41no—
114C2×D127—41no—
115C2×D127—42no—
116C2×D127—81no—
117C2×D127—81no—
118C2×D127—81no—
119C2×D127—81no—
120C4×C127—21no—
121C4×C127—21no—
122C4×C127—41no—
123Dic127—42no—
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#327—1271no—
143#323D12711yescharacteristic, complemented
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—1271no—
153#121—2541no—
154#120—2541no—
155C4×D4—2541no—
156C2×C22⋊C4—2541no—
157C2×C22⋊C4C2×D12711yescharacteristic
158#102—2541no—
159#102—2541no—
160C22⋊C8C2×D12711yescharacteristic
161C22⋊C8C2×D12711yescharacteristic
162#95—1271no—
163C24—2541no—
164C24D50811yescharacteristic
165C2×Q8—1271no—
166C2×D4—1271no—
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×D4—2541no—
176C22×C4—2541no—
177C22×C4—2541no—
178C22×C4D50811yescharacteristic
179C22×C4C22×D12711yescharacteristic
180SD16—5082no—
181D8—5082no—
182C2×C8—21no—
183C2×C8—21no—
184C4⋊C4—1271no—
185C4⋊C4—1271no—
186C4⋊C4—2541no—
187C4⋊C4—2541no—
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—5082no—
191C22⋊C4—2541no—
192C22⋊C4—2541no—
193C22⋊C4—2541no—
194C22⋊C4—2541no—
195C22⋊C4—2541no—
196C42—2541no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—41no—
201C23—41no—
202C23—5081no—
203C23—5081no—
204C23—5081no—
205C23—5081no—
206C23—5081no—
207C23—5081no—
208C23—2542no—
209C23—2541no—
210C23C2×D50811yescharacteristic
211Q8—5081no—
212D4—10161no—
213D4—10161no—
214D4—5081no—
215D4—5081no—
216D4—5081no—
217D4—5088no—
218D4—5081no—
219D4—5081no—
220D4—2542no—
221D4—2542no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—10161no—
226C2×C4—5081no—
227C2×C4—5081no—
228C2×C4—2542no—
229C2×C4—2541no—
230C2×C4—2541no—
231C2×C4—2541no—
232C2×C4D4×D12711yescharacteristic
233C2×C4D4×D12711yescharacteristic
234C8—41no—
235C8—41no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—81no—
243C22—10161no—
244C22—10161no—
245C22—10161no—
246C22—10161no—
247C22—5082no—
248C22—5082no—
249C22—5081no—
250C22—5081no—
251C22—5081no—
252C22—5082no—
253C22#1686117911yescharacteristic
254C22#1686123111yescharacteristic
255C22#1686123711yescharacteristic
256C4—21no—
257C4—21no—
258C4—41no—
259C4—5082no—
260C4—5081no—
261C4—5081no—
262C4—5081no—
263C2—21no—
264C2—21no—
265C2—81no—
266C2—10162no—
267C2—10161no—
268C2#2255002311yescharacteristic
269C2#2255004611yescharacteristic
270C2#2255006411yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#323×C127C2
socleC22×C127#120
hypercenter#323D127
hypocenterC127#1600
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
1. #22550464C2semidirect
2. #22550029C2semidirect
3. #22550012C2semidirect
4. #22549785C2semidirect
5. #22549614C2semidirect
6. #22549611C2semidirect
7. #323×C127C2semidirect
28. C2×Dic254D4semidirect
32. C2×D508D4semidirect
43. C4⋊Dic127D4semidirect
44. C4⋊Dic127D4semidirect
136. C127#1600semidirect
143. #323D127semidirect

Maximal subgroups

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

subgroupindex
#225504642
#225500292
#225500122
#225497852
#225496142
#225496112
#323×C1272
#1600127