Finite Group Databasebeta

← #30637663

Subgroups — #30637663

Order 16256.

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

normal subgroups only

This group has 310 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#22550453C211yesmaximal, characteristic, complemented
2#22550010C211yesmaximal, characteristic, complemented
3#22549790C211yesmaximal, characteristic, complemented
4#22549611C211yesmaximal, characteristic, complemented
5#22549610C211yesmaximal, characteristic, complemented
6#22549601C211yesmaximal, characteristic, complemented
7#447×C127C211yesmaximal, characteristic, complemented
8C2×#14523247—21no—
9C2×#14523233—21no—
10#16861187—21no—
11#16861186—21no—
12#16861183C2211yescharacteristic
13#16861182—21no—
14#16861181—21no—
15#16861181—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×D8×C127—21no—
24#121×C127C2211yescharacteristic
25#102×C127—21no—
26C22⋊C8×C127C2211yescharacteristic
27#14523247—42no—
28#14523247—42no—
29C2×Dic254—21no—
30C2×D508D411yescharacteristic
31C2×C22⋊D127—21no—
32C2×C22⋊D127—21no—
33C2×C22⋊D127—21no—
34C22×Dic127—21no—
35C22×Dic127—21no—
36C22×Dic127D411yescharacteristic
37C22×Dic127D411yescharacteristic
38C2×C4×D127—21no—
39#14523233—42no—
40C2×C127⋊C8—21no—
41C4⋊Dic127D411yescharacteristic
42Dic127⋊C4—21no—
43Dic127⋊C4—21no—
44Dic127⋊C4—21no—
45C22⋊Dic127—21no—
46C22⋊Dic127—42no—
47C22⋊Dic127—42no—
48(C2×D127)⋊C4—21no—
49(C2×D127)⋊C4—21no—
50C24×C127—21no—
51C2×D4×C127—21no—
52C2×D4×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127—42no—
55C2×D4×C127D411yescharacteristic
56C22×C4×C127C2311yescharacteristic
57D8×C127—42no—
58C2×C8×C127—21no—
59C4⋊C4×C127D411yescharacteristic
60C22⋊C4×C127—21no—
61C22×D127—21no—
62Dic254—42no—
63D508—41no—
64C22⋊D127—42no—
65C22⋊D127—41no—
66C22⋊D127—41no—
67C22⋊D127—81no—
68C2×Dic127—21no—
69C2×Dic127—21no—
70C2×Dic127—21no—
71C2×Dic127—21no—
72C2×Dic127—42no—
73C2×Dic127—41no—
74C2×Dic127—41no—
75C2×Dic127—41no—
76C2×Dic127—41no—
77C2×Dic127D812yescomplemented
78C4×D127—42no—
79C127⋊C8—41no—
80C23×C127—22no—
81C23×C127—21no—
82C23×C127—41no—
83C23×C127—41no—
84C23×C127—41no—
85C23×C127—41no—
86C23×C127—41no—
87C23×C127—41no—
88C23×C127C2×D411yescharacteristic
89D4×C127—22no—
90D4×C127—22no—
91D4×C127—41no—
92D4×C127—41no—
93D4×C127—41no—
94D4×C127—41no—
95D4×C127—41no—
96C2×C4×C127—21no—
97C2×C4×C127—41no—
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C8×C127—41no—
101C22×C127—21no—
102C22×C127—21no—
103C22×C127—42no—
104C22×C127—42no—
105C22×C127—41no—
106C22×C127—41no—
107C22×C127—42no—
108C22×C127—41no—
109C22×C127—81no—
110C22×C127—81no—
111C22×C127—81no—
112C22×C127—81no—
113C22×C127#12011yescharacteristic
114C22×C127C2×D811yescharacteristic
115C22×C127AGL(1,ℤ/8)11yescharacteristic
116C2×D127—41no—
117C2×D127—41no—
118C2×D127—41no—
119C4×C127—21no—
120C4×C127—21no—
121C4×C127—41no—
122Dic127—22no—
123Dic127—41no—
124Dic127—41no—
125Dic127—81no—
126C2×C127—21no—
127C2×C127—21no—
128C2×C127—81no—
129C2×C127—82no—
130C2×C127#44711yescharacteristic
131C2×C127#45411yescharacteristic
132C2×C127B(4,𝔽2)11yescharacteristic
133D127—81no—
134#1598—1271nomaximal, Sylow
135C127#159811yescharacteristic, complemented, Sylow
136#537—1271no—
137#447—1271no—
138#447D12711yescharacteristic, complemented
139#394—1271no—
140#327—1271no—
141#327—1271no—
142#324—1271no—
143C2×#41—2541no—
144C22×D4C2×D12711yescharacteristic
145C2×D8—21no—
146C2×D8—2541no—
147C22⋊Q8—2541no—
148C22⋊Q8—2541no—
149#121—1271no—
150#121—2541no—
151#121—2541no—
152#121—2541no—
153#121C2×D12711yescharacteristic
154C2×C4⋊C4—1271no—
155C2×C22⋊C4—2541no—
156C2×C22⋊C4—2541no—
157#102—21no—
158#102—2541no—
159C22⋊C8—1271no—
160C22⋊C8C2×D12711yescharacteristic
161#95—1271no—
162C24—21no—
163#41—5082no—
164#41—5082no—
165C2×Q8—2541no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—42no—
170C2×D4—1271no—
171C2×D4—2541no—
172C2×D4—2541no—
173C2×D4—2541no—
174C2×D4C22⋊D12711yescharacteristic
175C22×C4—1271no—
176C22×C4—1271no—
177C22×C4—2541no—
178C22×C4—2541no—
179C22×C4—2541no—
180C22×C4C22×D12711yescharacteristic
181D8—42no—
182D8—5082no—
183C2×C8—21no—
184C2×C8—2541no—
185C4⋊C4—1271no—
186C4⋊C4—2541no—
187C4⋊C4—2541no—
188C4⋊C4—2541no—
189C4⋊C4C22⋊D12711yescharacteristic
190C22⋊C4—21no—
191C22⋊C4—5082no—
192C22⋊C4—5082no—
193C22⋊C4—2541no—
194C22⋊C4—2541no—
195C22⋊C4—2541no—
196C23—22no—
197C23—21no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—2541no—
205C23D4×D12711yescharacteristic
206Q8—5082no—
207D4—22no—
208D4—22no—
209D4—41no—
210D4—41no—
211D4—41no—
212D4—41no—
213D4—41no—
214D4—10161no—
215D4—5081no—
216D4—5081no—
217D4—5081no—
218D4—5082no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—1272no—
222C2×C4—5082no—
223C2×C4—5081no—
224C2×C4—5081no—
225C2×C4—5081no—
226C2×C4—5082no—
227C2×C4—5081no—
228C2×C4—2541no—
229C2×C4—2541no—
230C2×C4—2541no—
231C2×C4—2541no—
232C2×C4D4×D12711yescharacteristic
233C2×C4C2×C22⋊D12711yescharacteristic
234C8—41no—
235C8—5081no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—42no—
240C22—41no—
241C22—42no—
242C22—41no—
243C22—41no—
244C22—81no—
245C22—81no—
246C22—81no—
247C22—81no—
248C22—5081no—
249C22—5081no—
250C22—5081no—
251C22#1686117711yescharacteristic
252C22D8×D12711yescharacteristic
253C22#1686123311yescharacteristic
254C4—21no—
255C4—21no—
256C4—41no—
257C4—10161no—
258C4—5081no—
259C4—5081no—
260C4—2542no—
261C2—21no—
262C2—21no—
263C2—82no—
264C2—81no—
265C2—10161no—
266C2#2255000811yescharacteristic
267C2#2255004711yescharacteristic
268C2#2255006211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#447×C127C2
socleC22×C127#120
hypercenter#447D127
hypocenterC127#1598
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #22550453C2semidirect
2. #22550010C2semidirect
3. #22549790C2semidirect
4. #22549611C2semidirect
5. #22549610C2semidirect
6. #22549601C2semidirect
7. #447×C127C2semidirect
77. C2×Dic127D8semidirect
135. C127#1598semidirect
138. #447D127semidirect

Maximal subgroups

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

subgroupindex
#225504532
#225500102
#225497902
#225496112
#225496102
#225496012
#447×C1272
#1598127