Finite Group Databasebeta

← #30637697

Subgroups — #30637697

Order 16256.

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

normal subgroups only

This group has 282 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
2#22550015C211yesmaximal, characteristic, complemented
3#22549795C211yesmaximal, characteristic, complemented
4#22549611C211yesmaximal, characteristic, complemented
5#22549610C211yesmaximal, characteristic, complemented
6#22549601C211yesmaximal, characteristic, complemented
7#448×C127C211yesmaximal, characteristic, complemented
8C2×#14523247—21no—
9C2×#14523235—21no—
10#16861187—21no—
11#16861186—21no—
12#16861183C2211yescharacteristic
13#16861182—21no—
14#16861181—21no—
15#16861181—21no—
16C2×Dic127⋊C4—21no—
17C2×Dic127⋊C4C2211yescharacteristic
18#16861133—21no—
19C22⋊(C127⋊C8)C2211yescharacteristic
20C127⋊#95—21no—
21C127⋊#95C2211yescharacteristic
22C22×Q8×C127C2211yescharacteristic
23C2×SD16×C127—21no—
24#121×C127C2211yescharacteristic
25#103×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#14523235—42no—
40C2×C127⋊C8—21no—
41C4⋊Dic127D411yescharacteristic
42Dic127⋊C4—21no—
43Dic127⋊C4—21no—
44Dic127⋊C4—21no—
45Dic127⋊C4—42no—
46C22⋊Dic127—21no—
47(C2×D127)⋊C4—21no—
48(C2×D127)⋊C4—21no—
49C2×Q8×C127—21no—
50C2×Q8×C127—21no—
51C2×Q8×C127—42no—
52C2×D4×C127—21no—
53C2×D4×C127D411yescharacteristic
54C22×C4×C127—21no—
55C22×C4×C127C2311yescharacteristic
56SD16×C127—42no—
57C2×C8×C127—21no—
58C4⋊C4×C127D411yescharacteristic
59C22⋊C4×C127—21no—
60C22×D127—21no—
61Dic254—42no—
62D508—41no—
63C22⋊D127—42no—
64C22⋊D127—41no—
65C22⋊D127—41no—
66C22⋊D127—81no—
67C2×Dic127—21no—
68C2×Dic127—21no—
69C2×Dic127—22no—
70C2×Dic127—21no—
71C2×Dic127—21no—
72C2×Dic127—42no—
73C2×Dic127—41no—
74C2×Dic127—41no—
75C2×Dic127—41no—
76C2×Dic127SD1612yescomplemented
77C4×D127—42no—
78C127⋊C8—41no—
79C23×C127—21no—
80C23×C127C2×D411yescharacteristic
81Q8×C127—22no—
82Q8×C127—22no—
83Q8×C127—41no—
84Q8×C127—41no—
85D4×C127—41no—
86D4×C127—41no—
87D4×C127—41no—
88C2×C4×C127—22no—
89C2×C4×C127—21no—
90C2×C4×C127—41no—
91C2×C4×C127—41no—
92C2×C4×C127—41no—
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D411yescharacteristic
95C8×C127—41no—
96C22×C127—21no—
97C22×C127—21no—
98C22×C127—41no—
99C22×C127—41no—
100C22×C127—41no—
101C22×C127#12011yescharacteristic
102C22×C127C2×SD1611yescharacteristic
103C22×C127AGL(1,ℤ/8)11yescharacteristic
104C2×D127—41no—
105C2×D127—41no—
106C2×D127—41no—
107C4×C127—21no—
108C4×C127—21no—
109C4×C127—41no—
110C4×C127—42no—
111Dic127—22no—
112Dic127—42no—
113Dic127—41no—
114Dic127—41no—
115C2×C127—21no—
116C2×C127—21no—
117C2×C127—81no—
118C2×C127#45011yescharacteristic
119C2×C127#45611yescharacteristic
120C2×C127B(4,𝔽2)11yescharacteristic
121D127—81no—
122#1604—1271nomaximal, Sylow
123C127#160411yescharacteristic, complemented, Sylow
124#537—1271no—
125#448—1271no—
126#448D12711yescharacteristic, complemented
127#395—1271no—
128#327—1271no—
129#327—1271no—
130#324—1271no—
131C2×#41—2541no—
132C22×Q8C2×D12711yescharacteristic
133C2×SD16—21no—
134C2×SD16—2541no—
135C22⋊Q8—2541no—
136C22⋊Q8—2541no—
137#121—1271no—
138#121—2541no—
139#121—2541no—
140#121—2541no—
141#121C2×D12711yescharacteristic
142C2×C4⋊C4—1271no—
143C2×C4⋊C4—2541no—
144#103—21no—
145#103—2541no—
146C22⋊C8—1271no—
147C22⋊C8C2×D12711yescharacteristic
148#95—1271no—
149#95—2541no—
150#41—5082no—
151#41—5082no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—42no—
155C2×Q8—2541no—
156C2×D4—21no—
157C2×D4—1271no—
158C2×D4—2541no—
159C2×D4—2541no—
160C2×D4—2541no—
161C2×D4C22⋊D12711yescharacteristic
162C22×C4—21no—
163C22×C4—1271no—
164C22×C4—1271no—
165C22×C4—2541no—
166C22×C4—2541no—
167C22×C4—2541no—
168C22×C4C22×D12711yescharacteristic
169SD16—42no—
170SD16—5082no—
171C2×C8—21no—
172C2×C8—2541no—
173C4⋊C4—1271no—
174C4⋊C4—5082no—
175C4⋊C4—2541no—
176C4⋊C4—2541no—
177C4⋊C4—2541no—
178C4⋊C4C22⋊D12711yescharacteristic
179C22⋊C4—21no—
180C22⋊C4—2541no—
181C22⋊C4—2541no—
182C22⋊C4—2541no—
183C23—21no—
184C23—2541no—
185C23D4×D12711yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
190Q8—5082no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—10161no—
195D4—5081no—
196D4—5081no—
197D4—5081no—
198D4—5082no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—1272no—
205C2×C4—5082no—
206C2×C4—5081no—
207C2×C4—5081no—
208C2×C4—5082no—
209C2×C4—5081no—
210C2×C4—2542no—
211C2×C4—2541no—
212C2×C4—2541no—
213C2×C4—2541no—
214C2×C4—2541no—
215C2×C4D4×D12711yescharacteristic
216C2×C4C2×C22⋊D12711yescharacteristic
217C8—41no—
218C8—5081no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—5081no—
225C22—5081no—
226C22—5081no—
227C22#1686117711yescharacteristic
228C22SD16×D12711yescharacteristic
229C22#1686123611yescharacteristic
230C4—21no—
231C4—21no—
232C4—41no—
233C4—42no—
234C4—5082no—
235C4—5081no—
236C4—5081no—
237C4—2542no—
238C2—21no—
239C2—21no—
240C2—81no—
241C2—10161no—
242C2#2255002511yescharacteristic
243C2#2255005711yescharacteristic
244C2#2255006211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#448×C127C2
socleC22×C127#120
prime radical#22550453C2
hypercenter#448D127
hypocenterC127#1604
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #22550015C2semidirect
3. #22549795C2semidirect
4. #22549611C2semidirect
5. #22549610C2semidirect
6. #22549601C2semidirect
7. #448×C127C2semidirect
76. C2×Dic127SD16semidirect
123. C127#1604semidirect
126. #448D127semidirect

Maximal subgroups

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

subgroupindex
#225504532
#225500152
#225497952
#225496112
#225496102
#225496012
#448×C1272
#1604127