Finite Group Databasebeta

← #30637713

Subgroups — #30637713

Order 16256.

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

normal subgroups only

This group has 260 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#22550466C211yesmaximal, characteristic
2#22550031C211yesmaximal, characteristic, complemented
3#22550017C211yesmaximal, characteristic, complemented
4#22549789C211yesmaximal, characteristic, complemented
5#22549609C211yesmaximal, characteristic, complemented
6#22549599C211yesmaximal, characteristic, complemented
7#328×C127C211yesmaximal, characteristic, complemented
8C22×Dic254C2211yescharacteristic
9C2×#14523238—21no—
10C2×#14523237—21no—
11#16861204—21no—
12#16861198—21no—
13#16861194—21no—
14#16861184—21no—
15#16861183C2211yescharacteristic
16#16861179—21no—
17(C22⋊D127)⋊C4—21no—
18C2×(C2×D127)⋊C4—21no—
19C2×(C2×D127)⋊C4C2211yescharacteristic
20#16861134—21no—
21#16861132—21no—
22C22⋊(C127⋊C8)C2211yescharacteristic
23C127⋊#95—21no—
24C22⋊Q8×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27C23×D127D411yescharacteristic
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254—42no—
31C2×D508—21no—
32C2×D508—21no—
33C2×D508D411yescharacteristic
34C2×C22⋊D127—21no—
35C22×Dic127—21no—
36C22×Dic127D411yescharacteristic
37C2×C4×D127—21no—
38#14523238—42no—
39#14523237—42no—
40C2×C127⋊C8—21no—
41C4⋊Dic127D411yescharacteristic
42Dic127⋊C4—21no—
43(C2×D127)⋊C4—21no—
44(C2×D127)⋊C4—21no—
45(C2×D127)⋊C4—21no—
46(C2×D127)⋊C4—21no—
47(C2×D127)⋊C4—21no—
48(C2×D127)⋊C4—21no—
49(C2×D127)⋊C4—42no—
50C4×Dic127—21no—
51C2×Q8×C127D411yescharacteristic
52C22×C4×C127—21no—
53C22×C4×C127C2311yescharacteristic
54C2×C8×C127—21no—
55C4⋊C4×C127—21no—
56C4⋊C4×C127D411yescharacteristic
57C22⋊C4×C127—21no—
58C22×D127—21no—
59C22×D127—21no—
60C22×D127—21no—
61C22×D127—21no—
62C22×D127—41no—
63C22×D127—41no—
64Dic254—22no—
65Dic254—22no—
66Dic254—41no—
67Dic254—41no—
68D508—41no—
69D508—81no—
70D508—81no—
71C22⋊D127—41no—
72C22⋊D127—41no—
73C2×Dic127—22no—
74C2×Dic127—21no—
75C2×Dic127—21no—
76C2×Dic127—41no—
77C2×Dic127—41no—
78C2×Dic127—41no—
79C4×D127—81no—
80C127⋊C8—41no—
81C23×C127C2×D411yescharacteristic
82Q8×C127—41no—
83C2×C4×C127—21no—
84C2×C4×C127—21no—
85C2×C4×C127—41no—
86C2×C4×C127—41no—
87C2×C4×C127—41no—
88C2×C4×C127—41no—
89C2×C4×C127C2×D411yescharacteristic
90C2×C4×C127C2×D411yescharacteristic
91C8×C127—41no—
92C22×C127—21no—
93C22×C127—21no—
94C22×C127#12011yescharacteristic
95C22×C127#13511yescharacteristic
96C22×C127#13711yescharacteristic
97C2×D127—42no—
98C2×D127—41no—
99C2×D127—41no—
100C2×D127—41no—
101C2×D127—41no—
102C2×D127—41no—
103C2×D127—41no—
104C2×D127—81no—
105C4×C127—21no—
106C4×C127—21no—
107C4×C127—41no—
108C4×C127—41no—
109C4×C127—81no—
110Dic127—41no—
111Dic127—41no—
112Dic127—42no—
113C2×C127—21no—
114C2×C127—21no—
115C2×C127#45211yescharacteristic
116C2×C127#45511yescharacteristic
117C2×C127B(4,𝔽2)11yescharacteristic
118D127—81no—
119D127—81no—
120#1606—1271nomaximal, Sylow
121C127#160611yescharacteristic, complemented, Sylow
122#538—1271no—
123#451—1271no—
124#448—1271no—
125#393—1271no—
126#328D12711yescharacteristic, complemented
127#327—1271no—
128#323—1271no—
129C22×Q8—1271no—
130C2×Q16—2541no—
131C2×SD16—2541no—
132#126—2541no—
133#124—2541no—
134#123—2541no—
135C22⋊Q8C2×D12711yescharacteristic
136#121—1271no—
137#121—2541no—
138#120—2541no—
139C4×D4—2541no—
140C2×C22⋊C4—1271no—
141C2×C22⋊C4—2541no—
142#103—2541no—
143#103—2541no—
144C22⋊C8—1271no—
145C22⋊C8C2×D12711yescharacteristic
146#95—2541no—
147#95C2×D12711yescharacteristic
148C24—1271no—
149C2×Q8—5082no—
150C2×Q8—2541no—
151C2×Q8—2541no—
152C2×Q8D50811yescharacteristic
153C2×D4—1271no—
154C2×D4—2541no—
155C2×D4—2541no—
156C2×D4—2541no—
157C22×C4—21no—
158C22×C4—1271no—
159C22×C4—2541no—
160C22×C4—2541no—
161C22×C4C22×D12711yescharacteristic
162Q16—5082no—
163SD16—5082no—
164C2×C8—21no—
165C2×C8—2541no—
166C4⋊C4—21no—
167C4⋊C4—1271no—
168C4⋊C4—2541no—
169C4⋊C4D50811yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—5082no—
172C22⋊C4—2541no—
173C22⋊C4—2541no—
174C22⋊C4—2541no—
175C22⋊C4—2541no—
176C22⋊C4—2541no—
177C22⋊C4—2541no—
178C42—2541no—
179C23—5081no—
180C23—5081no—
181C23—2541no—
182C23—2541no—
183C23—2541no—
184C23—2541no—
185C23D4×D12711yescharacteristic
186Q8—41no—
187Q8—5081no—
188Q8—5081no—
189Q8—2542no—
190Q8—2542no—
191D4—10161no—
192D4—10161no—
193D4—5081no—
194D4—5081no—
195D4—5081no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—10161no—
203C2×C4—5081no—
204C2×C4—5081no—
205C2×C4—5081no—
206C2×C4—2542no—
207C2×C4—2541no—
208C2×C4—2541no—
209C2×C4D4×D12711yescharacteristic
210C2×C4C2×D50811yescharacteristic
211C8—41no—
212C8—5081no—
213C22—21no—
214C22—21no—
215C22—10161no—
216C22—5082no—
217C22—5081no—
218C22—5081no—
219C22—5081no—
220C22—5081no—
221C22—5081no—
222C22—5081no—
223C22#1686117911yescharacteristic
224C22#1686123011yescharacteristic
225C22#1686124111yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—41no—
230C4—81no—
231C4—5081no—
232C4—5081no—
233C4—5082no—
234C2—21no—
235C2—21no—
236C2—10161no—
237C2—10161no—
238C2#2255004111yescharacteristic
239C2#2255005511yescharacteristic
240C2#2255006411yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#328×C127C2
socleC22×C127#120
prime radical#22550466C2
hypercenter#328D127
hypocenterC127#1606
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
2. #22550031C2semidirect
3. #22550017C2semidirect
4. #22549789C2semidirect
5. #22549609C2semidirect
6. #22549599C2semidirect
7. #328×C127C2semidirect
121. C127#1606semidirect
126. #328D127semidirect

Maximal subgroups

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

subgroupindex
#225504662
#225500312
#225500172
#225497892
#225496092
#225495992
#328×C1272
#1606127