Finite Group Databasebeta

← #30639756

Subgroups — #30639756

Order 16256.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 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#22550440C211yesmaximal, characteristic
2C2×#16861198C211yesmaximal, characteristic, complemented
3C2×#16861136C211yesmaximal, characteristic, complemented
4#22549885C211yesmaximal, characteristic, complemented
5C2×#16861125C211yesmaximal, characteristic, complemented
6#22549637C211yesmaximal, characteristic, complemented
7C2×#137×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10#16861205—21no—
11#16861198—22no—
12#16861198—21no—
13#16861198—22no—
14#16861183—22no—
15#16861177—22no—
16#16861168C2211yescharacteristic
17C2×(C2×D127)⋊C4—22no—
18C2×C4×Dic127C2211yescharacteristic
19#16861136—22no—
20#16861133—22no—
21#16861125—22no—
22C2×#41×C127C2211yescharacteristic
23C22×Q8×C127C2211yescharacteristic
24#137×C127—22no—
25#137×C127—22no—
26C2×Q16×C127—21no—
27C2×SD16×C127—21no—
28C2×M16×C127C2211yescharacteristic
29C23×D127D412yes—
30C2×D508—21no—
31C2×D508—21no—
32C2×D508—21no—
33C2×D508—21no—
34C2×D508—21no—
35C2×D508—21no—
36C2×C22⋊D127—24no—
37C22×Dic127—21no—
38#14523232—21no—
39#14523232—21no—
40C2×C127⋊C8—21no—
41C4⋊Dic127D412yes—
42C22⋊Dic127—21no—
43(C2×D127)⋊C4—28no—
44(C2×D127)⋊C4—24no—
45C4×Dic127—21no—
46C4×Dic127—21no—
47C4×Dic127D412yescomplemented
48#41×C127—21no—
49#41×C127—21no—
50#41×C127—21no—
51#41×C127—21no—
52C2×Q8×C127—21no—
53C2×Q8×C127—21no—
54C2×Q8×C127—21no—
55C2×Q8×C127—21no—
56C2×Q8×C127—21no—
57C2×Q8×C127D411yescharacteristic
58C2×Q8×C127D412yescomplemented
59C2×D4×C127—21no—
60C2×D4×C127D411yescharacteristic
61C22×C4×C127—21no—
62C22×C4×C127—21no—
63C22×C4×C127C2311yescharacteristic
64Q16×C127—24no—
65SD16×C127—24no—
66M16×C127—21no—
67M16×C127D412yescomplemented
68C2×C8×C127—21no—
69C22×D127—22no—
70C22×D127—22no—
71C22×D127—22no—
72C22×D127—44no—
73D508—41no—
74D508—41no—
75D508—41no—
76D508—41no—
77C22⋊D127—84no—
78C2×Dic127—22no—
79C2×Dic127—22no—
80C2×Dic127—42no—
81C127⋊C8—41no—
82C23×C127—21no—
83C23×C127C2×D411yescharacteristic
84Q8×C127—22no—
85Q8×C127—21no—
86Q8×C127—22no—
87Q8×C127—21no—
88Q8×C127—41no—
89Q8×C127—41no—
90D4×C127—21no—
91D4×C127—21no—
92D4×C127—41no—
93D4×C127—41no—
94C2×C4×C127—21no—
95C2×C4×C127—22no—
96C2×C4×C127—41no—
97C2×C4×C127—41no—
98C2×C4×C127—41no—
99C2×C4×C127—41no—
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127#4111yescharacteristic
106C8×C127—22no—
107C22×C127—22no—
108C22×C127—41no—
109C22×C127—42no—
110C22×C127#12011yescharacteristic
111C22×C127#12011yescharacteristic
112C22×C127#12111yescharacteristic
113C2×D127—44no—
114C2×D127—42no—
115C2×D127—44no—
116C2×D127—82no—
117C4×C127—41no—
118C4×C127—42no—
119C4×C127#12011yescharacteristic
120C4×C127#12111yescharacteristic
121C4×C127#12111yescharacteristic
122C4×C127#12711yescharacteristic
123Dic127—42no—
124Dic127—42no—
125C2×C127—21no—
126C2×C127—21no—
127C2×C127—81no—
128C2×C127#39211yescharacteristic
129C2×C127#45512yes—
130D127—82no—
131#2004—1271nomaximal, Sylow
132C127#200411yescharacteristic, complemented, Sylow
133C2×#137D12711yescharacteristic, complemented
134#535—1271no—
135C2×#124—1271no—
136C2×#104—1271no—
137#419—1271no—
138C2×#100—1271no—
139#339—1271no—
140C2×#41C2×D12711yescharacteristic
141C22×Q8C2×D12711yescharacteristic
142C22×D4—1271no—
143#137—22no—
144#137—22no—
145C2×Q16—21no—
146C2×SD16—21no—
147C2×M16—1271no—
148C2×M16C2×D12711yescharacteristic
149#127—2541no—
150#124—2542no—
151#124—2541no—
152#124—2542no—
153#121—2542no—
154#120—2542no—
155#117—1271no—
156C2×C22⋊C4—2542no—
157C2×C42—1271no—
158#104—2542no—
159#103—2542no—
160#100—2542no—
161C24—1272no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8C22⋊D12711yescharacteristic
172C2×Q8C22⋊D12712yescomplemented
173C2×D4—21no—
174C2×D4—2544no—
175C2×D4—2541no—
176C2×D4—2541no—
177C2×D4—2541no—
178C2×D4—2541no—
179C2×D4—2541no—
180C2×D4—2541no—
181C2×D4C22⋊D12711yescharacteristic
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—2541no—
185C22×C4C22×D12711yescharacteristic
186Q16—24no—
187SD16—24no—
188M16—21no—
189M16—2541no—
190M16—2541no—
191M16C22⋊D12712yescomplemented
192C2×C8—21no—
193C2×C8—2541no—
194C4⋊C4—1272no—
195C22⋊C4—2548no—
196C22⋊C4—2544no—
197C22⋊C4—2541no—
198C42—1272no—
199C42—2541no—
200C42—2541no—
201C23—21no—
202C23—5084no—
203C23—2542no—
204C23—2542no—
205C23—2542no—
206C23D4×D12711yescharacteristic
207Q8—22no—
208Q8—21no—
209Q8—22no—
210Q8—21no—
211Q8—41no—
212Q8—41no—
213D4—21no—
214D4—21no—
215D4—41no—
216D4—41no—
217D4—10164no—
218D4—5081no—
219D4—5081no—
220D4—5081no—
221D4—5081no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—5082no—
229C2×C4—2542no—
230C2×C4—2542no—
231C2×C4D4×D12711yescharacteristic
232C2×C4D4×D12711yescharacteristic
233C2×C4C2×C22⋊D12711yescharacteristic
234C2×C4C2×C22⋊D12711yescharacteristic
235C2×C4C2×C22⋊D12711yescharacteristic
236C2×C4#1452324711yescharacteristic
237C8—22no—
238C8—5081no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—10162no—
243C22—5084no—
244C22—5082no—
245C22—5084no—
246C22#1686117711yescharacteristic
247C22#1686117711yescharacteristic
248C22#1686118011yescharacteristic
249C4—41no—
250C4—42no—
251C4—5082no—
252C4—5082no—
253C4#1686117811yescharacteristic
254C4#1686118111yescharacteristic
255C4#1686118111yescharacteristic
256C4#1686120511yescharacteristic
257C2—21no—
258C2—21no—
259C2—81no—
260C2—10162no—
261C2#2254978411yescharacteristic
262C2#2255005212yes—
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#137×C127C2
socleC22×C127#120
prime radical#22550440C2
hypercenterC2×#137D127
hypocenterC127#2004
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#16861198C2semidirect
3. C2×#16861136C2semidirect
4. #22549885C2semidirect
5. C2×#16861125C2semidirect
6. #22549637C2semidirect
7. C2×#137×C127C2semidirect
47. C4×Dic127D4semidirect
58. C2×Q8×C127D4semidirect
67. M16×C127D4semidirect
132. C127#2004semidirect
133. C2×#137D127semidirect
172. C2×Q8C22⋊D127semidirect
191. M16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225504402
C2×#168611982
C2×#168611362
#225498852
C2×#168611252
#225496372
C2×#137×C1272
#2004127