Finite Group Databasebeta

← #30639758

Subgroups — #30639758

Order 16256.

394 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C2×#16861239C211yesmaximal, characteristic, complemented
2#22550440C211yesmaximal, characteristic
3C2×#16861197C211yesmaximal, characteristic, complemented
4#22549886C211yesmaximal, characteristic, complemented
5C2×#16861125C211yesmaximal, characteristic, complemented
6#22549636C211yesmaximal, characteristic, complemented
7C2×#104×C127C211yesmaximal, characteristic, complemented
8C22×Dic254C2211yescharacteristic
9C22×D508C2211yescharacteristic
10#16861239—22no—
11#16861239—22no—
12C2×#14523238—21no—
13C2×#14523236—21no—
14C2×#14523232C2211yescharacteristic
15#16861205—21no—
16#16861198—21no—
17#16861197—22no—
18#16861197—22no—
19#16861183—22no—
20#16861177—22no—
21#16861168C2211yescharacteristic
22C2×(C2×D127)⋊C4—22no—
23#16861134—22no—
24#16861125—22no—
25C2×#41×C127C2211yescharacteristic
26C2×M16×C127C2211yescharacteristic
27C2×C42×C127C2211yescharacteristic
28#104×C127—22no—
29C23×D127D412yes—
30C2×Dic254—21no—
31C2×Dic254—21no—
32C2×Dic254—21no—
33C2×Dic254—21no—
34C2×Dic254—21no—
35C2×Dic254D412yescomplemented
36C2×D508—21no—
37C2×D508—21no—
38C2×D508—21no—
39C2×D508—21no—
40C2×D508—21no—
41C2×D508—21no—
42C2×C22⋊D127—24no—
43C22×Dic127—21no—
44#14523238—24no—
45#14523236—24no—
46#14523232—21no—
47#14523232D412yescomplemented
48C2×C127⋊C8—21no—
49C4⋊Dic127D412yes—
50C22⋊Dic127—21no—
51(C2×D127)⋊C4—28no—
52(C2×D127)⋊C4—24no—
53C4×Dic127—21no—
54#41×C127—21no—
55#41×C127—21no—
56#41×C127—21no—
57#41×C127—21no—
58C2×Q8×C127D411yescharacteristic
59C2×D4×C127—21no—
60C2×D4×C127D411yescharacteristic
61C22×C4×C127—21no—
62C22×C4×C127—21no—
63C22×C4×C127C2311yescharacteristic
64M16×C127—21no—
65M16×C127—21no—
66C2×C8×C127—21no—
67C42×C127—21no—
68C42×C127D412yescomplemented
69C22×D127—22no—
70C22×D127—22no—
71C22×D127—22no—
72C22×D127—44no—
73Dic254—22no—
74Dic254—22no—
75Dic254—41no—
76Dic254—41no—
77D508—41no—
78D508—41no—
79D508—41no—
80D508—41no—
81C22⋊D127—84no—
82C2×Dic127—22no—
83C2×Dic127—22no—
84C2×Dic127—41no—
85C2×Dic127—41no—
86C127⋊C8—22no—
87C23×C127—21no—
88C23×C127C2×D411yescharacteristic
89Q8×C127—21no—
90Q8×C127—21no—
91D4×C127—21no—
92D4×C127—21no—
93D4×C127—41no—
94D4×C127—41no—
95C2×C4×C127—22no—
96C2×C4×C127—21no—
97C2×C4×C127—42no—
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—41no—
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—42no—
118C4×C127—41no—
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×#137—1271no—
134#535—1271no—
135C2×#124—1271no—
136C2×#104D12711yescharacteristic, complemented
137#419—1271no—
138C2×#100—1271no—
139#339—1271no—
140C2×#41C2×D12711yescharacteristic
141C22×Q8—1271no—
142C22×D4—1271no—
143#137—2542no—
144#137—2542no—
145C2×Q16—2541no—
146C2×SD16—2541no—
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×C42C2×D12711yescharacteristic
158#104—22no—
159#103—2542no—
160#100—2542no—
161C24—1272no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41—21no—
166C2×Q8—1272no—
167C2×Q8—2541no—
168C2×Q8—2541no—
169C2×Q8—2541no—
170C2×Q8—2541no—
171C2×Q8—2541no—
172C2×Q8D50811yescharacteristic
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×D4D50811yescharacteristic
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—2541no—
185C22×C4C22×D12711yescharacteristic
186Q16—2544no—
187SD16—2544no—
188M16—21no—
189M16—21no—
190M16—1272no—
191M16—2541no—
192C2×C8—21no—
193C2×C8—2541no—
194C4⋊C4—1272no—
195C22⋊C4—2548no—
196C22⋊C4—2544no—
197C22⋊C4—2541no—
198C42—21no—
199C42—2541no—
200C42C22⋊D12712yescomplemented
201C23—21no—
202C23—5084no—
203C23—2542no—
204C23—2542no—
205C23—2542no—
206C23D4×D12711yescharacteristic
207Q8—21no—
208Q8—21no—
209Q8—5081no—
210Q8—5081no—
211Q8—2542no—
212Q8—2542no—
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—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—5081no—
228C2×C4—5081no—
229C2×C4—2542no—
230C2×C4—2542no—
231C2×C4D4×D12711yescharacteristic
232C2×C4D4×D12711yescharacteristic
233C2×C4D4×D12711yescharacteristic
234C2×C4C2×C22⋊D12711yescharacteristic
235C2×C4C2×D50811yescharacteristic
236C2×C4#1452324811yescharacteristic
237C8—41no—
238C8—2542no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—10162no—
243C22—5084no—
244C22—5082no—
245C22—5084no—
246C22#1686117711yescharacteristic
247C22#1686117911yescharacteristic
248C22#1686118211yescharacteristic
249C4—42no—
250C4—41no—
251C4—5082no—
252C4—5082no—
253C4#1686117911yescharacteristic
254C4#1686118211yescharacteristic
255C4#1686118311yescharacteristic
256C4#1686120511yescharacteristic
257C2—21no—
258C2—21no—
259C2—81no—
260C2—10162no—
261C2#2254978511yescharacteristic
262C2#2255005512yes—
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#104×C127C2
socleC22×C127#120
prime radical#22550440C2
hypercenterC2×#104D127
hypocenterC127#2004
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
3. C2×#16861197C2semidirect
4. #22549886C2semidirect
5. C2×#16861125C2semidirect
6. #22549636C2semidirect
7. C2×#104×C127C2semidirect
35. C2×Dic254D4semidirect
47. #14523232D4semidirect
68. C42×C127D4semidirect
132. C127#2004semidirect
136. C2×#104D127semidirect
200. C42C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
#225504402
C2×#168611972
#225498862
C2×#168611252
#225496362
C2×#104×C1272
#2004127