Finite Group Databasebeta

← #30639170

Subgroups — #30639170

Order 16256.

442 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 87 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#22550383C211yesmaximal, characteristic
2C2×C42×D127C211yesmaximal, characteristic, complemented
3C2×#16861136C211yesmaximal, characteristic, complemented
4#22549839C211yesmaximal, characteristic, complemented
5#22549637C211yesmaximal, characteristic, complemented
6#22549636C211yesmaximal, characteristic, complemented
7C2×#104×C127C211yesmaximal, characteristic, complemented
8C22×C4×D127—21no—
9C22×C4×D127C2211yescharacteristic
10C2×#14523232C2211yescharacteristic
11#16861190—21no—
12#16861185C412yescomplemented
13#16861180C412yescomplemented
14#16861177—21no—
15C4×C22⋊D127—22no—
16#16861168C2211yescharacteristic
17C2×C4×Dic127—21no—
18C2×C4×Dic127C2211yescharacteristic
19C42×D127—22no—
20C42×D127—22no—
21#16861136—22no—
22(C2×D127)⋊C8—22no—
23C2×#41×C127C2211yescharacteristic
24C2×M16×C127C2211yescharacteristic
25C2×C42×C127C2211yescharacteristic
26#104×C127—22no—
27C23×D127D411yescharacteristic
28C2×C22⋊D127—22no—
29C22×Dic127—21no—
30C22×Dic127D411yescharacteristic
31C2×C4×D127—28no—
32C2×C4×D127—24no—
33C2×C4×D127—21no—
34C2×C4×D127—21no—
35C2×C4×D127—21no—
36C2×C4×D127—21no—
37C2×C4×D127C2×C412yes—
38C2×C4×D127D412yes—
39#14523232—21no—
40#14523232—21no—
41C2×C127⋊C8—21no—
42C4⋊Dic127C2×C412yescomplemented
43Dic127⋊C4—22no—
44C22⋊Dic127—22no—
45C22⋊Dic127—21no—
46(C2×D127)⋊C4—22no—
47C4×Dic127—24no—
48C4×Dic127—21no—
49C4×Dic127—21no—
50C4×Dic127D412yescomplemented
51#41×C127—21no—
52#41×C127—21no—
53#41×C127—21no—
54#41×C127—21no—
55C2×Q8×C127C2×C411yescharacteristic, complemented
56C2×D4×C127—21no—
57C2×D4×C127C2×C411yescharacteristic, complemented
58C22×C4×C127—21no—
59C22×C4×C127—21no—
60C22×C4×C127C2311yescharacteristic
61M16×C127—21no—
62M16×C127—21no—
63C2×C8×C127—21no—
64C42×C127—21no—
65C42×C127D412yescomplemented
66C22×D127—21no—
67C22×D127—21no—
68C22×D127—24no—
69C22×D127C22⋊C412yes—
70C22⋊D127—44no—
71C2×Dic127—24no—
72C2×Dic127—21no—
73C2×Dic127—22no—
74C2×Dic127—21no—
75C2×Dic127—22no—
76C2×Dic127C22⋊C412yes—
77C4×D127—216no—
78C4×D127—22no—
79C4×D127—22no—
80C4×D127—22no—
81C4×D127—22no—
82C127⋊C8—41no—
83C23×C127—21no—
84C23×C127C2×D411yescharacteristic
85Q8×C127—21no—
86Q8×C127—21no—
87D4×C127—21no—
88D4×C127—21no—
89D4×C127—41no—
90D4×C127—41no—
91C2×C4×C127—24no—
92C2×C4×C127—22no—
93C2×C4×C127—21no—
94C2×C4×C127—41no—
95C2×C4×C127—41no—
96C2×C4×C127C22×C411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127#4111yescharacteristic
101C2×C4×C127#4111yescharacteristic
102C8×C127—41no—
103C22×C127—22no—
104C22×C127—41no—
105C22×C127—42no—
106C22×C127C2×C22⋊C411yescharacteristic
107C22×C127#12011yescharacteristic
108C22×C127#12311yescharacteristic
109C2×D127—24no—
110C2×D127—24no—
111C2×D127—22no—
112C2×D127—22no—
113C2×D127#10414yescomplemented
114C4×C127—24no—
115C4×C127—41no—
116C4×C127C4×D411yescharacteristic
117C4×C127C4×D411yescharacteristic
118C4×C127#12111yescharacteristic
119C4×C127#12111yescharacteristic
120Dic127—24no—
121Dic127—22no—
122Dic127—42no—
123C2×C127—21no—
124C2×C127—21no—
125C2×C127—81no—
126C2×C127#38611yescharacteristic
127C2×C127C2×#10412yes—
128D127—24no—
129#1896—1271nomaximal, Sylow
130C127#189611yescharacteristic, complemented, Sylow
131#525—1271no—
132C22×C42—1271no—
133C2×#104—1271no—
134C2×#104D12711yescharacteristic, complemented
135#407—1271no—
136#339—1271no—
137#339—1271no—
138C2×#41C2×D12711yescharacteristic
139C23×C4—1271no—
140C23×C4—2541no—
141C2×M16—1271no—
142C2×M16C2×D12711yescharacteristic
143#123—2541no—
144C22⋊Q8—1272no—
145#121—1272no—
146#120—2541no—
147C4×D4—2542no—
148#117—1271no—
149C2×C42—1271no—
150C2×C42—2542no—
151C2×C42—2541no—
152C2×C42—2542no—
153C2×C42C2×D12711yescharacteristic
154#104—22no—
155#104—2542no—
156C22⋊C8—2542no—
157C24—1271no—
158#41—21no—
159#41—21no—
160#41—21no—
161#41—21no—
162C2×Q8C4×D12711yescharacteristic, complemented
163C2×D4—21no—
164C2×D4—2542no—
165C2×D4C4×D12711yescharacteristic, complemented
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—1271no—
169C22×C4—1272no—
170C22×C4—1272no—
171C22×C4—2544no—
172C22×C4—2548no—
173C22×C4—2541no—
174C22×C4—2541no—
175C22×C4—2541no—
176C22×C4—2541no—
177C22×C4—2541no—
178C22×C4C22×D12711yescharacteristic
179M16—21no—
180M16—21no—
181M16—2541no—
182M16—2541no—
183C2×C8—21no—
184C2×C8—2541no—
185C4⋊C4—1272no—
186C4⋊C4—2542no—
187C22⋊C4—2542no—
188C22⋊C4—2542no—
189C22⋊C4—2541no—
190C42—21no—
191C42—1272no—
192C42—2544no—
193C42—2541no—
194C42—2541no—
195C42C22⋊D12712yescomplemented
196C23—21no—
197C23—1272no—
198C23—2544no—
199C23—2541no—
200C23—2541no—
201C23D4×D12711yescharacteristic
202Q8—21no—
203Q8—21no—
204D4—21no—
205D4—21no—
206D4—41no—
207D4—41no—
208D4—5084no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—1272no—
215C2×C4—25416no—
216C2×C4—2542no—
217C2×C4—2544no—
218C2×C4—2541no—
219C2×C4—2542no—
220C2×C4—2541no—
221C2×C4—2542no—
222C2×C4—2542no—
223C2×C4—2542no—
224C2×C4—2542no—
225C2×C4C2×C4×D12711yescharacteristic
226C2×C4D4×D12711yescharacteristic
227C2×C4D4×D12711yescharacteristic
228C2×C4C2×C22⋊D12711yescharacteristic
229C2×C4#1452324711yescharacteristic
230C2×C4#1452324811yescharacteristic
231C8—41no—
232C8—5081no—
233C22—22no—
234C22—42no—
235C22—41no—
236C22—1274no—
237C22—2544no—
238C22—2544no—
239C22—2542no—
240C22—2542no—
241C22C22⋊C4×D12711yescharacteristic
242C22#1686117711yescharacteristic
243C22#1686119111yescharacteristic
244C4—24no—
245C4—41no—
246C4—5082no—
247C4—2544no—
248C4—2542no—
249C4(C22⋊D127)⋊C411yescharacteristic
250C4C4×C22⋊D12711yescharacteristic
251C4#1686118011yescharacteristic
252C4#1686118211yescharacteristic
253C2—21no—
254C2—21no—
255C2—81no—
256C2—2544no—
257C2#2254975211yescharacteristic
258C2#104×D12712yes—
subgroupG/H
centerC2×C4D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#104×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550383C2
cosocleC22×C4×C127C23
hypercenterC2×#104D127
hypocenterC127#1896
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×C42×D127C2semidirect
3. C2×#16861136C2semidirect
4. #22549839C2semidirect
5. #22549637C2semidirect
6. #22549636C2semidirect
7. C2×#104×C127C2semidirect
12. #16861185C4semidirect
13. #16861180C4semidirect
42. C4⋊Dic127C2×C4semidirect
50. C4×Dic127D4semidirect
55. C2×Q8×C127C2×C4semidirect
57. C2×D4×C127C2×C4semidirect
65. C42×C127D4semidirect
113. C2×D127#104semidirect
130. C127#1896semidirect
134. C2×#104D127semidirect
162. C2×Q8C4×D127semidirect
165. C2×D4C4×D127semidirect
195. C42C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225503832
C2×C42×D1272
C2×#168611362
#225498392
#225496372
#225496362
C2×#104×C1272
#1896127