Finite Group Databasebeta

← #30638941

Subgroups — #30638941

Order 16256.

324 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 83 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#22550350C211yesmaximal, characteristic
2#22549919C211yesmaximal, characteristic, complemented
3#22549918C211yesmaximal, characteristic, complemented
4C2×#16861125C211yesmaximal, characteristic
5#22549778C211yesmaximal, characteristic
6#22549636C211yesmaximal, characteristic, complemented
7#339×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C4×D508—21no—
11C4×D508—21no—
12#16861171—21no—
13#16861171—21no—
14#16861168C2211yescharacteristic
15C2×C4⋊Dic127C2211yescharacteristic
16C2×(C2×D127)⋊C4—22no—
17C22⋊C4×D127—22no—
18#16861143—21no—
19#16861142—21no—
20#16861141—21no—
21#16861140—21no—
22#16861125C414yescomplemented
23C2×M16×C127C2211yescharacteristic
24#117×C127C2211yescharacteristic
25C2×C42×C127C2211yescharacteristic
26C23×D127D412yes—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508C2×C412yes—
32C2×D508Q812yes—
33C22×Dic127—21no—
34C2×C4×D127—22no—
35C2×C4×D127—22no—
36#14523232—21no—
37#14523232C2×C412yescomplemented
38C2×C127⋊C8—21no—
39C4⋊Dic127—21no—
40C4⋊Dic127—21no—
41C4⋊Dic127D411yescharacteristic
42C4⋊Dic127D411yescharacteristic
43C22⋊Dic127—21no—
44(C2×D127)⋊C4—22no—
45(C2×D127)⋊C4—22no—
46(C2×D127)⋊C4—44no—
47C4×Dic127—21no—
48C22×C4×C127—21no—
49C22×C4×C127C2311yescharacteristic
50M16×C127—21no—
51M16×C127C2×C412yescomplemented
52C2×C8×C127—21no—
53C4⋊C4×C127D411yescharacteristic
54C4⋊C4×C127D411yescharacteristic
55C22⋊C4×C127—21no—
56C42×C127—21no—
57C42×C127—21no—
58C42×C127—21no—
59C22×D127—22no—
60C22×D127—22no—
61C22×D127—42no—
62C22×D127—42no—
63C22×D127C4⋊C414yes—
64D508—22no—
65D508—22no—
66D508—22no—
67D508—22no—
68C2×Dic127—21no—
69C2×Dic127—21no—
70C2×Dic127—42no—
71C2×Dic127—41no—
72C4×D127—82no—
73C4×D127—82no—
74C127⋊C8—22no—
75C23×C127C2×D411yescharacteristic
76C2×C4×C127—22no—
77C2×C4×C127—21no—
78C2×C4×C127—21no—
79C2×C4×C127—41no—
80C2×C4×C127—41no—
81C2×C4×C127C22×C411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×Q811yescharacteristic
85C2×C4×C127#4111yescharacteristic
86C2×C4×C127#4111yescharacteristic
87C8×C127—22no—
88C22×C127—21no—
89C22×C127—21no—
90C22×C127C2×C4⋊C411yescharacteristic
91C22×C127#12011yescharacteristic
92C22×C127#12311yescharacteristic
93C2×D127—24no—
94C2×D127—42no—
95C2×D127—42no—
96C2×D127—44no—
97C2×D127—82no—
98C4×C127—42no—
99C4×C127—41no—
100C4×C127—41no—
101C4×C127C4×D411yescharacteristic
102C4×C127C4×D411yescharacteristic
103C4×C127C22⋊Q811yescharacteristic
104C4×C127C22⋊Q811yescharacteristic
105Dic127—41no—
106Dic127—41no—
107Dic127—81no—
108C2×C127—21no—
109C2×C127—21no—
110C2×C127#38511yescharacteristic
111C2×C127#45311yescharacteristic
112C2×C127#45511yescharacteristic
113D127—44no—
114#1856—1271nomaximal, Sylow
115C127#185611yescharacteristic, complemented, Sylow
116#518—1271no—
117#428—1271no—
118#428—1271no—
119C2×#100—1271no—
120#390—1271no—
121#339—1271no—
122#339D12711yescharacteristic, complemented
123C22×D4—1271no—
124C2×M16—1271no—
125C2×M16C2×D12711yescharacteristic
126C4×D4—2541no—
127C4×D4—2541no—
128C4×D4—2541no—
129C4×D4—2541no—
130#117—1271no—
131#117C2×D12711yescharacteristic
132C2×C4⋊C4—1271no—
133C2×C22⋊C4—2542no—
134C2×C22⋊C4—2542no—
135C2×C42C2×D12711yescharacteristic
136#107—2541no—
137#107—2541no—
138#106—2541no—
139#106—2541no—
140#100—1274no—
141C24—1272no—
142C2×D4—1272no—
143C2×D4—1272no—
144C2×D4—2541no—
145C2×D4—2541no—
146C2×D4—2541no—
147C2×D4—2541no—
148C22×C4—21no—
149C22×C4—2542no—
150C22×C4—2542no—
151C22×C4—2541no—
152C22×C4C22×D12711yescharacteristic
153M16—21no—
154M16—1272no—
155M16—2541no—
156M16C4×D12712yescomplemented
157C2×C8—21no—
158C2×C8—2541no—
159C4⋊C4—1271no—
160C4⋊C4—1271no—
161C4⋊C4—2541no—
162C4⋊C4—2541no—
163C4⋊C4D50811yescharacteristic
164C4⋊C4D50811yescharacteristic
165C22⋊C4—21no—
166C22⋊C4—5084no—
167C22⋊C4—2542no—
168C22⋊C4—2542no—
169C22⋊C4—2541no—
170C42—21no—
171C42—21no—
172C42—21no—
173C42—2541no—
174C23—1274no—
175C23—5082no—
176C23—5082no—
177C23—2542no—
178C23—2542no—
179C23D4×D12711yescharacteristic
180D4—2542no—
181D4—2542no—
182D4—2542no—
183D4—2542no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—10162no—
190C2×C4—10162no—
191C2×C4—5082no—
192C2×C4—5081no—
193C2×C4—2541no—
194C2×C4—2541no—
195C2×C4C2×C4×D12711yescharacteristic
196C2×C4D4×D12711yescharacteristic
197C2×C4C2×D50811yescharacteristic
198C2×C4Q8×D12711yescharacteristic
199C2×C4#1452324711yescharacteristic
200C2×C4#1452324811yescharacteristic
201C8—22no—
202C8—2542no—
203C22—21no—
204C22—21no—
205C22—10162no—
206C22—5084no—
207C22—5082no—
208C22—5082no—
209C22—2544no—
210C22C4⋊C4×D12711yescharacteristic
211C22#1686117911yescharacteristic
212C22#1686119111yescharacteristic
213C4—42no—
214C4—41no—
215C4—41no—
216C4—10161no—
217C4—5081no—
218C4—5081no—
219C4(C22⋊D127)⋊C411yescharacteristic
220C4C4×D50811yescharacteristic
221C4#1686118711yescharacteristic
222C4#1686118911yescharacteristic
223C2—21no—
224C2—21no—
225C2—5084no—
226C2#2254975011yescharacteristic
227C2#2255004511yescharacteristic
228C2#2255005511yescharacteristic
subgroupG/H
centerC22#16861179
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#339×C127C2
socleC22×C127#120
prime radicalC22×D508C22
cosocleC22×C4×C127C23
hypercenter#339D127
hypocenterC127#1856
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #22549919C2semidirect
3. #22549918C2semidirect
6. #22549636C2semidirect
7. #339×C127C2semidirect
22. #16861125C4semidirect
37. #14523232C2×C4semidirect
51. M16×C127C2×C4semidirect
115. C127#1856semidirect
122. #339D127semidirect
156. M16C4×D127semidirect

Maximal subgroups

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

subgroupindex
#225503502
#225499192
#225499182
C2×#168611252
#225497782
#225496362
#339×C1272
#1856127