Finite Group Databasebeta

← #30638706

Subgroups — #30638706

Order 16256.

536 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16861256C211yesmaximal, characteristic
2#22549892C212yesmaximal, complemented
3#22549886C212yesmaximal, complemented
4#22549850C211yesmaximal, characteristic, complemented
5#413×C127C211yesmaximal, characteristic, complemented
6#16861256—24no—
7#16861256—24no—
8C2×#14523248C412yescomplemented
9C2×#14523248C2211yescharacteristic
10C2×#14523247—21no—
11C2×#14523247—22no—
12C22×Dic254C412yescomplemented
13C22×Dic254C2211yescharacteristic
14C2×Q8×D127—21no—
15#16861168C2212yes—
16#16861135—24no—
17#16861134—24no—
18#16861124—22no—
19C2×#41×C127C2211yescharacteristic
20C2×M16×C127C2212yes—
21#101×C127—21no—
22#100×C127—21no—
23#14523248—24no—
24#14523248—24no—
25#14523248—22no—
26#14523248D414yes—
27#14523247—216no—
28#14523247—28no—
29C2×Dic254—24no—
30C2×Dic254—24no—
31C2×Dic254—22no—
32C2×Dic254—21no—
33C2×Dic254—22no—
34C2×Dic254C2×C411yescharacteristic
35C2×Dic254C2×C412yes—
36C2×Dic254D414yes—
37Q8×D127—28no—
38C2×D508C2×C411yescharacteristic
39C2×C22⋊D127—22no—
40C2×C22⋊D127—21no—
41C22×Dic127—22no—
42C22×Dic127—21no—
43C2×C4×D127—22no—
44C2×C4×D127—21no—
45C4⋊Dic127—22no—
46C22⋊Dic127—22no—
47C4×Dic127—22no—
48#41×C127—22no—
49#41×C127—22no—
50C2×Q8×C127D411yescharacteristic
51C2×D4×C127D411yescharacteristic
52C2×D4×C127D412yes—
53C22×C4×C127C2×C412yes—
54C22×C4×C127C2311yescharacteristic
55M16×C127—22no—
56M16×C127—22no—
57C2×C8×C127—22no—
58C22×D127—21no—
59Dic254—28no—
60Dic254—21no—
61Dic254—22no—
62Dic254—24no—
63Dic254C22⋊C412yes—
64Dic254C22⋊C414yes—
65D508—21no—
66D508C22⋊C412yes—
67C22⋊D127—28no—
68C22⋊D127—24no—
69C2×Dic127—28no—
70C2×Dic127—22no—
71C2×Dic127—21no—
72C2×Dic127—24no—
73C2×Dic127—42no—
74C4×D127—28no—
75C4×D127—24no—
76C23×C127C22⋊C412yes—
77C23×C127C2×D411yescharacteristic
78Q8×C127—22no—
79D4×C127—24no—
80D4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—22no—
83C2×C4×C127C22⋊C412yes—
84C2×C4×C127C22×C411yescharacteristic
85C2×C4×C127C2×D412yes—
86C2×C4×C127C2×D412yes—
87C2×C4×C127C2×D411yescharacteristic
88C8×C127—42no—
89C22×C127—22no—
90C22×C127—41no—
91C22×C127—42no—
92C22×C127C2×C22⋊C411yescharacteristic
93C22×C127#12012yes—
94C2×D127—22no—
95C2×D127—42no—
96C4×C127—22no—
97C4×C127C2×C22⋊C411yescharacteristic
98C4×C127C2×C22⋊C411yescharacteristic
99C4×C127#12012yes—
100Dic127—24no—
101Dic127—22no—
102Dic127—82no—
103C2×C127—22no—
104C2×C127—21no—
105C2×C127—42no—
106C2×C127#37911yescharacteristic
107D127—42no—
108#1796—1271nomaximal, Sylow
109C127#179611yescharacteristic, complemented, Sylow
110C2×2−1+4—1271no—
111#421—1272no—
112#419—1272no—
113#413D12711yescharacteristic, complemented
114#410—1271no—
1152−1+4—2544no—
1162−1+4—2544no—
117C2×#41—1272no—
118C2×#41—1271no—
119C2×#41—2541no—
120C2×#41—2542no—
121C2×#41C2×D12711yescharacteristic
122C22×Q8—1272no—
123C22×Q8—1271no—
124C22×Q8—2541no—
125C2×M16C2×D12712yes—
126#117—1272no—
127#104—2544no—
128#103—2544no—
129#101—21no—
130#100—21no—
131#99—2542no—
132#41—22no—
133#41—22no—
134#41—1274no—
135#41—2544no—
136#41—2544no—
137#41—25416no—
138#41—2548no—
139#41—2542no—
140C2×Q8—1272no—
141C2×Q8—1271no—
142C2×Q8—1274no—
143C2×Q8—2544no—
144C2×Q8—2544no—
145C2×Q8—2548no—
146C2×Q8—2542no—
147C2×Q8—2542no—
148C2×Q8—2541no—
149C2×Q8C22⋊D12711yescharacteristic
150C2×D4—1271no—
151C2×D4—2542no—
152C2×D4—2541no—
153C2×D4C22⋊D12711yescharacteristic
154C2×D4D50812yes—
155C22×C4—2542no—
156C22×C4—2541no—
157C22×C4—2542no—
158C22×C4—2541no—
159C22×C4C4×D12712yes—
160C22×C4C22×D12711yescharacteristic
161M16—22no—
162M16—22no—
163C2×C8—22no—
164C4⋊C4—2542no—
165C22⋊C4—2542no—
166C42—2542no—
167C23—2541no—
168C23(C2×D127)⋊C412yes—
169C23C2×D50811yescharacteristic
170Q8—22no—
171Q8—1274no—
172Q8—1272no—
173Q8—2548no—
174Q8—2542no—
175Q8—2541no—
176Q8—2544no—
177D4—24no—
178D4—22no—
179D4—1272no—
180D4—2548no—
181D4—2544no—
182D4—2541no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—5082no—
186C2×C4—2548no—
187C2×C4—2548no—
188C2×C4—2541no—
189C2×C4—2542no—
190C2×C4—2544no—
191C2×C4—2544no—
192C2×C4(C2×D127)⋊C412yes—
193C2×C4C2×C4×D12711yescharacteristic
194C2×C4D4×D12712yes—
195C2×C4D4×D12712yes—
196C2×C4C2×C22⋊D12711yescharacteristic
197C8—42no—
198C22—22no—
199C22—41no—
200C22—42no—
201C22—5082no—
202C22—2542no—
203C22C2×(C2×D127)⋊C411yescharacteristic
204C22#1686117912yes—
205C4—22no—
206C4—10162no—
207C4—2544no—
208C4—2542no—
209C4C22⋊C4×D12711yescharacteristic
210C4C22⋊C4×D12711yescharacteristic
211C4#1686117712yes—
212C2—22no—
213C2—21no—
214C2—42no—
215C2—5082no—
216C2#2254973111yescharacteristic
subgroupG/H
centerC2#22549731
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#413×C127C2
socleC2×C127#379
prime radicalC2×#16861256C2
cosocleC22×C4×C127C23
hypercenter#413D127
hypocenterC127#1796
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #22549892C2semidirect
3. #22549886C2semidirect
4. #22549850C2semidirect
5. #413×C127C2semidirect
8. C2×#14523248C4semidirect
12. C22×Dic254C4semidirect
109. C127#1796semidirect
113. #413D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612562
#225498922
#225498862
#225498502
#413×C1272
#1796127