Finite Group Databasebeta

← #30640002

Subgroups — #30640002

Order 16256.

274 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 65 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×#16861235C211yesmaximal, characteristic
2#22549925C211yesmaximal, characteristic, complemented
3#22549892C211yesmaximal, characteristic, complemented
4#22549881C211yesmaximal, characteristic, complemented
5#22549858C211yesmaximal, characteristic, complemented
6#22549647C211yesmaximal, characteristic, complemented
7C2×#100×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861235—22no—
10#16861235—22no—
11C2×#14523236—21no—
12C2×#14523233—21no—
13C2×#14523232C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15#16861168C2211yescharacteristic
16#16861145—22no—
17#16861135—22no—
18#16861130—22no—
19#16861127—21no—
20#16861125—21no—
21C22×D4×C127C2211yescharacteristic
22C2×M16×C127C2211yescharacteristic
23C2×M16×C127C2211yescharacteristic
24#100×C127—22no—
25#14523248—21no—
26#14523248—21no—
27#14523248—21no—
28#14523248—21no—
29C2×Dic254D411yescharacteristic
30C2×D508D411yescharacteristic
31C2×C22⋊D127—21no—
32C2×C4×D127—21no—
33#14523236—24no—
34#14523233—24no—
35#14523232—21no—
36#14523232—21no—
37#14523232—21no—
38#14523232D412yes—
39C2×C127⋊C8—21no—
40C2×C127⋊C8—21no—
41C4⋊Dic127—21no—
42C22⋊Dic127—21no—
43C4×Dic127—21no—
44C24×C127—21no—
45C2×D4×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127—21no—
48C2×D4×C127—21no—
49C2×D4×C127—21no—
50C2×D4×C127D412yes—
51C22×C4×C127C2311yescharacteristic
52M16×C127—21no—
53M16×C127—21no—
54M16×C127—21no—
55M16×C127D412yes—
56C2×C8×C127—21no—
57C2×C8×C127—21no—
58C22×D127—21no—
59Dic254—21no—
60Dic254—21no—
61D508—21no—
62D508—21no—
63C22⋊D127—41no—
64C22⋊D127—41no—
65C2×Dic127—21no—
66C2×Dic127—41no—
67C4×D127—41no—
68C4×D127—41no—
69C127⋊C8—22no—
70C127⋊C8—41no—
71C23×C127—22no—
72C23×C127—41no—
73C23×C127—41no—
74C23×C127—82no—
75C23×C127#4111yescharacteristic
76D4×C127—22no—
77D4×C127—22no—
78D4×C127—41no—
79D4×C127—41no—
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127#4111yescharacteristic
85C2×C4×C127#4111yescharacteristic
86C8×C127—22no—
87C8×C127—41no—
88C22×C127—41no—
89C22×C127—42no—
90C22×C127—82no—
91C22×C127—82no—
92C22×C127—82no—
93C22×C127#12111yescharacteristic
94C22×C127#12311yescharacteristic
95C22×C127#12411yescharacteristic
96C2×D127—41no—
97C2×D127—81no—
98C4×C127#12011yescharacteristic
99C4×C127#12111yescharacteristic
100C4×C127#12111yescharacteristic
101C4×C127#12311yescharacteristic
102Dic127—41no—
103Dic127—81no—
104C2×C127—21no—
105C2×C127—21no—
106C2×C127—21no—
107C2×C127—82no—
108C2×C127#39411yescharacteristic
109D127—81no—
110#2048—1271nomaximal, Sylow
111C127#204811yescharacteristic, complemented, Sylow
112C2×AGL(1,ℤ/8)—1271no—
113#430—1271no—
114#421—1271no—
115#418—1271no—
116#413—1271no—
117C2×#100D12711yescharacteristic, complemented
118#344—1271no—
119C2×#41—1271no—
120C22×D4C2×D12711yescharacteristic
121AGL(1,ℤ/8)—2542no—
122AGL(1,ℤ/8)—2542no—
123C2×SD16—2541no—
124C2×D8—2541no—
125C2×M16—1271no—
126C2×M16—1271no—
127C2×M16C2×D12711yescharacteristic
128C2×M16C2×D12711yescharacteristic
129#117—1271no—
130#108—2542no—
131#104—2542no—
132#102—2542no—
133#101—2541no—
134#100—22no—
135#100—2541no—
136C24—21no—
137#41—2541no—
138#41—2541no—
139#41—2541no—
140#41—2541no—
141C2×Q8—1271no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—1271no—
148C2×D4—2541no—
149C2×D4D50812yes—
150C22×C4—2541no—
151C22×C4C22×D12711yescharacteristic
152SD16—2544no—
153D8—2544no—
154M16—21no—
155M16—21no—
156M16—21no—
157M16—1272no—
158M16—2541no—
159M16—2541no—
160M16—2541no—
161M16C22⋊D12712yes—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—2541no—
165C2×C8—2541no—
166C4⋊C4—2541no—
167C22⋊C4—2541no—
168C42—2541no—
169C23—22no—
170C23—41no—
171C23—41no—
172C23—82no—
173C23—2541no—
174C23#1452324811yescharacteristic
175Q8—2541no—
176Q8—2541no—
177D4—22no—
178D4—22no—
179D4—41no—
180D4—41no—
181D4—5081no—
182D4—5081no—
183D4—2541no—
184D4—2541no—
185C2×C4—5081no—
186C2×C4—5081no—
187C2×C4—5081no—
188C2×C4—2541no—
189C2×C4D4×D12711yescharacteristic
190C2×C4D4×D12711yescharacteristic
191C2×C4C2×C22⋊D12711yescharacteristic
192C2×C4C2×D50811yescharacteristic
193C2×C4#1452324711yescharacteristic
194C2×C4#1452324711yescharacteristic
195C8—22no—
196C8—41no—
197C8—5081no—
198C8—2542no—
199C22—41no—
200C22—42no—
201C22—82no—
202C22—82no—
203C22—82no—
204C22—10161no—
205C22—5081no—
206C22#1686118311yescharacteristic
207C22#1686119111yescharacteristic
208C22#1686119511yescharacteristic
209C4—10161no—
210C4—5081no—
211C4#1686117911yescharacteristic
212C4#1686118011yescharacteristic
213C4#1686118111yescharacteristic
214C4#1686119311yescharacteristic
215C2—21no—
216C2—21no—
217C2—21no—
218C2—82no—
219C2—10161no—
220C2#2254979211yescharacteristic
subgroupG/H
centerC2#22549792
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#100×C127C2
socleC2×C127#394
prime radicalC2×#16861235C2
hypercenterC2×#100D127
hypocenterC127#2048
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. #22549925C2semidirect
3. #22549892C2semidirect
4. #22549881C2semidirect
5. #22549858C2semidirect
6. #22549647C2semidirect
7. C2×#100×C127C2semidirect
111. C127#2048semidirect
117. C2×#100D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
#225499252
#225498922
#225498812
#225498582
#225496472
C2×#100×C1272
#2048127