Finite Group Databasebeta

← #30640022

Subgroups — #30640022

Order 16256.

282 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 67 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
2C2×#16861144C211yesmaximal, characteristic, complemented
3#22549878C211yesmaximal, characteristic, complemented
4C2×#16861130C211yesmaximal, characteristic, complemented
5#22549844C211yesmaximal, characteristic, complemented
6#22549644C211yesmaximal, characteristic, complemented
7C2×#100×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861235—22no—
10#16861235—22no—
11C2×#14523236—21no—
12C2×#14523233—21no—
13C2×#14523232C2211yescharacteristic
14C22×C127⋊C8C2211yescharacteristic
15C2×C4⋊Dic127C2211yescharacteristic
16#16861144—22no—
17#16861134—21no—
18#16861131—21no—
19#16861130—22no—
20(C2×D127)⋊C8—22no—
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—
32C22×Dic127—21no—
33C2×C4×D127—21no—
34#14523236—24no—
35#14523233—24no—
36#14523232—21no—
37#14523232D412yes—
38C2×C127⋊C8—21no—
39C2×C127⋊C8—21no—
40C2×C127⋊C8—21no—
41C2×C127⋊C8D412yes—
42C4⋊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×C127—21no—
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—
67C2×Dic127—81no—
68C4×D127—41no—
69C4×D127—41no—
70C127⋊C8—22no—
71C127⋊C8—22no—
72C23×C127—22no—
73C23×C127—41no—
74C23×C127—41no—
75C23×C127—81no—
76C23×C127—81no—
77C23×C127#4111yescharacteristic
78D4×C127—22no—
79D4×C127—22no—
80D4×C127—41no—
81D4×C127—41no—
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127#4111yescharacteristic
87C2×C4×C127#4111yescharacteristic
88C8×C127—41no—
89C8×C127—41no—
90C22×C127—21no—
91C22×C127—21no—
92C22×C127—42no—
93C22×C127—82no—
94C22×C127—82no—
95C22×C127—81no—
96C22×C127—81no—
97C22×C127#12111yescharacteristic
98C22×C127#12311yescharacteristic
99C22×C127#12411yescharacteristic
100C2×D127—41no—
101C2×D127—41no—
102C2×D127—41no—
103C4×C127#12011yescharacteristic
104C4×C127#12111yescharacteristic
105C4×C127#12111yescharacteristic
106C4×C127#12311yescharacteristic
107Dic127—41no—
108Dic127—81no—
109C2×C127—21no—
110C2×C127—21no—
111C2×C127—82no—
112C2×C127#39411yescharacteristic
113C2×C127#47111yescharacteristic
114C2×C127#47211yescharacteristic
115D127—81no—
116#2051—1271nomaximal, Sylow
117C127#205111yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)—1271no—
119C2×#108—1271no—
120#417—1271no—
121C2×#102—1271no—
122C2×#100D12711yescharacteristic, complemented
123#408—1271no—
124#343—1271no—
125C2×#41—1271no—
126C22×D4C2×D12711yescharacteristic
127AGL(1,ℤ/8)—2542no—
128AGL(1,ℤ/8)—2542no—
129C2×SD16—2541no—
130C2×D8—2541no—
131C2×M16—1271no—
132C2×M16C2×D12711yescharacteristic
133C2×M16C2×D12711yescharacteristic
134C22×C8—1271no—
135C2×C4⋊C4—1271no—
136#108—2542no—
137#103—2541no—
138#102—2541no—
139#102—2542no—
140#100—22no—
141C22⋊C8—2542no—
142C24—21no—
143#41—2541no—
144#41—2541no—
145#41—2541no—
146#41—2541no—
147C2×Q8—1271no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—1271no—
154C2×D4—2541no—
155C2×D4C22⋊D12712yes—
156C22×C4—2541no—
157C22×C4—2541no—
158C22×C4C22×D12711yescharacteristic
159SD16—2544no—
160D8—2544no—
161M16—21no—
162M16—21no—
163M16—21no—
164M16—21no—
165M16—1272no—
166M16—2541no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—1272no—
170C2×C8—2541no—
171C2×C8—2541no—
172C2×C8—2541no—
173C4⋊C4—2541no—
174C4⋊C4—2541no—
175C23—22no—
176C23—41no—
177C23—41no—
178C23—81no—
179C23—81no—
180C23—2541no—
181C23#1452324811yescharacteristic
182Q8—2541no—
183Q8—2541no—
184D4—22no—
185D4—22no—
186D4—41no—
187D4—41no—
188D4—5081no—
189D4—5081no—
190D4—2541no—
191D4—2541no—
192C2×C4—10161no—
193C2×C4—5081no—
194C2×C4—5081no—
195C2×C4—5081no—
196C2×C4—2541no—
197C2×C4D4×D12711yescharacteristic
198C2×C4D4×D12711yescharacteristic
199C2×C4D4×D12711yescharacteristic
200C2×C4C2×C22⋊D12711yescharacteristic
201C2×C4#1452324711yescharacteristic
202C2×C4#1452324811yescharacteristic
203C8—41no—
204C8—41no—
205C8—2542no—
206C8—2542no—
207C22—21no—
208C22—21no—
209C22—42no—
210C22—82no—
211C22—82no—
212C22—81no—
213C22—81no—
214C22—5081no—
215C22—5081no—
216C22—5081no—
217C22#1686118211yescharacteristic
218C22#1686119211yescharacteristic
219C22#1686119511yescharacteristic
220C4—10161no—
221C4—5081no—
222C4#1686117711yescharacteristic
223C4#1686118111yescharacteristic
224C4#1686118211yescharacteristic
225C4#1686119111yescharacteristic
226C2—21no—
227C2—21no—
228C2—82no—
229C2—10161no—
230C2#2254979111yescharacteristic
231C2#2255013811yescharacteristic
232C2#2255014511yescharacteristic
subgroupG/H
centerC22#16861182
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#100×C127C2
socleC22×C127#121
prime radicalC2×#16861235C2
hypercenterC2×#100D127
hypocenterC127#2051
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. C2×#16861144C2semidirect
3. #22549878C2semidirect
4. C2×#16861130C2semidirect
5. #22549844C2semidirect
6. #22549644C2semidirect
7. C2×#100×C127C2semidirect
117. C127#2051semidirect
122. C2×#100D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
C2×#168611442
#225498782
C2×#168611302
#225498442
#225496442
C2×#100×C1272
#2051127