Finite Group Databasebeta

← #30639219

Subgroups — #30639219

Order 16256.

346 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 93 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#22550351C211yesmaximal, characteristic, complemented
2#22549893C211yesmaximal, characteristic, complemented
3#22549881C211yesmaximal, characteristic, complemented
4#22549850C211yesmaximal, characteristic, complemented
5#22549647C211yesmaximal, characteristic, complemented
6#22549633C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9D4×Dic127—22no—
10D4×Dic127—22no—
11#16861168C2211yescharacteristic
12#16861168C2211yescharacteristic
13#16861168C2211yescharacteristic
14C2×C22⋊Dic127—22no—
15#16861136—22no—
16#16861130—22no—
17#16861124—22no—
18C2×#41×C127C2211yescharacteristic
19C22×D4×C127C2211yescharacteristic
20AGL(1,ℤ/8)×C127—22no—
21AGL(1,ℤ/8)×C127—22no—
22C2×SD16×C127C412yescomplemented
23C2×D8×C127C412yescomplemented
24C2×M16×C127C2211yescharacteristic
25C22×Dic127—24no—
26#14523232—21no—
27#14523232—21no—
28C2×C127⋊C8—21no—
29C4⋊Dic127—21no—
30C4⋊Dic127—21no—
31C4⋊Dic127—21no—
32C22⋊Dic127—24no—
33C22⋊Dic127—21no—
34C22⋊Dic127—21no—
35C22⋊Dic127D412yes—
36C4×Dic127—21no—
37C4×Dic127—21no—
38C4×Dic127D412yes—
39C24×C127—21no—
40#41×C127—21no—
41#41×C127—21no—
42#41×C127—21no—
43#41×C127—21no—
44C2×Q8×C127C2×C411yescharacteristic, complemented
45C2×D4×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127—21no—
48C2×D4×C127—21no—
49C2×D4×C127C2×C412yes—
50C2×D4×C127C2×C411yescharacteristic, complemented
51C2×D4×C127D412yes—
52C2×D4×C127D412yes—
53C22×C4×C127—21no—
54C22×C4×C127C2311yescharacteristic
55SD16×C127—24no—
56D8×C127—24no—
57M16×C127—21no—
58M16×C127—21no—
59C2×C8×C127C2×C412yescomplemented
60C2×Dic127—22no—
61C2×Dic127—22no—
62C2×Dic127—41no—
63C2×Dic127—84no—
64C127⋊C8—41no—
65C23×C127—22no—
66C23×C127—22no—
67C23×C127—22no—
68C23×C127—21no—
69C23×C127—82no—
70C23×C127C2×D411yescharacteristic
71Q8×C127—21no—
72Q8×C127—21no—
73D4×C127—21no—
74D4×C127—22no—
75D4×C127—21no—
76D4×C127—22no—
77D4×C127—22no—
78D4×C127—41no—
79D4×C127—41no—
80D4×C127C22⋊C414yes—
81C2×C4×C127—21no—
82C2×C4×C127—41no—
83C2×C4×C127—41no—
84C2×C4×C127C22×C411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127#4111yescharacteristic
89C2×C4×C127#4111yescharacteristic
90C8×C127—22no—
91C22×C127—24no—
92C22×C127—41no—
93C22×C127—44no—
94C22×C127—41no—
95C22×C127—82no—
96C22×C127—82no—
97C22×C127—81no—
98C22×C127C4×D411yescharacteristic
99C22×C127#12011yescharacteristic
100C22×C127#12111yescharacteristic
101C4×C127—41no—
102C4×C127C2×C22⋊C411yescharacteristic
103C4×C127C4×D411yescharacteristic
104C4×C127#12111yescharacteristic
105C4×C127#12311yescharacteristic
106Dic127—42no—
107Dic127—42no—
108Dic127—81no—
109C2×C127—21no—
110C2×C127—21no—
111C2×C127—21no—
112C2×C127—44no—
113C2×C127—81no—
114C2×C127#38611yescharacteristic
115#1904—1271nomaximal, Sylow
116C127#190411yescharacteristic, complemented, Sylow
117C2×AGL(1,ℤ/8)D12711yescharacteristic
118#518—1271no—
119#421—1271no—
120#418—1271no—
121#410—1271no—
122#344—1271no—
123H(3,ℤ/4)—1271no—
124C2×#41C2×D12711yescharacteristic
125C22×D4C2×D12711yescharacteristic
126AGL(1,ℤ/8)—22no—
127AGL(1,ℤ/8)—22no—
128C2×SD16Dic12712yescomplemented
129C2×D8Dic12712yescomplemented
130C2×M16—1271no—
131C2×M16C2×D12711yescharacteristic
132C4×D4—2542no—
133C4×D4—2542no—
134#117—1271no—
135#117—1271no—
136#117—1271no—
137C2×C22⋊C4—2542no—
138#104—2542no—
139#102—2542no—
140#99—2542no—
141C24—21no—
142#41—21no—
143#41—21no—
144#41—21no—
145#41—21no—
146C2×Q8C2×Dic12711yescharacteristic, complemented
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4C2×Dic12712yes—
152C2×D4C2×Dic12711yescharacteristic, complemented
153C2×D4C22⋊D12712yes—
154C2×D4C22⋊D12712yes—
155C22×C4—21no—
156C22×C4—2544no—
157C22×C4C22×D12711yescharacteristic
158SD16—24no—
159D8—24no—
160M16—21no—
161M16—21no—
162M16—2541no—
163M16—2541no—
164C2×C8—2541no—
165C2×C8C2×Dic12712yescomplemented
166C4⋊C4—2541no—
167C4⋊C4—2541no—
168C4⋊C4—2541no—
169C22⋊C4—1272no—
170C22⋊C4—2541no—
171C22⋊C4—2544no—
172C22⋊C4—2541no—
173C42—1272no—
174C42—2541no—
175C42—2541no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—21no—
180C23—82no—
181C23D4×D12711yescharacteristic
182Q8—21no—
183Q8—21no—
184D4—22no—
185D4—21no—
186D4—22no—
187D4—21no—
188D4—22no—
189D4—41no—
190D4—41no—
191D4C22⋊Dic12714yes—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—10164no—
196C2×C4—5081no—
197C2×C4—2542no—
198C2×C4—2542no—
199C2×C4C22×Dic12711yescharacteristic
200C2×C4D4×D12711yescharacteristic
201C2×C4C2×C22⋊D12711yescharacteristic
202C2×C4C2×C22⋊D12711yescharacteristic
203C2×C4#1452324711yescharacteristic
204C2×C4#1452324711yescharacteristic
205C8—22no—
206C8—5081no—
207C22—24no—
208C22—41no—
209C22—44no—
210C22—41no—
211C22—82no—
212C22—82no—
213C22—81no—
214C22D4×Dic12711yescharacteristic
215C22#1686117711yescharacteristic
216C22#1686118111yescharacteristic
217C4—41no—
218C4—10161no—
219C4—5082no—
220C4—5082no—
221C4C2×C22⋊Dic12711yescharacteristic
222C4D4×Dic12711yescharacteristic
223C4#1686118111yescharacteristic
224C4#1686119011yescharacteristic
225C2—21no—
226C2—21no—
227C2—21no—
228C2—44no—
229C2—81no—
230C2#2254975611yescharacteristic
subgroupG/H
centerC2#22549756
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting, prime radicalC2×AGL(1,ℤ/8)×C127C2
socleC2×C127#386
cosocleC22×C4×C127C23
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#1904
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #22550351C2semidirect
2. #22549893C2semidirect
3. #22549881C2semidirect
4. #22549850C2semidirect
5. #22549647C2semidirect
6. #22549633C2semidirect
22. C2×SD16×C127C4semidirect
23. C2×D8×C127C4semidirect
44. C2×Q8×C127C2×C4semidirect
50. C2×D4×C127C2×C4semidirect
59. C2×C8×C127C2×C4semidirect
116. C127#1904semidirect
128. C2×SD16Dic127semidirect
129. C2×D8Dic127semidirect
146. C2×Q8C2×Dic127semidirect
152. C2×D4C2×Dic127semidirect
165. C2×C8C2×Dic127semidirect

Maximal subgroups

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

subgroupindex
#225503512
#225498932
#225498812
#225498502
#225496472
#225496332
C2×AGL(1,ℤ/8)×C1272
#1904127