Finite Group Databasebeta

← #30639256

Subgroups — #30639256

Order 16256.

344 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 79 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#22550642C211yesmaximal, characteristic, complemented
2#22550445C211yesmaximal, characteristic
3#22549882C211yesmaximal, characteristic, complemented
4#22549880C211yesmaximal, characteristic, complemented
5#22549849C211yesmaximal, characteristic, complemented
6#22549632C211yesmaximal, characteristic, complemented
7H(3,ℤ/4)×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C22×D508C2211yescharacteristic
10#16861217—22no—
11#16861215—22no—
12C2×#14523232C2211yescharacteristic
13M16×D127—22no—
14#16861206C412yescomplemented
15#16861197C412yescomplemented
16#16861184—22no—
17#16861179—22no—
18#16861168C2211yescharacteristic
19#16861131—22no—
20#16861129—22no—
21#16861123—22no—
22C2×M16×C127C2211yescharacteristic
23#117×C127C2211yescharacteristic
24#117×C127C2211yescharacteristic
25C23×D127—21no—
26#14523248—21no—
27#14523248—21no—
28#14523248—21no—
29#14523248—21no—
30C2×Dic254C2×C411yescharacteristic
31C2×D508—24no—
32C2×D508—21no—
33C2×D508—21no—
34C2×D508—21no—
35C2×D508—21no—
36C2×D508—21no—
37C2×D508C2×C412yes—
38C2×D508C2×C411yescharacteristic
39C2×C22⋊D127D412yes—
40C2×C4×D127D412yes—
41#14523232—21no—
42#14523232—21no—
43#14523231—24no—
44C2×C127⋊C8D412yescomplemented
45C8×D127—24no—
46C4⋊Dic127—21no—
47C22⋊Dic127—21no—
48(C2×D127)⋊C4—24no—
49C4×Dic127—21no—
50C22×C4×C127C2311yescharacteristic
51M16×C127—21no—
52M16×C127—21no—
53C2×C8×C127D412yescomplemented
54C4⋊C4×C127—21no—
55C4⋊C4×C127—21no—
56C22⋊C4×C127—21no—
57C22⋊C4×C127—21no—
58C42×C127—21no—
59C42×C127C2×C412yes—
60C22×D127—22no—
61C22×D127—41no—
62C22×D127—41no—
63C22×D127—82no—
64C22×D127C22⋊C412yes—
65Dic254—21no—
66Dic254—21no—
67D508—21no—
68D508—21no—
69D508—22no—
70D508—22no—
71D508—41no—
72D508—41no—
73D508—84no—
74C22⋊D127—22no—
75C22⋊D127—22no—
76C2×Dic127—41no—
77C2×Dic127C22⋊C412yes—
78C4×D127—22no—
79C4×D127—22no—
80C127⋊C8—22no—
81C23×C127C2×D411yescharacteristic
82C2×C4×C127—22no—
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—41no—
92C22×C127C2×C22⋊C411yescharacteristic
93C22×C127#12111yescharacteristic
94C22×C127#12111yescharacteristic
95C2×D127—22no—
96C2×D127—42no—
97C2×D127—44no—
98C2×D127—42no—
99C2×D127—82no—
100C2×D127—82no—
101C4×C127—42no—
102C4×C127—81no—
103C4×C127C4×D411yescharacteristic
104C4×C127C4×D411yescharacteristic
105C4×C127#12011yescharacteristic
106C4×C127#12311yescharacteristic
107Dic127—22no—
108Dic127—81no—
109C2×C127—21no—
110C2×C127—21no—
111C2×C127—21no—
112C2×C127#38611yescharacteristic
113D127—42no—
114D127—82no—
115#1910—1271nomaximal, Sylow
116C127#191011yescharacteristic, complemented, Sylow
117#568—1271no—
118#535—1271no—
119#418—1271no—
120#418—1271no—
121#410—1271no—
122H(3,ℤ/4)—1271no—
123H(3,ℤ/4)D12711yescharacteristic, complemented
124C2×#41—1271no—
125C22×D4—1271no—
126#131—2542no—
127#131—2542no—
128C2×M16—1271no—
129C2×M16—2542no—
130C2×M16C2×D12711yescharacteristic
131#127—1272no—
132#124—1272no—
133#121—2542no—
134#120—2542no—
135#117—1271no—
136#117C2×D12711yescharacteristic
137#117C2×D12711yescharacteristic
138#102—2542no—
139#102—2542no—
140#99—2542no—
141C24—2541no—
142#41—2541no—
143#41—2541no—
144#41—2541no—
145#41—2541no—
146C2×Q8—1271no—
147C2×D4—1272no—
148C2×D4—1271no—
149C2×D4—1272no—
150C2×D4—2544no—
151C2×D4—2541no—
152C2×D4—2541no—
153C2×D4—2541no—
154C2×D4—2541no—
155C2×D4—2541no—
156C22×C4—1272no—
157C22×C4C22×D12711yescharacteristic
158M16—21no—
159M16—21no—
160M16—2544no—
161M16—2541no—
162M16—2541no—
163C2×C8—1272no—
164C2×C8—2544no—
165C2×C8D50812yescomplemented
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—2541no—
169C22⋊C4—21no—
170C22⋊C4—21no—
171C22⋊C4—2544no—
172C22⋊C4—2541no—
173C42—21no—
174C42—2541no—
175C42C4×D12712yes—
176C23—1272no—
177C23—10162no—
178C23—5081no—
179C23—5081no—
180C23—2542no—
181C23D4×D12711yescharacteristic
182Q8—2541no—
183Q8—2541no—
184D4—10164no—
185D4—5081no—
186D4—5081no—
187D4—2542no—
188D4—2542no—
189D4—2542no—
190D4—2541no—
191D4—2542no—
192D4—2541no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—1272no—
196C2×C4—5081no—
197C2×C4—2542no—
198C2×C4—2542no—
199C2×C4C2×C4×D12711yescharacteristic
200C2×C4D4×D12711yescharacteristic
201C2×C4D4×D12711yescharacteristic
202C2×C4C2×D50811yescharacteristic
203C2×C4#1452324811yescharacteristic
204C2×C4#1452324911yescharacteristic
205C8—22no—
206C8—2542no—
207C22—41no—
208C22—10162no—
209C22—10162no—
210C22—5084no—
211C22—5082no—
212C22—5082no—
213C22—2542no—
214C22C22⋊C4×D12711yescharacteristic
215C22#1686118211yescharacteristic
216C22#1686118411yescharacteristic
217C4—42no—
218C4—81no—
219C4—10161no—
220C4—2542no—
221C4#1686117111yescharacteristic
222C4C4×D50811yescharacteristic
223C4#1686117911yescharacteristic
224C4#1686119411yescharacteristic
225C2—21no—
226C2—21no—
227C2—21no—
228C2—10162no—
229C2—5082no—
230C2#2254975411yescharacteristic
subgroupG/H
centerC2#22549754
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingH(3,ℤ/4)×C127C2
socleC2×C127#386
prime radical#22550445C2
cosocleC22×C4×C127C23
hypercenterH(3,ℤ/4)D127
hypocenterC127#1910
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
1. #22550642C2semidirect
3. #22549882C2semidirect
4. #22549880C2semidirect
5. #22549849C2semidirect
6. #22549632C2semidirect
7. H(3,ℤ/4)×C127C2semidirect
14. #16861206C4semidirect
15. #16861197C4semidirect
44. C2×C127⋊C8D4semidirect
53. C2×C8×C127D4semidirect
116. C127#1910semidirect
123. H(3,ℤ/4)D127semidirect
165. C2×C8D508semidirect

Maximal subgroups

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

subgroupindex
#225506422
#225504452
#225498822
#225498802
#225498492
#225496322
H(3,ℤ/4)×C1272
#1910127