Finite Group Databasebeta

← #30639274

Subgroups — #30639274

Order 16256.

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

normal subgroups only

This group has 340 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#22550443C211yesmaximal, characteristic
3#22549858C211yesmaximal, characteristic, complemented
4C2×#16861126C211yesmaximal, characteristic, complemented
5#22549648C211yesmaximal, characteristic, complemented
6#22549647C211yesmaximal, characteristic, complemented
7C2×#100×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861217—22no—
10#16861215—22no—
11C2×#14523232C2211yescharacteristic
12C2×#14523232C2211yescharacteristic
13M16×D127—22no—
14#16861205—21no—
15#16861196—21no—
16#16861180—22no—
17#16861178C414yes—
18#16861168C2211yescharacteristic
19#16861127—21no—
20#16861126—22no—
21#16861125—21no—
22C22×D4×C127C2211yescharacteristic
23C2×M16×C127C2211yescharacteristic
24C2×M16×C127C2211yescharacteristic
25#100×C127—22no—
26#14523248—21no—
27#14523248—21no—
28#14523248—21no—
29#14523248—21no—
30C2×Dic254D411yescharacteristic
31C2×D508D411yescharacteristic
32C2×C22⋊D127—24no—
33C2×C22⋊D127C2×C412yes—
34C2×C4×D127D412yes—
35#14523232—21no—
36#14523232—21no—
37#14523232—21no—
38#14523232D412yescomplemented
39#14523231—24no—
40C2×C127⋊C8—21no—
41C2×C127⋊C8—21no—
42C8×D127—24no—
43C4⋊Dic127—21no—
44C22⋊Dic127—24no—
45C22⋊Dic127C2×C412yes—
46C4×Dic127—21no—
47C24×C127C2×C412yes—
48C2×D4×C127—21no—
49C2×D4×C127—21no—
50C2×D4×C127—21no—
51C2×D4×C127—21no—
52C2×D4×C127—21no—
53C2×D4×C127—21no—
54C22×C4×C127C2311yescharacteristic
55M16×C127—21no—
56M16×C127—21no—
57M16×C127—21no—
58M16×C127D412yescomplemented
59C2×C8×C127—21no—
60C2×C8×C127—21no—
61C22×D127C22⋊C412yes—
62Dic254—21no—
63Dic254—21no—
64D508—21no—
65D508—21no—
66C22⋊D127—22no—
67C22⋊D127—22no—
68C22⋊D127—84no—
69C2×Dic127—22no—
70C2×Dic127C22⋊C412yes—
71C4×D127—22no—
72C4×D127—22no—
73C127⋊C8—22no—
74C127⋊C8—41no—
75C23×C127—22no—
76C23×C127—22no—
77C23×C127—22no—
78C23×C127—82no—
79C23×C127C22×C411yescharacteristic
80D4×C127—41no—
81D4×C127—41no—
82D4×C127—41no—
83D4×C127—41no—
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127#4111yescharacteristic
89C2×C4×C127#4111yescharacteristic
90C8×C127—22no—
91C8×C127—41no—
92C22×C127—41no—
93C22×C127—44no—
94C22×C127—42no—
95C22×C127—82no—
96C22×C127—82no—
97C22×C127C2×C22⋊C411yescharacteristic
98C22×C127C4×D411yescharacteristic
99C22×C127C4×D411yescharacteristic
100C2×D127—22no—
101C2×D127—42no—
102C4×C127#12011yescharacteristic
103C4×C127#12111yescharacteristic
104C4×C127#12111yescharacteristic
105C4×C127#12311yescharacteristic
106Dic127—22no—
107Dic127—42no—
108C2×C127—21no—
109C2×C127—21no—
110C2×C127—21no—
111C2×C127—82no—
112C2×C127#38611yescharacteristic
113D127—42no—
114#1914—1271nomaximal, Sylow
115C127#191411yescharacteristic, complemented, Sylow
116#568—1271no—
117#535—1271no—
118#413—1271no—
119C2×#100—1271no—
120C2×#100D12711yescharacteristic, complemented
121#344—1271no—
122#344—1271no—
123C2×#41—1271no—
124C22×D4C2×D12711yescharacteristic
125#131—2542no—
126#131—2542no—
127C2×M16—1271no—
128C2×M16—1271no—
129C2×M16—2542no—
130C2×M16C2×D12711yescharacteristic
131C2×M16C2×D12711yescharacteristic
132#127—2541no—
133#124—2541no—
134#121—2542no—
135#120—1274no—
136#117—1271no—
137#101—2541no—
138#100—22no—
139#100—2541no—
140#100—2542no—
141C24C4×D12712yes—
142#41—2541no—
143#41—2541no—
144#41—2541no—
145#41—2541no—
146C2×Q8—1271no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—1272no—
154C2×D4—1271no—
155C2×D4—2544no—
156C22×C4—1272no—
157C22×C4C22×D12711yescharacteristic
158M16—21no—
159M16—21no—
160M16—21no—
161M16—1272no—
162M16—2544no—
163M16—2541no—
164M16—2541no—
165M16—2541no—
166M16C22⋊D12712yescomplemented
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—2544no—
170C2×C8—2541no—
171C2×C8—2541no—
172C4⋊C4—2541no—
173C22⋊C4—1272no—
174C22⋊C4—2544no—
175C42—2541no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—82no—
180C23—1272no—
181C23C2×C4×D12711yescharacteristic
182Q8—2541no—
183Q8—2541no—
184D4—41no—
185D4—41no—
186D4—41no—
187D4—41no—
188D4—10164no—
189D4—2542no—
190D4—2542no—
191D4—2541no—
192D4—2541no—
193C2×C4—1272no—
194C2×C4—2542no—
195C2×C4—2542no—
196C2×C4—2542no—
197C2×C4D4×D12711yescharacteristic
198C2×C4D4×D12711yescharacteristic
199C2×C4D4×D12711yescharacteristic
200C2×C4C2×C22⋊D12711yescharacteristic
201C2×C4#1452324711yescharacteristic
202C2×C4#1452324811yescharacteristic
203C8—22no—
204C8—41no—
205C8—5081no—
206C8—2542no—
207C22—41no—
208C22—44no—
209C22—42no—
210C22—82no—
211C22—82no—
212C22—5082no—
213C22—2542no—
214C22C22⋊C4×D12711yescharacteristic
215C22(C22⋊D127)⋊C411yescharacteristic
216C22C4×C22⋊D12711yescharacteristic
217C4—5082no—
218C4—2542no—
219C4#1686117711yescharacteristic
220C4#1686118011yescharacteristic
221C4#1686118211yescharacteristic
222C4#1686119111yescharacteristic
223C2—21no—
224C2—21no—
225C2—21no—
226C2—82no—
227C2—5082no—
228C2#2254975211yescharacteristic
subgroupG/H
centerC2#22549752
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#100×C127C2
socleC2×C127#386
prime radical#22550443C2
cosocleC22×C4×C127C23
hypercenterC2×#100D127
hypocenterC127#1914
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
1. #22550642C2semidirect
3. #22549858C2semidirect
4. C2×#16861126C2semidirect
5. #22549648C2semidirect
6. #22549647C2semidirect
7. C2×#100×C127C2semidirect
38. #14523232D4semidirect
58. M16×C127D4semidirect
115. C127#1914semidirect
120. C2×#100D127semidirect
166. M16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225506422
#225504432
#225498582
C2×#168611262
#225496482
#225496472
C2×#100×C1272
#1914127