Finite Group Databasebeta

← #30639247

Subgroups — #30639247

Order 16256.

312 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16861216C211yesmaximal, characteristic
2#22549877C211yesmaximal, characteristic
3C2×#16861133C211yesmaximal, characteristic, complemented
4C2×#16861128C211yesmaximal, characteristic, complemented
5#22549644C211yesmaximal, characteristic, complemented
6#22549640C211yesmaximal, characteristic, complemented
7C2×#137×C127C211yesmaximal, characteristic
8#16861216—22no—
9#16861216—22no—
10C2×#14523232—21no—
11C2×#14523232C2211yescharacteristic
12C2×#14523232C2211yescharacteristic
13C22×C127⋊C8—21no—
14C22×C127⋊C8C2211yescharacteristic
15C2×C4⋊Dic127C2211yescharacteristic
16#16861133—21no—
17#16861133—21no—
18#16861133—21no—
19#16861130—21no—
20#16861128—22no—
21C2×#41×C127C2211yescharacteristic
22C22×Q8×C127C2211yescharacteristic
23#137×C127—22no—
24#137×C127C414yescomplemented
25C2×Q16×C127—21no—
26C2×SD16×C127—21no—
27C2×M16×C127C2211yescharacteristic
28C22×Dic127—21no—
29#14523232—24no—
30#14523232—21no—
31#14523232—21no—
32#14523232—21no—
33#14523232D412yes—
34C2×C127⋊C8—24no—
35C2×C127⋊C8—21no—
36C2×C127⋊C8—21no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8D411yescharacteristic
40C2×C127⋊C8D411yescharacteristic
41C4⋊Dic127—21no—
42C4⋊Dic127—21no—
43#41×C127—21no—
44#41×C127—21no—
45#41×C127C2×C412yes—
46#41×C127D412yes—
47C2×Q8×C127—21no—
48C2×Q8×C127—21no—
49C2×Q8×C127—21no—
50C2×Q8×C127—21no—
51C2×Q8×C127—21no—
52C2×Q8×C127C2×C412yes—
53C2×Q8×C127D411yescharacteristic
54C2×D4×C127—21no—
55C2×D4×C127D411yescharacteristic
56C22×C4×C127—21no—
57C22×C4×C127—21no—
58C22×C4×C127C2311yescharacteristic
59Q16×C127—24no—
60SD16×C127—24no—
61M16×C127—21no—
62M16×C127C2×C412yes—
63C2×C8×C127—21no—
64C2×Dic127—41no—
65C2×Dic127—81no—
66C127⋊C8—21no—
67C127⋊C8—22no—
68C127⋊C8—21no—
69C127⋊C8—41no—
70C23×C127—21no—
71C23×C127#4111yescharacteristic
72Q8×C127—21no—
73Q8×C127—22no—
74Q8×C127—22no—
75Q8×C127—41no—
76Q8×C127—41no—
77Q8×C127C22⋊C412yes—
78D4×C127—22no—
79D4×C127—22no—
80D4×C127—21no—
81D4×C127C22⋊C412yes—
82C2×C4×C127—21no—
83C2×C4×C127—22no—
84C2×C4×C127—22no—
85C2×C4×C127—22no—
86C2×C4×C127—41no—
87C2×C4×C127—41no—
88C2×C4×C127C22×C411yescharacteristic
89C2×C4×C127C2×D411yescharacteristic
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127#4111yescharacteristic
94C8×C127—22no—
95C22×C127—21no—
96C22×C127—21no—
97C22×C127—22no—
98C22×C127—41no—
99C22×C127—41no—
100C22×C127C4×D411yescharacteristic
101C22×C127#12111yescharacteristic
102C22×C127#12311yescharacteristic
103C4×C127—22no—
104C4×C127—42no—
105C4×C127C2×C22⋊C411yescharacteristic
106C4×C127C4×D411yescharacteristic
107C4×C127#12011yescharacteristic
108C4×C127#12111yescharacteristic
109Dic127—81no—
110C2×C127—21no—
111C2×C127—21no—
112C2×C127—42no—
113C2×C127#38611yescharacteristic
114C2×C127#47111yescharacteristic
115C2×C127#47311yescharacteristic
116#1908—1271nomaximal, Sylow
117C127#190811yescharacteristic, complemented, Sylow
118C2×#137D12711yescharacteristic
119C2×#131—1271no—
120#417—1271no—
121C2×#103—1271no—
122C2×#101—1271no—
123#343—1271no—
124#341—1271no—
125C2×#41C2×D12711yescharacteristic
126C22×Q8C2×D12711yescharacteristic
127#137—22no—
128#137Dic12714yescomplemented
129C2×Q16—21no—
130C2×SD16—21no—
131#131—2542no—
132#131—2542no—
133C2×M16—1271no—
134C2×M16—1271no—
135C2×M16—2541no—
136C2×M16C2×D12711yescharacteristic
137C22×C8—1271no—
138C22×C8—2541no—
139C2×C4⋊C4—1271no—
140#103—2541no—
141#103—2541no—
142#103—2541no—
143#102—2541no—
144#101—2542no—
145#41—21no—
146#41—21no—
147#41C2×Dic12712yes—
148#41C22⋊D12712yes—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8C2×Dic12712yes—
155C2×Q8C22⋊D12711yescharacteristic
156C2×D4—21no—
157C2×D4C22⋊D12711yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—2541no—
161C22×C4C22×D12711yescharacteristic
162Q16—24no—
163SD16—24no—
164M16—21no—
165M16—1272no—
166M16—2541no—
167M16—2541no—
168M16—2544no—
169M16—2541no—
170M16C2×Dic12712yes—
171C2×C8—21no—
172C2×C8—1271no—
173C2×C8—1271no—
174C2×C8—2541no—
175C2×C8—2544no—
176C2×C8—2541no—
177C2×C8—2541no—
178C2×C8—2541no—
179C4⋊C4—2541no—
180C4⋊C4—2541no—
181C23—21no—
182C23#1452324711yescharacteristic
183Q8—21no—
184Q8—22no—
185Q8—22no—
186Q8—41no—
187Q8—41no—
188Q8C22⋊Dic12712yes—
189D4—22no—
190D4—22no—
191D4—21no—
192D4C22⋊Dic12712yes—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—10161no—
200C2×C4—5081no—
201C2×C4C22×Dic12711yescharacteristic
202C2×C4D4×D12711yescharacteristic
203C2×C4D4×D12711yescharacteristic
204C2×C4C2×C22⋊D12711yescharacteristic
205C2×C4C2×C22⋊D12711yescharacteristic
206C2×C4#1452324711yescharacteristic
207C8—22no—
208C8—5081no—
209C8—2541no—
210C8—2541no—
211C8—2542no—
212C22—21no—
213C22—21no—
214C22—22no—
215C22—41no—
216C22—41no—
217C22D4×Dic12711yescharacteristic
218C22#1686118111yescharacteristic
219C22#1686119011yescharacteristic
220C4—22no—
221C4—42no—
222C4—10161no—
223C4C2×C22⋊Dic12711yescharacteristic
224C4D4×Dic12711yescharacteristic
225C4#1686117711yescharacteristic
226C4#1686118111yescharacteristic
227C2—21no—
228C2—21no—
229C2—42no—
230C2#2254975611yescharacteristic
231C2#2255014111yescharacteristic
232C2#2255014911yescharacteristic
subgroupG/H
centerC22#16861181
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#137×C127C2
socleC22×C127#121
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#137D127
hypocenterC127#1908
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
3. C2×#16861133C2semidirect
4. C2×#16861128C2semidirect
5. #22549644C2semidirect
6. #22549640C2semidirect
24. #137×C127C4semidirect
117. C127#1908semidirect
128. #137Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612162
#225498772
C2×#168611332
C2×#168611282
#225496442
#225496402
C2×#137×C1272
#1908127