Finite Group Databasebeta

← #30639245

Subgroups — #30639245

Order 16256.

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

normal subgroups only

This group has 312 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
1C2×#16861239C211yesmaximal, characteristic
2#22549877C211yesmaximal, characteristic
3C2×#16861134C211yesmaximal, characteristic, complemented
4C2×#16861127C211yesmaximal, characteristic, complemented
5#22549644C211yesmaximal, characteristic, complemented
6#22549641C211yesmaximal, characteristic, complemented
7C2×#131×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9#16861239—22no—
10#16861239C414yescomplemented
11C2×#14523238—21no—
12C2×#14523236—21no—
13C2×#14523232C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15C2×C4⋊Dic127C2211yescharacteristic
16#16861134—21no—
17#16861134—21no—
18#16861133—21no—
19#16861130—21no—
20#16861127—22no—
21C2×#41×C127C2211yescharacteristic
22#131×C127—22no—
23#131×C127—22no—
24C2×M16×C127—21no—
25C2×M16×C127C2211yescharacteristic
26C22×C8×C127—21no—
27C22×C8×C127C2211yescharacteristic
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254—21no—
32C2×Dic254—21no—
33C2×Dic254C2×C412yes—
34C22×Dic127—21no—
35C22×Dic127—21no—
36#14523238—24no—
37#14523236—24no—
38#14523232—21no—
39#14523232—21no—
40#14523232—21no—
41#14523232C2×C412yes—
42C2×C127⋊C8—21no—
43C2×C127⋊C8—21no—
44C4⋊Dic127—21no—
45C4⋊Dic127—21no—
46#41×C127—21no—
47#41×C127—21no—
48#41×C127C2×C412yes—
49#41×C127D412yes—
50C2×Q8×C127D411yescharacteristic
51C2×D4×C127—21no—
52C2×D4×C127D411yescharacteristic
53C22×C4×C127—21no—
54C22×C4×C127C2311yescharacteristic
55M16×C127—24no—
56M16×C127—21no—
57M16×C127D412yes—
58C2×C8×C127—24no—
59C2×C8×C127—21no—
60C2×C8×C127—21no—
61C2×C8×C127—21no—
62C2×C8×C127D411yescharacteristic
63C2×C8×C127D411yescharacteristic
64Dic254—22no—
65Dic254—22no—
66Dic254—41no—
67Dic254—41no—
68C2×Dic127—22no—
69C2×Dic127—41no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C2×Dic127—81no—
73C127⋊C8—22no—
74C127⋊C8—41no—
75C23×C127—21no—
76C23×C127#4111yescharacteristic
77Q8×C127—21no—
78Q8×C127C22⋊C412yes—
79D4×C127—22no—
80D4×C127—22no—
81D4×C127—21no—
82D4×C127C22⋊C412yes—
83C2×C4×C127—21no—
84C2×C4×C127—22no—
85C2×C4×C127—22no—
86C2×C4×C127C22×C411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2×D411yescharacteristic
89C2×C4×C127C2×D411yescharacteristic
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127#4111yescharacteristic
92C8×C127—21no—
93C8×C127—22no—
94C8×C127—21no—
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×C127C2×C22⋊C411yescharacteristic
105C4×C127C4×D411yescharacteristic
106C4×C127#12011yescharacteristic
107C4×C127#12111yescharacteristic
108Dic127—42no—
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×#137—1271no—
119C2×#131D12711yescharacteristic
120#417—1271no—
121C2×#103—1271no—
122C2×#101—1271no—
123#343—1271no—
124#341—1271no—
125C2×#41C2×D12711yescharacteristic
126C22×Q8—1271no—
127#137—1274no—
128#137—2542no—
129C2×Q16—2541no—
130C2×SD16—2541no—
131#131—22no—
132#131—22no—
133C2×M16—21no—
134C2×M16—1271no—
135C2×M16—1271no—
136C2×M16C2×D12711yescharacteristic
137C22×C8—21no—
138C22×C8C2×D12711yescharacteristic
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#41C4×D12712yes—
148#41C22⋊D12712yes—
149C2×Q8—1272no—
150C2×Q8—2541no—
151C2×Q8—2541no—
152C2×Q8—2541no—
153C2×Q8—2541no—
154C2×Q8—2541no—
155C2×Q8D50811yescharacteristic
156C2×D4—21no—
157C2×D4D50811yescharacteristic
158C22×C4—21no—
159C22×C4—2541no—
160C22×C4—2541no—
161C22×C4C22×D12711yescharacteristic
162Q16—2544no—
163SD16—2544no—
164M16—24no—
165M16—21no—
166M16—1272no—
167M16—2541no—
168M16—2541no—
169M16—2541no—
170M16C22⋊D12712yes—
171C2×C8—24no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—2541no—
176C2×C8—2541no—
177C2×C8C22⋊D12711yescharacteristic
178C2×C8C22⋊D12711yescharacteristic
179C4⋊C4—2541no—
180C4⋊C4—2541no—
181C23—21no—
182C23#1452324811yescharacteristic
183Q8—21no—
184Q8—5081no—
185Q8—5081no—
186Q8—2542no—
187Q8—2542no—
188Q8(C2×D127)⋊C412yes—
189D4—22no—
190D4—22no—
191D4—21no—
192D4(C2×D127)⋊C412yes—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—10161no—
197C2×C4—5081no—
198C2×C4—5081no—
199C2×C4—5081no—
200C2×C4—2542no—
201C2×C4C2×C4×D12711yescharacteristic
202C2×C4C2×C22⋊D12711yescharacteristic
203C2×C4C2×C22⋊D12711yescharacteristic
204C2×C4C2×C22⋊D12711yescharacteristic
205C2×C4C2×D50811yescharacteristic
206C2×C4#1452324811yescharacteristic
207C8—21no—
208C8—22no—
209C8—21no—
210C8—5081no—
211C8—2542no—
212C22—21no—
213C22—21no—
214C22—22no—
215C22—41no—
216C22—41no—
217C22C4×C22⋊D12711yescharacteristic
218C22#1686118311yescharacteristic
219C22#1686119211yescharacteristic
220C4—22no—
221C4—10161no—
222C4—5082no—
223C4C2×(C2×D127)⋊C411yescharacteristic
224C4C4×C22⋊D12711yescharacteristic
225C4#1686117811yescharacteristic
226C4#1686118311yescharacteristic
227C2—21no—
228C2—21no—
229C2—42no—
230C2#2254975511yescharacteristic
231C2#2255013911yescharacteristic
232C2#2255015111yescharacteristic
subgroupG/H
centerC22#16861183
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#131×C127C2
socleC22×C127#121
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#131D127
hypocenterC127#1908
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
3. C2×#16861134C2semidirect
4. C2×#16861127C2semidirect
5. #22549644C2semidirect
6. #22549641C2semidirect
10. #16861239C4semidirect
117. C127#1908semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
#225498772
C2×#168611342
C2×#168611272
#225496442
#225496412
C2×#131×C1272
#1908127