Finite Group Databasebeta

← #30638762

Subgroups — #30638762

Order 16256.

316 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#14523231C211yesmaximal, characteristic
2C2×#16861142C211yesmaximal, characteristic, complemented
3C2×#16861140C211yesmaximal, characteristic, complemented
4#22549733C211yesmaximal, characteristic
5#22549733C211yesmaximal, characteristic
6#22549639C211yesmaximal, characteristic, complemented
7#340×C127C211yesmaximal, characteristic, complemented
8C22×C4×D127C2211yescharacteristic
9C2×#14523231—22no—
10C2×#14523231—22no—
11C2×#14523231C414yescomplemented
12C22×C127⋊C8C2211yescharacteristic
13C2×C4⋊Dic127C2211yescharacteristic
14C2×C4⋊Dic127C2211yescharacteristic
15C2×(C2×D127)⋊C4—21no—
16C2×(C2×D127)⋊C4—21no—
17#16861142—22no—
18#16861140—22no—
19C127⋊#95—21no—
20C127⋊#95—21no—
21C22×C8×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23C2×C4⋊C4×C127C2211yescharacteristic
24C23×D127D411yescharacteristic
25C22×Dic127—21no—
26C22×Dic127—21no—
27C22×Dic127D411yescharacteristic
28C2×C4×D127—21no—
29C2×C4×D127—21no—
30C2×C4×D127—21no—
31C2×C4×D127—21no—
32C2×C4×D127C2×C412yes—
33C2×C4×D127Q812yes—
34#14523231—28no—
35C2×C127⋊C8C2×C412yes—
36C2×C127⋊C8D412yes—
37C2×C127⋊C8D412yes—
38C4⋊Dic127—21no—
39C4⋊Dic127—21no—
40C4⋊Dic127—21no—
41C4⋊Dic127—21no—
42(C2×D127)⋊C4—42no—
43(C2×D127)⋊C4—42no—
44C22×C4×C127—21no—
45C22×C4×C127—21no—
46C22×C4×C127C2311yescharacteristic
47C2×C8×C127—21no—
48C2×C8×C127—21no—
49C2×C8×C127C2×C412yes—
50C4⋊C4×C127—21no—
51C4⋊C4×C127—21no—
52C4⋊C4×C127—21no—
53C4⋊C4×C127—21no—
54C22×D127—21no—
55C22×D127—21no—
56C22×D127—21no—
57C22×D127—21no—
58C22×D127—21no—
59C22×D127—21no—
60C22×D127C4⋊C412yes—
61C2×Dic127—21no—
62C2×Dic127—21no—
63C2×Dic127—41no—
64C2×Dic127—41no—
65C2×Dic127—41no—
66C2×Dic127—41no—
67C2×Dic127—41no—
68C2×Dic127—41no—
69C2×Dic127C4⋊C412yes—
70C4×D127—22no—
71C4×D127—22no—
72C4×D127—22no—
73C4×D127—22no—
74C127⋊C8C22⋊C414yescomplemented
75C23×C127C2×D411yescharacteristic
76C2×C4×C127—41no—
77C2×C4×C127—41no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127—41no—
81C2×C4×C127—41no—
82C2×C4×C127C22×C411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×Q811yescharacteristic
86C2×C4×C127#4111yescharacteristic
87C2×C4×C127#4111yescharacteristic
88C8×C127—22no—
89C22×C127C2×C4⋊C411yescharacteristic
90C22×C127#12111yescharacteristic
91C22×C127#12111yescharacteristic
92C22×C127AGL(1,ℤ/8)11yescharacteristic
93C22×C127AGL(1,ℤ/8)11yescharacteristic
94C22×C127#13711yescharacteristic
95C22×C127#13711yescharacteristic
96C2×D127—22no—
97C2×D127—22no—
98C2×D127—22no—
99C2×D127—41no—
100C2×D127—41no—
101C2×D127—41no—
102C2×D127—41no—
103C4×C127—81no—
104C4×C127—81no—
105C4×C127C2×C22⋊C411yescharacteristic
106C4×C127#11711yescharacteristic
107C4×C127C22⋊Q811yescharacteristic
108C4×C127C22⋊Q811yescharacteristic
109Dic127—22no—
110Dic127—81no—
111Dic127—81no—
112C2×C127#38011yescharacteristic
113C2×C127#42811yescharacteristic
114C2×C127#42811yescharacteristic
115C2×C127#46811yescharacteristic
116C2×C127#46811yescharacteristic
117C2×C127#46911yescharacteristic
118C2×C127#47011yescharacteristic
119D127—42no—
120#1809—1271nomaximal, Sylow
121C127#180911yescharacteristic, complemented, Sylow
122C22×M16—1271no—
123C2×#107—1271no—
124C2×#106—1271no—
125#380—1271no—
126#380—1271no—
127#340—1271no—
128#340D12711yescharacteristic, complemented
129C23×C4—1271no—
130C2×M16—1274no—
131C2×M16—2542no—
132C2×M16—2542no—
133C22×C8—1271no—
134C22×C8C2×D12711yescharacteristic
135C2×C4⋊C4—1271no—
136C2×C4⋊C4—1271no—
137C2×C4⋊C4C2×D12711yescharacteristic
138C2×C4⋊C4C2×D12711yescharacteristic
139C2×C22⋊C4—2541no—
140C2×C22⋊C4—2541no—
141#107—2542no—
142#106—2542no—
143#95—2541no—
144#95—2541no—
145C24—1271no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—1271no—
149C22×C4—1272no—
150C22×C4—1272no—
151C22×C4—2541no—
152C22×C4—2541no—
153C22×C4—2541no—
154C22×C4—2541no—
155C22×C4—2541no—
156C22×C4—2541no—
157C22×C4C22×D12711yescharacteristic
158M16—2548no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—1272no—
162C2×C8—1272no—
163C2×C8—1272no—
164C2×C8C4×D12712yes—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—2541no—
170C4⋊C4—2541no—
171C4⋊C4—2541no—
172C4⋊C4—2541no—
173C22⋊C4—5082no—
174C22⋊C4—5082no—
175C23—1272no—
176C23—2541no—
177C23—2541no—
178C23—2541no—
179C23—2541no—
180C23—2541no—
181C23—2541no—
182C23D4×D12711yescharacteristic
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—1272no—
190C2×C4—5081no—
191C2×C4—5081no—
192C2×C4—5081no—
193C2×C4—5081no—
194C2×C4—5081no—
195C2×C4—5081no—
196C2×C4—2541no—
197C2×C4—2541no—
198C2×C4—2542no—
199C2×C4—2542no—
200C2×C4—2542no—
201C2×C4—2542no—
202C2×C4C2×C4×D12711yescharacteristic
203C2×C4D4×D12711yescharacteristic
204C2×C4D4×D12711yescharacteristic
205C2×C4Q8×D12711yescharacteristic
206C2×C4#1452324811yescharacteristic
207C2×C4#1452324811yescharacteristic
208C8—22no—
209C8—1274no—
210C22—5081no—
211C22—5081no—
212C22—5081no—
213C22—5081no—
214C22—2542no—
215C22—2542no—
216C22—2542no—
217C22C4⋊C4×D12711yescharacteristic
218C22#1686118211yescharacteristic
219C22#1686118211yescharacteristic
220C22#1686123311yescharacteristic
221C22#1686123611yescharacteristic
222C22#1686123811yescharacteristic
223C22#1686124111yescharacteristic
224C4—81no—
225C4—81no—
226C4—10161no—
227C4—10161no—
228C4—2542no—
229C4C22⋊C4×D12711yescharacteristic
230C4#1686116611yescharacteristic
231C4#1686118711yescharacteristic
232C4#1686118711yescharacteristic
233C2—5082no—
234C2#2254973511yescharacteristic
235C2#2254991511yescharacteristic
236C2#2254991711yescharacteristic
237C2#2255012111yescharacteristic
238C2#2255012511yescharacteristic
239C2#2255012811yescharacteristic
240C2#2255013311yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#340×C127C2
socleC23×C127C2×D4
prime radicalC23×D127D4
cosocleC22×C4×C127C23
hypercenter#340D127
hypocenterC127#1809
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#16861142C2semidirect
3. C2×#16861140C2semidirect
6. #22549639C2semidirect
7. #340×C127C2semidirect
11. C2×#14523231C4semidirect
74. C127⋊C8C22⋊C4semidirect
121. C127#1809semidirect
128. #340D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232312
C2×#168611422
C2×#168611402
#225497332
#225496392
#340×C1272
#1809127