Finite Group Databasebeta

← #30638738

Subgroups — #30638738

Order 16256.

356 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 109 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×D4×Dic127C211yesmaximal, characteristic
2C2×Dic127⋊C8C211yesmaximal, characteristic, complemented
3C2×#16861130C211yesmaximal, characteristic
4#22549741C211yesmaximal, characteristic, complemented
5#22549639C211yesmaximal, characteristic, complemented
6#22549638C211yesmaximal, characteristic, complemented
7C2×#102×C127C211yesmaximal, characteristic
8C23×Dic127—21no—
9C22×C127⋊C8C2211yescharacteristic
10D4×Dic127—22no—
11D4×Dic127C414yescomplemented
12C2×C4⋊Dic127C2211yescharacteristic
13C2×C4⋊Dic127C2211yescharacteristic
14C2×C22⋊Dic127—21no—
15C2×C4×Dic127C2211yescharacteristic
16Dic127⋊C8—22no—
17#16861130—22no—
18C127⋊#95—21no—
19C127⋊#95—21no—
20C22×D4×C127C2211yescharacteristic
21C22×C8×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23#102×C127—22no—
24C22×Dic127—24no—
25C22×Dic127—21no—
26C22×Dic127—21no—
27C22×Dic127—42no—
28C22×Dic127—42no—
29C22×Dic127D411yescharacteristic
30C22×Dic127D411yescharacteristic
31C2×C127⋊C8—21no—
32C2×C127⋊C8—21no—
33C2×C127⋊C8—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127C2×C412yes—
38C22⋊Dic127—24no—
39C4×Dic127C2×C412yes—
40C4×Dic127D412yes—
41C24×C127—21no—
42C2×D4×C127—21no—
43C2×D4×C127—21no—
44C2×D4×C127—21no—
45C2×D4×C127C2×C412yes—
46C2×D4×C127D412yes—
47C2×D4×C127Q812yes—
48C22×C4×C127—21no—
49C22×C4×C127C2311yescharacteristic
50C2×C8×C127—21no—
51C2×C8×C127—21no—
52C2×C8×C127—21no—
53C4⋊C4×C127—21no—
54C4⋊C4×C127—21no—
55C2×Dic127—21no—
56C2×Dic127—21no—
57C2×Dic127—21no—
58C2×Dic127—21no—
59C2×Dic127—22no—
60C2×Dic127—48no—
61C2×Dic127—41no—
62C2×Dic127—41no—
63C2×Dic127—41no—
64C2×Dic127—41no—
65C2×Dic127—41no—
66C2×Dic127C22⋊C412yes—
67C2×Dic127C22⋊C412yes—
68C127⋊C8—41no—
69C23×C127—22no—
70C23×C127—22no—
71C23×C127—22no—
72C23×C127—41no—
73C23×C127—41no—
74C23×C127—41no—
75C23×C127—41no—
76C23×C127C2×D411yescharacteristic
77D4×C127—22no—
78D4×C127—22no—
79D4×C127—22no—
80D4×C127C4⋊C414yes—
81C2×C4×C127—41no—
82C2×C4×C127—41no—
83C2×C4×C127—41no—
84C2×C4×C127C22×C411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×Q811yescharacteristic
88C2×C4×C127#4111yescharacteristic
89C2×C4×C127#4111yescharacteristic
90C8×C127—41no—
91C22×C127—24no—
92C22×C127—42no—
93C22×C127—42no—
94C22×C127—42no—
95C22×C127—42no—
96C22×C127—42no—
97C22×C127—42no—
98C22×C127C2×C22⋊C411yescharacteristic
99C22×C127#12111yescharacteristic
100C22×C127C22⋊Q811yescharacteristic
101C22×C127#13511yescharacteristic
102C22×C127#13511yescharacteristic
103C22×C127AGL(1,ℤ/8)11yescharacteristic
104C22×C127AGL(1,ℤ/8)11yescharacteristic
105C4×C127—81no—
106C4×C127C2×C4⋊C411yescharacteristic
107C4×C127#11711yescharacteristic
108C4×C127#12111yescharacteristic
109C4×C127C22⋊Q811yescharacteristic
110Dic127—22no—
111Dic127—22no—
112Dic127—42no—
113Dic127—81no—
114C2×C127—44no—
115C2×C127#38011yescharacteristic
116C2×C127#41611yescharacteristic
117C2×C127#41811yescharacteristic
118C2×C127#46311yescharacteristic
119C2×C127#46311yescharacteristic
120C2×C127#47811yescharacteristic
121C2×C127#47811yescharacteristic
122#1804—1271nomaximal, Sylow
123C127#180411yescharacteristic, complemented, Sylow
124C2×C4×D4—1271no—
125C2×C4⋊C8—1271no—
126C2×#102—1271no—
127C2×#102D12711yescharacteristic
128#382—1271no—
129#340—1271no—
130#340—1271no—
131C22×D4C2×D12711yescharacteristic
132C23×C4—2541no—
133C22×C8—1271no—
134C22×C8C2×D12711yescharacteristic
135C4×D4—1274no—
136C4×D4—2542no—
137C2×C4⋊C4—1271no—
138C2×C4⋊C4—1271no—
139C2×C4⋊C4C2×D12711yescharacteristic
140C2×C22⋊C4—2541no—
141C2×C42—1271no—
142C4⋊C8—2542no—
143#102—22no—
144#102—2542no—
145#95—2541no—
146#95—2541no—
147C24—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4C4×D12712yes—
152C2×D4C22⋊D12712yes—
153C2×D4Dic25412yes—
154C22×C4—21no—
155C22×C4—1271no—
156C22×C4—1271no—
157C22×C4—5082no—
158C22×C4—5082no—
159C22×C4—2544no—
160C22×C4—2541no—
161C22×C4—2541no—
162C22×C4C22×D12711yescharacteristic
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—2541no—
167C2×C8—2541no—
168C2×C8—2541no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—1272no—
172C4⋊C4—2541no—
173C4⋊C4—2541no—
174C4⋊C4—2541no—
175C22⋊C4—2544no—
176C42—1272no—
177C42—1272no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23D4×D12711yescharacteristic
186D4—22no—
187D4—22no—
188D4—22no—
189D4Dic127⋊C414yes—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—1272no—
194C2×C4—1272no—
195C2×C4—5081no—
196C2×C4—5081no—
197C2×C4—5081no—
198C2×C4—5081no—
199C2×C4—5081no—
200C2×C4—5088no—
201C2×C4—2542no—
202C2×C4—2541no—
203C2×C4—2541no—
204C2×C4—2541no—
205C2×C4—2541no—
206C2×C4C2×C4×D12711yescharacteristic
207C2×C4D4×D12711yescharacteristic
208C2×C4C2×C22⋊D12711yescharacteristic
209C2×C4C2×Dic25411yescharacteristic
210C2×C4#1452324711yescharacteristic
211C2×C4#1452324711yescharacteristic
212C8—41no—
213C8—5081no—
214C22—24no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—42no—
221C22C22⋊C4×D12711yescharacteristic
222C22#1686118111yescharacteristic
223C22#1686118611yescharacteristic
224C22#1686122811yescharacteristic
225C22#1686123011yescharacteristic
226C22#1686123311yescharacteristic
227C22#1686123611yescharacteristic
228C4—81no—
229C4—10161no—
230C4—5082no—
231C4—2542no—
232C4—2542no—
233C4C2×Dic127⋊C411yescharacteristic
234C4#1686116511yescharacteristic
235C4#1686118111yescharacteristic
236C4#1686118611yescharacteristic
237C2—44no—
238C2#2254973411yescharacteristic
239C2#2254986911yescharacteristic
240C2#2254987911yescharacteristic
241C2#2255008911yescharacteristic
242C2#2255009311yescharacteristic
243C2#2255017411yescharacteristic
244C2#2255017811yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#102D127
hypocenterC127#1804
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×Dic127⋊C8C2semidirect
4. #22549741C2semidirect
5. #22549639C2semidirect
6. #22549638C2semidirect
11. D4×Dic127C4semidirect
123. C127#1804semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1272
C2×Dic127⋊C82
C2×#168611302
#225497412
#225496392
#225496382
C2×#102×C1272
#1804127