Finite Group Databasebeta

← #30639585

Subgroups — #30639585

Order 16256.

348 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 103 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×#14523236C211yesmaximal, characteristic
2C2×#16861140C211yesmaximal, characteristic, complemented
3C2×#16861134C211yesmaximal, characteristic, complemented
4C2×C8×Dic127C211yesmaximal, characteristic, complemented
5#22549781C211yesmaximal, characteristic, complemented
6#22549779C211yesmaximal, characteristic
7C2×#102×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523236—22no—
10C2×#14523236—22no—
11C2×#14523236C414yescomplemented
12C22×C127⋊C8C2211yescharacteristic
13C2×C4⋊Dic127C2211yescharacteristic
14C2×C22⋊Dic127—22no—
15C2×C4×Dic127C2211yescharacteristic
16#16861140—22no—
17#16861134—22no—
18C8×Dic127—22no—
19C127⋊#95—22no—
20C22×D4×C127C2211yescharacteristic
21C22×C8×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23#102×C127—22no—
24C2×Dic254—21no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254C2×C412yes—
30C22×Dic127—21no—
31C22×Dic127—21no—
32C22×Dic127D412yes—
33#14523236—28no—
34C2×C127⋊C8C2×C412yes—
35C2×C127⋊C8D412yes—
36C2×C127⋊C8D412yes—
37C4⋊Dic127—21no—
38C4⋊Dic127—21no—
39C22⋊Dic127—44no—
40C4×Dic127—21no—
41C4×Dic127—21no—
42C24×C127—21no—
43C2×D4×C127—21no—
44C2×D4×C127—21no—
45C2×D4×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127—21no—
48C2×D4×C127C2×C412yes—
49C22×C4×C127—21no—
50C22×C4×C127C2311yescharacteristic
51C2×C8×C127—21no—
52C2×C8×C127—21no—
53C2×C8×C127D412yes—
54C4⋊C4×C127—21no—
55C4⋊C4×C127—21no—
56Dic254—22no—
57Dic254—22no—
58Dic254—41no—
59Dic254—41no—
60C2×Dic127—22no—
61C2×Dic127—22no—
62C2×Dic127—22no—
63C2×Dic127—41no—
64C2×Dic127—41no—
65C2×Dic127—41no—
66C2×Dic127—42no—
67C2×Dic127SD1614yescomplemented
68C127⋊C8C22⋊C414yescomplemented
69C23×C127—22no—
70C23×C127—41no—
71C23×C127—41no—
72C23×C127—42no—
73C23×C127—42no—
74C23×C127C2×D411yescharacteristic
75D4×C127—22no—
76D4×C127—22no—
77D4×C127—41no—
78D4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127—42no—
81C2×C4×C127C22×C411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127#4111yescharacteristic
86C2×C4×C127#4111yescharacteristic
87C8×C127—22no—
88C22×C127—42no—
89C22×C127—44no—
90C22×C127—82no—
91C22×C127—81no—
92C22×C127—81no—
93C22×C127C4×D411yescharacteristic
94C22×C127#12111yescharacteristic
95C22×C127#12711yescharacteristic
96C22×C127C2×SD1612yes—
97C22×C127#13512yes—
98C4×C127—81no—
99C4×C127C2×C22⋊C411yescharacteristic
100C4×C127C4×D411yescharacteristic
101C4×C127#12111yescharacteristic
102C4×C127#12411yescharacteristic
103Dic127—24no—
104Dic127—42no—
105Dic127—81no—
106C2×C127—82no—
107C2×C127#39011yescharacteristic
108C2×C127C4×SD1612yes—
109C2×C127#46512yes—
110C2×C127#49211yescharacteristic
111C2×C127#49511yescharacteristic
112#1968—1271nomaximal, Sylow
113C127#196811yescharacteristic, complemented, Sylow
114C22×SD16—1271no—
115C2×#106—1271no—
116C2×#103—1271no—
117C2×#102D12711yescharacteristic
118C2×C4×C8—1271no—
119#391—1271no—
120#390—1271no—
121C22×Q8—1271no—
122C22×D4C2×D12711yescharacteristic
123C2×SD16—1274no—
124C2×SD16—2542no—
125C2×SD16—2542no—
126C22×C8—1271no—
127C22×C8C2×D12711yescharacteristic
128C2×C4⋊C4—1271no—
129C2×C4⋊C4C2×D12711yescharacteristic
130C2×C22⋊C4—2542no—
131C2×C42—1271no—
132#106—2542no—
133#103—2542no—
134#102—22no—
135C4×C8—2542no—
136#95—2542no—
137C24—21no—
138C2×Q8—1272no—
139C2×Q8—2541no—
140C2×Q8—2541no—
141C2×Q8—2541no—
142C2×Q8—2541no—
143C2×Q8—2541no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4C4×D12712yes—
150C22×C4—21no—
151C22×C4—1272no—
152C22×C4—2541no—
153C22×C4—2541no—
154C22×C4C22×D12711yescharacteristic
155SD16—2548no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—1272no—
159C2×C8—1272no—
160C2×C8—1272no—
161C2×C8C22⋊D12712yes—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—2541no—
165C4⋊C4—2541no—
166C22⋊C4—5084no—
167C42—2541no—
168C42—2541no—
169C23—22no—
170C23—41no—
171C23—42no—
172C23—42no—
173C23—41no—
174C23D4×D12711yescharacteristic
175Q8—5081no—
176Q8—5081no—
177Q8—2542no—
178Q8—2542no—
179D4—22no—
180D4—22no—
181D4—41no—
182D4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—1274no—
186C2×C4—5081no—
187C2×C4—5082no—
188C2×C4—5081no—
189C2×C4—5081no—
190C2×C4—2542no—
191C2×C4—2542no—
192C2×C4—2542no—
193C2×C4C2×C4×D12711yescharacteristic
194C2×C4D4×D12711yescharacteristic
195C2×C4D4×D12711yescharacteristic
196C2×C4C2×C22⋊D12711yescharacteristic
197C2×C4#1452324711yescharacteristic
198C2×C4#1452324811yescharacteristic
199C8—22no—
200C8—1274no—
201C22—42no—
202C22—44no—
203C22—82no—
204C22—81no—
205C22—81no—
206C22(C22⋊D127)⋊C411yescharacteristic
207C22#1686118211yescharacteristic
208C22#1686120511yescharacteristic
209C22SD16×D12712yes—
210C22#1686122812yes—
211C4—81no—
212C4—10161no—
213C4—5082no—
214C4—2544no—
215C4C22⋊C4×D12711yescharacteristic
216C4C4×C22⋊D12711yescharacteristic
217C4#1686118111yescharacteristic
218C4#1686119511yescharacteristic
219C2—82no—
220C2#2254977611yescharacteristic
221C2#2254996712yes—
222C2#2255010412yes—
223C2#2255024711yescharacteristic
224C2#2255025911yescharacteristic
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#1968
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×#16861140C2semidirect
3. C2×#16861134C2semidirect
4. C2×C8×Dic127C2semidirect
5. #22549781C2semidirect
11. C2×#14523236C4semidirect
67. C2×Dic127SD16semidirect
68. C127⋊C8C22⋊C4semidirect
113. C127#1968semidirect

Maximal subgroups

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

subgroupindex
C22×#145232362
C2×#168611402
C2×#168611342
C2×C8×Dic1272
#225497812
#225497792
C2×#102×C1272
#1968127