Finite Group Databasebeta

← #30637197

Subgroups — #30637197

Order 16256.

246 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 65 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
1#22550423C211yesmaximal, characteristic
2#22549932C211yesmaximal, characteristic, complemented
3#22549927C211yesmaximal, characteristic, complemented
4#22549605C211yesmaximal, characteristic, complemented
5D4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6#22549603C211yesmaximal, characteristic, complemented
7D4⋊C8×C127C211yesmaximal, characteristic, complemented
8C2×#14523248—21no—
9#16861208C411yescharacteristic
10#16861205C411yescharacteristic
11#16861195—21no—
12#16861184C411yescharacteristic
13#16861182—21no—
14#16861180C411yescharacteristic
15C4×D508C2211yescharacteristic
16C4×C22⋊D127—21no—
17C42×D127C2211yescharacteristic
18C4⋊(C127⋊C8)C2211yescharacteristic
19Dic127⋊C8—21no—
20(C2×D127)⋊C8—21no—
21(C2×D127)⋊C8—21no—
22#16861119—21no—
23C4×C127⋊C8C2211yescharacteristic
24C4×D4×C127C2211yescharacteristic
25C4⋊C8×C127C2211yescharacteristic
26C4×C8×C127C2211yescharacteristic
27#14523248—42no—
28#14523248—42no—
29C2×Dic254—21no—
30C2×D508—21no—
31C2×D508C2×C411yescharacteristic
32C2×C22⋊D127—21no—
33C2×C22⋊D127—21no—
34C2×C22⋊D127—21no—
35C2×C4×D127—21no—
36C2×C4×D127C2×C411yescharacteristic
37C2×C4×D127D411yescharacteristic
38C2×C4×D127D411yescharacteristic
39C2×C127⋊C8—21no—
40C2×C127⋊C8—21no—
41C4⋊Dic127C2×C411yescharacteristic
42Dic127⋊C4—21no—
43C22⋊Dic127—21no—
44(C2×D127)⋊C4—21no—
45(C2×D127)⋊C4—21no—
46C4×Dic127C2×C411yescharacteristic
47C4×Dic127D411yescharacteristic
48C4×Dic127D411yescharacteristic
49C2×D4×C127C2×C411yescharacteristic
50C22×C4×C127—21no—
51C2×C8×C127—21no—
52C2×C8×C127—21no—
53C4⋊C4×C127C2×C411yescharacteristic
54C22⋊C4×C127—21no—
55C42×C127C2311yescharacteristic
56C22×D127—21no—
57C22×D127C22⋊C411yescharacteristic
58Dic254—42no—
59D508—21no—
60D508—21no—
61D508—42no—
62C22⋊D127—42no—
63C22⋊D127—42no—
64C22⋊D127—42no—
65C2×Dic127—21no—
66C2×Dic127C22⋊C411yescharacteristic
67C2×Dic127C22⋊C411yescharacteristic
68C2×Dic127C22⋊C411yescharacteristic
69C4×D127—21no—
70C4×D127—21no—
71C4×D127—21no—
72C4×D127—21no—
73C4×D127—41no—
74C4×D127—41no—
75C4×D127—41no—
76C127⋊C8—41no—
77C127⋊C8—41no—
78C23×C127—21no—
79D4×C127—21no—
80D4×C127—21no—
81C2×C4×C127—21no—
82C2×C4×C127—41no—
83C2×C4×C127—41no—
84C2×C4×C127C22×C411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C8×C127—41no—
88C8×C127—41no—
89C22×C127—41no—
90C22×C127—41no—
91C22×C127—41no—
92C22×C127C2×C22⋊C411yescharacteristic
93C2×D127—21no—
94C2×D127—21no—
95C2×D127—21no—
96C2×D127—41no—
97C2×D127—41no—
98C2×D127—41no—
99C4×C127—21no—
100C4×C127—41no—
101C4×C127#13111yescharacteristic
102C4×C127#13111yescharacteristic
103C4×C127AGL(1,ℤ/8)11yescharacteristic
104C4×C127AGL(1,ℤ/8)11yescharacteristic
105Dic127—21no—
106Dic127—21no—
107Dic127—21no—
108Dic127—41no—
109C2×C127—81no—
110C2×C127#40811yescharacteristic
111C2×C127#41811yescharacteristic
112C2×C127#42111yescharacteristic
113D127—41no—
114D127—81no—
115#1496—1271nomaximal, Sylow
116C127#149611yescharacteristic, complemented, Sylow
117#532—1271no—
118#432—1271no—
119#431—1271no—
120D4⋊C8—1271no—
121D4⋊C8—1271no—
122D4⋊C8—1271no—
123D4⋊C8D12711yescharacteristic, complemented
124C2×#41—2541no—
125C4⋊Q8—1271no—
126#127—1271no—
127#124—2541no—
128#121—1271no—
129#121—1271no—
130#121—2541no—
131C4×D4—1271no—
132C4×D4—2541no—
133C4×D4C2×D12711yescharacteristic
134C2×C42—1271no—
135C4⋊C8—1271no—
136C4⋊C8—2541no—
137C4⋊C8C2×D12711yescharacteristic
138C22⋊C8—2541no—
139C22⋊C8—2541no—
140#97—2541no—
141C4×C8—1271no—
142C4×C8C2×D12711yescharacteristic
143#41—5082no—
144#41—5082no—
145C2×Q8—2541no—
146C2×D4—1271no—
147C2×D4—2541no—
148C2×D4—2541no—
149C2×D4—2541no—
150C2×D4—2541no—
151C2×D4C4×D12711yescharacteristic
152C22×C4—21no—
153C22×C4—1271no—
154C22×C4—1271no—
155C22×C4—1271no—
156C22×C4—2541no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—2541no—
160C2×C8—2541no—
161C4⋊C4—1271no—
162C4⋊C4—2541no—
163C4⋊C4C4×D12711yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—2541no—
166C22⋊C4—2541no—
167C22⋊C4—2541no—
168C42—1271no—
169C42—1271no—
170C42—1271no—
171C42C22×D12711yescharacteristic
172C23—21no—
173C23—1271no—
174C23—2541no—
175Q8—5082no—
176D4—21no—
177D4—21no—
178D4—5082no—
179D4—5082no—
180D4—5082no—
181D4—5082no—
182D4—2541no—
183D4—2541no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—1271no—
188C2×C4—1271no—
189C2×C4—1271no—
190C2×C4—5081no—
191C2×C4—5081no—
192C2×C4—5081no—
193C2×C4—2541no—
194C2×C4—2541no—
195C2×C4—2541no—
196C2×C4—2541no—
197C2×C4—2541no—
198C2×C4C2×C4×D12711yescharacteristic
199C2×C4D4×D12711yescharacteristic
200C2×C4D4×D12711yescharacteristic
201C8—41no—
202C8—41no—
203C8—5081no—
204C8—5081no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—5081no—
209C22—5081no—
210C22—5081no—
211C22—2541no—
212C22—2541no—
213C22—2541no—
214C22C22⋊C4×D12711yescharacteristic
215C4—21no—
216C4—41no—
217C4—5081no—
218C4—2541no—
219C4—2541no—
220C4—2541no—
221C4#1686121511yescharacteristic
222C4#1686121711yescharacteristic
223C4#1686123311yescharacteristic
224C4#1686123611yescharacteristic
225C2—81no—
226C2—10161no—
227C2—5081no—
228C2#2254984111yescharacteristic
229C2#2254987911yescharacteristic
230C2#2254989111yescharacteristic
subgroupG/H
centerC2×C4D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
FittingD4⋊C8×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550423C2
cosocleC42×C127C23
hypercenterD4⋊C8D127
hypocenterC127#1496
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #22549932C2semidirect
3. #22549927C2semidirect
4. #22549605C2semidirect
5. D4⋊(C127⋊C8)C2semidirect
6. #22549603C2semidirect
7. D4⋊C8×C127C2semidirect
116. C127#1496semidirect
123. D4⋊C8D127semidirect

Maximal subgroups

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

subgroupindex
#225504232
#225499322
#225499272
#225496052
D4⋊(C127⋊C8)2
#225496032
D4⋊C8×C1272
#1496127