Finite Group Databasebeta

← #30637287

Subgroups — #30637287

Order 16256.

284 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 73 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#22550387C211yesmaximal, characteristic
2#22549739C211yesmaximal, characteristic, complemented
3#22549614C212yesmaximal, complemented
4#22549611C212yesmaximal, complemented
5C2×C22⋊C8×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C411yescharacteristic, complemented
7#16861192—21no—
8#16861188C411yescharacteristic, complemented
9#16861188C2211yescharacteristic
10#16861183C411yescharacteristic, complemented
11#16861183C2211yescharacteristic
12#16861178—21no—
13C4×C22⋊D127—21no—
14C4×C22⋊D127—21no—
15#16861168C411yescharacteristic, complemented
16C2×C22⋊Dic127—21no—
17C127⋊#95—21no—
18C127⋊#95C2212yes—
19C23×C4×C127C2211yescharacteristic
20C22×C8×C127—21no—
21C22⋊C8×C127—21no—
22C22⋊C8×C127C2212yes—
23#14523248—42no—
24C2×Dic254C2×C411yescharacteristic
25C2×D508C2×C411yescharacteristic
26C2×C22⋊D127—21no—
27C2×C22⋊D127—21no—
28C2×C22⋊D127—21no—
29C22×Dic127—22no—
30C2×C4×D127—21no—
31C4⋊Dic127C2×C411yescharacteristic
32C4⋊Dic127C2×C411yescharacteristic
33Dic127⋊C4—21no—
34Dic127⋊C4—21no—
35C22⋊Dic127—21no—
36C22⋊Dic127—21no—
37C22⋊Dic127—21no—
38C22⋊Dic127—42no—
39(C2×D127)⋊C4—21no—
40(C2×D127)⋊C4—21no—
41C4×Dic127—21no—
42C24×C127D411yescharacteristic
43C22×C4×C127—42no—
44C22×C4×C127C2×C411yescharacteristic
45C22×C4×C127C2×C411yescharacteristic
46C22×C4×C127D411yescharacteristic
47C22×C4×C127D411yescharacteristic
48C22×C4×C127D411yescharacteristic
49C22×C4×C127C2311yescharacteristic
50C2×C8×C127—22no—
51C2×C8×C127—42no—
52C22×D127—21no—
53Dic254—41no—
54D508—41no—
55C22⋊D127—42no—
56C22⋊D127—42no—
57C22⋊D127—42no—
58C2×Dic127—21no—
59C2×Dic127—21no—
60C2×Dic127—21no—
61C2×Dic127—42no—
62C2×Dic127—42no—
63C2×Dic127—42no—
64C4×D127—42no—
65C23×C127—22no—
66C23×C127—22no—
67C23×C127—41no—
68C23×C127C22⋊C411yescharacteristic
69C23×C127C22⋊C411yescharacteristic
70C23×C127C2×D411yescharacteristic
71C2×C4×C127—22no—
72C2×C4×C127—22no—
73C2×C4×C127—41no—
74C2×C4×C127—41no—
75C2×C4×C127—41no—
76C2×C4×C127—41no—
77C2×C4×C127C22⋊C411yescharacteristic
78C2×C4×C127C22⋊C411yescharacteristic
79C2×C4×C127C22×C411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C8×C127—42no—
82C22×C127—21no—
83C22×C127—21no—
84C22×C127—21no—
85C22×C127—21no—
86C22×C127—21no—
87C22×C127—21no—
88C22×C127—21no—
89C22×C127—44no—
90C22×C127#9912yes—
91C22×C127C2×C22⋊C411yescharacteristic
92C22×C127#13512yes—
93C2×D127—41no—
94C2×D127—41no—
95C2×D127—41no—
96C4×C127—21no—
97C4×C127—21no—
98C4×C127—21no—
99C4×C127—21no—
100Dic127—41no—
101Dic127—41no—
102Dic127—41no—
103Dic127—82no—
104C2×C127—22no—
105C2×C127—22no—
106C2×C127—41no—
107C2×C127C2×#9911yescharacteristic
108C2×C127#41611yescharacteristic
109C2×C127#42111yescharacteristic
110D127—81no—
111#1516—1271nomaximal, Sylow
112C127#151611yescharacteristic, complemented, Sylow
113#525—1271no—
114C2×C22⋊C8D12711yescharacteristic, complemented
115#381—1271no—
116#328—1272no—
117#327—1272no—
118C2×#41—1271no—
119C23×C4C2×D12711yescharacteristic
120C22×C8—21no—
121#123—2541no—
122C22⋊Q8—1271no—
123C22⋊Q8—1271no—
124#121—1271no—
125#121—1271no—
126#120—2541no—
127C4×D4—2541no—
128C4×D4—2541no—
129#117—1271no—
130C2×C22⋊C4—2541no—
131C22⋊C8—21no—
132C22⋊C8C2×D12712yes—
133#95—1272no—
134#95—2541no—
135C24D50811yescharacteristic
136#41—5082no—
137C2×Q8—1271no—
138C2×D4—1271no—
139C2×D4—2541no—
140C2×D4—2541no—
141C2×D4—2541no—
142C22×C4—42no—
143C22×C4—2542no—
144C22×C4—2541no—
145C22×C4C4×D12711yescharacteristic
146C22×C4C4×D12711yescharacteristic
147C22×C4C22⋊D12711yescharacteristic
148C22×C4C22⋊D12711yescharacteristic
149C22×C4D50811yescharacteristic
150C22×C4C22×D12711yescharacteristic
151C2×C8—22no—
152C2×C8—42no—
153C4⋊C4—1271no—
154C4⋊C4—1271no—
155C4⋊C4—2541no—
156C4⋊C4—2541no—
157C22⋊C4—5082no—
158C22⋊C4—2541no—
159C22⋊C4—2541no—
160C22⋊C4—2541no—
161C22⋊C4—2541no—
162C22⋊C4—2541no—
163C42—2541no—
164C23—22no—
165C23—22no—
166C23—41no—
167C23—2541no—
168C23(C2×D127)⋊C411yescharacteristic
169C23(C2×D127)⋊C411yescharacteristic
170C23C2×D50811yescharacteristic
171Q8—5081no—
172D4—5081no—
173D4—5082no—
174D4—5082no—
175D4—5082no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—5082no—
183C2×C4—5082no—
184C2×C4—5082no—
185C2×C4—5082no—
186C2×C4—2541no—
187C2×C4—2541no—
188C2×C4—2541no—
189C2×C4(C2×D127)⋊C411yescharacteristic
190C2×C4(C2×D127)⋊C411yescharacteristic
191C2×C4C2×C4×D12711yescharacteristic
192C2×C4C2×C22⋊D12711yescharacteristic
193C8—42no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—5081no—
203C22—5081no—
204C22—5081no—
205C22#1686112312yes—
206C22C2×(C2×D127)⋊C411yescharacteristic
207C22#1686123112yes—
208C4—21no—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—10162no—
213C4—5081no—
214C4—5081no—
215C4—5081no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—10161no—
220C2C2×#1686112311yescharacteristic
221C2#2254987311yescharacteristic
222C2#2254989211yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×C22⋊C8×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550387C2
cosocleC22×C4×C127C23
hypercenterC2×C22⋊C8D127
hypocenterC127#1516
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. #22549739C2semidirect
3. #22549614C2semidirect
4. #22549611C2semidirect
5. C2×C22⋊C8×C127C2semidirect
6. C2×#14523248C4semidirect
8. #16861188C4semidirect
10. #16861183C4semidirect
15. #16861168C4semidirect
112. C127#1516semidirect
114. C2×C22⋊C8D127semidirect

Maximal subgroups

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

subgroupindex
#225503872
#225497392
#225496142
#225496112
C2×C22⋊C8×C1272
#1516127