Finite Group Databasebeta

← #30637222

Subgroups — #30637222

Order 16256.

304 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 77 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
2C4×C22⋊Dic127C211yesmaximal, 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—
17C2×C4×Dic127—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—24no—
36C22⋊Dic127—21no—
37C22⋊Dic127—21no—
38C22⋊Dic127—21no—
39(C2×D127)⋊C4—21no—
40(C2×D127)⋊C4—21no—
41C4×Dic127—21no—
42C4×Dic127—42no—
43C24×C127D411yescharacteristic
44C22×C4×C127—42no—
45C22×C4×C127C2×C411yescharacteristic
46C22×C4×C127C2×C411yescharacteristic
47C22×C4×C127D411yescharacteristic
48C22×C4×C127D411yescharacteristic
49C22×C4×C127D411yescharacteristic
50C22×C4×C127C2311yescharacteristic
51C2×C8×C127—22no—
52C2×C8×C127—42no—
53C22×D127—21no—
54Dic254—41no—
55D508—41no—
56C22⋊D127—42no—
57C22⋊D127—42no—
58C22⋊D127—42no—
59C2×Dic127—24no—
60C2×Dic127—21no—
61C2×Dic127—21no—
62C2×Dic127—21no—
63C2×Dic127—42no—
64C2×Dic127—42no—
65C4×D127—42no—
66C23×C127—22no—
67C23×C127—22no—
68C23×C127—41no—
69C23×C127C22⋊C411yescharacteristic
70C23×C127C22⋊C411yescharacteristic
71C23×C127C2×D411yescharacteristic
72C2×C4×C127—22no—
73C2×C4×C127—22no—
74C2×C4×C127—22no—
75C2×C4×C127—41no—
76C2×C4×C127—41no—
77C2×C4×C127—41no—
78C2×C4×C127C22⋊C411yescharacteristic
79C2×C4×C127C22⋊C411yescharacteristic
80C2×C4×C127C22×C411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C8×C127—42no—
83C22×C127—21no—
84C22×C127—21no—
85C22×C127—21no—
86C22×C127—21no—
87C22×C127—21no—
88C22×C127—21no—
89C22×C127—21no—
90C22×C127—44no—
91C22×C127#10412yes—
92C22×C127C2×C22⋊C411yescharacteristic
93C22×C127#13512yes—
94C2×D127—41no—
95C2×D127—41no—
96C2×D127—41no—
97C4×C127—21no—
98C4×C127—21no—
99C4×C127—21no—
100C4×C127#9912yes—
101Dic127—44no—
102Dic127—41no—
103Dic127—41no—
104Dic127—41no—
105C2×C127—22no—
106C2×C127—22no—
107C2×C127—41no—
108C2×C127C2×#9911yescharacteristic
109C2×C127#41611yescharacteristic
110C2×C127C2×#10411yescharacteristic
111D127—81no—
112#1501—1271nomaximal, Sylow
113C127#150111yescharacteristic, complemented, Sylow
114#525—1271no—
115C2×C22⋊C8D12711yescharacteristic, complemented
116C4×C22⋊C4—1271no—
117#328—1272no—
118#327—1272no—
119C2×#41—1271no—
120C23×C4C2×D12711yescharacteristic
121C22×C8—21no—
122#123—2541no—
123C22⋊Q8—1271no—
124C22⋊Q8—1271no—
125#121—1271no—
126#121—1271no—
127#120—2541no—
128C4×D4—2541no—
129C4×D4—2541no—
130#117—1271no—
131C2×C22⋊C4—2541no—
132C2×C42—2541no—
133C22⋊C8—21no—
134C22⋊C8C2×D12712yes—
135#95—1272no—
136C24D50811yescharacteristic
137#41—5082no—
138C2×Q8—1271no—
139C2×D4—1271no—
140C2×D4—2541no—
141C2×D4—2541no—
142C2×D4—2541no—
143C22×C4—42no—
144C22×C4—2542no—
145C22×C4—2541no—
146C22×C4C4×D12711yescharacteristic
147C22×C4C4×D12711yescharacteristic
148C22×C4C22⋊D12711yescharacteristic
149C22×C4C22⋊D12711yescharacteristic
150C22×C4D50811yescharacteristic
151C22×C4C22×D12711yescharacteristic
152C2×C8—22no—
153C2×C8—42no—
154C4⋊C4—1271no—
155C4⋊C4—1271no—
156C4⋊C4—2541no—
157C4⋊C4—2541no—
158C22⋊C4—2544no—
159C22⋊C4—2541no—
160C22⋊C4—2541no—
161C22⋊C4—2541no—
162C22⋊C4—2541no—
163C22⋊C4—2541no—
164C42—5082no—
165C42—2541no—
166C23—22no—
167C23—22no—
168C23—41no—
169C23—2541no—
170C23(C2×D127)⋊C411yescharacteristic
171C23(C2×D127)⋊C411yescharacteristic
172C23C2×D50811yescharacteristic
173Q8—5081no—
174D4—5081no—
175D4—5082no—
176D4—5082no—
177D4—5082no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—5082no—
185C2×C4—5082no—
186C2×C4—5082no—
187C2×C4—2544no—
188C2×C4—2541no—
189C2×C4—2541no—
190C2×C4—2541no—
191C2×C4(C2×D127)⋊C411yescharacteristic
192C2×C4(C2×D127)⋊C411yescharacteristic
193C2×C4C2×C4×D12711yescharacteristic
194C2×C4C2×C22⋊D12711yescharacteristic
195C8—42no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—44no—
204C22—5081no—
205C22—5081no—
206C22—5081no—
207C22#1686113512yes—
208C22C2×(C2×D127)⋊C411yescharacteristic
209C22#1686123112yes—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—5084no—
214C4—5081no—
215C4—5081no—
216C4—5081no—
217C4#1686112312yes—
218C2—22no—
219C2—22no—
220C2—41no—
221C2—10161no—
222C2C2×#1686112311yescharacteristic
223C2#2254987311yescharacteristic
224C2C2×#1686113511yescharacteristic
subgroupG/H
centerC2×C4(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#1501
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊Dic127C2semidirect
3. #22549614C2semidirect
4. #22549611C2semidirect
5. C2×C22⋊C8×C127C2semidirect
6. C2×#14523248C4semidirect
8. #16861188C4semidirect
10. #16861183C4semidirect
15. #16861168C4semidirect
113. C127#1501semidirect
115. C2×C22⋊C8D127semidirect

Maximal subgroups

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

subgroupindex
#225503872
C4×C22⋊Dic1272
#225496142
#225496112
C2×C22⋊C8×C1272
#1501127