Finite Group Databasebeta

← #30647061

Subgroups — #30647061

Order 16256.

688 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 189 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×#16861192C212yesmaximal, complemented
2C2×#16861183C211yesmaximal, characteristic
3#22549778C212yesmaximal, complemented
4#22549774C212yesmaximal, complemented
5#22549767C214yesmaximal, complemented
6#22549755C212yesmaximal, complemented
7#22549736C211yesmaximal, characteristic, complemented
8C22×C42×C127C211yesmaximal, characteristic, complemented
9C22×D508C2211yescharacteristic
10C22×C22⋊D127C2212yes—
11#16861192—28no—
12#16861183—24no—
13C2×C4⋊Dic127C2211yescharacteristic
14C2×C4⋊Dic127C2212yes—
15C2×Dic127⋊C4C2214yes—
16C2×C22⋊Dic127C2212yes—
17C2×(C2×D127)⋊C4C2212yes—
18C2×(C2×D127)⋊C4C2214yes—
19C2×(C2×D127)⋊C4C2214yes—
20C127⋊#95C2212yes—
21C127⋊#95C2214yes—
22C23×C4×C127C2211yescharacteristic
23C23×C4×C127C2211yescharacteristic
24C23×C4×C127C2211yescharacteristic
25C2×C42×C127—24no—
26C2×C42×C127C2212yes—
27C2×C42×C127C2212yes—
28C23×D127C2312yes—
29C2×D508—22no—
30C2×D508—24no—
31C2×C22⋊D127—24no—
32C2×C22⋊D127—24no—
33C2×C22⋊D127—24no—
34C22×Dic127C2312yes—
35C22×Dic127C2314yes—
36C4⋊Dic127—22no—
37C4⋊Dic127—24no—
38Dic127⋊C4—28no—
39C22⋊Dic127—24no—
40(C2×D127)⋊C4—28no—
41(C2×D127)⋊C4—24no—
42(C2×D127)⋊C4—28no—
43C24×C127C2311yescharacteristic
44C22×C4×C127—22no—
45C22×C4×C127—22no—
46C22×C4×C127—22no—
47C22×C4×C127—24no—
48C22×C4×C127—24no—
49C22×C4×C127D414yes—
50C22×C4×C127D414yescomplemented
51C22×C4×C127C2311yescharacteristic
52C22×C4×C127C2312yes—
53C22×C4×C127C2312yes—
54C22×C4×C127C2311yescharacteristic
55C42×C127—24no—
56C42×C127—24no—
57C22×D127—22no—
58C22×D127—22no—
59C22×D127—24no—
60C22×D127—22no—
61C22×D127—24no—
62D508—44no—
63C22⋊D127—48no—
64C2×Dic127—24no—
65C2×Dic127—22no—
66C2×Dic127—22no—
67C2×Dic127—24no—
68C2×Dic127—24no—
69C2×Dic127—22no—
70C23×C127—22no—
71C23×C127—22no—
72C23×C127C2×D412yes—
73C23×C127C2×D412yes—
74C23×C127#4112yes—
75C23×C127C2411yescharacteristic
76C2×C4×C127—28no—
77C2×C4×C127—28no—
78C2×C4×C127—22no—
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—22no—
83C2×C4×C127—28no—
84C2×C4×C127C2×D412yes—
85C2×C4×C127C2×D412yes—
86C2×C4×C127C2×D414yes—
87C2×C4×C127#4114yes—
88C2×C4×C127#4114yes—
89C22×C127—22no—
90C22×C127—22no—
91C22×C127—24no—
92C22×C127—24no—
93C22×C127#12114yes—
94C22×C127C22×D411yescharacteristic
95C22×C127C22×D411yescharacteristic
96C22×C127C2×#4111yescharacteristic
97C22×C127C2×#4112yes—
98C22×C127C2×#4112yes—
99C2×D127—42no—
100C2×D127—42no—
101C2×D127—42no—
102C2×D127—44no—
103C2×D127—44no—
104C4×C127—24no—
105C4×C127—22no—
106C4×C127—24no—
107C4×C127#12314yescomplemented
108Dic127—42no—
109Dic127—44no—
110C2×C127—24no—
111C2×C127C2×#12111yescharacteristic
112C2×C127C2×#12311yescharacteristic
113C2×C127#52511yescharacteristic
114C2×C127#52912yes—
115C2×C127#53212yes—
116D127—82no—
117#2854—1271nomaximal, Sylow
118C127#285411yescharacteristic, complemented, Sylow
119C2×#123—1272no—
120C2×#121—1271no—
121C22×C42D12711yescharacteristic, complemented
122#390—1272no—
123#389—1272no—
124#388—1274no—
125#386—1272no—
126#380—1271no—
127C22×D4—1272no—
128C22×D4—1271no—
129C23×C4C2×D12711yescharacteristic
130C23×C4C2×D12711yescharacteristic
131C23×C4C2×D12711yescharacteristic
132#123—2548no—
133#121—2544no—
134C2×C4⋊C4—1271no—
135C2×C4⋊C4—1274no—
136C2×C4⋊C4—1272no—
137C2×C22⋊C4—1272no—
138C2×C22⋊C4—1272no—
139C2×C22⋊C4—1274no—
140C2×C22⋊C4—1274no—
141C2×C42—24no—
142C2×C42C2×D12712yes—
143C2×C42C2×D12712yes—
144#95—1272no—
145#95—1274no—
146C24—1272no—
147C24C22×D12711yescharacteristic
148C2×D4—2544no—
149C2×D4—2544no—
150C2×D4—2544no—
151C2×D4—2542no—
152C2×D4—2544no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—1274no—
159C22×C4—1272no—
160C22×C4C22⋊D12714yes—
161C22×C4D50814yescomplemented
162C22×C4C22×D12711yescharacteristic
163C22×C4C22×D12712yes—
164C22×C4C22×D12712yes—
165C22×C4C22×D12711yescharacteristic
166C4⋊C4—2542no—
167C4⋊C4—2548no—
168C4⋊C4—2544no—
169C22⋊C4—2544no—
170C22⋊C4—2544no—
171C22⋊C4—2548no—
172C22⋊C4—2548no—
173C42—24no—
174C42—24no—
175C23—22no—
176C23—22no—
177C23—2542no—
178C23—2542no—
179C23—2544no—
180C23—2542no—
181C23—2544no—
182C23C2×C22⋊D12712yes—
183C23C2×D50812yes—
184C23#1452324812yes—
185C23C23×D12711yescharacteristic
186D4—5088no—
187D4—5084no—
188C2×C4—28no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—28no—
195C2×C4—28no—
196C2×C4—2542no—
197C2×C4—2544no—
198C2×C4—2542no—
199C2×C4—2544no—
200C2×C4—2544no—
201C2×C4—2542no—
202C2×C4C2×C22⋊D12712yes—
203C2×C4C2×C22⋊D12712yes—
204C2×C4C2×D50814yes—
205C2×C4#1452324814yes—
206C2×C4#1452324814yes—
207C22—22no—
208C22—22no—
209C22—24no—
210C22—24no—
211C22—5082no—
212C22—5082no—
213C22—5082no—
214C22—5084no—
215C22—5084no—
216C22#1686118314yes—
217C22C22×C22⋊D12711yescharacteristic
218C22C22×D50811yescharacteristic
219C22C2×#1452324811yescharacteristic
220C22C2×#1452324812yes—
221C22C2×#1452324812yes—
222C4—24no—
223C4—22no—
224C4—24no—
225C4—5082no—
226C4—5084no—
227C4#1686119214yescomplemented
228C2—24no—
229C2—10162no—
230C2C2×#1686118311yescharacteristic
231C2C2×#1686119211yescharacteristic
232C2#2255038711yescharacteristic
233C2#2255041012yes—
234C2#2255042612yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
FittingC22×C42×C127C2
prime radicalC2×#16861183C2
hypercenterC22×C42D127
hypocenterC127#2854
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
1. C2×#16861192C2semidirect
3. #22549778C2semidirect
4. #22549774C2semidirect
5. #22549767C2semidirect
6. #22549755C2semidirect
7. #22549736C2semidirect
8. C22×C42×C127C2semidirect
50. C22×C4×C127D4semidirect
107. C4×C127#123semidirect
118. C127#2854semidirect
121. C22×C42D127semidirect
161. C22×C4D508semidirect
227. C4#16861192semidirect

Maximal subgroups

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

subgroupindex
C2×#168611922
C2×#168611832
#225497782
#225497742
#225497672
#225497552
#225497362
C22×C42×C1272
#2854127