Finite Group Databasebeta

← #30648857

Subgroups — #30648857

Order 16256.

380 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 119 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#22550428C211yesmaximal, characteristic
2C2×#16861174C211yesmaximal, characteristic
3#22550206C211yesmaximal, characteristic
4#22550160C212yesmaximal, complemented
5#22550085C212yesmaximal, complemented
6#22550034C211yesmaximal, characteristic
7#22550031C211yesmaximal, characteristic
8#22549982C212yesmaximal, complemented
9#22549980C212yesmaximal, complemented
10#22549930C211yesmaximal, characteristic
11#483×C127C211yesmaximal, characteristic
12C22×Dic254C2211yescharacteristic
13C2×Dic508C2212yes—
14C2×#14523238C2212yes—
15#16861208C2211yescharacteristic
16#16861207C2211yescharacteristic
17#16861199—21no—
18#16861188C2211yescharacteristic
19#16861186—21no—
20Q8×Dic127C2211yescharacteristic
21#16861174—21no—
22#16861174—21no—
23#16861174—21no—
24#16861174C2212yes—
25#16861174C2212yes—
26#16861174C2211yescharacteristic
27#16861165—21no—
28C2×Dic127⋊C4—21no—
29C2×C4×Dic127—21no—
30C2×C4×Dic127C2211yescharacteristic
31#16861140C2212yes—
32Dic127⋊C8C2212yes—
33#16861134C2212yes—
34#16861132C2212yes—
35#16861132C2212yes—
36C22⋊(C127⋊C8)C2211yescharacteristic
37#16861119C2212yes—
38C22⋊Q8×C127C2211yescharacteristic
39C2×C4⋊C4×C127C2211yescharacteristic
40#106×C127C2212yes—
41#103×C127C2212yes—
42C22⋊C8×C127C2211yescharacteristic
43C2×Dic254—24no—
44C2×Dic254—21no—
45C2×Dic254—21no—
46C2×Dic254C2311yescharacteristic
47C2×Dic254C2312yes—
48C22×Dic127—21no—
49C22×Dic127D411yescharacteristic
50C22×Dic127D411yescharacteristic
51Dic508—24no—
52#14523238—24no—
53C2×C127⋊C8C2312yes—
54C4⋊Dic127—21no—
55C4⋊Dic127C2311yescharacteristic
56Dic127⋊C4—21no—
57Dic127⋊C4—21no—
58Dic127⋊C4—21no—
59Dic127⋊C4—21no—
60Dic127⋊C4—22no—
61Dic127⋊C4—22no—
62Dic127⋊C4—21no—
63C22⋊Dic127—21no—
64C4×Dic127—22no—
65C4×Dic127—22no—
66C4×Dic127—21no—
67C4×Dic127—21no—
68C4×Dic127D411yescharacteristic
69C4×Dic127D411yescharacteristic
70C4×Dic127C2311yescharacteristic
71C4×Dic127C2311yescharacteristic
72C2×Q8×C127C2311yescharacteristic
73C22×C4×C127—21no—
74C22×C4×C127C2311yescharacteristic
75C2×C8×C127C2312yes—
76C4⋊C4×C127—21no—
77C4⋊C4×C127—21no—
78C4⋊C4×C127C2311yescharacteristic
79C4⋊C4×C127C2312yes—
80C22⋊C4×C127—21no—
81Dic254—24no—
82Dic254—22no—
83Dic254—22no—
84Dic254—42no—
85Dic254#4114yes—
86C2×Dic127—24no—
87C2×Dic127—22no—
88C2×Dic127—22no—
89C2×Dic127—22no—
90C2×Dic127—21no—
91C2×Dic127—21no—
92C2×Dic127C2×D411yescharacteristic
93C2×Dic127C2×D411yescharacteristic
94C2×Dic127C2×D411yescharacteristic
95C2×Dic127C2×D411yescharacteristic
96C127⋊C8—22no—
97C23×C127C2×D411yescharacteristic
98Q8×C127—22no—
99C2×C4×C127—22no—
100C2×C4×C127—21no—
101C2×C4×C127—21no—
102C2×C4×C127—22no—
103C2×C4×C127—42no—
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127C2411yescharacteristic
106C8×C127—22no—
107C22×C127—21no—
108C22×C127—21no—
109C22×C127#13712yes—
110C22×C127C22×D411yescharacteristic
111C4×C127—21no—
112C4×C127—41no—
113C4×C127—42no—
114C4×C127—41no—
115C4×C127C2×#4112yes—
116Dic127—24no—
117Dic127—21no—
118Dic127—21no—
119Dic127—21no—
120Dic127—41no—
121Dic127—41no—
122Dic127#13712yes—
123C2×C127—22no—
124C2×C127#52511yescharacteristic
125C2×C127C2×#13711yescharacteristic
126C2×C127C2×#13711yescharacteristic
127#3113—1271nomaximal, Sylow
128C127#311311yescharacteristic, complemented, Sylow
129#533—1271no—
130C2×C4×Q8—1271no—
131#483—1271no—
132#483D12711yescharacteristic
133#475—1272no—
134#462—1272no—
135#451—1271no—
136#451—1271no—
137#441—1272no—
138#441—1272no—
139#432—1271no—
140C22×Q8—1271no—
141C2×Q16—1272no—
142C2×Q16—1272no—
143C4⋊Q8—1271no—
144C4⋊Q8—1271no—
145#125—2541no—
146C22⋊Q8—1271no—
147C22⋊Q8—2541no—
148C22⋊Q8C2×D12711yescharacteristic
149C4×Q8—1271no—
150C4×Q8—1272no—
151C4×Q8—1272no—
152C4×Q8—1271no—
153C4×Q8—2541no—
154C4×Q8—2541no—
155C4×Q8—2541no—
156#117—2541no—
157C2×C4⋊C4—2541no—
158C2×C4⋊C4C2×D12711yescharacteristic
159C2×C42—1271no—
160C2×C42—2541no—
161#106—1272no—
162#106C2×D12712yes—
163C4⋊C8—1272no—
164#103—1272no—
165#103—1272no—
166#103—1272no—
167#103C2×D12712yes—
168C22⋊C8—1271no—
169C22⋊C8C2×D12711yescharacteristic
170#97—1272no—
171C2×Q8—1272no—
172C2×Q8—1271no—
173C2×Q8—2541no—
174C2×Q8—2544no—
175C2×Q8—2541no—
176C2×Q8C22×D12711yescharacteristic
177C22×C4—21no—
178C22×C4—1271no—
179C22×C4—1271no—
180C22×C4—2541no—
181C22×C4C22×D12711yescharacteristic
182Q16—2544no—
183Q16—2544no—
184C2×C8—1272no—
185C2×C8C22×D12712yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—1271no—
189C4⋊C4—2542no—
190C4⋊C4—2542no—
191C4⋊C4—2541no—
192C4⋊C4—2541no—
193C4⋊C4—2541no—
194C4⋊C4—2541no—
195C4⋊C4—2541no—
196C4⋊C4—2541no—
197C4⋊C4C22×D12711yescharacteristic
198C4⋊C4C22×D12712yes—
199C22⋊C4—21no—
200C22⋊C4—2541no—
201C42—1271no—
202C42—1271no—
203C42—1271no—
204C42—1271no—
205C42—2542no—
206C42—2542no—
207C42—2541no—
208C42—2541no—
209C23D4×D12711yescharacteristic
210Q8—22no—
211Q8—1274no—
212Q8—5082no—
213Q8—2544no—
214Q8—2542no—
215Q8—2542no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—42no—
221C2×C4—1271no—
222C2×C4—1271no—
223C2×C4—1271no—
224C2×C4—1271no—
225C2×C4—2542no—
226C2×C4—2541no—
227C2×C4—2541no—
228C2×C4—2544no—
229C2×C4—2542no—
230C2×C4—2542no—
231C2×C4D4×D12711yescharacteristic
232C2×C4C23×D12711yescharacteristic
233C8—22no—
234C8—2542no—
235C22—21no—
236C22—21no—
237C22#1686123812yes—
238C22C2×D4×D12711yescharacteristic
239C4—21no—
240C4—41no—
241C4—41no—
242C4—42no—
243C4—1272no—
244C4—5081no—
245C4—5081no—
246C4—2544no—
247C4—2541no—
248C4—2541no—
249C4—2541no—
250C4#41×D12712yes—
251C2—22no—
252C2#2255038411yescharacteristic
253C2#137×D12711yescharacteristic
254C2C2×#1686123811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#483×C127C2
socleC22×C127C22×D4
prime radicalC23×C127C2×D4
hypercenter#483D127
hypocenterC127#3113
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
4. #22550160C2semidirect
5. #22550085C2semidirect
8. #22549982C2semidirect
9. #22549980C2semidirect
128. C127#3113semidirect

Maximal subgroups

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

subgroupindex
#225504282
C2×#168611742
#225502062
#225501602
#225500852
#225500342
#225500312
#225499822
#225499802
#225499302
#483×C1272
#3113127