Finite Group Databasebeta

← #30647067

Subgroups — #30647067

Order 16256.

564 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 169 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×#16861186C211yesmaximal, characteristic
2C22×C4×Dic127C211yesmaximal, characteristic
3#22549780C212yesmaximal, complemented
4#22549773C212yesmaximal, complemented
5#22549763C212yesmaximal, complemented
6#22549759C212yesmaximal, complemented
7#22549751C211yesmaximal, characteristic
8#22549748C211yesmaximal, characteristic
9#22549739C211yesmaximal, characteristic
10#22549734C211yesmaximal, characteristic
11#381×C127C211yesmaximal, characteristic
12C22×Dic254C2211yescharacteristic
13C23×Dic127C2211yescharacteristic
14C23×Dic127C2211yescharacteristic
15#16861186—24no—
16C2×C4⋊Dic127C2211yescharacteristic
17C2×C4⋊Dic127C2212yes—
18C2×Dic127⋊C4C2212yes—
19C2×Dic127⋊C4C2211yescharacteristic
20C2×Dic127⋊C4C2212yes—
21C2×Dic127⋊C4C2211yescharacteristic
22C2×C22⋊Dic127C2211yescharacteristic
23C2×C22⋊Dic127C2211yescharacteristic
24C2×C4×Dic127—24no—
25C2×C4×Dic127C2212yes—
26C2×C4×Dic127C2212yes—
27C127⋊#95C2212yes—
28C127⋊#95C2212yes—
29C127⋊#95C2212yes—
30C127⋊#95C2212yes—
31C127⋊#95C2212yes—
32C23×C4×C127C2211yescharacteristic
33C2×C22⋊C4×C127C2211yescharacteristic
34C2×C22⋊C4×C127C2211yescharacteristic
35#95×C127C2212yes—
36#95×C127C2212yes—
37C2×Dic254—22no—
38C2×Dic254—24no—
39C22×Dic127—24no—
40C22×Dic127—24no—
41C22×Dic127—22no—
42C22×Dic127D414yes—
43C22×Dic127C2311yescharacteristic
44C22×Dic127C2312yes—
45C22×Dic127C2311yescharacteristic
46C22×Dic127C2311yescharacteristic
47C22×Dic127C2312yes—
48C22×Dic127C2311yescharacteristic
49C4⋊Dic127—22no—
50C4⋊Dic127—24no—
51Dic127⋊C4—22no—
52Dic127⋊C4—22no—
53Dic127⋊C4—24no—
54Dic127⋊C4—24no—
55C22⋊Dic127—22no—
56C22⋊Dic127—22no—
57C4×Dic127—24no—
58C4×Dic127—24no—
59C24×C127C2311yescharacteristic
60C22×C4×C127—22no—
61C22×C4×C127—22no—
62C22×C4×C127Q814yescomplemented
63C22×C4×C127C2311yescharacteristic
64C22×C4×C127C2312yes—
65C22×C4×C127C2311yescharacteristic
66C22×C4×C127C2312yes—
67C22⋊C4×C127—22no—
68C22⋊C4×C127—22no—
69Dic254—44no—
70C2×Dic127—28no—
71C2×Dic127—22no—
72C2×Dic127—22no—
73C2×Dic127—22no—
74C2×Dic127—28no—
75C2×Dic127—21no—
76C2×Dic127—21no—
77C2×Dic127—22no—
78C2×Dic127—22no—
79C2×Dic127—22no—
80C2×Dic127—22no—
81C2×Dic127—22no—
82C2×Dic127C2×D412yes—
83C2×Dic127C2×D412yes—
84C2×Dic127#4114yes—
85C2×Dic127#4114yes—
86C23×C127—22no—
87C23×C127—22no—
88C23×C127C2×D412yes—
89C23×C127C2×Q812yes—
90C23×C127#4112yes—
91C23×C127C2411yescharacteristic
92C2×C4×C127—28no—
93C2×C4×C127—22no—
94C2×C4×C127—22no—
95C2×C4×C127—21no—
96C2×C4×C127—21no—
97C2×C4×C127—22no—
98C2×C4×C127—22no—
99C2×C4×C127—22no—
100C2×C4×C127—22no—
101C2×C4×C127—22no—
102C2×C4×C127C2×Q814yes—
103C22×C127—22no—
104C22×C127—22no—
105C22×C127—24no—
106C22×C127—24no—
107C22×C127C22⋊Q814yes—
108C22×C127C22×D411yescharacteristic
109C22×C127C22×Q811yescharacteristic
110C22×C127C2×#4112yes—
111C22×C127C2×#4112yes—
112C22×C127C2×#4111yescharacteristic
113C4×C127—24no—
114C4×C127—41no—
115C4×C127—41no—
116C4×C127—42no—
117Dic127—22no—
118Dic127—24no—
119Dic127—41no—
120Dic127—41no—
121Dic127—42no—
122Dic127#12314yes—
123C2×C127—24no—
124C2×C127C2×C22⋊Q811yescharacteristic
125C2×C127C2×#12311yescharacteristic
126C2×C127#52511yescharacteristic
127C2×C127#52912yes—
128C2×C127#53312yes—
129#2855—1271nomaximal, Sylow
130C127#285511yescharacteristic, complemented, Sylow
131C2×C22⋊Q8—1271no—
132C22×C42—1271no—
133#391—1272no—
134#389—1272no—
135#387—1272no—
136#387—1272no—
137#385—1271no—
138#385—1271no—
139#381—1271no—
140#381D12711yescharacteristic
141#380—1271no—
142C22×Q8—1271no—
143C23×C4—1271no—
144C23×C4—1271no—
145C23×C4C2×D12711yescharacteristic
146C22⋊Q8—2544no—
147C2×C4⋊C4—1271no—
148C2×C4⋊C4—1271no—
149C2×C4⋊C4—1271no—
150C2×C4⋊C4—1272no—
151C2×C4⋊C4—1272no—
152C2×C4⋊C4—1272no—
153C2×C22⋊C4—1271no—
154C2×C22⋊C4—1271no—
155C2×C22⋊C4C2×D12711yescharacteristic
156C2×C22⋊C4C2×D12711yescharacteristic
157C2×C42—1272no—
158C2×C42—1272no—
159C2×C42—2544no—
160#95—1272no—
161#95—1272no—
162#95—1272no—
163#95—1272no—
164#95—1272no—
165#95C2×D12712yes—
166#95C2×D12712yes—
167C24C22×D12711yescharacteristic
168C2×Q8—2542no—
169C2×Q8—2544no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—1271no—
173C22×C4—1274no—
174C22×C4—1271no—
175C22×C4—1272no—
176C22×C4—1272no—
177C22×C4—1271no—
178C22×C4—1271no—
179C22×C4—2544no—
180C22×C4—2544no—
181C22×C4—2542no—
182C22×C4Dic25414yescomplemented
183C22×C4C22×D12712yes—
184C22×C4C22×D12711yescharacteristic
185C22×C4C22×D12711yescharacteristic
186C22×C4C22×D12712yes—
187C4⋊C4—2544no—
188C4⋊C4—2544no—
189C4⋊C4—2542no—
190C4⋊C4—2542no—
191C4⋊C4—2544no—
192C4⋊C4—2542no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—2542no—
196C22⋊C4—2542no—
197C42—2544no—
198C42—2544no—
199C23—22no—
200C23—22no—
201C23D4×D12712yes—
202C23C2×Dic25412yes—
203C23#1452324712yes—
204C23C23×D12711yescharacteristic
205Q8—5084no—
206C2×C4—28no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—1272no—
217C2×C4—1274no—
218C2×C4—1272no—
219C2×C4—1274no—
220C2×C4—2548no—
221C2×C4—2541no—
222C2×C4—2541no—
223C2×C4—2542no—
224C2×C4—2542no—
225C2×C4—2542no—
226C2×C4—2542no—
227C2×C4—2542no—
228C2×C4—2548no—
229C2×C4—2542no—
230C2×C4—2542no—
231C2×C4—2542no—
232C2×C4C2×Dic25414yes—
233C22—22no—
234C22—22no—
235C22—24no—
236C22—24no—
237C22#1686118614yes—
238C22C2×D4×D12711yescharacteristic
239C22C22×Dic25411yescharacteristic
240C22#41×D12712yes—
241C22#41×D12712yes—
242C22C2×#1452324711yescharacteristic
243C4—24no—
244C4—41no—
245C4—41no—
246C4—42no—
247C4—1274no—
248C4—5081no—
249C4—5081no—
250C4—5082no—
251C4—2542no—
252C4—2544no—
253C2—24no—
254C2C2×#1686118611yescharacteristic
255C2#123×D12711yescharacteristic
256C2#2255038411yescharacteristic
257C2#2255040312yes—
258C2#2255042912yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#381×C127C2
prime radicalC24×C127C23
hypercenter#381D127
hypocenterC127#2855
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
3. #22549780C2semidirect
4. #22549773C2semidirect
5. #22549763C2semidirect
6. #22549759C2semidirect
62. C22×C4×C127Q8semidirect
130. C127#2855semidirect
182. C22×C4Dic254semidirect

Maximal subgroups

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

subgroupindex
C2×#168611862
C22×C4×Dic1272
#225497802
#225497732
#225497632
#225497592
#225497512
#225497482
#225497392
#225497342
#381×C1272
#2855127