Finite Group Databasebeta

← #30639107

Subgroups — #30639107

Order 16256.

370 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16861237C211yesmaximal, characteristic
2#22550356C211yesmaximal, characteristic, complemented
3C2×#16861137C211yesmaximal, characteristic, complemented
4#22549880C211yesmaximal, characteristic, complemented
5#22549778C211yesmaximal, characteristic, complemented
6#22549644C211yesmaximal, characteristic, complemented
7#339×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C22×D508C2211yescharacteristic
10#16861237—22no—
11#16861237C414yescomplemented
12C2×#14523237—21no—
13C2×D1016—21no—
14C2×#14523232C2211yescharacteristic
15C4×Dic254—21no—
16C4×D508—21no—
17(C22⋊D127)⋊C4—22no—
18#16861167—21no—
19C2×C4⋊Dic127C2211yescharacteristic
20C4⋊C4×D127—21no—
21C2×(C2×D127)⋊C4—22no—
22#16861137—22no—
23#16861129—21no—
24#16861129—21no—
25C2×M16×C127C2211yescharacteristic
26#117×C127C2211yescharacteristic
27C2×C42×C127C2211yescharacteristic
28C23×D127—21no—
29#14523248—21no—
30#14523248—21no—
31#14523248C2×C412yes—
32#14523248D412yes—
33C2×Dic254D411yescharacteristic
34C2×D508—21no—
35C2×D508—21no—
36C2×D508—21no—
37C2×D508—21no—
38C2×D508—21no—
39C2×D508C2×C412yescomplemented
40C2×D508D411yescharacteristic
41C2×C22⋊D127—21no—
42C22×Dic127D412yes—
43C2×C4×D127—22no—
44C2×C4×D127—21no—
45#14523237—24no—
46D1016—24no—
47#14523232—21no—
48#14523232—21no—
49C2×C127⋊C8—21no—
50C4⋊Dic127—21no—
51C4⋊Dic127—21no—
52Dic127⋊C4—22no—
53(C2×D127)⋊C4—22no—
54(C2×D127)⋊C4—44no—
55C4×Dic127—22no—
56C22×C4×C127—21no—
57C22×C4×C127C2311yescharacteristic
58M16×C127—21no—
59M16×C127C2×C412yescomplemented
60C2×C8×C127—21no—
61C4⋊C4×C127D411yescharacteristic
62C4⋊C4×C127D411yescharacteristic
63C22⋊C4×C127—21no—
64C42×C127—21no—
65C42×C127—21no—
66C42×C127—21no—
67C22×D127—21no—
68C22×D127—22no—
69C22×D127—42no—
70C22×D127—42no—
71C22×D127—41no—
72C22×D127—41no—
73Dic254—21no—
74Dic254C22⋊C412yescomplemented
75D508—22no—
76D508—22no—
77D508—21no—
78D508—41no—
79D508—41no—
80D508C22⋊C412yescomplemented
81C22⋊D127—22no—
82C22⋊D127—22no—
83C2×Dic127—21no—
84C2×Dic127—22no—
85C2×Dic127—42no—
86C4×D127—22no—
87C4×D127—22no—
88C4×D127—82no—
89C127⋊C8—41no—
90C23×C127C2×D411yescharacteristic
91C2×C4×C127—22no—
92C2×C4×C127—21no—
93C2×C4×C127—21no—
94C2×C4×C127—41no—
95C2×C4×C127—41no—
96C2×C4×C127C22×C411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127#4111yescharacteristic
101C2×C4×C127#4111yescharacteristic
102C8×C127—22no—
103C22×C127—21no—
104C22×C127—21no—
105C22×C127C4×D411yescharacteristic
106C22×C127#12011yescharacteristic
107C22×C127#12111yescharacteristic
108C2×D127—22no—
109C2×D127—41no—
110C2×D127—41no—
111C2×D127—44no—
112C2×D127—42no—
113C2×D127—82no—
114C2×D127—81no—
115C2×D127—81no—
116C4×C127—42no—
117C4×C127—41no—
118C4×C127—41no—
119C4×C127C2×C22⋊C411yescharacteristic
120C4×C127C4×D411yescharacteristic
121C4×C127#12111yescharacteristic
122C4×C127#12311yescharacteristic
123Dic127—22no—
124Dic127—42no—
125C2×C127—21no—
126C2×C127—21no—
127C2×C127#38611yescharacteristic
128C2×C127#45311yescharacteristic
129C2×C127#45411yescharacteristic
130D127—42no—
131D127—82no—
132#1883—1271nomaximal, Sylow
133C127#188311yescharacteristic, complemented, Sylow
134C2×AGL(1,ℤ/8)—1271no—
135#520—1271no—
136C2×#104—1271no—
137#418—1271no—
138#390—1271no—
139#343—1271no—
140#339D12711yescharacteristic, complemented
141C2×#41—1271no—
142C22×D4—1271no—
143AGL(1,ℤ/8)—1274no—
144AGL(1,ℤ/8)—2542no—
145C2×SD16—2541no—
146C2×D8—2541no—
147C2×M16—1271no—
148C2×M16C2×D12711yescharacteristic
149C4×Q8—2541no—
150C4×D4—2541no—
151C4×D4—2542no—
152#117—2541no—
153#117C2×D12711yescharacteristic
154C2×C4⋊C4—1271no—
155C2×C4⋊C4—2541no—
156C2×C22⋊C4—2542no—
157C2×C42C2×D12711yescharacteristic
158#104—2542no—
159#102—2541no—
160#102—2541no—
161C24—2541no—
162#41—1272no—
163#41—1272no—
164#41—2541no—
165#41—2541no—
166C2×Q8—1271no—
167C2×D4—1272no—
168C2×D4—1271no—
169C2×D4—2541no—
170C2×D4—2541no—
171C2×D4—2541no—
172C2×D4—2541no—
173C2×D4—2541no—
174C2×D4—2541no—
175C22×C4—21no—
176C22×C4—1272no—
177C22×C4—2542no—
178C22×C4—2541no—
179C22×C4C22×D12711yescharacteristic
180SD16—2544no—
181D8—2544no—
182M16—21no—
183M16—2541no—
184M16—2541no—
185M16C4×D12712yescomplemented
186C2×C8—21no—
187C2×C8—2541no—
188C4⋊C4—2542no—
189C4⋊C4—2541no—
190C4⋊C4—2541no—
191C4⋊C4D50811yescharacteristic
192C4⋊C4D50811yescharacteristic
193C22⋊C4—21no—
194C22⋊C4—5084no—
195C22⋊C4—2542no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—2542no—
200C23—5082no—
201C23—5082no—
202C23—5081no—
203C23—5081no—
204C23—2542no—
205C23—2541no—
206C23D4×D12711yescharacteristic
207Q8—1272no—
208Q8—2541no—
209D4—1272no—
210D4—5081no—
211D4—5081no—
212D4—2542no—
213D4—2542no—
214D4—2541no—
215D4—2542no—
216D4—2542no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—10162no—
223C2×C4—5082no—
224C2×C4—2542no—
225C2×C4—2541no—
226C2×C4—2542no—
227C2×C4—2542no—
228C2×C4C2×C4×D12711yescharacteristic
229C2×C4D4×D12711yescharacteristic
230C2×C4D4×D12711yescharacteristic
231C2×C4C2×D50811yescharacteristic
232C2×C4#1452324811yescharacteristic
233C2×C4#1452324911yescharacteristic
234C8—22no—
235C8—5081no—
236C22—21no—
237C22—21no—
238C22—10162no—
239C22—10161no—
240C22—10161no—
241C22—5084no—
242C22—5082no—
243C22—5081no—
244C22—5081no—
245C22—2542no—
246C22#1686117111yescharacteristic
247C22#1686117911yescharacteristic
248C22#1686118211yescharacteristic
249C4—42no—
250C4—41no—
251C4—41no—
252C4—5082no—
253C4—2542no—
254C4C22⋊C4×D12711yescharacteristic
255C4C4×D50811yescharacteristic
256C4#1686118411yescharacteristic
257C4#1686119411yescharacteristic
258C2—21no—
259C2—21no—
260C2—10162no—
261C2—5082no—
262C2#2254975411yescharacteristic
263C2#2255004511yescharacteristic
264C2#2255005011yescharacteristic
subgroupG/H
centerC22#16861179
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#339×C127C2
socleC22×C127#120
prime radicalC2×#16861237C2
cosocleC22×C4×C127C23
hypercenter#339D127
hypocenterC127#1883
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #22550356C2semidirect
3. C2×#16861137C2semidirect
4. #22549880C2semidirect
5. #22549778C2semidirect
6. #22549644C2semidirect
7. #339×C127C2semidirect
11. #16861237C4semidirect
39. C2×D508C2×C4semidirect
59. M16×C127C2×C4semidirect
74. Dic254C22⋊C4semidirect
80. D508C22⋊C4semidirect
133. C127#1883semidirect
140. #339D127semidirect
185. M16C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612372
#225503562
C2×#168611372
#225498802
#225497782
#225496442
#339×C1272
#1883127