Finite Group Databasebeta

← #30639086

Subgroups — #30639086

Order 16256.

374 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#14523237C211yesmaximal, characteristic
2C2×#16861174C211yesmaximal, characteristic, complemented
3C2×#16861132C211yesmaximal, characteristic, complemented
4C2×#16861129C211yesmaximal, characteristic
5#22549777C211yesmaximal, characteristic
6#22549630C211yesmaximal, characteristic, complemented
7#340×C127C211yesmaximal, characteristic, complemented
8C22×Dic254C2211yescharacteristic
9C22×D508C2211yescharacteristic
10C2×#14523237—22no—
11C2×#14523237—22no—
12C2×#14523237C414yescomplemented
13C22×C127⋊C8C2211yescharacteristic
14#16861174—22no—
15#16861174—22no—
16C2×Dic127⋊C4—21no—
17C2×(C2×D127)⋊C4—21no—
18C2×(C2×D127)⋊C4—21no—
19C2×C4×Dic127—21no—
20C2×C4×Dic127C2211yescharacteristic
21#16861132—22no—
22#16861129—22no—
23C22×C8×C127C2211yescharacteristic
24C2×C4⋊C4×C127C2211yescharacteristic
25C2×C4⋊C4×C127C2211yescharacteristic
26C23×D127—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254C2×C412yes—
31C2×Dic254D412yes—
32C2×Dic254D412yes—
33C2×D508—21no—
34C2×D508—21no—
35C2×D508—21no—
36C2×D508—21no—
37C2×D508—21no—
38C2×D508C2×C412yes—
39C22×Dic127—21no—
40C22×Dic127D411yescharacteristic
41C22×Dic127D411yescharacteristic
42#14523237—28no—
43C2×C127⋊C8—21no—
44C2×C127⋊C8—21no—
45C2×C127⋊C8—21no—
46Dic127⋊C4—24no—
47(C2×D127)⋊C4—42no—
48(C2×D127)⋊C4—42no—
49C4×Dic127—24no—
50C4×Dic127—21no—
51C4×Dic127—21no—
52C22×C4×C127—21no—
53C22×C4×C127—21no—
54C22×C4×C127C2311yescharacteristic
55C2×C8×C127—21no—
56C2×C8×C127—21no—
57C2×C8×C127C2×C412yes—
58C4⋊C4×C127—21no—
59C4⋊C4×C127—21no—
60C4⋊C4×C127—21no—
61C4⋊C4×C127D412yes—
62C22×D127—22no—
63C22×D127—41no—
64C22×D127—41no—
65C22×D127—41no—
66C22×D127—41no—
67C22×D127—41no—
68C22×D127—41no—
69Dic254—22no—
70Dic254—22no—
71Dic254—22no—
72Dic254C22⋊C414yescomplemented
73D508—22no—
74D508—22no—
75D508—41no—
76D508—41no—
77C2×Dic127—22no—
78C2×Dic127—21no—
79C2×Dic127—22no—
80C2×Dic127—22no—
81C2×Dic127—21no—
82C2×Dic127—21no—
83C2×Dic127—21no—
84C2×Dic127—21no—
85C2×Dic127SD1612yescomplemented
86C127⋊C8—41no—
87C23×C127C2×D411yescharacteristic
88C2×C4×C127—22no—
89C2×C4×C127—41no—
90C2×C4×C127—41no—
91C2×C4×C127—41no—
92C2×C4×C127—41no—
93C2×C4×C127—41no—
94C2×C4×C127C22×C411yescharacteristic
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127#4111yescharacteristic
99C2×C4×C127#4111yescharacteristic
100C8×C127—22no—
101C22×C127C4×D411yescharacteristic
102C22×C127#12011yescharacteristic
103C22×C127#12111yescharacteristic
104C22×C127C2×SD1611yescharacteristic
105C22×C127#13511yescharacteristic
106C22×C127AGL(1,ℤ/8)11yescharacteristic
107C22×C127#13711yescharacteristic
108C2×D127—42no—
109C2×D127—42no—
110C2×D127—42no—
111C2×D127—81no—
112C2×D127—81no—
113C2×D127—81no—
114C2×D127—81no—
115C4×C127—42no—
116C4×C127—81no—
117C4×C127C2×C22⋊C411yescharacteristic
118C4×C127C4×D411yescharacteristic
119C4×C127#12111yescharacteristic
120C4×C127#12311yescharacteristic
121Dic127—24no—
122Dic127—22no—
123Dic127—41no—
124C2×C127#38611yescharacteristic
125C2×C127C4×SD1611yescharacteristic
126C2×C127#44011yescharacteristic
127C2×C127#44811yescharacteristic
128C2×C127#44911yescharacteristic
129C2×C127#46011yescharacteristic
130C2×C127#46411yescharacteristic
131D127—82no—
132#1880—1271nomaximal, Sylow
133C127#188011yescharacteristic, complemented, Sylow
134C22×SD16—1271no—
135C2×C4×Q8—1271no—
136C2×#103—1271no—
137C2×#102—1271no—
138#390—1271no—
139#340D12711yescharacteristic, complemented
140#336—1271no—
141C22×Q8—1271no—
142C22×D4—1271no—
143C2×SD16—1274no—
144C2×SD16—2542no—
145C2×SD16—2542no—
146C22×C8—1271no—
147C22×C8C2×D12711yescharacteristic
148C4×Q8—2542no—
149C4×Q8—2542no—
150C2×C4⋊C4—2541no—
151C2×C4⋊C4C2×D12711yescharacteristic
152C2×C4⋊C4C2×D12711yescharacteristic
153C2×C22⋊C4—2541no—
154C2×C22⋊C4—2541no—
155C2×C42—1271no—
156C2×C42—2541no—
157#103—2542no—
158#102—2542no—
159C24—2541no—
160C2×Q8—1272no—
161C2×Q8—1272no—
162C2×Q8—1272no—
163C2×Q8—2541no—
164C2×Q8—2541no—
165C2×Q8—2541no—
166C2×D4—1272no—
167C2×D4—2541no—
168C2×D4—2541no—
169C2×D4—2541no—
170C2×D4—2541no—
171C2×D4—2541no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—1271no—
175C22×C4—1271no—
176C22×C4—2541no—
177C22×C4C22×D12711yescharacteristic
178SD16—2548no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—2541no—
182C2×C8—2541no—
183C2×C8—2541no—
184C2×C8C4×D12712yes—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—2544no—
189C4⋊C4D50812yes—
190C22⋊C4—5082no—
191C22⋊C4—5082no—
192C42—2544no—
193C42—2541no—
194C42—2541no—
195C23—5081no—
196C23—5081no—
197C23—5081no—
198C23—5081no—
199C23—5081no—
200C23—5081no—
201C23—2542no—
202C23D4×D12711yescharacteristic
203Q8—1274no—
204Q8—2542no—
205Q8—2542no—
206Q8—2542no—
207D4—5081no—
208D4—5081no—
209D4—2542no—
210D4—2542no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—1272no—
218C2×C4—2541no—
219C2×C4—2541no—
220C2×C4—2541no—
221C2×C4—2541no—
222C2×C4—2542no—
223C2×C4—2541no—
224C2×C4—2542no—
225C2×C4—2542no—
226C2×C4C2×C4×D12711yescharacteristic
227C2×C4D4×D12711yescharacteristic
228C2×C4D4×D12711yescharacteristic
229C2×C4C2×D50811yescharacteristic
230C2×C4#1452324811yescharacteristic
231C2×C4#1452324911yescharacteristic
232C8—22no—
233C8—5081no—
234C22—10161no—
235C22—10161no—
236C22—10161no—
237C22—10161no—
238C22—5082no—
239C22—5082no—
240C22—5082no—
241C22#1686117111yescharacteristic
242C22#1686117911yescharacteristic
243C22#1686118211yescharacteristic
244C22SD16×D12711yescharacteristic
245C22#1686123011yescharacteristic
246C22#1686123311yescharacteristic
247C22#1686124111yescharacteristic
248C4—42no—
249C4—81no—
250C4—5081no—
251C4—2542no—
252C4—2544no—
253C4C22⋊C4×D12711yescharacteristic
254C4C4×D50811yescharacteristic
255C4#1686118411yescharacteristic
256C4#1686119411yescharacteristic
257C2—10162no—
258C2#2254975411yescharacteristic
259C2#2254996311yescharacteristic
260C2#2254997511yescharacteristic
261C2#2255001611yescharacteristic
262C2#2255002211yescharacteristic
263C2#2255007511yescharacteristic
264C2#2255010111yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#340×C127C2
socleC23×C127C2×D4
prime radicalC22×D508C22
cosocleC22×C4×C127C23
hypercenter#340D127
hypocenterC127#1880
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#16861174C2semidirect
3. C2×#16861132C2semidirect
6. #22549630C2semidirect
7. #340×C127C2semidirect
12. C2×#14523237C4semidirect
72. Dic254C22⋊C4semidirect
85. C2×Dic127SD16semidirect
133. C127#1880semidirect
139. #340D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232372
C2×#168611742
C2×#168611322
C2×#168611292
#225497772
#225496302
#340×C1272
#1880127