Finite Group Databasebeta

← #30639747

Subgroups — #30639747

Order 16256.

478 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 73 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, complemented
2#22550443C211yesmaximal, characteristic, complemented
3C2×#16861205C211yesmaximal, characteristic, complemented
4C2×#16861135C211yesmaximal, characteristic, complemented
5#22549882C211yesmaximal, characteristic, complemented
6#22549637C211yesmaximal, characteristic, complemented
7C2×#100×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C22×D508C2211yescharacteristic
10C22×C22⋊D127—22no—
11#16861237—22no—
12#16861237—22no—
13C2×#14523237—21no—
14C2×D1016—21no—
15#16861205—22no—
16#16861205—21no—
17#16861205—22no—
18#16861196—21no—
19#16861180—22no—
20#16861178—22no—
21#16861168C2211yescharacteristic
22C2×C4×Dic127C2211yescharacteristic
23#16861135—22no—
24#16861131—22no—
25C22×D4×C127C2211yescharacteristic
26C2×M16×C127C2211yescharacteristic
27C2×M16×C127C2211yescharacteristic
28#100×C127—22no—
29C23×D127—21no—
30#14523248—21no—
31#14523248—21no—
32#14523248—21no—
33#14523248—21no—
34C2×Dic254D411yescharacteristic, complemented
35C2×D508—21no—
36C2×D508—21no—
37C2×D508—21no—
38C2×D508—21no—
39C2×D508—21no—
40C2×D508D411yescharacteristic, complemented
41C2×D508D412yescomplemented
42C2×C22⋊D127—28no—
43C2×C22⋊D127—24no—
44C2×C22⋊D127—21no—
45C2×C22⋊D127—48no—
46C22×Dic127—21no—
47C2×C4×D127—21no—
48#14523237—24no—
49D1016—24no—
50C4⋊Dic127D412yescomplemented
51C22⋊Dic127—24no—
52C22⋊Dic127—21no—
53C4×Dic127—21no—
54C4×Dic127—21no—
55C4×Dic127D412yescomplemented
56C24×C127D412yes—
57C2×D4×C127—21no—
58C2×D4×C127—21no—
59C2×D4×C127—21no—
60C2×D4×C127—21no—
61C2×D4×C127—21no—
62C2×D4×C127—21no—
63C22×C4×C127C2311yescharacteristic
64M16×C127—21no—
65M16×C127—21no—
66M16×C127—21no—
67M16×C127D412yescomplemented
68C2×C8×C127—21no—
69C2×C8×C127—21no—
70C22×D127—21no—
71C22×D127—22no—
72C22×D127—44no—
73C22×D127—41no—
74C22×D127—41no—
75Dic254—21no—
76Dic254—21no—
77D508—21no—
78D508—22no—
79D508—22no—
80D508—21no—
81D508—41no—
82D508—41no—
83C22⋊D127—416no—
84C22⋊D127—41no—
85C22⋊D127—41no—
86C22⋊D127—84no—
87C2×Dic127—24no—
88C2×Dic127—22no—
89C2×Dic127—22no—
90C2×Dic127—21no—
91C4×D127—41no—
92C4×D127—41no—
93C23×C127—22no—
94C23×C127—22no—
95C23×C127—22no—
96C23×C127—44no—
97C23×C127C2×D411yescharacteristic
98D4×C127—41no—
99D4×C127—41no—
100D4×C127—41no—
101D4×C127—41no—
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127#4111yescharacteristic
108C8×C127—22no—
109C8×C127—41no—
110C22×C127—22no—
111C22×C127—44no—
112C22×C127—42no—
113C22×C127—44no—
114C22×C127—82no—
115C22×C127#12011yescharacteristic
116C22×C127#12011yescharacteristic
117C22×C127#12111yescharacteristic
118C2×D127—44no—
119C2×D127—41no—
120C2×D127—42no—
121C2×D127—42no—
122C2×D127—82no—
123C2×D127—82no—
124C4×C127#12011yescharacteristic
125C4×C127#12111yescharacteristic
126C4×C127#12111yescharacteristic
127C4×C127#12711yescharacteristic
128Dic127—24no—
129Dic127—41no—
130Dic127—42no—
131C2×C127—21no—
132C2×C127—21no—
133C2×C127—82no—
134C2×C127#39211yescharacteristic
135C2×C127#45312yes—
136D127—81no—
137D127—82no—
138#2002—1271nomaximal, Sylow
139C127#200211yescharacteristic, complemented, Sylow
140C2×AGL(1,ℤ/8)—1271no—
141#535—1271no—
142C2×#127—1271no—
143C2×#104—1271no—
144#418—1271no—
145C2×#100D12711yescharacteristic, complemented
146#339—1271no—
147C2×#41—1271no—
148C22×D4—1271no—
149C22×D4—2542no—
150C22×D4C2×D12711yescharacteristic
151AGL(1,ℤ/8)—2542no—
152AGL(1,ℤ/8)—2542no—
153C2×SD16—2541no—
154C2×D8—2541no—
155C2×M16C2×D12711yescharacteristic
156C2×M16C2×D12711yescharacteristic
157#127—2542no—
158#127—2542no—
159#127—2541no—
160#124—2541no—
161#121—2542no—
162#120—2542no—
163#117—1271no—
164C2×C42—1271no—
165#104—2542no—
166#102—2542no—
167#100—22no—
168C24—2541no—
169C24D50812yes—
170#41—2541no—
171#41—2541no—
172#41—2541no—
173#41—2541no—
174C2×Q8—1271no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—1271no—
182C2×D4—1272no—
183C2×D4—5088no—
184C2×D4—2544no—
185C2×D4—2548no—
186C2×D4—2541no—
187C2×D4—2541no—
188C2×D4—2541no—
189C2×D4—2541no—
190C2×D4—2541no—
191C2×D4—2541no—
192C22×C4—2541no—
193C22×C4—2541no—
194C22×C4C22×D12711yescharacteristic
195SD16—2544no—
196D8—2544no—
197M16—21no—
198M16—21no—
199M16—21no—
200M16C22⋊D12712yescomplemented
201C2×C8—21no—
202C2×C8—21no—
203C4⋊C4—1272no—
204C22⋊C4—2544no—
205C22⋊C4—2541no—
206C42—1272no—
207C42—2541no—
208C42—2541no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—44no—
213C23—5084no—
214C23—5081no—
215C23—5081no—
216C23—2542no—
217C23—2541no—
218C23C2×D50811yescharacteristic
219Q8—2541no—
220Q8—2541no—
221D4—41no—
222D4—41no—
223D4—41no—
224D4—41no—
225D4—10164no—
226D4—5081no—
227D4—5081no—
228D4—50816no—
229D4—5081no—
230D4—5081no—
231D4—2541no—
232D4—2542no—
233D4—2542no—
234D4—2541no—
235C2×C4—5081no—
236C2×C4—5081no—
237C2×C4—2542no—
238C2×C4—2541no—
239C2×C4—2544no—
240C2×C4—2542no—
241C2×C4D4×D12711yescharacteristic
242C2×C4D4×D12711yescharacteristic
243C2×C4D4×D12711yescharacteristic
244C2×C4D4×D12711yescharacteristic
245C2×C4C2×C22⋊D12711yescharacteristic
246C2×C4#1452324811yescharacteristic
247C8—22no—
248C8—41no—
249C22—22no—
250C22—44no—
251C22—42no—
252C22—44no—
253C22—82no—
254C22—10162no—
255C22—10162no—
256C22—5084no—
257C22—5081no—
258C22—5082no—
259C22—5082no—
260C22#1686117911yescharacteristic
261C22#1686117911yescharacteristic
262C22#1686118311yescharacteristic
263C4—5081no—
264C4—5082no—
265C4—2544no—
266C4#1686117711yescharacteristic
267C4#1686118211yescharacteristic
268C4#1686118211yescharacteristic
269C4#1686120511yescharacteristic
270C2—21no—
271C2—21no—
272C2—82no—
273C2—10162no—
274C2—10161no—
275C2#2254978511yescharacteristic
276C2#2255004612yes—
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#100×C127C2
socleC22×C127#120
hypercenterC2×#100D127
hypocenterC127#2002
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
1. C2×#16861237C2semidirect
2. #22550443C2semidirect
3. C2×#16861205C2semidirect
4. C2×#16861135C2semidirect
5. #22549882C2semidirect
6. #22549637C2semidirect
7. C2×#100×C127C2semidirect
34. C2×Dic254D4semidirect
40. C2×D508D4semidirect
41. C2×D508D4semidirect
50. C4⋊Dic127D4semidirect
55. C4×Dic127D4semidirect
67. M16×C127D4semidirect
139. C127#2002semidirect
145. C2×#100D127semidirect
200. M16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612372
#225504432
C2×#168612052
C2×#168611352
#225498822
#225496372
C2×#100×C1272
#2002127