Finite Group Databasebeta

← #30639786

Subgroups — #30639786

Order 16256.

380 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 71 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#22550676C211yesmaximal, characteristic, complemented
2C2×#16861237C211yesmaximal, characteristic, complemented
3#22550445C211yesmaximal, characteristic, complemented
4#22549892C211yesmaximal, characteristic, complemented
5#22549882C211yesmaximal, characteristic, complemented
6#22549849C211yesmaximal, characteristic, complemented
7#344×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C2×#14523248C2211yescharacteristic
10C22×D508C2211yescharacteristic
11#16861242—21no—
12#16861242—21no—
13#16861237—22no—
14#16861237—21no—
15#16861237—21no—
16#16861237—22no—
17#16861231—22no—
18C2×#14523237—21no—
19C2×D1016—21no—
20#16861206—21no—
21#16861197—21no—
22#16861184—22no—
23#16861179—22no—
24#16861168C2211yescharacteristic
25#16861135—22no—
26#16861131—22no—
27#16861123—22no—
28C2×M16×C127C2211yescharacteristic
29C2×M16×C127C2211yescharacteristic
30#117×C127C2211yescharacteristic
31C23×D127—21no—
32#14523248—21no—
33#14523248—21no—
34#14523248—21no—
35#14523248—21no—
36#14523248—21no—
37#14523248—21no—
38#14523248—21no—
39#14523248—21no—
40C2×Dic254D411yescharacteristic
41C2×Dic254D411yescharacteristic, complemented
42C2×D508—24no—
43C2×D508—21no—
44C2×D508—21no—
45C2×D508—21no—
46C2×D508—21no—
47C2×D508—21no—
48C2×D508D412yes—
49C2×D508D411yescharacteristic
50C2×D508D411yescharacteristic, complemented
51C2×C22⋊D127—21no—
52C2×C22⋊D127—21no—
53C2×C4×D127—21no—
54C2×C4×D127—21no—
55Dic508—22no—
56#14523237—24no—
57#14523237—22no—
58#14523237—22no—
59D1016—24no—
60D1016—22no—
61C4⋊Dic127—21no—
62C22⋊Dic127—21no—
63(C2×D127)⋊C4—24no—
64C4×Dic127—21no—
65C22×C4×C127C2311yescharacteristic
66M16×C127—21no—
67M16×C127—21no—
68M16×C127—21no—
69M16×C127D412yescomplemented
70C2×C8×C127—21no—
71C2×C8×C127D412yescomplemented
72C4⋊C4×C127—21no—
73C22⋊C4×C127D412yes—
74C42×C127—21no—
75C22×D127—22no—
76C22×D127—21no—
77C22×D127—21no—
78C22×D127—41no—
79C22×D127—41no—
80C22×D127—82no—
81Dic254—21no—
82Dic254—21no—
83Dic254—21no—
84Dic254—21no—
85D508—21no—
86D508—22no—
87D508—21no—
88D508—21no—
89D508—22no—
90D508—21no—
91D508—41no—
92D508—41no—
93D508—84no—
94C22⋊D127—41no—
95C22⋊D127—41no—
96C22⋊D127—41no—
97C22⋊D127—41no—
98C2×Dic127—21no—
99C2×Dic127—21no—
100C2×Dic127—41no—
101C4×D127—41no—
102C4×D127—41no—
103C4×D127—41no—
104C4×D127—41no—
105C23×C127C2×D411yescharacteristic
106C2×C4×C127—22no—
107C2×C4×C127C2×D411yescharacteristic
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127#4111yescharacteristic
113C8×C127—22no—
114C8×C127—22no—
115C22×C127—41no—
116C22×C127#12011yescharacteristic
117C22×C127#12111yescharacteristic
118C22×C127#12711yescharacteristic
119C2×D127—42no—
120C2×D127—44no—
121C2×D127—41no—
122C2×D127—41no—
123C2×D127—42no—
124C2×D127—82no—
125C2×D127—82no—
126C2×D127—81no—
127C4×C127—42no—
128C4×C127#12011yescharacteristic
129C4×C127#12011yescharacteristic
130C4×C127#12111yescharacteristic
131C4×C127#12111yescharacteristic
132Dic127—41no—
133Dic127—41no—
134Dic127—81no—
135C2×C127—21no—
136C2×C127—21no—
137C2×C127—21no—
138C2×C127#39211yescharacteristic
139D127—81no—
140D127—82no—
141D127—81no—
142#2008—1271nomaximal, Sylow
143C127#200811yescharacteristic, complemented, Sylow
144#575—1271no—
145C2×AGL(1,ℤ/8)—1271no—
146#535—1271no—
147#421—1271no—
148#418—1271no—
149#410—1271no—
150#344D12711yescharacteristic, complemented
151C2×#41—1271no—
152C2×#41—1271no—
153C22×D4—1271no—
154#137—2541no—
155#137—2541no—
156AGL(1,ℤ/8)—2542no—
157AGL(1,ℤ/8)—2541no—
158AGL(1,ℤ/8)—2541no—
159AGL(1,ℤ/8)—2542no—
160#135—2542no—
161C2×SD16—2541no—
162C2×D8—2541no—
163C2×M16C2×D12711yescharacteristic
164C2×M16C2×D12711yescharacteristic
165#127—2541no—
166#124—2541no—
167#121—2542no—
168#120—2542no—
169#117—1271no—
170#117C2×D12711yescharacteristic
171#104—2542no—
172#102—2542no—
173#99—2542no—
174C24—2541no—
175#41—2541no—
176#41—2541no—
177#41—2541no—
178#41—2541no—
179#41—2541no—
180#41—2541no—
181#41—2541no—
182#41—2541no—
183C2×Q8—1271no—
184C2×Q8—1271no—
185C2×D4—1272no—
186C2×D4—1271no—
187C2×D4—1271no—
188C2×D4—2544no—
189C2×D4—2541no—
190C2×D4—2541no—
191C2×D4—2541no—
192C2×D4—2541no—
193C2×D4—2541no—
194C2×D4—2541no—
195C2×D4—2541no—
196C22×C4—2541no—
197C22×C4—2541no—
198C22×C4C22×D12711yescharacteristic
199Q16—2542no—
200SD16—2544no—
201SD16—2542no—
202SD16—2542no—
203D8—2544no—
204D8—2542no—
205M16—21no—
206M16—21no—
207M16—21no—
208M16D50812yescomplemented
209C2×C8—21no—
210C2×C8D50812yescomplemented
211C4⋊C4—21no—
212C4⋊C4—2541no—
213C22⋊C4—2544no—
214C22⋊C4—2541no—
215C22⋊C4D50812yes—
216C42—21no—
217C42—2541no—
218C23—10162no—
219C23—5081no—
220C23—5081no—
221C23—2541no—
222C23—2542no—
223C23—2541no—
224C23C2×D50811yescharacteristic
225Q8—2541no—
226Q8—2541no—
227Q8—2541no—
228Q8—2541no—
229D4—10164no—
230D4—5081no—
231D4—5081no—
232D4—5081no—
233D4—5081no—
234D4—5081no—
235D4—5081no—
236D4—2542no—
237D4—2541no—
238D4—2541no—
239D4—2542no—
240D4—2541no—
241D4—2541no—
242C2×C4—22no—
243C2×C4—5081no—
244C2×C4—5081no—
245C2×C4—5081no—
246C2×C4—5081no—
247C2×C4—5081no—
248C2×C4—2541no—
249C2×C4—2541no—
250C2×C4D4×D12711yescharacteristic
251C2×C4D4×D12711yescharacteristic
252C2×C4D4×D12711yescharacteristic
253C2×C4C2×D50811yescharacteristic
254C2×C4C2×D50811yescharacteristic
255C2×C4#1452324911yescharacteristic
256C8—22no—
257C8—22no—
258C22—41no—
259C22—10161no—
260C22—10162no—
261C22—10162no—
262C22—5082no—
263C22—5084no—
264C22—5081no—
265C22—5081no—
266C22—5082no—
267C22#1686117911yescharacteristic
268C22#1686118411yescharacteristic
269C22#1686120611yescharacteristic
270C4—42no—
271C4—10161no—
272C4—5081no—
273C4—5081no—
274C4#1686117911yescharacteristic
275C4#1686117911yescharacteristic
276C4#1686118411yescharacteristic
277C4#1686118411yescharacteristic
278C2—21no—
279C2—21no—
280C2—21no—
281C2—10162no—
282C2—10161no—
283C2—10161no—
284C2#2254978611yescharacteristic
subgroupG/H
centerC2#22549786
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#344×C127C2
socleC2×C127#392
hypercenter#344D127
hypocenterC127#2008
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
1. #22550676C2semidirect
2. C2×#16861237C2semidirect
3. #22550445C2semidirect
4. #22549892C2semidirect
5. #22549882C2semidirect
6. #22549849C2semidirect
7. #344×C127C2semidirect
41. C2×Dic254D4semidirect
50. C2×D508D4semidirect
69. M16×C127D4semidirect
71. C2×C8×C127D4semidirect
143. C127#2008semidirect
150. #344D127semidirect
208. M16D508semidirect
210. C2×C8D508semidirect

Maximal subgroups

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

subgroupindex
#225506762
C2×#168612372
#225504452
#225498922
#225498822
#225498492
#344×C1272
#2008127