Finite Group Databasebeta

← #30639781

Subgroups — #30639781

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#22550674C211yesmaximal, characteristic, complemented
2#22550443C211yesmaximal, characteristic, complemented
3#22549892C211yesmaximal, characteristic, complemented
4#22549881C211yesmaximal, characteristic, complemented
5#22549850C211yesmaximal, characteristic, complemented
6#22549648C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861240—21no—
10#16861239—21no—
11#16861235—21no—
12#16861234—21no—
13#16861229—22no—
14C2×#14523232C2211yescharacteristic
15#16861205—21no—
16#16861196—21no—
17#16861180—22no—
18#16861178—22no—
19#16861168C2211yescharacteristic
20#16861168C2211yescharacteristic
21#16861135—22no—
22#16861130—22no—
23#16861124—22no—
24C2×#41×C127C2211yescharacteristic
25C22×D4×C127C2211yescharacteristic
26AGL(1,ℤ/8)×C127—22no—
27AGL(1,ℤ/8)×C127—22no—
28C2×SD16×C127—21no—
29C2×D8×C127—21no—
30C2×M16×C127C2211yescharacteristic
31#14523248—21no—
32#14523248—21no—
33#14523248—21no—
34#14523248—21no—
35C2×Dic254D411yescharacteristic
36C2×D508D411yescharacteristic
37C2×C22⋊D127—24no—
38C2×C22⋊D127—21no—
39C2×C4×D127—21no—
40#14523238—22no—
41#14523236—22no—
42#14523235—22no—
43#14523233—22no—
44#14523232—21no—
45#14523232—21no—
46C2×C127⋊C8D412yescomplemented
47C4⋊Dic127—21no—
48C4⋊Dic127—21no—
49C22⋊Dic127—24no—
50C22⋊Dic127—21no—
51C22⋊Dic127D412yes—
52C4×Dic127—21no—
53C4×Dic127—21no—
54C24×C127—21no—
55#41×C127—21no—
56#41×C127—21no—
57#41×C127—21no—
58#41×C127—21no—
59C2×Q8×C127D411yescharacteristic, complemented
60C2×D4×C127—21no—
61C2×D4×C127—21no—
62C2×D4×C127—21no—
63C2×D4×C127—21no—
64C2×D4×C127—21no—
65C2×D4×C127—21no—
66C2×D4×C127D412yes—
67C2×D4×C127D411yescharacteristic, complemented
68C22×C4×C127—21no—
69C22×C4×C127C2311yescharacteristic
70SD16×C127—24no—
71D8×C127—24no—
72M16×C127—21no—
73M16×C127D412yescomplemented
74C2×C8×C127—21no—
75C22×D127—21no—
76Dic254—21no—
77Dic254—21no—
78D508—21no—
79D508—21no—
80C22⋊D127—41no—
81C22⋊D127—41no—
82C22⋊D127—84no—
83C2×Dic127—22no—
84C2×Dic127—21no—
85C2×Dic127—41no—
86C4×D127—41no—
87C4×D127—41no—
88C127⋊C8—22no—
89C23×C127—22no—
90C23×C127—21no—
91C23×C127—41no—
92C23×C127—41no—
93C23×C127—82no—
94C23×C127C2×D411yescharacteristic
95Q8×C127—21no—
96Q8×C127—21no—
97D4×C127—21no—
98D4×C127—22no—
99D4×C127—21no—
100D4×C127—22no—
101D4×C127—41no—
102D4×C127—41no—
103D4×C127—41no—
104D4×C127—41no—
105C2×C4×C127—21no—
106C2×C4×C127—41no—
107C2×C4×C127—41no—
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2×D411yescharacteristic
113C2×C4×C127#4111yescharacteristic
114C8×C127—22no—
115C22×C127—41no—
116C22×C127—44no—
117C22×C127—41no—
118C22×C127—42no—
119C22×C127—81no—
120C22×C127—82no—
121C22×C127—82no—
122C22×C127#12011yescharacteristic
123C22×C127#12111yescharacteristic
124C22×C127#12711yescharacteristic
125C2×D127—42no—
126C2×D127—41no—
127C4×C127—41no—
128C4×C127#12011yescharacteristic
129C4×C127#12011yescharacteristic
130C4×C127#12111yescharacteristic
131C4×C127#12111yescharacteristic
132Dic127—42no—
133Dic127—41no—
134Dic127—81no—
135C2×C127—21no—
136C2×C127—21no—
137C2×C127—21no—
138C2×C127—82no—
139C2×C127—81no—
140C2×C127#39211yescharacteristic
141D127—81no—
142#2008—1271nomaximal, Sylow
143C127#200811yescharacteristic, complemented, Sylow
144#575—1271no—
145C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
146#535—1271no—
147#421—1271no—
148#418—1271no—
149#410—1271no—
150#344—1271no—
151C2×#41—1271no—
152C2×#41C2×D12711yescharacteristic
153C22×D4C2×D12711yescharacteristic
154#137—2541no—
155#137—2541no—
156AGL(1,ℤ/8)—22no—
157AGL(1,ℤ/8)—22no—
158AGL(1,ℤ/8)—2541no—
159AGL(1,ℤ/8)—2541no—
160#135—2542no—
161C2×SD16—21no—
162C2×D8—21no—
163C2×M16—1271no—
164C2×M16C2×D12711yescharacteristic
165#127—2541no—
166#124—2541no—
167#121—2542no—
168#120—2542no—
169#117—1271no—
170#117—1271no—
171#104—2542no—
172#102—2542no—
173#99—2542no—
174C24—21no—
175#41—21no—
176#41—21no—
177#41—21no—
178#41—21no—
179#41—2541no—
180#41—2541no—
181#41—2541no—
182#41—2541no—
183C2×Q8—1271no—
184C2×Q8C22⋊D12711yescharacteristic, complemented
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—1271no—
192C2×D4—2544no—
193C2×D4—2541no—
194C2×D4C22⋊D12712yes—
195C2×D4C22⋊D12711yescharacteristic, complemented
196C22×C4—21no—
197C22×C4—2541no—
198C22×C4C22×D12711yescharacteristic
199Q16—2542no—
200SD16—24no—
201SD16—2542no—
202SD16—2542no—
203D8—24no—
204D8—2542no—
205M16—21no—
206M16—2541no—
207M16—2541no—
208M16C22⋊D12712yescomplemented
209C2×C8—21no—
210C2×C8—1272no—
211C4⋊C4—2541no—
212C4⋊C4—2541no—
213C22⋊C4—1272no—
214C22⋊C4—2541no—
215C22⋊C4—2544no—
216C42—2541no—
217C42—2541no—
218C23—22no—
219C23—21no—
220C23—41no—
221C23—41no—
222C23—82no—
223C23—2541no—
224C23D4×D12711yescharacteristic
225Q8—21no—
226Q8—21no—
227Q8—2541no—
228Q8—2541no—
229D4—21no—
230D4—21no—
231D4—22no—
232D4—22no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—10164no—
238D4—5081no—
239D4—5081no—
240D4—2541no—
241D4—2541no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—5081no—
246C2×C4—5081no—
247C2×C4—5081no—
248C2×C4—2542no—
249C2×C4—2541no—
250C2×C4D4×D12711yescharacteristic
251C2×C4D4×D12711yescharacteristic
252C2×C4C2×C22⋊D12711yescharacteristic
253C2×C4C2×C22⋊D12711yescharacteristic
254C2×C4C2×C22⋊D12711yescharacteristic
255C2×C4#1452324711yescharacteristic
256C8—22no—
257C8—2542no—
258C22—41no—
259C22—44no—
260C22—41no—
261C22—42no—
262C22—81no—
263C22—82no—
264C22—82no—
265C22—5082no—
266C22—5081no—
267C22#1686117711yescharacteristic
268C22#1686118111yescharacteristic
269C22#1686120511yescharacteristic
270C4—41no—
271C4—10161no—
272C4—5082no—
273C4—5081no—
274C4#1686117711yescharacteristic
275C4#1686117811yescharacteristic
276C4#1686118011yescharacteristic
277C4#1686118111yescharacteristic
278C2—21no—
279C2—21no—
280C2—21no—
281C2—82no—
282C2—81no—
283C2—10161no—
284C2#2254978411yescharacteristic
subgroupG/H
centerC2#22549784
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×AGL(1,ℤ/8)×C127C2
socleC2×C127#392
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#2008
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #22550674C2semidirect
2. #22550443C2semidirect
3. #22549892C2semidirect
4. #22549881C2semidirect
5. #22549850C2semidirect
6. #22549648C2semidirect
7. C2×AGL(1,ℤ/8)×C127C2semidirect
46. C2×C127⋊C8D4semidirect
59. C2×Q8×C127D4semidirect
67. C2×D4×C127D4semidirect
73. M16×C127D4semidirect
143. C127#2008semidirect
145. C2×AGL(1,ℤ/8)D127semidirect
184. C2×Q8C22⋊D127semidirect
195. C2×D4C22⋊D127semidirect
208. M16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225506742
#225504432
#225498922
#225498812
#225498502
#225496482
C2×AGL(1,ℤ/8)×C1272
#2008127