Finite Group Databasebeta

← #30639785

Subgroups — #30639785

Order 16256.

380 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 69 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×#16861235C211yesmaximal, characteristic, complemented
3#22550443C211yesmaximal, characteristic, complemented
4#22549894C211yesmaximal, characteristic, complemented
5#22549849C211yesmaximal, characteristic, complemented
6#22549647C211yesmaximal, characteristic, complemented
7#418×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C2×#14523248C2211yescharacteristic
10#16861242—21no—
11#16861242—21no—
12#16861237—21no—
13#16861237—21no—
14#16861235—22no—
15#16861235—22no—
16#16861231—22no—
17C2×#14523236—21no—
18C2×#14523233—21no—
19C2×#14523232C2211yescharacteristic
20#16861205—21no—
21#16861196—21no—
22#16861180—22no—
23#16861178—22no—
24#16861168C2211yescharacteristic
25#16861137—22no—
26#16861123—22no—
27C22×D4×C127C2211yescharacteristic
28C2×M16×C127C2211yescharacteristic
29#117×C127C2211yescharacteristic
30#102×C127—22no—
31#14523248—21no—
32#14523248—21no—
33#14523248—21no—
34#14523248—21no—
35#14523248—21no—
36#14523248—21no—
37#14523248—21no—
38#14523248—21no—
39C2×Dic254D411yescharacteristic
40C2×Dic254D411yescharacteristic, complemented
41C2×D508D411yescharacteristic
42C2×D508D411yescharacteristic, complemented
43C2×C22⋊D127—24no—
44C2×C22⋊D127—21no—
45C2×C22⋊D127—21no—
46C2×C4×D127—21no—
47C2×C4×D127—21no—
48Dic508—22no—
49#14523237—22no—
50#14523237—22no—
51#14523236—24no—
52D1016—22no—
53#14523233—24no—
54#14523232—21no—
55#14523232D412yescomplemented
56C2×C127⋊C8—21no—
57C4⋊Dic127—21no—
58C22⋊Dic127—24no—
59C22⋊Dic127D412yes—
60C4×Dic127—21no—
61C24×C127—21no—
62C2×D4×C127—21no—
63C2×D4×C127—21no—
64C2×D4×C127—21no—
65C2×D4×C127—21no—
66C2×D4×C127—21no—
67C2×D4×C127D412yes—
68C22×C4×C127C2311yescharacteristic
69M16×C127—21no—
70M16×C127—21no—
71C2×C8×C127D412yescomplemented
72C4⋊C4×C127—21no—
73C22⋊C4×C127—21no—
74C42×C127—21no—
75C22×D127—21no—
76C22×D127—21no—
77Dic254—21no—
78Dic254—21no—
79Dic254—21no—
80Dic254—21no—
81D508—21no—
82D508—21no—
83D508—21no—
84D508—21no—
85C22⋊D127—41no—
86C22⋊D127—41no—
87C22⋊D127—41no—
88C22⋊D127—41no—
89C22⋊D127—84no—
90C2×Dic127—22no—
91C2×Dic127—21no—
92C2×Dic127—21no—
93C4×D127—41no—
94C4×D127—41no—
95C4×D127—41no—
96C4×D127—41no—
97C127⋊C8—22no—
98C23×C127—22no—
99C23×C127—41no—
100C23×C127—41no—
101C23×C127—82no—
102C23×C127C2×D411yescharacteristic
103D4×C127—22no—
104D4×C127—22no—
105D4×C127—41no—
106D4×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—42no—
118C22×C127—82no—
119C22×C127—82no—
120C22×C127#12011yescharacteristic
121C22×C127#12111yescharacteristic
122C22×C127#12711yescharacteristic
123C2×D127—42no—
124C2×D127—41no—
125C2×D127—41no—
126C2×D127—81no—
127C4×C127—81no—
128C4×C127#12011yescharacteristic
129C4×C127#12011yescharacteristic
130C4×C127#12111yescharacteristic
131C4×C127#12111yescharacteristic
132Dic127—42no—
133Dic127—41no—
134Dic127—41no—
135C2×C127—21no—
136C2×C127—21no—
137C2×C127—21no—
138C2×C127—82no—
139C2×C127#39211yescharacteristic
140D127—81no—
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#418D12711yescharacteristic, complemented
149#410—1271no—
150#344—1271no—
151C2×#41—1271no—
152C2×#41—1271no—
153C22×D4C2×D12711yescharacteristic
154#137—2541no—
155#137—2541no—
156AGL(1,ℤ/8)—2541no—
157AGL(1,ℤ/8)—2541no—
158AGL(1,ℤ/8)—2542no—
159AGL(1,ℤ/8)—2542no—
160#135—2542no—
161C2×SD16—2541no—
162C2×D8—2541no—
163C2×M16—1271no—
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—22no—
173#99—2542no—
174C24—21no—
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—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—1271no—
191C2×D4—1271no—
192C2×D4—2544no—
193C2×D4—2541no—
194C2×D4—2541no—
195C2×D4D50812yes—
196C22×C4—2541no—
197C22×C4—2541no—
198C22×C4C22×D12711yescharacteristic
199Q16—2542no—
200SD16—2542no—
201SD16—2542no—
202SD16—2544no—
203D8—2542no—
204D8—2544no—
205M16—21no—
206M16—21no—
207M16—1272no—
208M16—2541no—
209C2×C8—2541no—
210C2×C8C22⋊D12712yescomplemented
211C4⋊C4—21no—
212C4⋊C4—2541no—
213C22⋊C4—21no—
214C22⋊C4—1272no—
215C22⋊C4—2544no—
216C42—21no—
217C42—2541no—
218C23—22no—
219C23—41no—
220C23—41no—
221C23—82no—
222C23—2541no—
223C23—2541no—
224C23D4×D12711yescharacteristic
225Q8—2541no—
226Q8—2541no—
227Q8—2541no—
228Q8—2541no—
229D4—22no—
230D4—22no—
231D4—41no—
232D4—41no—
233D4—10164no—
234D4—5081no—
235D4—5081no—
236D4—5081no—
237D4—5081no—
238D4—2541no—
239D4—2541no—
240D4—2541no—
241D4—2541no—
242C2×C4—41no—
243C2×C4—5081no—
244C2×C4—5081no—
245C2×C4—5081no—
246C2×C4—5081no—
247C2×C4—2542no—
248C2×C4—2541no—
249C2×C4—2541no—
250C2×C4D4×D12711yescharacteristic
251C2×C4D4×D12711yescharacteristic
252C2×C4D4×D12711yescharacteristic
253C2×C4C2×C22⋊D12711yescharacteristic
254C2×C4C2×D50811yescharacteristic
255C2×C4#1452324811yescharacteristic
256C8—22no—
257C8—2542no—
258C22—41no—
259C22—44no—
260C22—42no—
261C22—82no—
262C22—82no—
263C22—10161no—
264C22—5082no—
265C22—5081no—
266C22—5081no—
267C22#1686117911yescharacteristic
268C22#1686118211yescharacteristic
269C22#1686120511yescharacteristic
270C4—81no—
271C4—5082no—
272C4—5081no—
273C4—5081no—
274C4#1686117711yescharacteristic
275C4#1686117911yescharacteristic
276C4#1686118211yescharacteristic
277C4#1686118311yescharacteristic
278C2—21no—
279C2—21no—
280C2—21no—
281C2—82no—
282C2—10161no—
283C2—10161no—
284C2#2254978511yescharacteristic
subgroupG/H
centerC2#22549785
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#418×C127C2
socleC2×C127#392
hypercenter#418D127
hypocenterC127#2008
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #22550676C2semidirect
2. C2×#16861235C2semidirect
3. #22550443C2semidirect
4. #22549894C2semidirect
5. #22549849C2semidirect
6. #22549647C2semidirect
7. #418×C127C2semidirect
40. C2×Dic254D4semidirect
42. C2×D508D4semidirect
55. #14523232D4semidirect
71. C2×C8×C127D4semidirect
143. C127#2008semidirect
148. #418D127semidirect
210. C2×C8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225506762
C2×#168612352
#225504432
#225498942
#225498492
#225496472
#418×C1272
#2008127