Finite Group Databasebeta

← #30639782

Subgroups — #30639782

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#22550674C211yesmaximal, characteristic, complemented
2C2×#16861234C211yesmaximal, characteristic, complemented
3#22550445C211yesmaximal, characteristic, complemented
4#22549894C211yesmaximal, characteristic, complemented
5#22549880C211yesmaximal, characteristic, complemented
6#22549646C211yesmaximal, characteristic, complemented
7#410×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C22×D508C2211yescharacteristic
10#16861240—21no—
11#16861239—21no—
12#16861235—21no—
13#16861234—22no—
14#16861234—21no—
15#16861234—22no—
16#16861229—22no—
17C2×#14523235—21no—
18C2×#14523233—21no—
19C2×#14523232C2211yescharacteristic
20C2×#14523232C2211yescharacteristic
21#16861206—21no—
22#16861197—21no—
23#16861184—22no—
24#16861179—22no—
25#16861137—22no—
26#16861129—22no—
27C2×#41×C127C2211yescharacteristic
28#117×C127C2211yescharacteristic
29#117×C127C2211yescharacteristic
30#99×C127—22no—
31C23×D127—21no—
32#14523248—21no—
33#14523248—21no—
34#14523248—21no—
35#14523248—21no—
36C2×Dic254D411yescharacteristic
37C2×D508—24no—
38C2×D508—21no—
39C2×D508—21no—
40C2×D508—21no—
41C2×D508—21no—
42C2×D508—21no—
43C2×D508D412yes—
44C2×D508D411yescharacteristic
45C2×C22⋊D127—21no—
46C2×C4×D127—21no—
47#14523238—22no—
48#14523236—22no—
49#14523235—24no—
50#14523235—22no—
51#14523233—24no—
52#14523233—22no—
53#14523232—21no—
54#14523232—21no—
55#14523232—21no—
56#14523232D412yescomplemented
57C2×C127⋊C8—21no—
58C2×C127⋊C8D412yescomplemented
59(C2×D127)⋊C4—24no—
60#41×C127—21no—
61#41×C127—21no—
62#41×C127—21no—
63#41×C127—21no—
64C2×Q8×C127D411yescharacteristic, complemented
65C2×D4×C127—21no—
66C2×D4×C127D411yescharacteristic, complemented
67C22×C4×C127—21no—
68C22×C4×C127C2311yescharacteristic
69C4⋊C4×C127—21no—
70C4⋊C4×C127—21no—
71C22⋊C4×C127—21no—
72C22⋊C4×C127D412yes—
73C42×C127—21no—
74C42×C127—21no—
75C22×D127—22no—
76C22×D127—21no—
77C22×D127—41no—
78C22×D127—41no—
79C22×D127—82no—
80Dic254—21no—
81Dic254—21no—
82D508—22no—
83D508—21no—
84D508—22no—
85D508—21no—
86D508—41no—
87D508—41no—
88D508—84no—
89C22⋊D127—41no—
90C22⋊D127—41no—
91C2×Dic127—21no—
92C4×D127—41no—
93C4×D127—41no—
94C127⋊C8—22no—
95C127⋊C8—22no—
96C23×C127—21no—
97C23×C127C2×D411yescharacteristic
98Q8×C127—21no—
99Q8×C127—21no—
100D4×C127—21no—
101D4×C127—21no—
102D4×C127—41no—
103D4×C127—41no—
104C2×C4×C127—22no—
105C2×C4×C127—21no—
106C2×C4×C127—41no—
107C2×C4×C127—41no—
108C2×C4×C127—41no—
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2×D411yescharacteristic
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127#4111yescharacteristic
115C22×C127—41no—
116C22×C127—41no—
117C22×C127—81no—
118C22×C127#12011yescharacteristic
119C22×C127#12111yescharacteristic
120C22×C127#12711yescharacteristic
121C2×D127—42no—
122C2×D127—44no—
123C2×D127—41no—
124C2×D127—42no—
125C2×D127—82no—
126C2×D127—82no—
127C4×C127—42no—
128C4×C127—41no—
129C4×C127—81no—
130C4×C127#12011yescharacteristic
131C4×C127#12011yescharacteristic
132C4×C127#12111yescharacteristic
133C4×C127#12111yescharacteristic
134Dic127—41no—
135C2×C127—21no—
136C2×C127—21no—
137C2×C127—21no—
138C2×C127—81no—
139C2×C127#39211yescharacteristic
140D127—81no—
141D127—82no—
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#410D12711yescharacteristic, complemented
150#344—1271no—
151C2×#41—1271no—
152C2×#41C2×D12711yescharacteristic
153C22×D4—1271no—
154#137—2541no—
155#137—2541no—
156AGL(1,ℤ/8)—2542no—
157AGL(1,ℤ/8)—2542no—
158AGL(1,ℤ/8)—2541no—
159AGL(1,ℤ/8)—2541no—
160#135—2542no—
161C2×SD16—2541no—
162C2×D8—2541no—
163C2×M16—1271no—
164C2×M16—1271no—
165#127—2541no—
166#124—2541no—
167#121—2542no—
168#120—2542no—
169#117C2×D12711yescharacteristic
170#117C2×D12711yescharacteristic
171#104—2542no—
172#102—2542no—
173#99—22no—
174C24—2541no—
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×Q8D50811yescharacteristic, complemented
185C2×D4—21no—
186C2×D4—1272no—
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×D4D50811yescharacteristic, complemented
196C22×C4—21no—
197C22×C4—2541no—
198C22×C4C22×D12711yescharacteristic
199Q16—2542no—
200SD16—2542no—
201SD16—2544no—
202SD16—2542no—
203D8—2544no—
204D8—2542no—
205M16—1272no—
206M16—2541no—
207M16—2541no—
208M16—2541no—
209C2×C8—1272no—
210C2×C8—2541no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—2544no—
215C22⋊C4C22⋊D12712yes—
216C42—21no—
217C42—21no—
218C23—21no—
219C23—10162no—
220C23—5081no—
221C23—5081no—
222C23—2542no—
223C23—2541no—
224C23C2×C22⋊D12711yescharacteristic
225Q8—21no—
226Q8—21no—
227Q8—2541no—
228Q8—2541no—
229D4—21no—
230D4—21no—
231D4—41no—
232D4—41no—
233D4—10164no—
234D4—5081no—
235D4—5081no—
236D4—5081no—
237D4—5081no—
238D4—2542no—
239D4—2541no—
240D4—2542no—
241D4—2541no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—5081no—
248C2×C4—5081no—
249C2×C4—2541no—
250C2×C4D4×D12711yescharacteristic
251C2×C4D4×D12711yescharacteristic
252C2×C4D4×D12711yescharacteristic
253C2×C4D4×D12711yescharacteristic
254C2×C4C2×D50811yescharacteristic
255C2×C4#1452324811yescharacteristic
256C8—2542no—
257C8—2542no—
258C22—41no—
259C22—41no—
260C22—81no—
261C22—10162no—
262C22—10162no—
263C22—5082no—
264C22—5084no—
265C22—5081no—
266C22—5082no—
267C22#1686117711yescharacteristic
268C22#1686118311yescharacteristic
269C22#1686120511yescharacteristic
270C4—42no—
271C4—41no—
272C4—81no—
273C4—5081no—
274C4#1686117911yescharacteristic
275C4#1686117911yescharacteristic
276C4#1686118211yescharacteristic
277C4#1686118211yescharacteristic
278C2—21no—
279C2—21no—
280C2—21no—
281C2—81no—
282C2—10162no—
283C2—10161no—
284C2#2254978511yescharacteristic
subgroupG/H
centerC2#22549785
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#410×C127C2
socleC2×C127#392
hypercenter#410D127
hypocenterC127#2008
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #22550674C2semidirect
2. C2×#16861234C2semidirect
3. #22550445C2semidirect
4. #22549894C2semidirect
5. #22549880C2semidirect
6. #22549646C2semidirect
7. #410×C127C2semidirect
56. #14523232D4semidirect
58. C2×C127⋊C8D4semidirect
64. C2×Q8×C127D4semidirect
66. C2×D4×C127D4semidirect
143. C127#2008semidirect
149. #410D127semidirect
184. C2×Q8D508semidirect
195. C2×D4D508semidirect

Maximal subgroups

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

subgroupindex
#225506742
C2×#168612342
#225504452
#225498942
#225498802
#225496462
#410×C1272
#2008127