Finite Group Databasebeta

← #30639773

Subgroups — #30639773

Order 16256.

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

normal subgroups only

This group has 406 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
2C2×#16861234C211yesmaximal, characteristic, complemented
3#22550454C211yesmaximal, characteristic, complemented
4C2×#16861137C211yesmaximal, characteristic, complemented
5#22549778C211yesmaximal, characteristic, complemented
6#22549644C211yesmaximal, characteristic, complemented
7C2×#104×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C22×D508C2211yescharacteristic
10#16861237—22no—
11#16861237—22no—
12#16861234—22no—
13#16861234—22no—
14C2×#14523237—21no—
15C2×#14523235—21no—
16C2×D1016—21no—
17C2×#14523233—21no—
18C2×#14523232C2211yescharacteristic
19#16861188—21no—
20#16861185—21no—
21#16861183—21no—
22#16861181—22no—
23#16861180—21no—
24C2×C4⋊Dic127C2211yescharacteristic
25C2×(C2×D127)⋊C4—22no—
26#16861137—22no—
27C2×#41×C127C2211yescharacteristic
28C2×M16×C127C2211yescharacteristic
29C2×C42×C127C2211yescharacteristic
30#104×C127—22no—
31C23×D127—21no—
32#14523248—21no—
33#14523248—21no—
34#14523248—21no—
35#14523248—21no—
36C2×Dic254D411yescharacteristic
37C2×D508—21no—
38C2×D508—21no—
39C2×D508—21no—
40C2×D508—21no—
41C2×D508—21no—
42C2×D508D411yescharacteristic
43C2×D508D412yescomplemented
44C2×C22⋊D127—22no—
45C2×C22⋊D127—21no—
46C22×Dic127D412yes—
47C2×C4×D127—21no—
48#14523237—24no—
49#14523235—24no—
50D1016—24no—
51#14523233—24no—
52#14523232—21no—
53#14523232D412yescomplemented
54C2×C127⋊C8—21no—
55C4⋊Dic127—21no—
56C4⋊Dic127—21no—
57Dic127⋊C4—22no—
58C22⋊Dic127—22no—
59(C2×D127)⋊C4—22no—
60(C2×D127)⋊C4—44no—
61#41×C127—21no—
62#41×C127—21no—
63#41×C127—21no—
64#41×C127—21no—
65C2×Q8×C127D411yescharacteristic
66C2×D4×C127—21no—
67C2×D4×C127D411yescharacteristic
68C22×C4×C127—21no—
69C22×C4×C127—21no—
70C22×C4×C127C2311yescharacteristic
71M16×C127—21no—
72M16×C127D412yescomplemented
73C2×C8×C127—21no—
74C42×C127—21no—
75C42×C127—21no—
76C22×D127—21no—
77C22×D127—22no—
78C22×D127—42no—
79C22×D127—42no—
80C22×D127—41no—
81C22×D127—41no—
82Dic254—21no—
83Dic254—21no—
84D508—22no—
85D508—21no—
86D508—22no—
87D508—21no—
88D508—41no—
89D508—41no—
90C22⋊D127—41no—
91C22⋊D127—41no—
92C22⋊D127—82no—
93C2×Dic127—22no—
94C2×Dic127—21no—
95C2×Dic127—42no—
96C4×D127—41no—
97C4×D127—41no—
98C127⋊C8—22no—
99C23×C127—21no—
100C23×C127C2×D411yescharacteristic
101Q8×C127—21no—
102Q8×C127—21no—
103D4×C127—21no—
104D4×C127—21no—
105D4×C127—41no—
106D4×C127—41no—
107C2×C4×C127—22no—
108C2×C4×C127—21no—
109C2×C4×C127—41no—
110C2×C4×C127—41no—
111C2×C4×C127—41no—
112C2×C4×C127—41no—
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127C2×D411yescharacteristic
118C2×C4×C127#4111yescharacteristic
119C8×C127—22no—
120C22×C127—21no—
121C22×C127—21no—
122C22×C127—41no—
123C22×C127—41no—
124C22×C127—41no—
125C22×C127#12011yescharacteristic
126C22×C127#12111yescharacteristic
127C22×C127#12711yescharacteristic
128C2×D127—44no—
129C2×D127—41no—
130C2×D127—42no—
131C2×D127—41no—
132C2×D127—41no—
133C2×D127—82no—
134C2×D127—81no—
135C2×D127—81no—
136C4×C127—42no—
137C4×C127—41no—
138C4×C127#12011yescharacteristic
139C4×C127#12011yescharacteristic
140C4×C127#12111yescharacteristic
141C4×C127#12111yescharacteristic
142Dic127—41no—
143Dic127—42no—
144C2×C127—21no—
145C2×C127—21no—
146C2×C127—81no—
147C2×C127#39211yescharacteristic
148C2×C127#45311yescharacteristic
149C2×C127#45411yescharacteristic
150D127—81no—
151D127—82no—
152#2006—1271nomaximal, Sylow
153C127#200611yescharacteristic, complemented, Sylow
154C2×AGL(1,ℤ/8)—1271no—
155C2×AGL(1,ℤ/8)—1271no—
156#537—1271no—
157C2×#104—1271no—
158C2×#104D12711yescharacteristic, complemented
159#390—1271no—
160#343—1271no—
161C2×#41—1271no—
162C2×#41C2×D12711yescharacteristic
163C22×D4—1271no—
164AGL(1,ℤ/8)—2542no—
165AGL(1,ℤ/8)—2542no—
166AGL(1,ℤ/8)—2542no—
167AGL(1,ℤ/8)—2542no—
168C2×SD16—2541no—
169C2×SD16—2541no—
170C2×D8—2541no—
171C2×D8—2541no—
172C2×M16—1271no—
173C2×M16C2×D12711yescharacteristic
174C22⋊Q8—2541no—
175C22⋊Q8—2541no—
176#121—2541no—
177#121—2542no—
178#121—2541no—
179C2×C4⋊C4—1271no—
180C2×C22⋊C4—2542no—
181C2×C42C2×D12711yescharacteristic
182#104—22no—
183#104—2542no—
184C24—2541no—
185#41—21no—
186#41—21no—
187#41—21no—
188#41—21no—
189#41—2541no—
190#41—2541no—
191#41—2541no—
192#41—2541no—
193C2×Q8—1271no—
194C2×Q8D50811yescharacteristic
195C2×D4—21no—
196C2×D4—1271no—
197C2×D4—1272no—
198C2×D4—2542no—
199C2×D4—2541no—
200C2×D4—2541no—
201C2×D4—2541no—
202C2×D4—2541no—
203C2×D4—2541no—
204C2×D4—2541no—
205C2×D4D50811yescharacteristic
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—1272no—
209C22×C4—2541no—
210C22×C4C22×D12711yescharacteristic
211SD16—2544no—
212SD16—2544no—
213D8—2544no—
214D8—2544no—
215M16—21no—
216M16—1272no—
217M16—2541no—
218M16C22⋊D12712yescomplemented
219C2×C8—21no—
220C2×C8—2541no—
221C4⋊C4—2542no—
222C4⋊C4—2541no—
223C4⋊C4—2541no—
224C22⋊C4—5084no—
225C22⋊C4—2542no—
226C22⋊C4—2542no—
227C42—21no—
228C42—21no—
229C23—21no—
230C23—5082no—
231C23—5082no—
232C23—5081no—
233C23—5081no—
234C23—2541no—
235C23—2542no—
236C23D4×D12711yescharacteristic
237Q8—21no—
238Q8—21no—
239Q8—2541no—
240Q8—2541no—
241D4—21no—
242D4—21no—
243D4—41no—
244D4—41no—
245D4—10162no—
246D4—5081no—
247D4—5081no—
248D4—5081no—
249D4—5081no—
250D4—2541no—
251D4—2542no—
252D4—2542no—
253D4—2541no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—5081no—
261C2×C4—5081no—
262C2×C4—5082no—
263C2×C4—2542no—
264C2×C4—2541no—
265C2×C4D4×D12711yescharacteristic
266C2×C4D4×D12711yescharacteristic
267C2×C4D4×D12711yescharacteristic
268C2×C4C2×C22⋊D12711yescharacteristic
269C2×C4C2×D50811yescharacteristic
270C2×C4#1452324811yescharacteristic
271C8—22no—
272C8—2542no—
273C22—21no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—10162no—
279C22—10161no—
280C22—10161no—
281C22—5084no—
282C22—5081no—
283C22—5082no—
284C22—5081no—
285C22—5081no—
286C22#1686117911yescharacteristic
287C22#1686118211yescharacteristic
288C22#1686120511yescharacteristic
289C4—42no—
290C4—41no—
291C4—5081no—
292C4—5082no—
293C4#1686117711yescharacteristic
294C4#1686117911yescharacteristic
295C4#1686118211yescharacteristic
296C4#1686118311yescharacteristic
297C2—21no—
298C2—21no—
299C2—81no—
300C2—10162no—
301C2—10161no—
302C2#2254978511yescharacteristic
303C2#2255004511yescharacteristic
304C2#2255005011yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#104×C127C2
socleC22×C127#120
hypercenterC2×#104D127
hypocenterC127#2006
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
1. C2×#16861237C2semidirect
2. C2×#16861234C2semidirect
3. #22550454C2semidirect
4. C2×#16861137C2semidirect
5. #22549778C2semidirect
6. #22549644C2semidirect
7. C2×#104×C127C2semidirect
43. C2×D508D4semidirect
53. #14523232D4semidirect
72. M16×C127D4semidirect
153. C127#2006semidirect
158. C2×#104D127semidirect
218. M16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612372
C2×#168612342
#225504542
C2×#168611372
#225497782
#225496442
C2×#104×C1272
#2006127