Finite Group Databasebeta

← #30639789

Subgroups — #30639789

Order 16256.

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

normal subgroups only

This group has 348 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
2C2×#16861239C211yesmaximal, characteristic, complemented
3#22550452C211yesmaximal, characteristic, complemented
4#22549894C211yesmaximal, characteristic, complemented
5#22549884C211yesmaximal, characteristic, complemented
6#22549646C211yesmaximal, characteristic, complemented
7#410×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C22×Dic254C2211yescharacteristic
10#16861240—21no—
11#16861239—22no—
12#16861239—21no—
13#16861239—22no—
14#16861235—21no—
15#16861234—21no—
16#16861229—22no—
17C2×#14523238—21no—
18C2×#14523236—21no—
19C2×#14523232C2211yescharacteristic
20C2×#14523232C2211yescharacteristic
21#16861209—21no—
22#16861197—21no—
23#16861193—22no—
24#16861189—22no—
25#16861137—22no—
26#16861132—22no—
27C2×#41×C127C2211yescharacteristic
28#117×C127C2211yescharacteristic
29#117×C127C2211yescharacteristic
30#99×C127—22no—
31#14523248—21no—
32#14523248—21no—
33#14523248—21no—
34#14523248—21no—
35C2×Dic254—21no—
36C2×Dic254—21no—
37C2×Dic254—21no—
38C2×Dic254—21no—
39C2×Dic254—21no—
40C2×Dic254D412yes—
41C2×Dic254D411yescharacteristic
42C2×D508D411yescharacteristic
43C2×C22⋊D127—21no—
44C22×Dic127—21no—
45C2×C4×D127—21no—
46#14523238—24no—
47#14523238—22no—
48#14523236—24no—
49#14523236—22no—
50#14523235—22no—
51#14523233—22no—
52#14523232—21no—
53#14523232—21no—
54#14523232—21no—
55#14523232D412yes—
56C2×C127⋊C8—21no—
57C2×C127⋊C8D412yes—
58C4⋊Dic127—24no—
59(C2×D127)⋊C4—24no—
60#41×C127—21no—
61#41×C127—21no—
62#41×C127—21no—
63#41×C127—21no—
64C2×Q8×C127D411yescharacteristic
65C2×D4×C127—21no—
66C2×D4×C127D411yescharacteristic
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—21no—
76Dic254—22no—
77Dic254—21no—
78Dic254—22no—
79Dic254—21no—
80Dic254—41no—
81Dic254—41no—
82D508—21no—
83D508—21no—
84C22⋊D127—41no—
85C22⋊D127—41no—
86C2×Dic127—22no—
87C2×Dic127—21no—
88C2×Dic127—41no—
89C2×Dic127—41no—
90C4×D127—41no—
91C4×D127—41no—
92C127⋊C8—22no—
93C127⋊C8—22no—
94C23×C127—21no—
95C23×C127C2×D411yescharacteristic
96Q8×C127—21no—
97Q8×C127—21no—
98D4×C127—21no—
99D4×C127—21no—
100D4×C127—41no—
101D4×C127—41no—
102C2×C4×C127—22no—
103C2×C4×C127—21no—
104C2×C4×C127—41no—
105C2×C4×C127—41no—
106C2×C4×C127—41no—
107C2×C4×C127C2×D411yescharacteristic
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127#4111yescharacteristic
113C22×C127—41no—
114C22×C127—41no—
115C22×C127—81no—
116C22×C127#12011yescharacteristic
117C22×C127#12111yescharacteristic
118C22×C127#12711yescharacteristic
119C2×D127—42no—
120C2×D127—41no—
121C4×C127—42no—
122C4×C127—41no—
123C4×C127—81no—
124C4×C127#12011yescharacteristic
125C4×C127#12011yescharacteristic
126C4×C127#12111yescharacteristic
127C4×C127#12111yescharacteristic
128Dic127—41no—
129Dic127—42no—
130C2×C127—21no—
131C2×C127—21no—
132C2×C127—21no—
133C2×C127—81no—
134C2×C127#39211yescharacteristic
135D127—81no—
136#2009—1271nomaximal, Sylow
137C127#200911yescharacteristic, complemented, Sylow
138#575—1271no—
139C2×#137—1271no—
140#536—1271no—
141#421—1271no—
142#419—1271no—
143#410D12711yescharacteristic, complemented
144#344—1271no—
145C2×#41—1271no—
146C2×#41C2×D12711yescharacteristic
147C22×Q8—1271no—
148#137—2542no—
149#137—2542no—
150#137—2541no—
151#137—2541no—
152AGL(1,ℤ/8)—2541no—
153AGL(1,ℤ/8)—2541no—
154#135—2542no—
155C2×Q16—2541no—
156C2×SD16—2541no—
157C2×M16—1271no—
158C2×M16—1271no—
159C4⋊Q8—2541no—
160#124—2541no—
161#123—2542no—
162C22⋊Q8—2542no—
163#117C2×D12711yescharacteristic
164#117C2×D12711yescharacteristic
165#104—2542no—
166#103—2542no—
167#99—22no—
168#41—21no—
169#41—21no—
170#41—21no—
171#41—21no—
172#41—2541no—
173#41—2541no—
174#41—2541no—
175#41—2541no—
176C2×Q8—1272no—
177C2×Q8—1271no—
178C2×Q8—2541no—
179C2×Q8—2541no—
180C2×Q8—2541no—
181C2×Q8—2541no—
182C2×Q8—2541no—
183C2×Q8D50811yescharacteristic
184C2×D4—21no—
185C2×D4—1271no—
186C2×D4—2541no—
187C2×D4D50811yescharacteristic
188C22×C4—21no—
189C22×C4—2541no—
190C22×C4—2541no—
191C22×C4C22×D12711yescharacteristic
192Q16—2544no—
193Q16—2542no—
194SD16—2544no—
195SD16—2542no—
196SD16—2542no—
197D8—2542no—
198M16—1272no—
199M16—2541no—
200M16—2541no—
201M16—2541no—
202C2×C8—1272no—
203C2×C8—2541no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—2544no—
207C22⋊C4—21no—
208C22⋊C4—2544no—
209C22⋊C4C22⋊D12712yes—
210C42—21no—
211C42—21no—
212C23—21no—
213C23—2541no—
214C23C2×C22⋊D12711yescharacteristic
215Q8—21no—
216Q8—21no—
217Q8—5081no—
218Q8—5081no—
219Q8—2542no—
220Q8—2541no—
221Q8—2542no—
222Q8—2541no—
223D4—21no—
224D4—21no—
225D4—41no—
226D4—41no—
227D4—5081no—
228D4—5081no—
229D4—2541no—
230D4—2541no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—5081no—
237C2×C4—5081no—
238C2×C4—5081no—
239C2×C4—5081no—
240C2×C4—2542no—
241C2×C4—2541no—
242C2×C4D4×D12711yescharacteristic
243C2×C4D4×D12711yescharacteristic
244C2×C4D4×D12711yescharacteristic
245C2×C4D4×D12711yescharacteristic
246C2×C4C2×D50811yescharacteristic
247C2×C4#1452324811yescharacteristic
248C8—2542no—
249C8—2542no—
250C22—41no—
251C22—41no—
252C22—81no—
253C22—5082no—
254C22—5081no—
255C22#1686117711yescharacteristic
256C22#1686118311yescharacteristic
257C22#1686120511yescharacteristic
258C4—42no—
259C4—41no—
260C4—81no—
261C4—5081no—
262C4—5082no—
263C4#1686117911yescharacteristic
264C4#1686117911yescharacteristic
265C4#1686118211yescharacteristic
266C4#1686118211yescharacteristic
267C2—21no—
268C2—21no—
269C2—21no—
270C2—81no—
271C2—10161no—
272C2#2254978511yescharacteristic
subgroupG/H
centerC2#22549785
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#410×C127C2
socleC2×C127#392
prime radical#22550674C2
hypercenter#410D127
hypocenterC127#2009
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
2. C2×#16861239C2semidirect
3. #22550452C2semidirect
4. #22549894C2semidirect
5. #22549884C2semidirect
6. #22549646C2semidirect
7. #410×C127C2semidirect
137. C127#2009semidirect
143. #410D127semidirect

Maximal subgroups

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

subgroupindex
#225506742
C2×#168612392
#225504522
#225498942
#225498842
#225496462
#410×C1272
#2009127