Finite Group Databasebeta

← #30648446

Subgroups — #30648446

Order 16256.

452 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 137 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#22550674C212yesmaximal, complemented
2C2×#16861215C211yesmaximal, characteristic, complemented
3#22550148C212yesmaximal, complemented
4#22550143C212yesmaximal, complemented
5#22550136C214yesmaximal, complemented
6C2×#16861145C211yesmaximal, characteristic, complemented
7#22549860C212yesmaximal, complemented
8C2×#135×C127C211yesmaximal, characteristic, complemented
9C2×#14523248C2211yescharacteristic
10#16861240C2214yes—
11#16861239—22no—
12#16861235C2214yes—
13#16861234—22no—
14#16861229—24no—
15#16861215—21no—
16#16861215—21no—
17#16861215C2212yes—
18#16861215C2212yes—
19C2×#14523232C2211yescharacteristic
20C2×#14523232C2212yes—
21C2×#14523231—21no—
22C2×C8×D127—21no—
23#16861145C2212yescomplemented
24#16861145C2212yescomplemented
25#16861128C2214yescomplemented
26#16861126C2214yescomplemented
27C2×#41×C127C2212yes—
28#135×C127—21no—
29#135×C127—21no—
30#135×C127—21no—
31#135×C127—21no—
32C2×Q16×C127C2211yescharacteristic, complemented
33C2×SD16×C127C2212yescomplemented
34C2×D8×C127C2211yescharacteristic, complemented
35C22×C8×C127C2211yescharacteristic
36#14523248—21no—
37#14523248—21no—
38#14523248D412yes—
39#14523248C2312yes—
40C2×Dic254D411yescharacteristic
41C2×D508D411yescharacteristic
42C2×C22⋊D127—21no—
43C2×C4×D127—21no—
44#14523238—24no—
45#14523236—24no—
46#14523235—24no—
47#14523233—24no—
48#14523232—21no—
49#14523232—22no—
50#14523232C2312yes—
51#14523232C2314yes—
52#14523231—22no—
53#14523231—22no—
54C2×C127⋊C8—21no—
55C2×C127⋊C8—22no—
56C8×D127—22no—
57C8×D127—22no—
58#41×C127—22no—
59#41×C127—22no—
60#41×C127—22no—
61#41×C127—22no—
62C2×Q8×C127C2312yes—
63C2×D4×C127—22no—
64C2×D4×C127C2312yes—
65C22×C4×C127—22no—
66C22×C4×C127C2311yescharacteristic
67Q16×C127—21no—
68Q16×C127—21no—
69SD16×C127—22no—
70SD16×C127—22no—
71D8×C127—21no—
72D8×C127—21no—
73C2×C8×C127D411yescharacteristic
74C2×C8×C127D411yescharacteristic
75C2×C8×C127D411yescharacteristic
76C2×C8×C127D411yescharacteristic
77C2×C8×C127C2311yescharacteristic
78C2×C8×C127C2311yescharacteristic
79C22×D127—21no—
80Dic254—21no—
81Dic254C2×D412yes—
82D508—21no—
83D508C2×D412yes—
84C22⋊D127—22no—
85C22⋊D127—22no—
86C2×Dic127—21no—
87C4×D127—22no—
88C4×D127—22no—
89C127⋊C8—22no—
90C127⋊C8—24no—
91C23×C127—22no—
92C23×C127#4111yescharacteristic
93Q8×C127—22no—
94Q8×C127—22no—
95D4×C127—22no—
96D4×C127—22no—
97D4×C127—42no—
98D4×C127—42no—
99C2×C4×C127—22no—
100C2×C4×C127—42no—
101C2×C4×C127—42no—
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127#4111yescharacteristic
107C2×C4×C127C2411yescharacteristic
108C8×C127C2×D411yescharacteristic
109C8×C127C2×D411yescharacteristic
110C8×C127C2×D411yescharacteristic
111C8×C127C2×D411yescharacteristic
112C22×C127—41no—
113C22×C127—42no—
114C22×C127—82no—
115C22×C127#12111yescharacteristic
116C22×C127#12111yescharacteristic
117C22×C127C2×#4111yescharacteristic
118C2×D127—22no—
119C2×D127—42no—
120C4×C127—42no—
121C4×C127#12111yescharacteristic
122C4×C127#12111yescharacteristic
123C4×C127C22×D411yescharacteristic
124C4×C127C22×D411yescharacteristic
125Dic127—22no—
126C2×C127—21no—
127C2×C127—21no—
128C2×C127—21no—
129C2×C127—82no—
130C2×C127C2×#12111yescharacteristic
131D127—42no—
132#3067—1271nomaximal, Sylow
133C127#306711yescharacteristic, complemented, Sylow
134#575—1272no—
135C2×#135D12711yescharacteristic, complemented
136C2×#131—1271no—
137#473—1272no—
138#472—1272no—
139#471—1274no—
140C2×#108—1271no—
141#413—1272no—
142C2×#41—1271no—
143C2×#41C2×D12712yes—
144#137—1274no—
145#137—2542no—
146AGL(1,ℤ/8)—1274no—
147AGL(1,ℤ/8)—2542no—
148#135—21no—
149#135—21no—
150#135—21no—
151#135—21no—
152#135—2544no—
153C2×Q16C2×D12711yescharacteristic, complemented
154C2×SD16C2×D12712yescomplemented
155C2×D8C2×D12711yescharacteristic, complemented
156#131—1272no—
157#131—1272no—
158#131—2541no—
159#131—2541no—
160C2×M16—1271no—
161C2×M16—1272no—
162C2×M16—2541no—
163C22×C8—2541no—
164C22×C8C2×D12711yescharacteristic
165#108—1272no—
166#108—1272no—
167#101—1274no—
168#100—1274no—
169#41—22no—
170#41—22no—
171#41—22no—
172#41—22no—
173#41—1272no—
174#41—1272no—
175#41—2541no—
176#41—2541no—
177C2×Q8—1271no—
178C2×Q8C22×D12712yes—
179C2×D4—22no—
180C2×D4—1271no—
181C2×D4—2541no—
182C2×D4C22×D12712yes—
183C22×C4—22no—
184C22×C4—2541no—
185C22×C4C22×D12711yescharacteristic
186Q16—21no—
187Q16—21no—
188Q16—2544no—
189SD16—22no—
190SD16—22no—
191SD16—2544no—
192SD16—2544no—
193D8—21no—
194D8—21no—
195D8—2544no—
196M16—1272no—
197M16—1274no—
198M16—2542no—
199M16—2541no—
200M16—2542no—
201M16—2542no—
202C2×C8—2542no—
203C2×C8—2541no—
204C2×C8—2542no—
205C2×C8—2542no—
206C2×C8C22⋊D12711yescharacteristic
207C2×C8C22⋊D12711yescharacteristic
208C2×C8C22⋊D12711yescharacteristic
209C2×C8C22⋊D12711yescharacteristic
210C2×C8C22×D12711yescharacteristic
211C2×C8C22×D12711yescharacteristic
212C23—22no—
213C23—2541no—
214C23#1452324711yescharacteristic
215Q8—22no—
216Q8—22no—
217Q8—1272no—
218Q8—2541no—
219D4—22no—
220D4—22no—
221D4—42no—
222D4—42no—
223D4—1272no—
224D4—2542no—
225D4—2542no—
226D4—2541no—
227C2×C4—22no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—2541no—
231C2×C4—2542no—
232C2×C4—2542no—
233C2×C4D4×D12711yescharacteristic
234C2×C4D4×D12711yescharacteristic
235C2×C4C2×C22⋊D12711yescharacteristic
236C2×C4C2×C22⋊D12711yescharacteristic
237C2×C4#1452324711yescharacteristic
238C2×C4C23×D12711yescharacteristic
239C8—2544no—
240C8—2542no—
241C8C2×C22⋊D12711yescharacteristic
242C8C2×C22⋊D12711yescharacteristic
243C8C2×C22⋊D12711yescharacteristic
244C8C2×C22⋊D12711yescharacteristic
245C22—41no—
246C22—42no—
247C22—82no—
248C22—5082no—
249C22—2542no—
250C22#1686118011yescharacteristic
251C22#1686118011yescharacteristic
252C22C2×#1452324711yescharacteristic
253C4—42no—
254C4—2542no—
255C4#1686118011yescharacteristic
256C4#1686118011yescharacteristic
257C4C2×D4×D12711yescharacteristic
258C4C22×C22⋊D12711yescharacteristic
259C2—21no—
260C2—21no—
261C2—21no—
262C2—82no—
263C2—5082no—
264C2C2×#1686118011yescharacteristic
subgroupG/H
centerC4#16861180
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#135×C127C2
socleC2×C127C2×#121
hypercenterC2×#135D127
hypocenterC127#3067
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. #22550674C2semidirect
2. C2×#16861215C2semidirect
3. #22550148C2semidirect
4. #22550143C2semidirect
5. #22550136C2semidirect
6. C2×#16861145C2semidirect
7. #22549860C2semidirect
8. C2×#135×C127C2semidirect
23. #16861145C22semidirect
24. #16861145C22semidirect
25. #16861128C22semidirect
26. #16861126C22semidirect
32. C2×Q16×C127C22semidirect
33. C2×SD16×C127C22semidirect
34. C2×D8×C127C22semidirect
133. C127#3067semidirect
135. C2×#135D127semidirect
153. C2×Q16C2×D127semidirect
154. C2×SD16C2×D127semidirect
155. C2×D8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506742
C2×#168612152
#225501482
#225501432
#225501362
C2×#168611452
#225498602
C2×#135×C1272
#3067127