Finite Group Databasebeta

← #30639790

Subgroups — #30639790

Order 16256.

348 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 71 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×#16861240C211yesmaximal, characteristic, complemented
3#22549893C211yesmaximal, characteristic, complemented
4#22549885C211yesmaximal, characteristic, complemented
5#22549849C211yesmaximal, characteristic, complemented
6#22549646C211yesmaximal, characteristic, complemented
7#536×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861240—22no—
10#16861240—22no—
11#16861240—21no—
12#16861239—21no—
13#16861235—21no—
14#16861234—21no—
15#16861229—22no—
16C2×#14523238—21no—
17C2×#14523235—21no—
18C2×#14523232C2211yescharacteristic
19C2×#14523232C2211yescharacteristic
20#16861168C2211yescharacteristic
21#16861136—22no—
22#16861133—22no—
23#16861123—22no—
24C2×#41×C127C2211yescharacteristic
25C22×Q8×C127C2211yescharacteristic
26C4⋊Q8×C127—21no—
27#124×C127—21no—
28#123×C127—22no—
29C22⋊Q8×C127—22no—
30#117×C127C2211yescharacteristic
31#14523248—21no—
32#14523248—21no—
33#14523248—21no—
34#14523248—21no—
35C2×Dic254D411yescharacteristic
36C2×D508D411yescharacteristic
37C2×C22⋊D127—21no—
38C2×C4×D127—21no—
39#14523238—24no—
40#14523238—22no—
41#14523236—22no—
42#14523235—24no—
43#14523235—22no—
44#14523233—22no—
45#14523232—21no—
46#14523232—21no—
47#14523232—21no—
48#14523232D412yes—
49C2×C127⋊C8—21no—
50C2×C127⋊C8D412yes—
51C4⋊Dic127—21no—
52C22⋊Dic127—21no—
53C4×Dic127—21no—
54#41×C127—21no—
55#41×C127—21no—
56#41×C127—21no—
57#41×C127—21no—
58C2×Q8×C127—21no—
59C2×Q8×C127—21no—
60C2×Q8×C127—21no—
61C2×Q8×C127—21no—
62C2×Q8×C127—21no—
63C2×Q8×C127D412yes—
64C2×Q8×C127D411yescharacteristic
65C2×D4×C127—21no—
66C2×D4×C127D411yescharacteristic
67C22×C4×C127—21no—
68C22×C4×C127—21no—
69C22×C4×C127C2311yescharacteristic
70C4⋊C4×C127—24no—
71C4⋊C4×C127—21no—
72C22⋊C4×C127—24no—
73C22⋊C4×C127D412yes—
74C42×C127—21no—
75C22×D127—21no—
76Dic254—21no—
77Dic254—21no—
78D508—21no—
79D508—21no—
80C22⋊D127—41no—
81C22⋊D127—41no—
82C2×Dic127—21no—
83C2×Dic127—41no—
84C4×D127—41no—
85C4×D127—41no—
86C127⋊C8—22no—
87C127⋊C8—22no—
88C23×C127—21no—
89C23×C127C2×D411yescharacteristic
90Q8×C127—22no—
91Q8×C127—21no—
92Q8×C127—22no—
93Q8×C127—21no—
94Q8×C127—41no—
95Q8×C127—41no—
96D4×C127—21no—
97D4×C127—21no—
98D4×C127—41no—
99D4×C127—41no—
100C2×C4×C127—22no—
101C2×C4×C127—22no—
102C2×C4×C127—21no—
103C2×C4×C127—41no—
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—42no—
115C22×C127—41no—
116C22×C127#12011yescharacteristic
117C22×C127#12111yescharacteristic
118C22×C127#12711yescharacteristic
119C2×D127—41no—
120C2×D127—81no—
121C4×C127—42no—
122C4×C127—41no—
123C4×C127—42no—
124C4×C127#12011yescharacteristic
125C4×C127#12011yescharacteristic
126C4×C127#12111yescharacteristic
127C4×C127#12111yescharacteristic
128Dic127—41no—
129Dic127—81no—
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#536D12711yescharacteristic, complemented
141#421—1271no—
142#419—1271no—
143#410—1271no—
144#344—1271no—
145C2×#41—1271no—
146C2×#41C2×D12711yescharacteristic
147C22×Q8C2×D12711yescharacteristic
148#137—2542no—
149#137—2541no—
150#137—2542no—
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—21no—
160#124—21no—
161#123—22no—
162C22⋊Q8—22no—
163#117—1271no—
164#117C2×D12711yescharacteristic
165#104—2542no—
166#103—2542no—
167#99—2542no—
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—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—1271no—
182C2×Q8C22⋊D12712yes—
183C2×Q8C22⋊D12711yescharacteristic
184C2×D4—21no—
185C2×D4—1271no—
186C2×D4—2541no—
187C2×D4C22⋊D12711yescharacteristic
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—2541no—
191C22×C4C22×D12711yescharacteristic
192Q16—2544no—
193Q16—2542no—
194SD16—2542no—
195SD16—2544no—
196SD16—2542no—
197D8—2542no—
198M16—1272no—
199M16—2541no—
200M16—2541no—
201M16—2541no—
202C2×C8—1272no—
203C2×C8—2541no—
204C4⋊C4—24no—
205C4⋊C4—21no—
206C4⋊C4—2541no—
207C22⋊C4—24no—
208C22⋊C4—2541no—
209C22⋊C4C22⋊D12712yes—
210C42—21no—
211C42—2541no—
212C23—21no—
213C23—2541no—
214C23C2×C22⋊D12711yescharacteristic
215Q8—22no—
216Q8—21no—
217Q8—22no—
218Q8—21no—
219Q8—41no—
220Q8—41no—
221Q8—2541no—
222Q8—2541no—
223D4—21no—
224D4—21no—
225D4—41no—
226D4—41no—
227D4—5081no—
228D4—5081no—
229D4—2541no—
230D4—2541no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—5081no—
239C2×C4—5081no—
240C2×C4—5081no—
241C2×C4—2541no—
242C2×C4D4×D12711yescharacteristic
243C2×C4D4×D12711yescharacteristic
244C2×C4D4×D12711yescharacteristic
245C2×C4C2×C22⋊D12711yescharacteristic
246C2×C4C2×C22⋊D12711yescharacteristic
247C2×C4#1452324711yescharacteristic
248C8—2542no—
249C8—2542no—
250C22—41no—
251C22—42no—
252C22—41no—
253C22—10161no—
254C22—5081no—
255C22#1686117811yescharacteristic
256C22#1686118011yescharacteristic
257C22#1686120511yescharacteristic
258C4—42no—
259C4—41no—
260C4—42no—
261C4—10161no—
262C4—5081no—
263C4#1686117711yescharacteristic
264C4#1686117711yescharacteristic
265C4#1686118111yescharacteristic
266C4#1686118111yescharacteristic
267C2—21no—
268C2—21no—
269C2—21no—
270C2—81no—
271C2—10161no—
272C2#2254978411yescharacteristic
subgroupG/H
centerC2#22549784
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#536×C127C2
socleC2×C127#392
prime radical#22550674C2
hypercenter#536D127
hypocenterC127#2009
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#16861240C2semidirect
3. #22549893C2semidirect
4. #22549885C2semidirect
5. #22549849C2semidirect
6. #22549646C2semidirect
7. #536×C127C2semidirect
137. C127#2009semidirect
140. #536D127semidirect

Maximal subgroups

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

subgroupindex
#225506742
C2×#168612402
#225498932
#225498852
#225498492
#225496462
#536×C1272
#2009127