Finite Group Databasebeta

← #30648469

Subgroups — #30648469

Order 16256.

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

normal subgroups only

This group has 446 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
1C2×#16861239C211yesmaximal, characteristic, complemented
2C2×#16861239C211yesmaximal, characteristic, complemented
3#22550643C211yesmaximal, characteristic, complemented
4#22550149C214yesmaximal, complemented
5#22550137C214yesmaximal, complemented
6#22549924C211yesmaximal, characteristic, complemented
7#22549860C211yesmaximal, characteristic, complemented
8C2×#16861127C211yesmaximal, characteristic, complemented
9#575×C127C211yesmaximal, characteristic
10C22×Dic254C2211yescharacteristic
11#16861239—22no—
12#16861239—22no—
13#16861239—22no—
14#16861239C2214yes—
15C2×#14523238—21no—
16C2×#14523238C2212yescomplemented
17C2×#14523236—21no—
18C2×#14523236C2212yescomplemented
19#16861216—22no—
20#16861216C2214yes—
21C2×#14523232—22no—
22C2×#14523232C2211yescharacteristic
23C2×#14523232C2211yescharacteristic
24C2×#14523232C2211yescharacteristic
25#16861144C2214yescomplemented
26#16861128C2212yescomplemented
27#16861127C2214yescomplemented
28#16861126C2212yescomplemented
29C2×#41×C127C2211yescharacteristic
30C2×#41×C127C2211yescharacteristic
31#137×C127—21no—
32#137×C127C2212yes—
33AGL(1,ℤ/8)×C127—21no—
34AGL(1,ℤ/8)×C127C2212yes—
35#135×C127—22no—
36C2×M16×C127C2211yescharacteristic
37C2×Dic254—21no—
38C2×Dic254—21no—
39C2×Dic254—21no—
40C2×Dic254—21no—
41C2×Dic254—21no—
42C2×Dic254C2312yes—
43C22×Dic127—21no—
44#14523238—24no—
45#14523238—24no—
46#14523236—24no—
47#14523236—24no—
48#14523232—24no—
49#14523232—21no—
50#14523232—21no—
51#14523232D412yes—
52#14523232D412yes—
53#14523232C2312yes—
54#14523232C2312yes—
55C2×C127⋊C8—24no—
56C2×C127⋊C8—21no—
57C2×C127⋊C8—21no—
58C2×C127⋊C8C2312yes—
59#41×C127—21no—
60#41×C127—21no—
61#41×C127—21no—
62#41×C127—21no—
63#41×C127—21no—
64#41×C127—21no—
65#41×C127D412yes—
66#41×C127C2312yes—
67C2×Q8×C127D411yescharacteristic
68C2×Q8×C127C2311yescharacteristic
69C2×D4×C127—21no—
70C2×D4×C127—21no—
71C2×D4×C127D411yescharacteristic
72C2×D4×C127C2311yescharacteristic
73C22×C4×C127—21no—
74C22×C4×C127—21no—
75C22×C4×C127C2311yescharacteristic
76Q16×C127—22no—
77SD16×C127—22no—
78SD16×C127—22no—
79D8×C127—22no—
80M16×C127—21no—
81M16×C127C2312yes—
82C2×C8×C127—21no—
83Dic254—22no—
84Dic254—22no—
85Dic254—41no—
86Dic254—41no—
87C2×Dic127—22no—
88C2×Dic127—41no—
89C2×Dic127—41no—
90C127⋊C8—22no—
91C127⋊C8—22no—
92C127⋊C8C2×D414yes—
93C23×C127—21no—
94C23×C127—21no—
95C23×C127#4111yescharacteristic
96Q8×C127—21no—
97Q8×C127—21no—
98Q8×C127—21no—
99Q8×C127C2×D412yes—
100D4×C127—22no—
101D4×C127—22no—
102D4×C127—21no—
103D4×C127—21no—
104D4×C127—21no—
105D4×C127—41no—
106D4×C127—41no—
107D4×C127C2×D412yes—
108C2×C4×C127—21no—
109C2×C4×C127—22no—
110C2×C4×C127—22no—
111C2×C4×C127—21no—
112C2×C4×C127—41no—
113C2×C4×C127—41no—
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127C2×D411yescharacteristic
118C2×C4×C127#4111yescharacteristic
119C2×C4×C127C2411yescharacteristic
120C8×C127—22no—
121C22×C127—22no—
122C22×C127—41no—
123C22×C127—41no—
124C22×C127—81no—
125C22×C127—81no—
126C22×C127#12111yescharacteristic
127C22×C127#12111yescharacteristic
128C22×C127C2×#4111yescharacteristic
129C4×C127—22no—
130C4×C127—41no—
131C4×C127#12111yescharacteristic
132C4×C127#12111yescharacteristic
133C4×C127C22×D411yescharacteristic
134C4×C127C22×D411yescharacteristic
135Dic127—42no—
136C2×C127—21no—
137C2×C127—21no—
138C2×C127—21no—
139C2×C127—42no—
140C2×C127—81no—
141C2×C127C2×#12111yescharacteristic
142#3069—1271nomaximal, Sylow
143C127#306911yescharacteristic, complemented, Sylow
144#575D12711yescharacteristic
145C2×#137—1271no—
146C2×#137—1271no—
147#568—1271no—
148#473—1274no—
149#471—1274no—
150#430—1271no—
151#413—1271no—
152C2×#101—1271no—
153C2×#41C2×D12711yescharacteristic
154C2×#41C2×D12711yescharacteristic
155C22×Q8—1271no—
156#137—21no—
157#137—1274no—
158#137—2542no—
159#137—2542no—
160#137—2542no—
161#137C2×D12712yes—
162AGL(1,ℤ/8)—21no—
163AGL(1,ℤ/8)C2×D12712yes—
164#135—22no—
165C2×Q16—1272no—
166C2×Q16—2541no—
167C2×SD16—1272no—
168C2×SD16—2541no—
169#131—1274no—
170#131—2542no—
171C2×M16—1271no—
172C2×M16—1271no—
173C2×M16—1271no—
174C2×M16—2542no—
175C2×M16C2×D12711yescharacteristic
176#108—1274no—
177#101—1272no—
178#101—1274no—
179#100—1272no—
180#41—21no—
181#41—21no—
182#41—21no—
183#41—21no—
184#41—21no—
185#41—21no—
186#41C22⋊D12712yes—
187#41C22×D12712yes—
188C2×Q8—1272no—
189C2×Q8—2541no—
190C2×Q8—2541no—
191C2×Q8—2541no—
192C2×Q8—2541no—
193C2×Q8—2541no—
194C2×Q8C22⋊D12711yescharacteristic
195C2×Q8C22×D12711yescharacteristic
196C2×D4—21no—
197C2×D4—21no—
198C2×D4C22⋊D12711yescharacteristic
199C2×D4C22×D12711yescharacteristic
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—2541no—
203C22×C4C22×D12711yescharacteristic
204Q16—22no—
205Q16—2544no—
206Q16—2544no—
207SD16—22no—
208SD16—22no—
209SD16—2544no—
210SD16—2544no—
211D8—22no—
212M16—21no—
213M16—1272no—
214M16—1272no—
215M16—1272no—
216M16—1272no—
217M16—2544no—
218M16—2541no—
219M16—2541no—
220M16C22×D12712yes—
221C2×C8—21no—
222C2×C8—1272no—
223C2×C8—2544no—
224C2×C8—2541no—
225C2×C8—2541no—
226C23—21no—
227C23—21no—
228C23#1452324711yescharacteristic
229Q8—21no—
230Q8—21no—
231Q8—21no—
232Q8—5081no—
233Q8—5081no—
234Q8—2542no—
235Q8—2542no—
236Q8C2×C22⋊D12712yes—
237D4—22no—
238D4—22no—
239D4—21no—
240D4—21no—
241D4—21no—
242D4—41no—
243D4—41no—
244D4C2×C22⋊D12712yes—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—5081no—
252C2×C4—5081no—
253C2×C4—2542no—
254C2×C4D4×D12711yescharacteristic
255C2×C4D4×D12711yescharacteristic
256C2×C4C2×C22⋊D12711yescharacteristic
257C2×C4C2×C22⋊D12711yescharacteristic
258C2×C4#1452324711yescharacteristic
259C2×C4C23×D12711yescharacteristic
260C8—22no—
261C8—1274no—
262C8—2542no—
263C8—2542no—
264C22—22no—
265C22—41no—
266C22—41no—
267C22—81no—
268C22—81no—
269C22#1686118111yescharacteristic
270C22#1686118111yescharacteristic
271C22C2×#1452324711yescharacteristic
272C4—22no—
273C4—41no—
274C4—5082no—
275C4#1686118111yescharacteristic
276C4#1686118111yescharacteristic
277C4C2×D4×D12711yescharacteristic
278C4C22×C22⋊D12711yescharacteristic
279C2—21no—
280C2—21no—
281C2—21no—
282C2—42no—
283C2—81no—
284C2C2×#1686118111yescharacteristic
subgroupG/H
centerC2C2×#16861181
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#575×C127C2
socleC2×C127C2×#121
hypercenter#575D127
hypocenterC127#3069
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
2. C2×#16861239C2semidirect
3. #22550643C2semidirect
4. #22550149C2semidirect
5. #22550137C2semidirect
6. #22549924C2semidirect
7. #22549860C2semidirect
8. C2×#16861127C2semidirect
16. C2×#14523238C22semidirect
18. C2×#14523236C22semidirect
25. #16861144C22semidirect
26. #16861128C22semidirect
27. #16861127C22semidirect
28. #16861126C22semidirect
143. C127#3069semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
#225506432
#225501492
#225501372
#225499242
#225498602
C2×#168611272
#575×C1272
#3069127