Finite Group Databasebeta

← #30648421

Subgroups — #30648421

Order 16256.

472 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861242C211yesmaximal, characteristic, complemented
2#22550454C211yesmaximal, characteristic
3#22550447C211yesmaximal, characteristic, complemented
4#22550134C212yesmaximal, complemented
5#22550122C212yesmaximal, complemented
6#22550117C212yesmaximal, complemented
7#22550105C212yesmaximal, complemented
8#22549918C211yesmaximal, characteristic, complemented
9#22549873C211yesmaximal, characteristic, complemented
10C2×#16861134C211yesmaximal, characteristic, complemented
11C2×#131×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×Dic254C2211yescharacteristic
14#16861242—24no—
15C2×Dic508C2212yes—
16C2×#14523237C2212yes—
17#16861207—21no—
18#16861196—21no—
19#16861190—22no—
20#16861188C2212yes—
21#16861188C2212yes—
22#16861188C2212yes—
23#16861185—21no—
24#16861183C2212yes—
25#16861181—22no—
26#16861180—21no—
27#16861168C2211yescharacteristic
28C2×C4⋊Dic127C2211yescharacteristic
29#16861143C2212yes—
30#16861141C2212yes—
31#16861134C2212yes—
32#16861134C2211yescharacteristic
33#16861134C2212yes—
34#16861134C2211yescharacteristic
35#16861131C2211yescharacteristic
36#16861131C2211yescharacteristic
37C2×#41×C127C2211yescharacteristic
38#131×C127—24no—
39C2×M16×C127C2212yes—
40C2×M16×C127C2211yescharacteristic
41C22×C8×C127C2212yes—
42C22×C8×C127C2211yescharacteristic
43#14523248—22no—
44#14523248—22no—
45C2×Dic254—21no—
46C2×Dic254—22no—
47C2×Dic254—22no—
48C2×Dic254C2312yes—
49C2×Dic254C2311yescharacteristic
50C2×D508C2311yescharacteristic
51C2×C22⋊D127—21no—
52C2×C22⋊D127—22no—
53C22×Dic127—21no—
54C22×Dic127—21no—
55C2×C4×D127—21no—
56Dic508—24no—
57#14523237—24no—
58C4⋊Dic127—21no—
59C4⋊Dic127C2311yescharacteristic
60C4⋊Dic127C2311yescharacteristic
61C4⋊Dic127C2312yes—
62Dic127⋊C4—22no—
63Dic127⋊C4—22no—
64Dic127⋊C4—22no—
65C22⋊Dic127—22no—
66C22⋊Dic127—22no—
67C22⋊Dic127—22no—
68C22⋊Dic127—21no—
69(C2×D127)⋊C4—22no—
70C4×Dic127—21no—
71#41×C127—22no—
72#41×C127—22no—
73C2×Q8×C127D411yescharacteristic
74C2×D4×C127D412yes—
75C2×D4×C127D411yescharacteristic
76C22×C4×C127C2312yes—
77C22×C4×C127C2311yescharacteristic
78M16×C127—24no—
79M16×C127D414yescomplemented
80C2×C8×C127—22no—
81C2×C8×C127—24no—
82C2×C8×C127C2311yescharacteristic
83C2×C8×C127C2311yescharacteristic
84C2×C8×C127C2312yes—
85C22×D127—21no—
86Dic254—22no—
87Dic254—21no—
88Dic254—22no—
89Dic254—21no—
90Dic254—42no—
91D508—21no—
92D508—21no—
93C22⋊D127—42no—
94C22⋊D127—82no—
95C2×Dic127—21no—
96C2×Dic127—21no—
97C2×Dic127—22no—
98C2×Dic127—21no—
99C2×Dic127—22no—
100C2×Dic127—42no—
101C2×Dic127—81no—
102C4×D127—42no—
103C23×C127C2×D412yes—
104C23×C127C2×D411yescharacteristic
105Q8×C127—22no—
106D4×C127—24no—
107D4×C127—22no—
108C2×C4×C127—22no—
109C2×C4×C127—22no—
110C2×C4×C127C2×D412yes—
111C2×C4×C127C2×D412yes—
112C2×C4×C127C2×D411yescharacteristic
113C2×C4×C127#4112yes—
114C2×C4×C127C2411yescharacteristic
115C8×C127—21no—
116C8×C127—21no—
117C8×C127C2×D414yes—
118C22×C127—21no—
119C22×C127—21no—
120C22×C127—22no—
121C22×C127—22no—
122C22×C127—22no—
123C22×C127#12112yes—
124C22×C127C22×D411yescharacteristic
125C2×D127—41no—
126C2×D127—41no—
127C2×D127—41no—
128C4×C127—22no—
129C4×C127#12112yes—
130C4×C127C22×D411yescharacteristic
131C4×C127C2×#4111yescharacteristic
132Dic127—41no—
133Dic127—42no—
134Dic127—41no—
135Dic127—42no—
136Dic127—41no—
137C2×C127—22no—
138C2×C127—42no—
139C2×C127C2×#12111yescharacteristic
140C2×C127#57711yescharacteristic
141C2×C127#57811yescharacteristic
142D127—81no—
143#3063—1271nomaximal, Sylow
144C127#306311yescharacteristic, complemented, Sylow
145C2×#137—1271no—
146C2×#131D12711yescharacteristic, complemented
147#537—1271no—
148#536—1271no—
149#470—1272no—
150#468—1272no—
151#467—1272no—
152#465—1272no—
153#428—1271no—
154#416—1271no—
155C2×#103—1271no—
156C2×#41—1271no—
157C2×#41C2×D12711yescharacteristic
158C22×Q8—1271no—
159#137—2544no—
160C2×Q16—1272no—
161C2×SD16—1272no—
162#131—24no—
163C2×M16C2×D12712yes—
164C2×M16C2×D12711yescharacteristic
165C22×C8C2×D12711yescharacteristic
166C22×C8C2×D12712yes—
167C4⋊Q8—2541no—
168#124—2541no—
169#123—2542no—
170C22⋊Q8—1272no—
171C22⋊Q8—1272no—
172C22⋊Q8—1272no—
173C22⋊Q8—2541no—
174#121—1272no—
175#121—2541no—
176#121—2542no—
177#117—1271no—
178C2×C4⋊C4—1271no—
179#107—1272no—
180#106—1272no—
181#103—1272no—
182#103—1272no—
183#103—1271no—
184#103—1271no—
185#102—1271no—
186#102—1271no—
187#41—22no—
188#41—22no—
189#41—2542no—
190#41—2542no—
191C2×Q8—1272no—
192C2×Q8—1271no—
193C2×Q8—2542no—
194C2×Q8—2542no—
195C2×Q8—2541no—
196C2×Q8D50811yescharacteristic
197C2×D4—1271no—
198C2×D4—2542no—
199C2×D4—2541no—
200C2×D4D50812yes—
201C2×D4D50811yescharacteristic
202C22×C4—2541no—
203C22×C4—2541no—
204C22×C4—2541no—
205C22×C4C22×D12712yes—
206C22×C4C22×D12711yescharacteristic
207Q16—2544no—
208SD16—2544no—
209M16—24no—
210M16C22⋊D12714yescomplemented
211C2×C8—22no—
212C2×C8—24no—
213C2×C8C22×D12711yescharacteristic
214C2×C8C22×D12711yescharacteristic
215C2×C8C22×D12712yes—
216C4⋊C4—1271no—
217C4⋊C4—1271no—
218C4⋊C4—1272no—
219C4⋊C4—2542no—
220C4⋊C4—2542no—
221C4⋊C4—2542no—
222C4⋊C4—2541no—
223C22⋊C4—2541no—
224C22⋊C4—2542no—
225C22⋊C4—2542no—
226C22⋊C4—2542no—
227C22⋊C4—2542no—
228C42—2541no—
229C23—2541no—
230C23C2×D50812yes—
231C23C2×D50811yescharacteristic
232Q8—22no—
233Q8—5082no—
234Q8—2542no—
235Q8—2541no—
236Q8—2542no—
237Q8—2541no—
238D4—24no—
239D4—22no—
240D4—10162no—
241D4—5082no—
242D4—2541no—
243D4—2541no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—10161no—
247C2×C4—5082no—
248C2×C4—5082no—
249C2×C4—2541no—
250C2×C4—2541no—
251C2×C4—2542no—
252C2×C4—2541no—
253C2×C4—2542no—
254C2×C4C2×C22⋊D12712yes—
255C2×C4C2×D50812yes—
256C2×C4C2×D50811yescharacteristic
257C2×C4#1452324812yes—
258C2×C4C23×D12711yescharacteristic
259C8—21no—
260C8—21no—
261C8C2×C22⋊D12714yes—
262C22—21no—
263C22—21no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—5081no—
268C22—5081no—
269C22—5081no—
270C22#1686118312yes—
271C22C22×D50811yescharacteristic
272C4—22no—
273C4—5081no—
274C4—5082no—
275C4—5082no—
276C4—5081no—
277C4—5081no—
278C4#1686118312yes—
279C4C22×C22⋊D12711yescharacteristic
280C4C2×#1452324811yescharacteristic
281C2—22no—
282C2—42no—
283C2—10161no—
284C2C2×#1686118311yescharacteristic
285C2#2255068811yescharacteristic
286C2#2255069311yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#131×C127C2
socleC22×C127C22×D4
prime radical#22550454C2
hypercenterC2×#131D127
hypocenterC127#3063
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
1. C2×#16861242C2semidirect
3. #22550447C2semidirect
4. #22550134C2semidirect
5. #22550122C2semidirect
6. #22550117C2semidirect
7. #22550105C2semidirect
8. #22549918C2semidirect
9. #22549873C2semidirect
10. C2×#16861134C2semidirect
11. C2×#131×C127C2semidirect
79. M16×C127D4semidirect
144. C127#3063semidirect
146. C2×#131D127semidirect
210. M16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612422
#225504542
#225504472
#225501342
#225501222
#225501172
#225501052
#225499182
#225498732
C2×#168611342
C2×#131×C1272
#3063127