Finite Group Databasebeta

← #30651328

Subgroups — #30651328

Order 16256.

386 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 115 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#22550591C211yesmaximal, characteristic, complemented
2#22550529C211yesmaximal, characteristic, complemented
3#22550272C211yesmaximal, characteristic, complemented
4#22550203C212yesmaximal, complemented
5#22550193C212yesmaximal, complemented
6#22550167C211yesmaximal, characteristic, complemented
7#22550128C212yesmaximal, complemented
8C4×#14523233C211yesmaximal, characteristic, complemented
9#22549953C211yesmaximal, characteristic, complemented
10#22549917C212yesmaximal, complemented
11#474×C127C211yesmaximal, characteristic, complemented
12C2×#14523249—21no—
13C2×#14523233C2211yescharacteristic
14C2×#14523231C2212yes—
15#16861209C2211yescharacteristic
16#16861195—22no—
17#16861194—22no—
18#16861185—21no—
19#16861184C2212yes—
20#16861183—21no—
21#16861180C2212yes—
22C4×D508C2211yescharacteristic
23C4×D508C2211yescharacteristic
24#16861168—21no—
25#16861167C2212yes—
26C4⋊C4×D127C2212yes—
27#16861142C2211yescharacteristic
28#16861141C2212yes—
29#16861140C2212yes—
30#16861140C2212yes—
31#16861131C2212yes—
32#16861130C2211yescharacteristic
33#16861129C2211yescharacteristic
34(C2×D127)⋊C8C2212yes—
35C4×C127⋊C8C2211yescharacteristic
36C4⋊Q8×C127C2211yescharacteristic
37C4×D4×C127C2211yescharacteristic
38#107×C127C2212yes—
39C4⋊C8×C127C2211yescharacteristic
40#102×C127C2212yes—
41#14523249—44no—
42C2×Dic254—21no—
43C2×D508—22no—
44C2×D508D411yescharacteristic
45C2×D508C2311yescharacteristic
46C2×C22⋊D127—22no—
47C2×C4×D127—22no—
48C2×C4×D127—21no—
49C2×C4×D127C2312yes—
50#14523233—22no—
51#14523231—24no—
52C2×C127⋊C8C2311yescharacteristic
53C2×C127⋊C8C2311yescharacteristic
54C4⋊Dic127—21no—
55C4⋊Dic127D411yescharacteristic
56C4⋊Dic127C2312yes—
57C4⋊Dic127C2311yescharacteristic
58Dic127⋊C4—22no—
59C22⋊Dic127—22no—
60(C2×D127)⋊C4—22no—
61(C2×D127)⋊C4—22no—
62(C2×D127)⋊C4—21no—
63(C2×D127)⋊C4D412yes—
64C4×Dic127—22no—
65C2×Q8×C127—21no—
66C2×D4×C127C2311yescharacteristic
67C22×C4×C127—21no—
68C2×C8×C127C2312yes—
69C4⋊C4×C127—21no—
70C4⋊C4×C127C2312yes—
71C4⋊C4×C127C2311yescharacteristic
72C22⋊C4×C127—21no—
73C42×C127C2311yescharacteristic
74C22×D127—21no—
75C22×D127C2×D412yes—
76Dic254—81no—
77D508—22no—
78D508—21no—
79D508—21no—
80D508—44no—
81C22⋊D127—82no—
82C2×Dic127—22no—
83C2×Dic127—21no—
84C2×Dic127C2×D412yes—
85C4×D127—22no—
86C4×D127—22no—
87C4×D127—44no—
88C4×D127—42no—
89C127⋊C8—21no—
90C127⋊C8#4112yescomplemented
91C23×C127—21no—
92Q8×C127—42no—
93D4×C127—21no—
94D4×C127—21no—
95C2×C4×C127—21no—
96C2×C4×C127—22no—
97C2×C4×C127—42no—
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2411yescharacteristic
101C8×C127—22no—
102C22×C127—41no—
103C22×C127—41no—
104C22×C127—41no—
105C22×C127C22×D411yescharacteristic
106C2×D127—22no—
107C2×D127—22no—
108C2×D127—22no—
109C2×D127—41no—
110C2×D127—41no—
111C2×D127—41no—
112C4×C127—21no—
113C4×C127—41no—
114C4×C127—42no—
115C4×C127AGL(1,ℤ/8)12yes—
116C4×C127C2×#4111yescharacteristic
117C4×C1272−1+411yescharacteristic
118Dic127—22no—
119Dic127—41no—
120Dic127—42no—
121C2×C127—81no—
122C2×C127#54711yescharacteristic
123C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
124C2×C127#57711yescharacteristic
125D127—42no—
126D127—81no—
127#3354—1271nomaximal, Sylow
128C127#335411yescharacteristic, complemented, Sylow
129#555—1271no—
130#547—1271no—
131#498—1271no—
132#482—1272no—
133#481—1272no—
134#476—1271no—
135#474D12711yescharacteristic, complemented
136#469—1272no—
137C4×D8—1271no—
138#436—1271no—
139#428—1272no—
140C2×#41—2541no—
141C2×D8—1271no—
142C2×M16—1272no—
143C4⋊Q8—1271no—
144C4⋊Q8C2×D12711yescharacteristic
145#124—2542no—
146#123—2542no—
147C22⋊Q8—2541no—
148#121—1272no—
149#121—1272no—
150#121—2541no—
151C4×D4—1271no—
152C4×D4—1271no—
153C4×D4C2×D12711yescharacteristic
154#117—1272no—
155#117—2541no—
156C2×C4⋊C4—1272no—
157#107—1271no—
158#107C2×D12712yes—
159#106—1272no—
160#106—1272no—
161#106—1272no—
162C4⋊C8C2×D12711yescharacteristic
163#102—1271no—
164#102—1272no—
165#102—1271no—
166#102C2×D12712yes—
167C22⋊C8—1272no—
168C4×C8—1271no—
169#41—5084no—
170C2×Q8—21no—
171C2×Q8—2541no—
172C2×D4—1271no—
173C2×D4—1271no—
174C2×D4—2542no—
175C2×D4—2542no—
176C2×D4C22×D12711yescharacteristic
177C22×C4—21no—
178C22×C4—1272no—
179C22×C4—2542no—
180C22×C4—2541no—
181D8—2542no—
182M16—2544no—
183C2×C8—1271no—
184C2×C8—1271no—
185C2×C8C22×D12712yes—
186C4⋊C4—21no—
187C4⋊C4—1271no—
188C4⋊C4—1272no—
189C4⋊C4—1271no—
190C4⋊C4—2542no—
191C4⋊C4—2541no—
192C4⋊C4C22×D12711yescharacteristic
193C4⋊C4C22×D12712yes—
194C22⋊C4—21no—
195C22⋊C4—1272no—
196C22⋊C4—2542no—
197C22⋊C4—2542no—
198C22⋊C4—2542no—
199C22⋊C4—2541no—
200C42—2542no—
201C42C22×D12711yescharacteristic
202C23—21no—
203C23—1272no—
204C23—2541no—
205Q8—42no—
206Q8—10161no—
207D4—21no—
208D4—21no—
209D4—10162no—
210D4—5084no—
211D4—2542no—
212D4—2541no—
213D4—2541no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—42no—
217C2×C4—1272no—
218C2×C4—5084no—
219C2×C4—5082no—
220C2×C4—2542no—
221C2×C4—2541no—
222C2×C4—2542no—
223C2×C4—2542no—
224C2×C4D4×D12711yescharacteristic
225C2×C4D4×D12711yescharacteristic
226C2×C4C23×D12711yescharacteristic
227C8—22no—
228C8—1272no—
229C8—2541no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—5081no—
234C22—5081no—
235C22—5081no—
236C22—2542no—
237C22—2542no—
238C22—2542no—
239C22C2×D4×D12711yescharacteristic
240C4—21no—
241C4—41no—
242C4—42no—
243C4—5081no—
244C4—5082no—
245C4—2542no—
246C4#1686123312yes—
247C4#41×D12711yescharacteristic
248C4#1686125711yescharacteristic
249C2—81no—
250C2—10161no—
251C2—5082no—
252C2#2255052811yescharacteristic
253C2C2×#1686123311yescharacteristic
254C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#474×C127C2
socleC22×C127C22×D4
hypercenter#474D127
hypocenterC127#3354
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #22550591C2semidirect
2. #22550529C2semidirect
3. #22550272C2semidirect
4. #22550203C2semidirect
5. #22550193C2semidirect
6. #22550167C2semidirect
7. #22550128C2semidirect
8. C4×#14523233C2semidirect
9. #22549953C2semidirect
10. #22549917C2semidirect
11. #474×C127C2semidirect
90. C127⋊C8#41semidirect
128. C127#3354semidirect
135. #474D127semidirect

Maximal subgroups

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

subgroupindex
#225505912
#225505292
#225502722
#225502032
#225501932
#225501672
#225501282
C4×#145232332
#225499532
#225499172
#474×C1272
#3354127