Finite Group Databasebeta

← #30650598

Subgroups — #30650598

Order 16256.

420 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 123 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#22550541C211yesmaximal, characteristic, complemented
2#22550531C211yesmaximal, characteristic
3#22550226C211yesmaximal, characteristic, complemented
4#22550205C212yesmaximal, complemented
5#22550162C211yesmaximal, characteristic, complemented
6#22550102C212yesmaximal, complemented
7#22550013C212yesmaximal, complemented
8C4×#14523236C211yesmaximal, characteristic, complemented
9#22549941C211yesmaximal, characteristic, complemented
10#103×D127C212yesmaximal, complemented
11#461×C127C211yesmaximal, characteristic, complemented
12C2×#14523248—21no—
13C2×Q8×D127C2212yes—
14C2×#14523236C2211yescharacteristic
15C2×C8×D127C2212yes—
16#16861209C2211yescharacteristic
17#16861208—21no—
18#16861191—22no—
19#16861188—21no—
20#16861187—22no—
21#16861185C2212yes—
22#16861180C2212yes—
23C4×Dic254C2211yescharacteristic
24C4×D508C2211yescharacteristic
25#16861171—21no—
26C4⋊C4×D127C2212yes—
27#16861141C2212yes—
28#16861140C2211yescharacteristic
29#16861134C2212yes—
30#16861133C2212yes—
31#16861133C2212yes—
32#16861132C2211yescharacteristic
33#16861130C2211yescharacteristic
34(C2×D127)⋊C8C2212yes—
35C4×C127⋊C8C2211yescharacteristic
36C2×SD16×C127C2212yes—
37C4⋊Q8×C127C2211yescharacteristic
38C4×D4×C127C2211yescharacteristic
39C4⋊C8×C127C2211yescharacteristic
40#103×C127C2212yes—
41#14523248—44no—
42C2×Dic254—22no—
43C2×Dic254—21no—
44C2×Dic254C2311yescharacteristic
45Q8×D127—24no—
46Q8×D127—42no—
47C2×D508D411yescharacteristic
48C2×C22⋊D127—22no—
49C2×C4×D127—22no—
50C2×C4×D127—22no—
51C2×C4×D127C2312yes—
52#14523236—22no—
53C2×C127⋊C8C2311yescharacteristic
54C2×C127⋊C8C2311yescharacteristic
55C8×D127—42no—
56C4⋊Dic127—21no—
57C4⋊Dic127D411yescharacteristic
58C4⋊Dic127C2311yescharacteristic
59C4⋊Dic127C2312yes—
60Dic127⋊C4—22no—
61Dic127⋊C4—22no—
62Dic127⋊C4—21no—
63C22⋊Dic127—22no—
64(C2×D127)⋊C4—22no—
65(C2×D127)⋊C4D412yes—
66C4×Dic127—21no—
67C2×Q8×C127C2312yes—
68C2×D4×C127C2311yescharacteristic
69C22×C4×C127—21no—
70SD16×C127—42no—
71C2×C8×C127C2312yes—
72C4⋊C4×C127—21no—
73C4⋊C4×C127—21no—
74C4⋊C4×C127C2311yescharacteristic
75C22⋊C4×C127—21no—
76C42×C127C2311yescharacteristic
77C22×D127C2×D412yes—
78Dic254—21no—
79Dic254—44no—
80Dic254—42no—
81Dic254#4112yescomplemented
82D508SD1614yescomplemented
83C22⋊D127—44no—
84C2×Dic127—21no—
85C2×Dic127—21no—
86C2×Dic127—22no—
87C2×Dic127C2×D412yes—
88C4×D127—22no—
89C4×D127—22no—
90C4×D127—44no—
91C4×D127—44no—
92C127⋊C8—21no—
93C127⋊C8—21no—
94C23×C127—21no—
95Q8×C127—22no—
96Q8×C127—22no—
97D4×C127—21no—
98D4×C127—21no—
99C2×C4×C127—21no—
100C2×C4×C127—22no—
101C2×C4×C127—42no—
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2411yescharacteristic
105C8×C127—22no—
106C22×C127—41no—
107C22×C127—41no—
108C22×C127—41no—
109C22×C127C22×D411yescharacteristic
110C2×D127—22no—
111C2×D127—22no—
112C2×D127C2×SD1614yes—
113C4×C127—21no—
114C4×C127—42no—
115C4×C127—41no—
116C4×C127C2×SD1612yes—
117C4×C127C2×#4111yescharacteristic
118C4×C127H(5,𝔽2)11yescharacteristic
119Dic127—22no—
120Dic127—22no—
121Dic127—42no—
122Dic127—41no—
123C2×C127—81no—
124C2×C127#54611yescharacteristic
125C2×C127C22×SD1611yescharacteristic
126C2×C127#57811yescharacteristic
127D127—24no—
128#3298—1271nomaximal, Sylow
129C127#329811yescharacteristic, complemented, Sylow
130D4×Q8—1271no—
131#547—1271no—
132#487—1271no—
133#483—1272no—
134#475—1271no—
135#465—1272no—
136#461D12711yescharacteristic, complemented
137#448—1272no—
138C4×SD16—1271no—
139C8×D4—1271no—
140C2×#103—1272no—
141C2×#41—2541no—
142C22×Q8—1272no—
143C2×SD16—1271no—
144C2×SD16C2×D12712yes—
145C22×C8—1272no—
146C4⋊Q8—1271no—
147C4⋊Q8—2541no—
148C4⋊Q8C2×D12711yescharacteristic
149#123—2542no—
150C22⋊Q8—1272no—
151C22⋊Q8—2542no—
152C22⋊Q8—2541no—
153#121—1272no—
154C4×Q8—1271no—
155C4×D4—1271no—
156C4×D4—2541no—
157C4×D4C2×D12711yescharacteristic
158C2×C4⋊C4—1272no—
159#106—1272no—
160#106—1271no—
161C4⋊C8C2×D12711yescharacteristic
162#103—1272no—
163#103—1272no—
164#103—1272no—
165#103—1271no—
166#103C2×D12712yes—
167#102—1271no—
168C22⋊C8—1272no—
169C4×C8—1271no—
170#41—5084no—
171C2×Q8—1271no—
172C2×Q8—5082no—
173C2×Q8—2542no—
174C2×Q8—2544no—
175C2×Q8—2541no—
176C2×Q8C22×D12712yes—
177C2×D4—1271no—
178C2×D4—2542no—
179C2×D4C22×D12711yescharacteristic
180C22×C4—21no—
181C22×C4—1272no—
182C22×C4—2542no—
183C22×C4—2542no—
184SD16—42no—
185SD16—2542no—
186C2×C8—1271no—
187C2×C8—1271no—
188C2×C8—5082no—
189C2×C8C22×D12712yes—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—1271no—
193C4⋊C4—1271no—
194C4⋊C4—1272no—
195C4⋊C4—2542no—
196C4⋊C4—2542no—
197C4⋊C4—2541no—
198C4⋊C4—2541no—
199C4⋊C4C22×D12711yescharacteristic
200C22⋊C4—21no—
201C22⋊C4—1272no—
202C22⋊C4—2542no—
203C22⋊C4—2542no—
204C42—2541no—
205C42C22×D12711yescharacteristic
206C23—21no—
207C23—1272no—
208Q8—22no—
209Q8—22no—
210Q8—1272no—
211Q8—5084no—
212Q8—5082no—
213Q8—2541no—
214D4—21no—
215D4—21no—
216D4—1274no—
217D4—5084no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—1272no—
222C2×C4—5084no—
223C2×C4—5084no—
224C2×C4—2541no—
225C2×C4—2542no—
226C2×C4—2541no—
227C2×C4—2542no—
228C2×C4—2542no—
229C2×C4D4×D12711yescharacteristic
230C2×C4D4×D12711yescharacteristic
231C2×C4C23×D12711yescharacteristic
232C8—22no—
233C8—2541no—
234C8—2541no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—1274no—
239C22—2542no—
240C22—2542no—
241C22C2×D4×D12711yescharacteristic
242C4—21no—
243C4—41no—
244C4—42no—
245C4—5082no—
246C4—5081no—
247C4—2542no—
248C4—2542no—
249C4SD16×D12712yes—
250C4#41×D12711yescharacteristic
251C4#1686125411yescharacteristic
252C2—81no—
253C2—2544no—
254C2#2255052311yescharacteristic
255C2C2×SD16×D12711yescharacteristic
256C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#461×C127C2
socleC22×C127C22×D4
prime radical#22550531C2
hypercenter#461D127
hypocenterC127#3298
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #22550541C2semidirect
3. #22550226C2semidirect
4. #22550205C2semidirect
5. #22550162C2semidirect
6. #22550102C2semidirect
7. #22550013C2semidirect
8. C4×#14523236C2semidirect
9. #22549941C2semidirect
10. #103×D127C2semidirect
11. #461×C127C2semidirect
81. Dic254#41semidirect
82. D508SD16semidirect
129. C127#3298semidirect
136. #461D127semidirect

Maximal subgroups

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

subgroupindex
#225505412
#225505312
#225502262
#225502052
#225501622
#225501022
#225500132
C4×#145232362
#225499412
#103×D1272
#461×C1272
#3298127