Finite Group Databasebeta

← #30651430

Subgroups — #30651430

Order 16256.

348 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 117 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#22550612C211yesmaximal, characteristic
2#22550605C211yesmaximal, characteristic, complemented
3#22550229C212yesmaximal, complemented
4#22550222C211yesmaximal, characteristic
5#22550178C212yesmaximal, complemented
6#22550162C211yesmaximal, characteristic, complemented
7#22550096C212yesmaximal, complemented
8#22550000C211yesmaximal, characteristic, complemented
9C4×#14523238C211yesmaximal, characteristic, complemented
10SD16×Dic127C212yesmaximal, complemented
11#460×C127C211yesmaximal, characteristic
12C2×#14523238C2211yescharacteristic
13#16861209—21no—
14#16861209C2211yescharacteristic
15#16861200—22no—
16#16861200C2212yes—
17#16861196—21no—
18#16861196C2212yes—
19#16861190—22no—
20Q8×Dic127C2212yes—
21C4×Dic254—21no—
22C4×Dic254C2211yescharacteristic
23C4×Dic254C2211yescharacteristic
24D4×Dic127—21no—
25D4×Dic127C2212yes—
26#16861142C2211yescharacteristic
27#16861141C2212yes—
28Dic127⋊C8C2212yes—
29#16861134C2212yes—
30#16861133C2211yescharacteristic
31#16861132C2211yescharacteristic
32#16861130C2212yes—
33#16861130C2212yes—
34C4×C127⋊C8C2211yescharacteristic
35C8×Dic127C2212yes—
36C2×SD16×C127C2212yes—
37#127×C127C2211yescharacteristic
38C4×Q8×C127C2211yescharacteristic
39C4⋊C8×C127C2211yescharacteristic
40#102×C127C2212yes—
41C2×Dic254—21no—
42C2×Dic254D411yescharacteristic
43C2×Dic254C2311yescharacteristic
44C22×Dic127—22no—
45#14523238—22no—
46C2×C127⋊C8C2311yescharacteristic
47C2×C127⋊C8C2311yescharacteristic
48C4⋊Dic127—22no—
49C4⋊Dic127—22no—
50C4⋊Dic127—21no—
51C4⋊Dic127D411yescharacteristic
52C4⋊Dic127C2312yes—
53C4⋊Dic127C2311yescharacteristic
54Dic127⋊C4—22no—
55Dic127⋊C4—21no—
56Dic127⋊C4D412yes—
57C22⋊Dic127—22no—
58C22⋊Dic127—22no—
59C22⋊Dic127—22no—
60C4×Dic127—22no—
61C4×Dic127—21no—
62C4×Dic127C2312yes—
63C2×Q8×C127C2311yescharacteristic
64C2×D4×C127—21no—
65C2×D4×C127—21no—
66C2×D4×C127C2312yes—
67SD16×C127—24no—
68C2×C8×C127C2312yes—
69C4⋊C4×C127—21no—
70C4⋊C4×C127C2311yescharacteristic
71C42×C127—21no—
72C42×C127C2311yescharacteristic
73Dic254—22no—
74Dic254—21no—
75Dic254—21no—
76Dic254—42no—
77C2×Dic127—21no—
78C2×Dic127—21no—
79C2×Dic127—22no—
80C2×Dic127—82no—
81C2×Dic127C2×D412yes—
82C2×Dic127C2×D412yes—
83C127⋊C8—21no—
84C127⋊C8—21no—
85C23×C127—22no—
86Q8×C127—21no—
87Q8×C127#4112yes—
88D4×C127—22no—
89D4×C127—22no—
90D4×C127—42no—
91D4×C127—81no—
92C2×C4×C127—21no—
93C2×C4×C127—21no—
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2411yescharacteristic
97C8×C127—22no—
98C22×C127—42no—
99C22×C127—42no—
100C22×C127—42no—
101C22×C127C22×D411yescharacteristic
102C4×C127—22no—
103C4×C127—21no—
104C4×C127—41no—
105C4×C127#13512yes—
106C4×C127C2×#4111yescharacteristic
107C4×C1272−1+411yescharacteristic
108Dic127—22no—
109Dic127—22no—
110Dic127—41no—
111Dic127—42no—
112Dic127—41no—
113C2×C127—82no—
114C2×C127#54811yescharacteristic
115C2×C127C2×#13511yescharacteristic
116C2×C127#57711yescharacteristic
117#3365—1271nomaximal, Sylow
118C127#336511yescharacteristic, complemented, Sylow
119#559—1271no—
120#557—1271no—
121#488—1272no—
122#486—1271no—
123#478—1272no—
124#475—1271no—
125#464—1272no—
126#460D12711yescharacteristic
127C8×Q8—1271no—
128C4×Q16—1271no—
129C4×SD16—1272no—
130C2×Q16—1271no—
131C2×SD16C2×D12712yes—
132C4⋊Q8—1271no—
133C4⋊Q8—2541no—
134#127C2×D12711yescharacteristic
135#125—1272no—
136#125—2542no—
137#124—1272no—
138#124—2541no—
139#123—2542no—
140C4×Q8—1271no—
141C4×Q8—1272no—
142C4×Q8—1271no—
143C4×Q8—2541no—
144C4×Q8C2×D12711yescharacteristic
145C4×D4—1272no—
146C4×D4—2541no—
147#107—1271no—
148#106—1272no—
149C4⋊C8—1272no—
150C4⋊C8C2×D12711yescharacteristic
151#103—1271no—
152#103—1272no—
153#103—1271no—
154#102—1272no—
155#102—1272no—
156#102C2×D12712yes—
157C4×C8—1272no—
158C4×C8—1271no—
159C2×Q8—1271no—
160C2×Q8—1271no—
161C2×Q8—2541no—
162C2×Q8C22×D12711yescharacteristic
163C2×D4—21no—
164C2×D4—21no—
165C2×D4C22×D12712yes—
166C22×C4—2542no—
167Q16—2542no—
168SD16—24no—
169C2×C8—1271no—
170C2×C8—1271no—
171C2×C8C22×D12712yes—
172C4⋊C4—21no—
173C4⋊C4—1272no—
174C4⋊C4—1271no—
175C4⋊C4—1272no—
176C4⋊C4—1271no—
177C4⋊C4—2542no—
178C4⋊C4—2542no—
179C4⋊C4—2542no—
180C4⋊C4—2541no—
181C4⋊C4—2541no—
182C4⋊C4C22×D12711yescharacteristic
183C22⋊C4—2542no—
184C22⋊C4—2542no—
185C22⋊C4—2542no—
186C42—21no—
187C42—1272no—
188C42—2542no—
189C42—2541no—
190C42C22×D12711yescharacteristic
191C23—22no—
192Q8—21no—
193Q8—5082no—
194Q8—2541no—
195Q8—2541no—
196Q8—2542no—
197Q8#1452324712yes—
198D4—22no—
199D4—22no—
200D4—42no—
201D4—81no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—1272no—
205C2×C4—1272no—
206C2×C4—10162no—
207C2×C4—2541no—
208C2×C4—2541no—
209C2×C4—2542no—
210C2×C4D4×D12711yescharacteristic
211C2×C4D4×D12711yescharacteristic
212C2×C4C23×D12711yescharacteristic
213C8—22no—
214C8—2541no—
215C8—2541no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22C2×D4×D12711yescharacteristic
220C4—22no—
221C4—21no—
222C4—41no—
223C4—5081no—
224C4—5082no—
225C4—5081no—
226C4—2542no—
227C4—2542no—
228C4#1686123012yes—
229C4C2×#1452324711yescharacteristic
230C4#1686125611yescharacteristic
231C2—82no—
232C2#2255053211yescharacteristic
233C2C2×#1686123011yescharacteristic
234C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#460×C127C2
socleC22×C127C22×D4
prime radical#127×C127C22
hypercenter#460D127
hypocenterC127#3365
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
2. #22550605C2semidirect
3. #22550229C2semidirect
5. #22550178C2semidirect
6. #22550162C2semidirect
7. #22550096C2semidirect
8. #22550000C2semidirect
9. C4×#14523238C2semidirect
10. SD16×Dic127C2semidirect
118. C127#3365semidirect

Maximal subgroups

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

subgroupindex
#225506122
#225506052
#225502292
#225502222
#225501782
#225501622
#225500962
#225500002
C4×#145232382
SD16×Dic1272
#460×C1272
#3365127