Finite Group Databasebeta

← #30651474

Subgroups — #30651474

Order 16256.

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

normal subgroups only

This group has 338 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
1Q8×Dic254C211yesmaximal, characteristic, complemented
2#22550600C211yesmaximal, characteristic
3#22550237C212yesmaximal
4#22550226C211yesmaximal, characteristic, complemented
5#22550167C211yesmaximal, characteristic
6#22550159C212yesmaximal, complemented
7#22550100C212yesmaximal
8#22550003C211yesmaximal, characteristic, complemented
9#22549983C212yesmaximal, complemented
10C4×#14523235C211yesmaximal, characteristic
11#462×C127C211yesmaximal, characteristic, complemented
12C2×#14523235C2211yescharacteristic
13#16861209—21no—
14#16861209C2211yescharacteristic
15#16861208—22no—
16#16861208C2212yes—
17#16861204—22no—
18#16861198C2212yes—
19#16861189—21no—
20Q8×Dic127C2212yes—
21Q8×Dic127C2212yes—
22C4×Dic254—21no—
23C4×Dic254C2211yescharacteristic
24C4×D508C2211yescharacteristic
25#16861167—21no—
26#16861141C2212yes—
27#16861140C2211yescharacteristic
28Dic127⋊C8C2212yes—
29#16861133C2212yes—
30#16861133C2211yescharacteristic
31#16861133C2212yes—
32#16861131C2212yes—
33#16861129C2211yescharacteristic
34#16861119C2212yes—
35C4×C127⋊C8C2211yescharacteristic
36C2×Q16×C127C2212yes—
37C4⋊Q8×C127C2211yescharacteristic
38C4×Q8×C127C2211yescharacteristic
39C4⋊C8×C127C2211yescharacteristic
40#103×C127C2212yes—
41C2×Dic254—22no—
42C2×Dic254—21no—
43C2×Dic254D411yescharacteristic
44C2×D508C2311yescharacteristic
45C2×C4×D127—21no—
46#14523235—22no—
47C2×C127⋊C8C2311yescharacteristic
48C2×C127⋊C8C2311yescharacteristic
49C4⋊Dic127—22no—
50C4⋊Dic127—22no—
51C4⋊Dic127—21no—
52C4⋊Dic127D411yescharacteristic
53C4⋊Dic127C2312yes—
54C4⋊Dic127C2311yescharacteristic
55Dic127⋊C4—22no—
56Dic127⋊C4D412yes—
57(C2×D127)⋊C4—22no—
58(C2×D127)⋊C4—22no—
59(C2×D127)⋊C4—21no—
60C4×Dic127—22no—
61C4×Dic127—22no—
62C4×Dic127C2312yes—
63C2×Q8×C127C2312yes—
64C2×Q8×C127C2311yescharacteristic
65Q16×C127—24no—
66C2×C8×C127C2312yes—
67C4⋊C4×C127—21no—
68C4⋊C4×C127—21no—
69C4⋊C4×C127—21no—
70C4⋊C4×C127C2311yescharacteristic
71C42×C127—21no—
72C42×C127C2311yescharacteristic
73C22×D127—21no—
74Dic254—22no—
75Dic254—82no—
76Dic254—81no—
77D508—21no—
78D508—21no—
79C2×Dic127—21no—
80C2×Dic127—22no—
81C2×Dic127C2×D412yes—
82C2×Dic127C2×D412yes—
83C4×D127—42no—
84C127⋊C8—21no—
85C127⋊C8—21no—
86Q8×C127—21no—
87Q8×C127—22no—
88Q8×C127—22no—
89Q8×C127#4112yes—
90C2×C4×C127—21no—
91C2×C4×C127—21no—
92C2×C4×C127—22no—
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2411yescharacteristic
96C8×C127—22no—
97C22×C127C22×D411yescharacteristic
98C2×D127—41no—
99C2×D127—41no—
100C2×D127—41no—
101C4×C127—22no—
102C4×C127—21no—
103C4×C127—42no—
104C4×C127—41no—
105C4×C127#13712yes—
106C4×C127C2×#4111yescharacteristic
107C4×C1272−1+411yescharacteristic
108Dic127—22no—
109Dic127—22no—
110Dic127—41no—
111Dic127—42no—
112C2×C127#54811yescharacteristic
113C2×C127C2×#13711yescharacteristic
114C2×C127#57711yescharacteristic
115D127—81no—
116#3369—1271nomaximal, Sylow
117C127#336911yescharacteristic, complemented, Sylow
118Q82—1271no—
119#556—1271no—
120#489—1272no—
121#487—1271no—
122#476—1271no—
123#475—1272no—
124#464—1272no—
125#462D12711yescharacteristic, complemented
126#446—1271no—
127#441—1272no—
128C4×SD16—1271no—
129C2×Q16C2×D12712yes—
130C2×SD16—1271no—
131C4⋊Q8—1272no—
132C4⋊Q8—1271no—
133C4⋊Q8—2541no—
134C4⋊Q8—2542no—
135C4⋊Q8C2×D12711yescharacteristic
136#126—2542no—
137#124—1272no—
138C22⋊Q8—2541no—
139C4×Q8—1271no—
140C4×Q8—1272no—
141C4×Q8—1272no—
142C4×Q8—2541no—
143C4×Q8C2×D12711yescharacteristic
144C4×D4—1271no—
145#117—2541no—
146#106—1272no—
147#106—1271no—
148C4⋊C8—1272no—
149C4⋊C8C2×D12711yescharacteristic
150#103—1271no—
151#103—1272no—
152#103—1272no—
153#103C2×D12712yes—
154#102—1272no—
155#102—1271no—
156#97—1272no—
157C4×C8—1271no—
158C2×Q8—1271no—
159C2×Q8—2542no—
160C2×Q8—2541no—
161C2×Q8C22×D12712yes—
162C2×Q8C22×D12711yescharacteristic
163C2×D4—1271no—
164C22×C4—2541no—
165Q16—24no—
166SD16—2542no—
167C2×C8—1271no—
168C2×C8—1271no—
169C2×C8C22×D12712yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
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⋊C4C22×D12711yescharacteristic
182C22⋊C4—2542no—
183C22⋊C4—2542no—
184C22⋊C4—2541no—
185C42—21no—
186C42—1272no—
187C42—2542no—
188C42—2542no—
189C42C22×D12711yescharacteristic
190C23—2541no—
191Q8—21no—
192Q8—22no—
193Q8—22no—
194Q8—10161no—
195Q8—10162no—
196Q8—2542no—
197Q8#1452324712yes—
198D4—2541no—
199D4—2541no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—1272no—
204C2×C4—1272no—
205C2×C4—5082no—
206C2×C4—2541no—
207C2×C4—2542no—
208C2×C4D4×D12711yescharacteristic
209C2×C4D4×D12711yescharacteristic
210C2×C4C23×D12711yescharacteristic
211C8—22no—
212C8—2541no—
213C8—2541no—
214C22—5081no—
215C22—5081no—
216C22—5081no—
217C22C2×D4×D12711yescharacteristic
218C4—22no—
219C4—21no—
220C4—42no—
221C4—41no—
222C4—5081no—
223C4—5082no—
224C4—2542no—
225C4—2542no—
226C4#1686124112yes—
227C4C2×#1452324711yescharacteristic
228C4#1686125611yescharacteristic
229C2—10161no—
230C2#2255053211yescharacteristic
231C2C2×#1686124111yescharacteristic
232C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#462×C127C2
socleC22×C127C22×D4
prime radicalC2×D508C23
hypercenter#462D127
hypocenterC127#3369
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. Q8×Dic254C2semidirect
4. #22550226C2semidirect
6. #22550159C2semidirect
8. #22550003C2semidirect
9. #22549983C2semidirect
11. #462×C127C2semidirect
117. C127#3369semidirect
125. #462D127semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic2542
#225506002
#225502372
#225502262
#225501672
#225501592
#225501002
#225500032
#225499832
C4×#145232352
#462×C1272
#3369127