Finite Group Databasebeta

← #30651367

Subgroups — #30651367

Order 16256.

370 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550601C211yesmaximal, characteristic, complemented
2D4×Dic254C211yesmaximal, characteristic
3#22550226C211yesmaximal, characteristic, complemented
4#22550225C212yesmaximal, complemented
5#22550162C211yesmaximal, characteristic, complemented
6#22550159C212yesmaximal, complemented
7#22550079C212yesmaximal
8#22550000C211yesmaximal, characteristic, complemented
9SD16×Dic127C212yesmaximal
10C4×#14523236C211yesmaximal, characteristic
11#461×C127C211yesmaximal, characteristic
12C22×Dic254—21no—
13C2×#14523236C2211yescharacteristic
14#16861209C2211yescharacteristic
15#16861208C2212yes—
16#16861207C2212yes—
17#16861200—21no—
18#16861199—22no—
19#16861188—21no—
20#16861186—22no—
21Q8×Dic127C2212yes—
22C4×Dic254C2211yescharacteristic
23C4×Dic254C2211yescharacteristic
24#16861174—21no—
25D4×Dic127C2212yes—
26#16861141C2212yes—
27#16861140C2211yescharacteristic
28Dic127⋊C8C2212yes—
29#16861134C2212yes—
30#16861133C2212yes—
31#16861133C2212yes—
32#16861132C2211yescharacteristic
33#16861130C2211yescharacteristic
34C4×C127⋊C8C2211yescharacteristic
35C8×Dic127C2212yes—
36C2×SD16×C127C2212yes—
37C4⋊Q8×C127C2211yescharacteristic
38C4×D4×C127C2211yescharacteristic
39C4⋊C8×C127C2211yescharacteristic
40#103×C127C2212yes—
41C2×Dic254—22no—
42C2×Dic254—21no—
43C2×Dic254—44no—
44C2×Dic254D411yescharacteristic
45C2×Dic254C2311yescharacteristic
46C22×Dic127—22no—
47#14523236—22no—
48C2×C127⋊C8C2311yescharacteristic
49C2×C127⋊C8C2311yescharacteristic
50C4⋊Dic127—22no—
51C4⋊Dic127—21no—
52C4⋊Dic127D411yescharacteristic
53C4⋊Dic127C2312yes—
54C4⋊Dic127C2311yescharacteristic
55Dic127⋊C4—22no—
56Dic127⋊C4—22no—
57Dic127⋊C4—22no—
58Dic127⋊C4—21no—
59Dic127⋊C4D412yes—
60C22⋊Dic127—22no—
61C4×Dic127—22no—
62C4×Dic127—21no—
63C4×Dic127C2312yes—
64C2×Q8×C127C2312yes—
65C2×D4×C127C2311yescharacteristic
66C22×C4×C127—21no—
67SD16×C127—24no—
68C2×C8×C127C2312yes—
69C4⋊C4×C127—21no—
70C4⋊C4×C127—21no—
71C4⋊C4×C127C2311yescharacteristic
72C22⋊C4×C127—21no—
73C42×C127C2311yescharacteristic
74Dic254—21no—
75Dic254—21no—
76Dic254—42no—
77Dic254—44no—
78Dic254SD1614yescomplemented
79C2×Dic127—21no—
80C2×Dic127—21no—
81C2×Dic127—22no—
82C2×Dic127—44no—
83C2×Dic127C2×D412yes—
84C2×Dic127C2×D412yes—
85C127⋊C8—21no—
86C127⋊C8—21no—
87C23×C127—21no—
88Q8×C127—22no—
89Q8×C127—22no—
90D4×C127—21no—
91D4×C127#4112yes—
92C2×C4×C127—21no—
93C2×C4×C127—22no—
94C2×C4×C127—42no—
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2411yescharacteristic
98C8×C127—22no—
99C22×C127—22no—
100C22×C127—41no—
101C22×C127—41no—
102C22×C127C22×D411yescharacteristic
103C4×C127—21no—
104C4×C127—42no—
105C4×C127—41no—
106C4×C127C2×SD1612yes—
107C4×C127C2×#4111yescharacteristic
108C4×C1272−1+411yescharacteristic
109Dic127—22no—
110Dic127—41no—
111Dic127—42no—
112Dic127—41no—
113Dic127C2×SD1614yes—
114C2×C127—42no—
115C2×C127#54811yescharacteristic
116C2×C127C22×SD1611yescharacteristic
117C2×C127#57811yescharacteristic
118#3359—1271nomaximal, Sylow
119C127#335911yescharacteristic, complemented, Sylow
120#557—1271no—
121D4×Q8—1271no—
122#487—1271no—
123#487—1272no—
124#475—1271no—
125#475—1272no—
126#461—1272no—
127#461D12711yescharacteristic
128C8×Q8—1271no—
129C4×SD16—1272no—
130C4×SD16—1271no—
131C22×Q8—2541no—
132C2×SD16—1271no—
133C2×SD16C2×D12712yes—
134C4⋊Q8—1272no—
135C4⋊Q8—1272no—
136C4⋊Q8—1271no—
137C4⋊Q8C2×D12711yescharacteristic
138#125—2542no—
139#125—2541no—
140C22⋊Q8—2542no—
141C22⋊Q8—2541no—
142C4×Q8—1271no—
143C4×Q8—1272no—
144C4×Q8—1271no—
145C4×Q8—2541no—
146C4×D4—1272no—
147C4×D4C2×D12711yescharacteristic
148#106—1272no—
149#106—1271no—
150C4⋊C8—1272no—
151C4⋊C8C2×D12711yescharacteristic
152#103—1272no—
153#103—1272no—
154#103—1272no—
155#103—1271no—
156#103C2×D12712yes—
157#102—1271no—
158C4×C8—1272no—
159C4×C8—1271no—
160C2×Q8—1271no—
161C2×Q8—1271no—
162C2×Q8—5084no—
163C2×Q8—2542no—
164C2×Q8—2541no—
165C2×Q8C22×D12712yes—
166C2×D4C22×D12711yescharacteristic
167C22×C4—21no—
168C22×C4—2542no—
169SD16—24no—
170SD16—2542no—
171C2×C8—1271no—
172C2×C8—1271no—
173C2×C8C22×D12712yes—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—1272no—
177C4⋊C4—1271no—
178C4⋊C4—1271no—
179C4⋊C4—1272no—
180C4⋊C4—2542no—
181C4⋊C4—2542no—
182C4⋊C4—2542no—
183C4⋊C4—2542no—
184C4⋊C4—2541no—
185C4⋊C4—2541no—
186C4⋊C4C22×D12711yescharacteristic
187C22⋊C4—21no—
188C22⋊C4—2542no—
189C42—1272no—
190C42—2542no—
191C42—2541no—
192C42C22×D12711yescharacteristic
193C23—21no—
194Q8—22no—
195Q8—22no—
196Q8—1274no—
197Q8—5084no—
198Q8—5082no—
199Q8—2541no—
200Q8—2541no—
201D4—21no—
202D4#1452324712yes—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4—1272no—
207C2×C4—1272no—
208C2×C4—5084no—
209C2×C4—2541no—
210C2×C4—2541no—
211C2×C4—2542no—
212C2×C4D4×D12711yescharacteristic
213C2×C4D4×D12711yescharacteristic
214C2×C4C23×D12711yescharacteristic
215C8—22no—
216C8—2541no—
217C8—2541no—
218C22—22no—
219C22—41no—
220C22—41no—
221C22C2×D4×D12711yescharacteristic
222C4—21no—
223C4—42no—
224C4—41no—
225C4—1274no—
226C4—5081no—
227C4—5082no—
228C4—5081no—
229C4—2542no—
230C4SD16×D12712yes—
231C4C2×#1452324711yescharacteristic
232C4#1686125611yescharacteristic
233C2—42no—
234C2#2255053211yescharacteristic
235C2C2×SD16×D12711yescharacteristic
236C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#461×C127C2
socleC22×C127C22×D4
prime radicalC2×D4×C127C23
hypercenter#461D127
hypocenterC127#3359
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. #22550601C2semidirect
3. #22550226C2semidirect
4. #22550225C2semidirect
5. #22550162C2semidirect
6. #22550159C2semidirect
8. #22550000C2semidirect
78. Dic254SD16semidirect
119. C127#3359semidirect

Maximal subgroups

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

subgroupindex
#225506012
D4×Dic2542
#225502262
#225502252
#225501622
#225501592
#225500792
#225500002
SD16×Dic1272
C4×#145232362
#461×C1272
#3359127