Finite Group Databasebeta

← #30651425

Subgroups — #30651425

Order 16256.

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

normal subgroups only

This group has 348 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#22550611C211yesmaximal, characteristic
2#22550601C211yesmaximal, characteristic, complemented
3#22550232C212yesmaximal, complemented
4#22550224C211yesmaximal, characteristic
5#22550175C212yesmaximal, complemented
6#22550101C212yesmaximal, complemented
7#22550077C211yesmaximal, characteristic
8#22550000C211yesmaximal, characteristic, complemented
9C4×#14523238C211yesmaximal, characteristic, complemented
10#22549963C212yesmaximal, complemented
11#475×C127C211yesmaximal, characteristic, complemented
12C2×#14523238C2211yescharacteristic
13C2×#14523237C2212yes—
14#16861207—21no—
15#16861206C2211yescharacteristic
16#16861200C2212yes—
17#16861199—22no—
18#16861198C2212yes—
19#16861197—21no—
20#16861194—22no—
21Q8×Dic127—21no—
22C4×Dic254C2211yescharacteristic
23C4×Dic254C2211yescharacteristic
24#16861174C2212yes—
25C4×D508—21no—
26#16861171C2212yes—
27#16861142C2211yescharacteristic
28Dic127⋊C8C2212yes—
29#16861133C2211yescharacteristic
30#16861132C2211yescharacteristic
31#16861131C2212yes—
32#16861129C2212yes—
33#16861129C2212yes—
34C4×C127⋊C8C2211yescharacteristic
35C8×Dic127C2212yes—
36C4⋊Q8×C127C2211yescharacteristic
37C4×Q8×C127C2211yescharacteristic
38#106×C127C2212yes—
39C4⋊C8×C127C2211yescharacteristic
40#103×C127C2212yes—
41C2×Dic254D411yescharacteristic
42C2×Dic254C2311yescharacteristic
43C2×D508—21no—
44C2×D508—21no—
45C2×D508C2312yes—
46C2×C4×D127—22no—
47#14523238—22no—
48#14523237—24no—
49C2×C127⋊C8C2311yescharacteristic
50C2×C127⋊C8C2311yescharacteristic
51C4⋊Dic127—22no—
52C4⋊Dic127D411yescharacteristic
53C4⋊Dic127C2311yescharacteristic
54Dic127⋊C4—22no—
55Dic127⋊C4—22no—
56Dic127⋊C4—21no—
57Dic127⋊C4D412yes—
58(C2×D127)⋊C4—22no—
59(C2×D127)⋊C4—22no—
60(C2×D127)⋊C4—22no—
61C4×Dic127—22no—
62C4×Dic127—21no—
63C4×Dic127C2312yes—
64C2×Q8×C127—21no—
65C2×Q8×C127C2311yescharacteristic
66C2×C8×C127C2312yes—
67C4⋊C4×C127—21no—
68C4⋊C4×C127—21no—
69C4⋊C4×C127C2312yes—
70C4⋊C4×C127C2311yescharacteristic
71C42×C127—21no—
72C42×C127C2311yescharacteristic
73C22×D127—22no—
74Dic254—22no—
75Dic254—21no—
76Dic254#4112yes—
77D508—22no—
78D508—22no—
79D508—42no—
80D508—81no—
81C2×Dic127—21no—
82C2×Dic127—21no—
83C2×Dic127C2×D412yes—
84C2×Dic127C2×D412yes—
85C4×D127—82no—
86C127⋊C8—21no—
87C127⋊C8—21no—
88Q8×C127—21no—
89Q8×C127—21no—
90Q8×C127—42no—
91C2×C4×C127—21no—
92C2×C4×C127—21no—
93C2×C4×C127—22no—
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2411yescharacteristic
97C8×C127—22no—
98C22×C127C22×D411yescharacteristic
99C2×D127—42no—
100C2×D127—42no—
101C2×D127—42no—
102C4×C127—21no—
103C4×C127—41no—
104C4×C127—42no—
105C4×C127—41no—
106C4×C127#13512yes—
107C4×C127C2×#4111yescharacteristic
108C4×C1272−1+411yescharacteristic
109Dic127—22no—
110Dic127—22no—
111Dic127—22no—
112Dic127—41no—
113C2×C127#54811yescharacteristic
114C2×C127C2×#13511yescharacteristic
115C2×C127#57711yescharacteristic
116D127—82no—
117#3365—1271nomaximal, Sylow
118C127#336511yescharacteristic, complemented, Sylow
119#559—1271no—
120#557—1271no—
121#488—1272no—
122#486—1271no—
123#478—1272no—
124#475D12711yescharacteristic, complemented
125#464—1272no—
126#460—1271no—
127C8×Q8—1271no—
128C4×Q16—1271no—
129C4×SD16—1272no—
130C2×Q16—1271no—
131C2×SD16—1272no—
132C4⋊Q8—2541no—
133C4⋊Q8C2×D12711yescharacteristic
134#127—1271no—
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#106C2×D12712yes—
149C4⋊C8—1272no—
150C4⋊C8C2×D12711yescharacteristic
151#103—1271no—
152#103—1271no—
153#103C2×D12712yes—
154#102—1272no—
155#102—1272no—
156#102—1272no—
157C4×C8—1272no—
158C4×C8—1271no—
159C2×Q8—21no—
160C2×Q8—1271no—
161C2×Q8—1271no—
162C2×Q8C22×D12711yescharacteristic
163C2×D4—1272no—
164C2×D4—2541no—
165C2×D4—2541no—
166C22×C4—2542no—
167Q16—2542no—
168SD16—2544no—
169C2×C8—1271no—
170C2×C8—1271no—
171C2×C8C22×D12712yes—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—1272no—
175C4⋊C4—1271no—
176C4⋊C4—1271no—
177C4⋊C4—2542no—
178C4⋊C4—2542no—
179C4⋊C4—2542no—
180C4⋊C4—2541no—
181C4⋊C4C22×D12712yes—
182C4⋊C4C22×D12711yescharacteristic
183C22⋊C4—2542no—
184C22⋊C4—2542no—
185C22⋊C4—2542no—
186C42—21no—
187C42—1272no—
188C42—2542no—
189C42—2541no—
190C42C22×D12711yescharacteristic
191C23—2542no—
192Q8—21no—
193Q8—21no—
194Q8—42no—
195Q8—1272no—
196Q8—2542no—
197Q8—2541no—
198D4—10161no—
199D4—5082no—
200D4—2542no—
201D4—2542no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—1272no—
206C2×C4—1272no—
207C2×C4—10162no—
208C2×C4—2541no—
209C2×C4—2541no—
210C2×C4D4×D12711yescharacteristic
211C2×C4D4×D12711yescharacteristic
212C2×C4C23×D12711yescharacteristic
213C8—22no—
214C8—2541no—
215C8—2541no—
216C22—5082no—
217C22—5082no—
218C22—5082no—
219C22C2×D4×D12711yescharacteristic
220C4—21no—
221C4—41no—
222C4—42no—
223C4—41no—
224C4—5081no—
225C4—2542no—
226C4—2542no—
227C4—2542no—
228C4#1686123012yes—
229C4#41×D12711yescharacteristic
230C4#1686125711yescharacteristic
231C2—10162no—
232C2#2255053611yescharacteristic
233C2C2×#1686123011yescharacteristic
234C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#475×C127C2
socleC22×C127C22×D4
prime radical#16861206C22
hypercenter#475D127
hypocenterC127#3365
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #22550601C2semidirect
3. #22550232C2semidirect
5. #22550175C2semidirect
6. #22550101C2semidirect
8. #22550000C2semidirect
9. C4×#14523238C2semidirect
10. #22549963C2semidirect
11. #475×C127C2semidirect
118. C127#3365semidirect
124. #475D127semidirect

Maximal subgroups

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

subgroupindex
#225506112
#225506012
#225502322
#225502242
#225501752
#225501012
#225500772
#225500002
C4×#145232382
#225499632
#475×C1272
#3365127