Finite Group Databasebeta

← #30649285

Subgroups — #30649285

Order 16256.

412 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 145 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#22550433C211yesmaximal, characteristic, complemented
2#22550272C211yesmaximal, characteristic, complemented
3#22550248C211yesmaximal, characteristic, complemented
4#22550228C211yesmaximal, characteristic, complemented
5#22550222C211yesmaximal, characteristic, complemented
6#22550103C214yesmaximal, complemented
7C4×#14523236C214yesmaximal, complemented
8C2×C4×C127⋊C8C211yesmaximal, characteristic, complemented
9#532×C127C211yesmaximal, characteristic
10C2×#14523236C2214yes—
11C22×C127⋊C8C2212yes—
12#16861209C2211yescharacteristic
13#16861209C2211yescharacteristic
14#16861201—21no—
15#16861188—21no—
16#16861188C2212yes—
17C4×Dic254—21no—
18C4×Dic254C2212yes—
19#16861168—21no—
20#16861140C2214yes—
21#16861132C2214yes—
22#16861130C2214yes—
23C4×C127⋊C8C2211yescharacteristic
24C4×C127⋊C8C2212yes—
25C4×C127⋊C8C2211yescharacteristic
26C2×#41×C127—21no—
27C4⋊Q8×C127C2211yescharacteristic
28#127×C127C2211yescharacteristic
29#124×C127—21no—
30#121×C127—21no—
31#121×C127C2212yes—
32C4×D4×C127—21no—
33C4×D4×C127C2212yes—
34C2×C42×C127C2211yescharacteristic
35C2×Dic254—21no—
36C2×Dic254C2312yes—
37#14523236—28no—
38C2×C127⋊C8—24no—
39C2×C127⋊C8C2312yes—
40C2×C127⋊C8C2312yes—
41C4⋊Dic127—21no—
42C4⋊Dic127—21no—
43C4⋊Dic127—21no—
44C4⋊Dic127C2312yes—
45Dic127⋊C4—22no—
46Dic127⋊C4—22no—
47C22⋊Dic127—22no—
48C4×Dic127—22no—
49#41×C127—44no—
50C2×Q8×C127—21no—
51C2×D4×C127—22no—
52C2×D4×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127C2312yes—
55C22×C4×C127—22no—
56C22×C4×C127D411yescharacteristic
57C22×C4×C127D411yescharacteristic
58C22×C4×C127C2311yescharacteristic
59C4⋊C4×C127—21no—
60C4⋊C4×C127C2312yes—
61C22⋊C4×C127—22no—
62C22⋊C4×C127—22no—
63C42×C127D411yescharacteristic
64C42×C127D411yescharacteristic
65C42×C127C2311yescharacteristic
66C42×C127C2311yescharacteristic
67Dic254—24no—
68Dic254—42no—
69C2×Dic127—22no—
70C2×Dic127—22no—
71C127⋊C8—22no—
72C127⋊C8#4114yescomplemented
73C23×C127—22no—
74C23×C127C2×D411yescharacteristic
75Q8×C127—42no—
76D4×C127—24no—
77D4×C127—44no—
78D4×C127—42no—
79D4×C127—42no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—22no—
83C2×C4×C127—44no—
84C2×C4×C127SD1614yescomplemented
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2×D411yescharacteristic
89C2×C4×C127C2×D411yescharacteristic
90C2×C4×C127C2411yescharacteristic
91C22×C127—21no—
92C22×C127—21no—
93C22×C127—42no—
94C22×C127—42no—
95C22×C127—42no—
96C22×C127C2×SD1612yes—
97C22×C127C22×D411yescharacteristic
98C4×C127—21no—
99C4×C127—21no—
100C4×C127—42no—
101C4×C127C2×SD1612yes—
102C4×C127C2×SD1612yes—
103C4×C127#13512yes—
104C4×C127C2×#4112yes—
105Dic127—42no—
106Dic127—42no—
107C2×C127—22no—
108C2×C127—82no—
109C2×C127#53211yescharacteristic
110C2×C127C22×SD1611yescharacteristic
111C2×C127C2×#13511yescharacteristic
112#3167—1271nomaximal, Sylow
113C127#316711yescharacteristic, complemented, Sylow
114#533—1271no—
115#532D12711yescharacteristic
116#498—1271no—
117#492—1271no—
118#487—1271no—
119#486—1271no—
120#465—1274no—
121C4×SD16—1274no—
122C2×C4×C8—1271no—
123C2×#41—21no—
124C2×SD16—1274no—
125C22×C8—1272no—
126C4⋊Q8—1271no—
127C4⋊Q8—1271no—
128C4⋊Q8C2×D12711yescharacteristic
129#127C2×D12711yescharacteristic
130#125—2541no—
131#124—21no—
132C22⋊Q8—1272no—
133C22⋊Q8—2541no—
134#121—21no—
135#121C2×D12712yes—
136C4×Q8—1272no—
137C4×Q8—2541no—
138C4×D4—21no—
139C4×D4C2×D12712yes—
140#117—2541no—
141C2×C42C2×D12711yescharacteristic
142#106—1274no—
143#103—1274no—
144#102—1274no—
145C4×C8—1272no—
146C4×C8—1271no—
147C4×C8—1271no—
148#41—44no—
149C2×Q8—21no—
150C2×Q8—1272no—
151C2×Q8—2541no—
152C2×D4—22no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4C22×D12712yes—
156C22×C4—22no—
157C22×C4C22⋊D12711yescharacteristic
158C22×C4C22⋊D12711yescharacteristic
159C22×C4C22×D12711yescharacteristic
160SD16—2548no—
161C2×C8—1272no—
162C2×C8—1272no—
163C2×C8—2544no—
164C4⋊C4—21no—
165C4⋊C4—1272no—
166C4⋊C4—2542no—
167C4⋊C4—2542no—
168C4⋊C4—2541no—
169C4⋊C4—2541no—
170C4⋊C4—2541no—
171C4⋊C4C22×D12712yes—
172C22⋊C4—22no—
173C22⋊C4—22no—
174C22⋊C4—2542no—
175C42—2542no—
176C42C22⋊D12711yescharacteristic
177C42C22⋊D12711yescharacteristic
178C42C22×D12711yescharacteristic
179C42C22×D12711yescharacteristic
180C23—22no—
181C23C2×C22⋊D12711yescharacteristic
182Q8—42no—
183Q8—5082no—
184Q8—2544no—
185D4—24no—
186D4—44no—
187D4—42no—
188D4—42no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—44no—
193C2×C4—2542no—
194C2×C4—2542no—
195C2×C4#1452323614yescomplemented
196C2×C4C2×C22⋊D12711yescharacteristic
197C2×C4C2×C22⋊D12711yescharacteristic
198C2×C4C2×C22⋊D12711yescharacteristic
199C2×C4C2×C22⋊D12711yescharacteristic
200C2×C4C2×C22⋊D12711yescharacteristic
201C2×C4C23×D12711yescharacteristic
202C8—1274no—
203C8—2542no—
204C22—21no—
205C22—21no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22C2×#1452323612yes—
210C22C22×C22⋊D12711yescharacteristic
211C4—21no—
212C4—21no—
213C4—42no—
214C4—5082no—
215C4—5082no—
216C4C2×#1452323612yes—
217C4C2×#1452323612yes—
218C4#1686122912yes—
219C4#41×D12712yes—
220C2—22no—
221C2—82no—
222C2#2255042211yescharacteristic
223C2C22×#1452323611yescharacteristic
224C2C2×#1686122911yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#532×C127C2
socleC22×C127C22×D4
hypercenter#532D127
hypocenterC127#3167
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #22550433C2semidirect
2. #22550272C2semidirect
3. #22550248C2semidirect
4. #22550228C2semidirect
5. #22550222C2semidirect
6. #22550103C2semidirect
7. C4×#14523236C2semidirect
8. C2×C4×C127⋊C8C2semidirect
72. C127⋊C8#41semidirect
84. C2×C4×C127SD16semidirect
113. C127#3167semidirect
195. C2×C4#14523236semidirect

Maximal subgroups

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

subgroupindex
#225504332
#225502722
#225502482
#225502282
#225502222
#225501032
C4×#145232362
C2×C4×C127⋊C82
#532×C1272
#3167127