Finite Group Databasebeta

← #30648396

Subgroups — #30648396

Order 16256.

484 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#16861215C211yesmaximal, characteristic, complemented
2#22550454C212yesmaximal, complemented
3#22550119C214yesmaximal, complemented
4#22550114C212yesmaximal, complemented
5#22550109C212yesmaximal, complemented
6C2×#16861143C211yesmaximal, characteristic, complemented
7#22549877C212yesmaximal, complemented
8C2×#135×C127C211yesmaximal, characteristic, complemented
9C2×#14523248C2211yescharacteristic
10#16861215—24no—
11C2×#14523232C2211yescharacteristic
12C2×#14523231C2212yes—
13C2×C8×D127C2212yes—
14#16861188—22no—
15#16861185C2214yes—
16#16861183—22no—
17#16861181—24no—
18#16861180C2214yes—
19C2×C4⋊Dic127C2212yes—
20#16861143C2212yes—
21#16861143C2212yes—
22#16861133C2214yes—
23#16861130C2214yes—
24C2×#41×C127C2212yes—
25#135×C127—24no—
26C2×Q16×C127C2211yescharacteristic
27C2×SD16×C127C2212yes—
28C2×D8×C127C2211yescharacteristic
29C22×C8×C127C2211yescharacteristic
30#14523248—22no—
31#14523248—22no—
32C2×Dic254D411yescharacteristic
33C2×D508D411yescharacteristic
34C2×C22⋊D127—24no—
35C2×C22⋊D127D412yes—
36C22×Dic127—22no—
37C2×C4×D127C2312yes—
38#14523232—22no—
39#14523231—24no—
40C2×C127⋊C8C2312yes—
41C8×D127—24no—
42C4⋊Dic127—22no—
43C4⋊Dic127C2314yes—
44Dic127⋊C4—24no—
45C22⋊Dic127—24no—
46(C2×D127)⋊C4—24no—
47#41×C127—24no—
48#41×C127—24no—
49C2×Q8×C127C2312yes—
50C2×D4×C127—22no—
51C2×D4×C127C2312yes—
52C22×C4×C127—22no—
53C22×C4×C127C2311yescharacteristic
54Q16×C127—22no—
55SD16×C127—24no—
56D8×C127—22no—
57C2×C8×C127D414yescomplemented
58C2×C8×C127C2311yescharacteristic
59C2×C8×C127C2311yescharacteristic
60C22×D127C2×D412yes—
61Dic254—22no—
62D508—22no—
63C22⋊D127—24no—
64C22⋊D127—84no—
65C2×Dic127—24no—
66C2×Dic127—82no—
67C2×Dic127C2×D412yes—
68C4×D127—22no—
69C4×D127—22no—
70C127⋊C8—22no—
71C23×C127—22no—
72C23×C127C2×D411yescharacteristic
73Q8×C127—22no—
74Q8×C127—22no—
75D4×C127—22no—
76D4×C127—22no—
77D4×C127—44no—
78C2×C4×C127—22no—
79C2×C4×C127—44no—
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D412yes—
82C2×C4×C127#4112yes—
83C2×C4×C127C2411yescharacteristic
84C8×C127C2×D412yes—
85C8×C127C2×D412yes—
86C22×C127—21no—
87C22×C127—21no—
88C22×C127—42no—
89C22×C127—42no—
90C22×C127—42no—
91C22×C127#12112yes—
92C22×C127C22×D411yescharacteristic
93C2×D127—22no—
94C2×D127—22no—
95C2×D127—22no—
96C4×C127—42no—
97C4×C127#12112yes—
98C4×C127C22×D411yescharacteristic
99C4×C127C2×#4111yescharacteristic
100Dic127—22no—
101Dic127—44no—
102C2×C127—22no—
103C2×C127—82no—
104C2×C127C2×#12111yescharacteristic
105C2×C127#57611yescharacteristic
106C2×C127#57811yescharacteristic
107D127—42no—
108#3061—1271nomaximal, Sylow
109C127#306111yescharacteristic, complemented, Sylow
110C2×#135D12711yescharacteristic, complemented
111C2×#131—1271no—
112#537—1272no—
113#468—1274no—
114#467—1272no—
115#466—1272no—
116C2×#107—1271no—
117#417—1272no—
118C2×#41—1271no—
119C2×#41C2×D12712yes—
120#135—24no—
121C2×Q16C2×D12711yescharacteristic
122C2×SD16C2×D12712yes—
123C2×D8C2×D12711yescharacteristic
124#131—2544no—
125C2×M16—1271no—
126C2×M16—1272no—
127C22×C8—1272no—
128C22×C8C2×D12711yescharacteristic
129C22⋊Q8—1274no—
130C22⋊Q8—2542no—
131#121—1274no—
132#121—2542no—
133#121—2544no—
134C2×C4⋊C4—1272no—
135#107—1272no—
136#107—1272no—
137#103—1274no—
138#102—1274no—
139#41—24no—
140#41—24no—
141#41—2542no—
142#41—2542no—
143C2×Q8—1271no—
144C2×Q8C22×D12712yes—
145C2×D4—22no—
146C2×D4—1272no—
147C2×D4—1271no—
148C2×D4—2544no—
149C2×D4C22×D12712yes—
150C22×C4—22no—
151C22×C4—1272no—
152C22×C4—2542no—
153C22×C4C22×D12711yescharacteristic
154Q16—22no—
155SD16—24no—
156D8—22no—
157M16—2542no—
158M16—2544no—
159C2×C8—1272no—
160C2×C8—2544no—
161C2×C8C22⋊D12714yescomplemented
162C2×C8C22×D12711yescharacteristic
163C2×C8C22×D12711yescharacteristic
164C4⋊C4—1274no—
165C4⋊C4—2544no—
166C4⋊C4—2542no—
167C22⋊C4—2544no—
168C22⋊C4—2544no—
169C23—22no—
170C23—1272no—
171C23D4×D12711yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—2542no—
175D4—22no—
176D4—22no—
177D4—44no—
178D4—10164no—
179D4—2544no—
180D4—2542no—
181C2×C4—22no—
182C2×C4—44no—
183C2×C4—1272no—
184C2×C4—10162no—
185C2×C4—2544no—
186C2×C4—2542no—
187C2×C4—2542no—
188C2×C4D4×D12711yescharacteristic
189C2×C4C2×C22⋊D12712yes—
190C2×C4#1452324712yes—
191C2×C4C23×D12711yescharacteristic
192C8—2542no—
193C8C2×C22⋊D12712yes—
194C8C2×C22⋊D12712yes—
195C22—21no—
196C22—21no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—2542no—
201C22—2542no—
202C22—2542no—
203C22#1686118012yes—
204C22C2×D4×D12711yescharacteristic
205C4—42no—
206C4—5084no—
207C4—2542no—
208C4#1686118012yes—
209C4C22×C22⋊D12711yescharacteristic
210C4C2×#1452324711yescharacteristic
211C2—22no—
212C2—82no—
213C2—5082no—
214C2C2×#1686118011yescharacteristic
215C2#2255067811yescharacteristic
216C2#2255069011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#135×C127C2
socleC22×C127C22×D4
hypercenterC2×#135D127
hypocenterC127#3061
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#16861215C2semidirect
2. #22550454C2semidirect
3. #22550119C2semidirect
4. #22550114C2semidirect
5. #22550109C2semidirect
6. C2×#16861143C2semidirect
7. #22549877C2semidirect
8. C2×#135×C127C2semidirect
57. C2×C8×C127D4semidirect
109. C127#3061semidirect
110. C2×#135D127semidirect
161. C2×C8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612152
#225504542
#225501192
#225501142
#225501092
C2×#168611432
#225498772
C2×#135×C1272
#3061127