Finite Group Databasebeta

← #30638024

Subgroups — #30638024

Order 16256.

230 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 45 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#22550620C211yesmaximal, characteristic, complemented
2#22550266C211yesmaximal, characteristic, complemented
3#22550265C211yesmaximal, characteristic, complemented
4#22550241C211yesmaximal, characteristic, complemented
5#22549617C211yesmaximal, characteristic, complemented
6#22549615C211yesmaximal, characteristic, complemented
7#329×C127C211yesmaximal, characteristic, complemented
8C2×#14523237—21no—
9C2×#14523235—21no—
10C2×D1016—21no—
11C2×#14523233—21no—
12#16861206C2211yescharacteristic
13#16861201C2211yescharacteristic
14#16861197C2211yescharacteristic
15#16861193—21no—
16#16861191—21no—
17#16861186—21no—
18#16861185—21no—
19#16861182—21no—
20#16861181—21no—
21#16861131—21no—
22#16861131—21no—
23#16861120C2211yescharacteristic
24#124×C127C2211yescharacteristic
25#97×C127C2211yescharacteristic
26#97×C127C2211yescharacteristic
27C2×Dic254D411yescharacteristic
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508D411yescharacteristic
32C2×C22⋊D127—21no—
33C2×C22⋊D127—21no—
34C22×Dic127—21no—
35C2×C4×D127—21no—
36#14523237—42no—
37#14523235—42no—
38D1016—42no—
39#14523233—42no—
40C2×C127⋊C8—21no—
41C4⋊Dic127—21no—
42C4⋊Dic127D411yescharacteristic
43C4⋊Dic127D411yescharacteristic
44Dic127⋊C4—21no—
45Dic127⋊C4—21no—
46Dic127⋊C4—21no—
47C22⋊Dic127—21no—
48(C2×D127)⋊C4—21no—
49(C2×D127)⋊C4—21no—
50(C2×D127)⋊C4—21no—
51C2×Q8×C127D411yescharacteristic
52C2×D4×C127D411yescharacteristic
53C2×C8×C127—21no—
54C2×C8×C127—21no—
55C22⋊C4×C127—21no—
56C22⋊C4×C127—21no—
57C42×C127C2311yescharacteristic
58C22×D127—21no—
59C22×D127—41no—
60Dic254—41no—
61D508—41no—
62D508—41no—
63D508—41no—
64D508—41no—
65D508—41no—
66D508—42no—
67C22⋊D127—81no—
68C22⋊D127—81no—
69C2×Dic127—21no—
70C2×Dic127—21no—
71C2×Dic127—21no—
72C2×Dic127—81no—
73C4×D127—81no—
74C127⋊C8—22no—
75C23×C127—21no—
76Q8×C127—41no—
77D4×C127—41no—
78C2×C4×C127—21no—
79C2×C4×C127C2×D411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C8×C127—22no—
83C8×C127—41no—
84C22×C127—41no—
85C22×C127—41no—
86C22×C127—41no—
87C22×C127#12011yescharacteristic
88C2×D127—41no—
89C2×D127—41no—
90C2×D127—41no—
91C2×D127—81no—
92C2×D127—81no—
93C2×D127—81no—
94C4×C127—21no—
95C4×C127—21no—
96C4×C127—21no—
97C4×C127—41no—
98Dic127—41no—
99Dic127—41no—
100Dic127—41no—
101C2×C127—81no—
102C2×C127#45311yescharacteristic
103C2×C127#45311yescharacteristic
104C2×C127#45411yescharacteristic
105D127—81no—
106D127—161no—
107#1656—1271nomaximal, Sylow
108C127#165611yescharacteristic, complemented, Sylow
109#562—1271no—
110#496—1271no—
111#496—1271no—
112#490—1271no—
113#330—1271no—
114#329—1271no—
115#329D12711yescharacteristic, complemented
116C2×SD16—2541no—
117C2×SD16—2541no—
118C2×D8—2541no—
119C2×D8—2541no—
120#127—1271no—
121#125—1271no—
122#124—1271no—
123#124C2×D12711yescharacteristic
124#123—2541no—
125#123—2541no—
126C22⋊Q8—2541no—
127C22⋊Q8—2541no—
128#121—2541no—
129#121—2541no—
130#102—2541no—
131#102—2541no—
132#97—1271no—
133#97C2×D12711yescharacteristic
134#97C2×D12711yescharacteristic
135C2×Q8—1271no—
136C2×Q8D50811yescharacteristic
137C2×D4—1271no—
138C2×D4—2541no—
139C2×D4—2541no—
140C2×D4—2541no—
141C2×D4—2541no—
142C2×D4—2541no—
143C2×D4D50811yescharacteristic
144C22×C4—2541no—
145C22×C4—2541no—
146SD16—5082no—
147SD16—5082no—
148D8—5082no—
149D8—5082no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—2541no—
153C4⋊C4—1271no—
154C4⋊C4—1271no—
155C4⋊C4—2541no—
156C4⋊C4—2541no—
157C4⋊C4—2541no—
158C4⋊C4—2541no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—2541no—
162C22⋊C4—2541no—
163C22⋊C4—2541no—
164C22⋊C4—2541no—
165C42C22×D12711yescharacteristic
166C23—21no—
167C23—5081no—
168C23—2541no—
169Q8—41no—
170Q8—5081no—
171D4—41no—
172D4—10161no—
173D4—10161no—
174D4—5081no—
175D4—5081no—
176D4—5081no—
177D4—5081no—
178D4—5082no—
179D4—5081no—
180C2×C4—21no—
181C2×C4—10161no—
182C2×C4—10161no—
183C2×C4—2541no—
184C2×C4—2541no—
185C2×C4—2541no—
186C2×C4D4×D12711yescharacteristic
187C2×C4D4×D12711yescharacteristic
188C2×C4C2×D50811yescharacteristic
189C8—22no—
190C8—41no—
191C8—2542no—
192C22—41no—
193C22—41no—
194C22—41no—
195C22—10161no—
196C22—10161no—
197C22—10161no—
198C22—5081no—
199C22—5081no—
200C22—5081no—
201C22#1686117911yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—41no—
206C4—5081no—
207C4—5081no—
208C4—5081no—
209C2—81no—
210C2—20321no—
211C2—10161no—
212C2#2255004511yescharacteristic
213C2#2255004611yescharacteristic
214C2#2255005011yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#329×C127C2
socleC22×C127#120
hypercenter#329D127
hypocenterC127#1656
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
1. #22550620C2semidirect
2. #22550266C2semidirect
3. #22550265C2semidirect
4. #22550241C2semidirect
5. #22549617C2semidirect
6. #22549615C2semidirect
7. #329×C127C2semidirect
108. C127#1656semidirect
115. #329D127semidirect

Maximal subgroups

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

subgroupindex
#225506202
#225502662
#225502652
#225502412
#225496172
#225496152
#329×C1272
#1656127