Finite Group Databasebeta

← #30637910

Subgroups — #30637910

Order 16256.

308 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550511C211yesmaximal, characteristic
2#22550222C211yesmaximal, characteristic, complemented
3#22550157C211yesmaximal, characteristic, complemented
4#22550082C211yesmaximal, characteristic, complemented
5#22549603C211yesmaximal, characteristic, complemented
6#22549603C211yesmaximal, characteristic, complemented
7#331×C127C211yesmaximal, characteristic, complemented
8C2×D4×D127—21no—
9C2×D4×D127—21no—
10C2×#14523237—21no—
11#16861209C2211yescharacteristic, complemented
12#16861180—21no—
13#16861180—21no—
14#16861177—22no—
15C4×D508C2211yescharacteristic
16C4×D508C2211yescharacteristic
17#16861143—21no—
18#16861134—21no—
19#16861131—21no—
20#16861130—22no—
21C4×C127⋊C8C2211yescharacteristic, complemented
22#127×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic, complemented
24C4⋊C8×C127C2211yescharacteristic, complemented
25C23×D127—22no—
26C2×Dic254—21no—
27C2×D508D411yescharacteristic
28C2×D508D411yescharacteristic
29C2×C22⋊D127—22no—
30C2×C22⋊D127—22no—
31D4×D127—42no—
32D4×D127—42no—
33D4×D127—42no—
34D4×D127—42no—
35C2×C4×D127—21no—
36C2×C4×D127—21no—
37#14523237—42no—
38C2×C127⋊C8—21no—
39C4⋊Dic127—21no—
40C4⋊Dic127—21no—
41C4⋊Dic127D411yescharacteristic
42C4⋊Dic127D411yescharacteristic
43C22⋊Dic127—22no—
44(C2×D127)⋊C4—21no—
45(C2×D127)⋊C4—21no—
46C2×D4×C127—21no—
47C2×D4×C127—21no—
48C2×D4×C127D412yes—
49C2×C8×C127—21no—
50C2×C8×C127—21no—
51C42×C127C2311yescharacteristic
52C22×D127—21no—
53C22×D127—21no—
54C22×D127—44no—
55C22×D127—44no—
56C22×D127—82no—
57Dic254—41no—
58Dic254—41no—
59D508—21no—
60D508—21no—
61D508D812yes—
62D508SD1612yescomplemented
63C22⋊D127—44no—
64C22⋊D127—44no—
65C2×Dic127—21no—
66C2×Dic127—21no—
67C2×Dic127—41no—
68C4×D127—41no—
69C4×D127—41no—
70C4×D127—41no—
71C4×D127—41no—
72C127⋊C8—41no—
73C23×C127—22no—
74D4×C127—42no—
75D4×C127—41no—
76D4×C127—41no—
77D4×C127—41no—
78D4×C127—41no—
79C2×C4×C127C2×D411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C8×C127—41no—
83C8×C127—41no—
84C22×C127—42no—
85C22×C127—42no—
86C22×C127—42no—
87C22×C127#12011yescharacteristic
88C2×D127—22no—
89C2×D127—22no—
90C2×D127—41no—
91C2×D127—41no—
92C2×D127—41no—
93C2×D127—41no—
94C2×D127—88no—
95C4×C127—21no—
96C4×C127C2×D811yescharacteristic
97C4×C127C2×SD1611yescharacteristic
98C4×C127AGL(1,ℤ/8)11yescharacteristic
99C4×C127AGL(1,ℤ/8)11yescharacteristic
100Dic127—41no—
101Dic127—41no—
102Dic127—81no—
103C2×C127—82no—
104C2×C127#44711yescharacteristic
105C2×C127#45011yescharacteristic
106C2×C127#45511yescharacteristic
107D127—42no—
108D127—42no—
109#1639—1271nomaximal, Sylow
110C127#163911yescharacteristic, complemented, Sylow
111D42—1271no—
112#486—1271no—
113#474—1271no—
114#461—1271no—
115#331D12711yescharacteristic, complemented
116D4⋊C8—1271no—
117D4⋊C8—1271no—
118C22×D4—2541no—
119C22×D4—2541no—
120C2×SD16—2541no—
121C4⋊Q8—1271no—
122#127C2×D12711yescharacteristic
123#121—2541no—
124#121—2541no—
125#120—2542no—
126C4×D4—1271no—
127C4×D4—1271no—
128#107—2541no—
129C4⋊C8C2×D12711yescharacteristic, complemented
130C4⋊C8C2×D12711yescharacteristic, complemented
131#103—2541no—
132#102—2541no—
133#102—2542no—
134C4×C8—1271no—
135C24—2542no—
136C2×Q8—2541no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—1271no—
140C2×D4—1271no—
141C2×D4—5082no—
142C2×D4—5082no—
143C2×D4—5082no—
144C2×D4—5082no—
145C2×D4—2542no—
146C2×D4—2542no—
147C2×D4D50812yes—
148C22×C4—2541no—
149C22×C4—2541no—
150SD16—5082no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—2541no—
154C4⋊C4—1271no—
155C4⋊C4—1271no—
156C4⋊C4—2541no—
157C4⋊C4—2541no—
158C22⋊C4—2542no—
159C22⋊C4—2541no—
160C22⋊C4—2541no—
161C42C22×D12711yescharacteristic
162C23—22no—
163C23—10162no—
164C23—5084no—
165C23—5084no—
166C23—2541no—
167C23—2541no—
168Q8—5081no—
169Q8—5081no—
170D4—42no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—1272no—
176D4—1272no—
177D4—5084no—
178D4—5084no—
179D4—2541no—
180D4—2541no—
181C2×C4—5081no—
182C2×C4—5081no—
183C2×C4—5081no—
184C2×C4—5081no—
185C2×C4—5081no—
186C2×C4—2541no—
187C2×C4—2541no—
188C2×C4D4×D12711yescharacteristic
189C2×C4D4×D12711yescharacteristic
190C2×C4C2×D50811yescharacteristic
191C8—41no—
192C8—41no—
193C8—5081no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—10168no—
198C22—5081no—
199C22—5081no—
200C22—5081no—
201C22—5081no—
202C22—2542no—
203C22—2542no—
204C22#1686117911yescharacteristic
205C4—21no—
206C4—10161no—
207C4—5081no—
208C4—5081no—
209C4D8×D12711yescharacteristic
210C4SD16×D12711yescharacteristic
211C4#1686123311yescharacteristic
212C4#1686123611yescharacteristic
213C2—82no—
214C2—5082no—
215C2—5082no—
216C2#2255001111yescharacteristic
217C2#2255002811yescharacteristic
218C2#2255005611yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#331×C127C2
socleC22×C127#120
prime radical#22550511C2
hypercenter#331D127
hypocenterC127#1639
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #22550222C2semidirect
3. #22550157C2semidirect
4. #22550082C2semidirect
5. #22549603C2semidirect
6. #22549603C2semidirect
7. #331×C127C2semidirect
11. #16861209C22semidirect
21. C4×C127⋊C8C22semidirect
23. C4⋊C8×C127C22semidirect
24. C4⋊C8×C127C22semidirect
62. D508SD16semidirect
110. C127#1639semidirect
115. #331D127semidirect
129. C4⋊C8C2×D127semidirect
130. C4⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505112
#225502222
#225501572
#225500822
#225496032
#331×C1272
#1639127