Finite Group Databasebeta

← #30637909

Subgroups — #30637909

Order 16256.

308 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 65 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#22550222C211yesmaximal, characteristic, complemented
2#22550156C211yesmaximal, characteristic, complemented
3#22550081C211yesmaximal, characteristic, complemented
4#22549620C211yesmaximal, characteristic, complemented
5D4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
7D42×C127C211yesmaximal, characteristic
8C2×#14523236—21no—
9#16861209C2211yescharacteristic, complemented
10#16861142—21no—
11C4⋊(C127⋊C8)C2211yescharacteristic, complemented
12C4⋊(C127⋊C8)C2211yescharacteristic, complemented
13#16861132—21no—
14#16861130—21no—
15#16861130—22no—
16C4×C127⋊C8C2211yescharacteristic, complemented
17C22×D4×C127—21no—
18C22×D4×C127—21no—
19#127×C127C2211yescharacteristic
20#121×C127—21no—
21#121×C127—21no—
22#120×C127—22no—
23C4×D4×C127C2211yescharacteristic
24C4×D4×C127C2211yescharacteristic
25C2×Dic254—21no—
26#14523236—42no—
27C2×C127⋊C8—21no—
28C2×C127⋊C8—21no—
29C2×C127⋊C8—21no—
30C4⋊Dic127—21no—
31C4⋊Dic127—21no—
32C24×C127—22no—
33C2×D4×C127—22no—
34C2×D4×C127—22no—
35C2×D4×C127—21no—
36C2×D4×C127—21no—
37C2×D4×C127—42no—
38C2×D4×C127—42no—
39C2×D4×C127—42no—
40C2×D4×C127—42no—
41C2×D4×C127D411yescharacteristic
42C2×D4×C127D411yescharacteristic
43C2×D4×C127D412yes—
44C22×C4×C127—21no—
45C22×C4×C127—21no—
46C4⋊C4×C127D411yescharacteristic
47C4⋊C4×C127D411yescharacteristic
48C22⋊C4×C127—22no—
49C22⋊C4×C127—21no—
50C22⋊C4×C127—21no—
51C42×C127C2311yescharacteristic
52Dic254—41no—
53Dic254—41no—
54C2×Dic127—41no—
55C127⋊C8—41no—
56C127⋊C8—41no—
57C127⋊C8—41no—
58C23×C127—21no—
59C23×C127—21no—
60C23×C127—22no—
61C23×C127—44no—
62C23×C127—44no—
63C23×C127—82no—
64D4×C127—21no—
65D4×C127—21no—
66D4×C127—42no—
67D4×C127—44no—
68D4×C127—44no—
69D4×C127—41no—
70D4×C127—41no—
71D4×C127—41no—
72D4×C127—41no—
73D4×C127D812yes—
74D4×C127SD1612yescomplemented
75C2×C4×C127—21no—
76C2×C4×C127—21no—
77C2×C4×C127—41no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127—41no—
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C22×C127—22no—
85C22×C127—22no—
86C22×C127—41no—
87C22×C127—41no—
88C22×C127—41no—
89C22×C127—41no—
90C22×C127—42no—
91C22×C127—42no—
92C22×C127—42no—
93C22×C127—88no—
94C22×C127#12011yescharacteristic
95C4×C127—21no—
96C4×C127—41no—
97C4×C127—41no—
98C4×C127C2×D811yescharacteristic
99C4×C127C2×SD1611yescharacteristic
100C4×C127AGL(1,ℤ/8)11yescharacteristic
101C4×C127AGL(1,ℤ/8)11yescharacteristic
102Dic127—81no—
103C2×C127—42no—
104C2×C127—42no—
105C2×C127—82no—
106C2×C127#44711yescharacteristic
107C2×C127#45011yescharacteristic
108C2×C127#45511yescharacteristic
109#1639—1271nomaximal, Sylow
110C127#163911yescharacteristic, complemented, Sylow
111D42D12711yescharacteristic
112#486—1271no—
113#474—1271no—
114#461—1271no—
115#331—1271no—
116D4⋊C8—1271no—
117D4⋊C8—1271no—
118C22×D4—21no—
119C22×D4—21no—
120C2×SD16—2541no—
121C4⋊Q8—1271no—
122#127C2×D12711yescharacteristic
123#121—21no—
124#121—21no—
125#120—22no—
126C4×D4C2×D12711yescharacteristic
127C4×D4C2×D12711yescharacteristic
128#107—2541no—
129C4⋊C8—1271no—
130C4⋊C8—1271no—
131#103—2541no—
132#102—2541no—
133#102—2542no—
134C4×C8—1271no—
135C24—22no—
136C2×Q8—2541no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4C22⋊D12711yescharacteristic
146C2×D4C22⋊D12711yescharacteristic
147C2×D4C22⋊D12712yes—
148C22×C4—21no—
149C22×C4—21no—
150SD16—5082no—
151C2×C8—2541no—
152C2×C8—2541no—
153C2×C8—2541no—
154C4⋊C4—2541no—
155C4⋊C4—2541no—
156C4⋊C4C22⋊D12711yescharacteristic
157C4⋊C4C22⋊D12711yescharacteristic
158C22⋊C4—22no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C42C22×D12711yescharacteristic
162C23—21no—
163C23—21no—
164C23—22no—
165C23—44no—
166C23—44no—
167C23—82no—
168Q8—5081no—
169Q8—5081no—
170D4—21no—
171D4—21no—
172D4—42no—
173D4—44no—
174D4—44no—
175D4—41no—
176D4—41no—
177D4—41no—
178D4—41no—
179D4#1452323312yes—
180D4#1452323612yescomplemented
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—5081no—
188C2×C4C2×C22⋊D12711yescharacteristic
189C2×C4C2×C22⋊D12711yescharacteristic
190C2×C4C2×C22⋊D12711yescharacteristic
191C8—5081no—
192C8—5081no—
193C8—5081no—
194C22—22no—
195C22—22no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—41no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—88no—
204C22#1686117811yescharacteristic
205C4—21no—
206C4—41no—
207C4—41no—
208C4—10161no—
209C4C2×#1452323311yescharacteristic
210C4C2×#1452323611yescharacteristic
211C4#1686123511yescharacteristic
212C4#1686123511yescharacteristic
213C2—42no—
214C2—42no—
215C2—82no—
216C2#2255001011yescharacteristic
217C2#2255002711yescharacteristic
218C2#2255005411yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting, prime radicalD42×C127C2
socleC22×C127#120
hypercenterD42D127
hypocenterC127#1639
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #22550222C2semidirect
2. #22550156C2semidirect
3. #22550081C2semidirect
4. #22549620C2semidirect
5. D4⋊(C127⋊C8)C2semidirect
6. D4⋊(C127⋊C8)C2semidirect
9. #16861209C22semidirect
11. C4⋊(C127⋊C8)C22semidirect
12. C4⋊(C127⋊C8)C22semidirect
16. C4×C127⋊C8C22semidirect
74. D4×C127SD16semidirect
110. C127#1639semidirect
180. D4#14523236semidirect

Maximal subgroups

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

subgroupindex
#225502222
#225501562
#225500812
#225496202
D4⋊(C127⋊C8)2
D42×C1272
#1639127