Finite Group Databasebeta

← #30640749

Subgroups — #30640749

Order 16256.

216 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 41 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#22550625C211yesmaximal, characteristic, complemented
2#22550059C211yesmaximal, characteristic, complemented
3#22550049C211yesmaximal, characteristic
4#22549675C211yesmaximal, characteristic, complemented
5#22549674C211yesmaximal, characteristic, complemented
6#22549672C211yesmaximal, characteristic, complemented
7#456×C127C211yesmaximal, characteristic, complemented
8#16861239—21no—
9#16861234—21no—
10#16861207C2211yescharacteristic
11#16861204—21no—
12#16861203—21no—
13#16861199—21no—
14#16861198C2211yescharacteristic
15#16861194—21no—
16#16861187—21no—
17#16861185—21no—
18#16861136—21no—
19#16861136—21no—
20#16861128C2211yescharacteristic
21#16861128C2211yescharacteristic
222−1+4×C127C2211yescharacteristic
23#137×C127—21no—
24C4⋊Q8×C127C2211yescharacteristic
25#104×C127—21no—
26#101×C127C2211yescharacteristic
27C2×Dic254D411yescharacteristic
28C2×D508D411yescharacteristic
29C2×C4×D127—21no—
30#14523238—41no—
31#14523236—41no—
32#14523235—41no—
33#14523233—41no—
34#14523232—21no—
35#14523232—21no—
36C4⋊Dic127—21no—
37Dic127⋊C4—21no—
38Dic127⋊C4—21no—
39Dic127⋊C4—21no—
40Dic127⋊C4—21no—
41Dic127⋊C4—21no—
42(C2×D127)⋊C4—21no—
43(C2×D127)⋊C4—21no—
44(C2×D127)⋊C4—21no—
45(C2×D127)⋊C4—21no—
46(C2×D127)⋊C4—21no—
47C4×Dic127D411yescharacteristic
48C4×Dic127D411yescharacteristic
49#41×C127—21no—
50#41×C127—21no—
51#41×C127—21no—
52#41×C127—41no—
53C2×Q8×C127—41no—
54C2×Q8×C127D411yescharacteristic
55C2×Q8×C127C2311yescharacteristic
56Q16×C127—41no—
57SD16×C127—41no—
58M16×C127—21no—
59C4⋊C4×C127—21no—
60C4⋊C4×C127—21no—
61C42×C127D411yescharacteristic
62C22×D127—21no—
63Dic254—41no—
64D508—41no—
65C2×Dic127—21no—
66C2×Dic127—21no—
67C2×Dic127—21no—
68C4×D127—81no—
69C127⋊C8—41no—
70C127⋊C8—41no—
71Q8×C127—21no—
72Q8×C127—21no—
73Q8×C127—21no—
74Q8×C127—41no—
75Q8×C127—41no—
76D4×C127—21no—
77D4×C127—21no—
78D4×C127—21no—
79D4×C127—41no—
80C2×C4×C127—21no—
81C2×C4×C127—21no—
82C2×C4×C127—41no—
83C2×C4×C127—41no—
84C2×C4×C127—41no—
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C8×C127—41no—
89C22×C127—41no—
90C22×C127#12011yescharacteristic
91C2×D127—42no—
92C2×D127—41no—
93C4×C127—21no—
94C4×C127—21no—
95C4×C127—21no—
96C4×C127—41no—
97C4×C127—41no—
98C4×C127—41no—
99Dic127—41no—
100Dic127—41no—
101Dic127—41no—
102C2×C127—21no—
103C2×C127—81no—
104C2×C127B(4,𝔽2)11yescharacteristic
105D127—81no—
106#2203—1271nomaximal, Sylow
107C127#220311yescharacteristic, complemented, Sylow
108#563—1271no—
109#456—1271no—
110#456D12711yescharacteristic, complemented
111#454—1271no—
112#356—1271no—
113#356—1271no—
114#355—1271no—
1152−1+4C2×D12711yescharacteristic
116#137—21no—
117#137—2541no—
118AGL(1,ℤ/8)—2541no—
119C4⋊Q8—1271no—
120C4⋊Q8C2×D12711yescharacteristic
121#126—2541no—
122#126—2541no—
123#125—2541no—
124#124—1271no—
125#123—2541no—
126C22⋊Q8—2541no—
127C22⋊Q8—2541no—
128#104—21no—
129#104—2541no—
130#104—2541no—
131#101—1271no—
132#101—1271no—
133#101C2×D12711yescharacteristic
134#41—21no—
135#41—21no—
136#41—21no—
137#41—41no—
138C2×Q8—41no—
139C2×Q8—1271no—
140C2×Q8C22⋊D12711yescharacteristic
141C2×Q8C22×D12711yescharacteristic
142C2×D4—1271no—
143C22×C4—2541no—
144Q16—41no—
145Q16—5081no—
146SD16—41no—
147SD16—5081no—
148SD16—5081no—
149D8—5081no—
150M16—21no—
151M16—2541no—
152M16—2541no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—2541no—
156C4⋊C4—2541no—
157C4⋊C4—2541no—
158C4⋊C4—2541no—
159C4⋊C4—2541no—
160C4⋊C4—2541no—
161C22⋊C4—2541no—
162C22⋊C4—2541no—
163C22⋊C4—2541no—
164C22⋊C4—2541no—
165C22⋊C4—2541no—
166C42—1271no—
167C42—1271no—
168C42C22⋊D12711yescharacteristic
169C23—2541no—
170Q8—21no—
171Q8—21no—
172Q8—21no—
173Q8—41no—
174Q8—41no—
175Q8—5081no—
176D4—21no—
177D4—21no—
178D4—21no—
179D4—41no—
180D4—5081no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—10161no—
187C2×C4—2541no—
188C2×C4—2541no—
189C2×C4—2541no—
190C2×C4D4×D12711yescharacteristic
191C2×C4D4×D12711yescharacteristic
192C2×C4C2×C22⋊D12711yescharacteristic
193C8—41no—
194C8—5081no—
195C8—5081no—
196C22—41no—
197C22—5082no—
198C22—5081no—
199C22#1686117711yescharacteristic
200C4—21no—
201C4—21no—
202C4—21no—
203C4—41no—
204C4—41no—
205C4—41no—
206C4—5081no—
207C4—5081no—
208C4—5081no—
209C2—21no—
210C2—81no—
211C2—10161no—
212C2#2255006211yescharacteristic
subgroupG/H
centerC2#22550062
commutator, cosocleC2×Q8×C127C23
FrattiniC2×Q8C22×D127
Fitting#456×C127C2
socleC2×C127B(4,𝔽2)
prime radical#22550049C2
hypercenter#456D127
hypocenterC127#2203
LCSG ▷ C2×Q8×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #22550625C2semidirect
2. #22550059C2semidirect
4. #22549675C2semidirect
5. #22549674C2semidirect
6. #22549672C2semidirect
7. #456×C127C2semidirect
107. C127#2203semidirect
110. #456D127semidirect

Maximal subgroups

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

subgroupindex
#225506252
#225500592
#225500492
#225496752
#225496742
#225496722
#456×C1272
#2203127