Finite Group Databasebeta

← #30637958

Subgroups — #30637958

Order 16256.

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

normal subgroups only

This group has 212 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#22550226C211yesmaximal, characteristic, complemented
2#22550156C211yesmaximal, characteristic, complemented
3#22550081C211yesmaximal, characteristic, complemented
4#22549625C211yesmaximal, characteristic, complemented
5D4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
7#555×C127C211yesmaximal, characteristic
8C2×#14523236—21no—
9#16861209C2211yescharacteristic
10#16861142—21no—
11C4⋊(C127⋊C8)C2211yescharacteristic
12C4⋊(C127⋊C8)C2211yescharacteristic
13#16861133—22no—
14#16861132—21no—
15#16861130—21no—
16C4×C127⋊C8C2211yescharacteristic
17C4⋊Q8×C127C2211yescharacteristic
18#124×C127—22no—
19#121×C127—21no—
20#121×C127—21no—
21C4×D4×C127C2211yescharacteristic
22C4×D4×C127C2211yescharacteristic
23#117×C127—21no—
24#117×C127—21no—
25C2×Dic254—21no—
26#14523236—42no—
27C2×C127⋊C8—21no—
28C2×C127⋊C8—21no—
29C2×C127⋊C8—21no—
30C4⋊Dic127—21no—
31C4⋊Dic127—21no—
32C2×Q8×C127D412yes—
33C2×D4×C127—22no—
34C2×D4×C127D411yescharacteristic
35C2×D4×C127D411yescharacteristic
36C22×C4×C127—21no—
37C22×C4×C127—21no—
38C4⋊C4×C127—21no—
39C4⋊C4×C127—21no—
40C4⋊C4×C127D411yescharacteristic
41C4⋊C4×C127D411yescharacteristic
42C22⋊C4×C127—22no—
43C22⋊C4×C127—22no—
44C22⋊C4×C127—21no—
45C22⋊C4×C127—21no—
46C42×C127—22no—
47C42×C127C2311yescharacteristic
48Dic254—41no—
49Dic254—41no—
50C2×Dic127—41no—
51C127⋊C8—41no—
52C127⋊C8—41no—
53C127⋊C8—41no—
54C23×C127—21no—
55C23×C127—21no—
56Q8×C127—42no—
57D4×C127—21no—
58D4×C127—21no—
59D4×C127—21no—
60D4×C127—21no—
61D4×C127—82no—
62C2×C4×C127—21no—
63C2×C4×C127—21no—
64C2×C4×C127—22no—
65C2×C4×C127—41no—
66C2×C4×C127—41no—
67C2×C4×C127—41no—
68C2×C4×C127—41no—
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C22×C127—41no—
73C22×C127—41no—
74C22×C127—41no—
75C22×C127—41no—
76C22×C127—41no—
77C22×C127—41no—
78C22×C127#12011yescharacteristic
79C4×C127—21no—
80C4×C127—41no—
81C4×C127—41no—
82C4×C127—42no—
83C4×C127#13511yescharacteristic
84C4×C127#13511yescharacteristic
85C4×C127AGL(1,ℤ/8)11yescharacteristic
86C4×C127#13711yescharacteristic
87Dic127—81no—
88C2×C127—81no—
89C2×C127—81no—
90C2×C127#44911yescharacteristic
91C2×C127#45211yescharacteristic
92C2×C127#45511yescharacteristic
93#1646—1271nomaximal, Sylow
94C127#164611yescharacteristic, complemented, Sylow
95#555D12711yescharacteristic
96#487—1271no—
97#474—1271no—
98#461—1271no—
99#333—1271no—
100D4⋊C8—1271no—
101D4⋊C8—1271no—
102C2×SD16—2541no—
103C4⋊Q8—1271no—
104C4⋊Q8C2×D12711yescharacteristic
105#124—22no—
106#121—21no—
107#121—21no—
108C4×D4C2×D12711yescharacteristic
109C4×D4C2×D12711yescharacteristic
110#117—21no—
111#117—21no—
112#107—2541no—
113C4⋊C8—1271no—
114C4⋊C8—1271no—
115#103—2542no—
116#103—2541no—
117#102—2541no—
118C4×C8—1271no—
119C2×Q8—2541no—
120C2×Q8C22⋊D12712yes—
121C2×D4—22no—
122C2×D4C22⋊D12711yescharacteristic
123C2×D4C22⋊D12711yescharacteristic
124C22×C4—21no—
125C22×C4—21no—
126SD16—5082no—
127C2×C8—2541no—
128C2×C8—2541no—
129C2×C8—2541no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—2541no—
133C4⋊C4—2541no—
134C4⋊C4C22⋊D12711yescharacteristic
135C4⋊C4C22⋊D12711yescharacteristic
136C22⋊C4—22no—
137C22⋊C4—22no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C42—22no—
141C42C22×D12711yescharacteristic
142C23—21no—
143C23—21no—
144Q8—42no—
145Q8—5081no—
146Q8—5081no—
147D4—21no—
148D4—21no—
149D4—21no—
150D4—21no—
151D4—82no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—5081no—
160C2×C4C2×C22⋊D12711yescharacteristic
161C2×C4C2×C22⋊D12711yescharacteristic
162C2×C4C2×C22⋊D12711yescharacteristic
163C8—5081no—
164C8—5081no—
165C8—5081no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22#1686117811yescharacteristic
173C4—21no—
174C4—41no—
175C4—41no—
176C4—42no—
177C4—10161no—
178C4#1686122911yescharacteristic
179C4#1686122911yescharacteristic
180C4#1686123411yescharacteristic
181C4#1686123911yescharacteristic
182C2—81no—
183C2—81no—
184C2#2255002111yescharacteristic
185C2#2255003811yescharacteristic
186C2#2255005411yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting, prime radical#555×C127C2
socleC22×C127#120
hypercenter#555D127
hypocenterC127#1646
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #22550226C2semidirect
2. #22550156C2semidirect
3. #22550081C2semidirect
4. #22549625C2semidirect
5. D4⋊(C127⋊C8)C2semidirect
6. D4⋊(C127⋊C8)C2semidirect
94. C127#1646semidirect

Maximal subgroups

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

subgroupindex
#225502262
#225501562
#225500812
#225496252
D4⋊(C127⋊C8)2
#555×C1272
#1646127