Finite Group Databasebeta

← #27140844

Subgroups — #27140844

Order 12416.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261455C211yesmaximal, characteristic, complemented
2#20261081C211yesmaximal, characteristic, complemented
3#20260932C211yesmaximal, characteristic, complemented
4#20260482C211yesmaximal, characteristic, complemented
5#20260465C211yesmaximal, characteristic, complemented
6D4⋊#11715895C211yesmaximal, characteristic, complemented
7#459×C97C211yesmaximal, characteristic
8C2×#13095499—21no—
9#15690810C2211yescharacteristic
10#15690803—22no—
11#15690797—21no—
12#15690789—21no—
13C4×Dic194C2211yescharacteristic
14#15690775—21no—
15#15690769—21no—
16C4⋊#11715895C2211yescharacteristic
17#15690731—21no—
18#15690731—22no—
19C4×#11715895C2211yescharacteristic
20C2×D8×C97—21no—
21#127×C97C2211yescharacteristic
22C4×D4×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24#102×C97—21no—
25C2×Dic194—21no—
26C2×Dic194D411yescharacteristic
27#13095499—42no—
28C2×#11715895—21no—
29C2×#11715895—21no—
30C4⋊Dic97—21no—
31C4⋊Dic97D411yescharacteristic
32C4⋊Dic97D412yes—
33#13095490—22no—
34#13095490—21no—
35C22⋊Dic97—22no—
36C22⋊Dic97—22no—
37C4×Dic97—22no—
38C4×Dic97—21no—
39C2×D4×C97—21no—
40C2×D4×C97—21no—
41C2×D4×C97—21no—
42C2×D4×C97D411yescharacteristic
43C22×C4×C97—21no—
44D8×C97—42no—
45C2×C8×C97—21no—
46C4⋊C4×C97D411yescharacteristic
47C22⋊C4×C97—21no—
48C42×C97C2311yescharacteristic
49C23×C97—21no—
50C23×C97—41no—
51D4×C97—21no—
52D4×C97—21no—
53D4×C97—42no—
54D4×C97—41no—
55D4×C97—41no—
56D4×C97—41no—
57D4×C97—41no—
58C2×C4×C97—21no—
59C2×C4×C97—41no—
60C2×C4×C97—41no—
61C2×C4×C97C2×D411yescharacteristic
62C2×C4×C97C2×D411yescharacteristic
63C2×C4×C97C2×D411yescharacteristic
64C2×Dic97—21no—
65C2×Dic97—21no—
66C2×Dic97—22no—
67Dic194—21no—
68Dic194—21no—
69Dic194—41no—
70Dic194—41no—
71C8×C97—41no—
72#11715895—41no—
73#11715895—41no—
74C22×C97—41no—
75C22×C97—41no—
76C22×C97—41no—
77C22×C97—81no—
78C22×C97—81no—
79C22×C97—81no—
80C22×C97#12011yescharacteristic
81C4×C97—21no—
82C4×C97—41no—
83C4×C97#13511yescharacteristic
84C4×C97#13511yescharacteristic
85C4×C97AGL(1,ℤ/8)11yescharacteristic
86C4×C97AGL(1,ℤ/8)11yescharacteristic
87Dic97—41no—
88Dic97—42no—
89Dic97—41no—
90C2×C97—81no—
91C2×C97—161no—
92C2×C97#44911yescharacteristic
93C2×C97#44911yescharacteristic
94C2×C97#45411yescharacteristic
95#1628—971nomaximal, Sylow
96C97#162811yescharacteristic, complemented, Sylow
97#556—971no—
98#486—971no—
99#460—971no—
100#459D9711yescharacteristic
101#332—971no—
102Q8⋊C8—971no—
103D4⋊C8—971no—
104C2×SD16—1941no—
105C2×D8—21no—
106C4⋊Q8—971no—
107#127C2×D9711yescharacteristic
108#126—1942no—
109#124—1941no—
110C22⋊Q8—1941no—
111C4×Q8—971no—
112C4×Q8—1941no—
113C4×D4C2×D9711yescharacteristic
114#117—1941no—
115C4⋊C8—971no—
116C4⋊C8C2×D9711yescharacteristic
117#102—21no—
118#102—1941no—
119#102—1942no—
120C4×C8—971no—
121C2×Q8—971no—
122C2×Q8—1941no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4C22⋊D9711yescharacteristic
127C22×C4—21no—
128SD16—3882no—
129D8—42no—
130C2×C8—21no—
131C2×C8—1941no—
132C2×C8—1941no—
133C4⋊C4—971no—
134C4⋊C4—972no—
135C4⋊C4—1942no—
136C4⋊C4—1941no—
137C4⋊C4—1941no—
138C4⋊C4C22⋊D9711yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—1942no—
141C22⋊C4—1942no—
142C42—1942no—
143C42—1941no—
144C42C22×D9711yescharacteristic
145C23—21no—
146C23—41no—
147Q8—3881no—
148Q8—3881no—
149Q8—1941no—
150Q8—1941no—
151D4—21no—
152D4—21no—
153D4—42no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—41no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—1941no—
162C2×C4—1942no—
163C2×C4—1941no—
164C2×C4D4×D9711yescharacteristic
165C2×C4D4×D9711yescharacteristic
166C2×C4C2×C22⋊D9711yescharacteristic
167C8—41no—
168C8—3881no—
169C8—3881no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—81no—
174C22—81no—
175C22—81no—
176C22#1569077811yescharacteristic
177C4—21no—
178C4—41no—
179C4—3881no—
180C4—3882no—
181C4—3881no—
182C4#1569082911yescharacteristic
183C4#1569083011yescharacteristic
184C4#1569083411yescharacteristic
185C4#1569083611yescharacteristic
186C2—81no—
187C2—161no—
188C2#2026087711yescharacteristic
189C2#2026087811yescharacteristic
190C2#2026090611yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting, prime radical#459×C97C2
socleC22×C97#120
hypercenter#459D97
hypocenterC97#1628
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #20261455C2semidirect
2. #20261081C2semidirect
3. #20260932C2semidirect
4. #20260482C2semidirect
5. #20260465C2semidirect
6. D4⋊#11715895C2semidirect
96. C97#1628semidirect

Maximal subgroups

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

subgroupindex
#202614552
#202610812
#202609322
#202604822
#202604652
D4⋊#117158952
#459×C972
#162897