Finite Group Databasebeta

← #27152027

Subgroups — #27152027

Order 12416.

382 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 125 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#20261340C211yesmaximal, characteristic
2#20261059C212yesmaximal, complemented
3#20261042C212yesmaximal, complemented
4#20260935C212yesmaximal, complemented
5#20260874C212yesmaximal, complemented
6#20260833C212yesmaximal, complemented
7C4×#13095498C212yesmaximal, complemented
8#20260796C211yesmaximal, characteristic, complemented
9#519×C97C211yesmaximal, characteristic, complemented
10C2×#13095498C2212yes—
11C2×#13095498C2212yes—
12#15690807C2211yescharacteristic
13#15690802C2211yescharacteristic
14#15690795—22no—
15#15690784C2212yes—
16#15690783—22no—
17C4×D388C2212yes—
18#15690743C2212yes—
19#15690741C2212yes—
20C4⋊#11715895C2212yes—
21#15690734C2212yes—
22#15690734C2212yes—
23#15690730C2212yes—
24#15690730C2212yes—
25C22⋊#11715895C2212yes—
26#15690721C2211yescharacteristic
27C4×#11715895C2211yescharacteristic
28C22×Q8×C97C2211yescharacteristic
29C4×Q8×C97—22no—
30C4×Q8×C97C2212yes—
31C4×Q8×C97C2212yes—
32#117×C97—22no—
33#117×C97C2211yescharacteristic
34#117×C97C2211yescharacteristic
35C2×D388—22no—
36C2×D388C2312yes—
37C2×C22⋊D97—22no—
38C2×C4×D97—22no—
39#13095498—24no—
40#13095498—24no—
41C2×#11715895C2312yes—
42C2×#11715895C2312yes—
43C4⋊Dic97C2312yes—
44#13095490—22no—
45#13095488—22no—
46#13095488—22no—
47C2×Q8×C97—24no—
48C2×Q8×C97—21no—
49C2×Q8×C97C2312yes—
50C22×C4×C97—21no—
51C22×C4×C97C2311yescharacteristic
52C4⋊C4×C97—22no—
53C4⋊C4×C97—22no—
54C4⋊C4×C97D412yes—
55C4⋊C4×C97C2312yes—
56C22⋊C4×C97—21no—
57C22⋊C4×C97D412yes—
58C42×C97—22no—
59C42×C97—22no—
60C42×C97—21no—
61C42×C97C2311yescharacteristic
62C42×C97C2311yescharacteristic
63C23×C97C2×D411yescharacteristic
64C22×D97—22no—
65Q8×C97—24no—
66Q8×C97—22no—
67Q8×C97#4114yes—
68C2×C4×C97—24no—
69C2×C4×C97—22no—
70C2×C4×C97—22no—
71C2×C4×C97—22no—
72C2×C4×C97C2×D412yes—
73C2×C4×C97C2×D412yes—
74C2×C4×C97C2×D411yescharacteristic
75C2×C4×C97C2411yescharacteristic
76C22⋊D97—82no—
77C2×Dic97—22no—
78D388—22no—
79D388—22no—
80D388—82no—
81C4×D97—82no—
82#11715895—22no—
83#11715895—22no—
84C22×C97—21no—
85C22×C97—22no—
86C22×C97C22×D411yescharacteristic
87C2×D97—42no—
88C2×D97—42no—
89C2×D97—42no—
90C4×C97—24no—
91C4×C97—21no—
92C4×C97—22no—
93C4×C97—22no—
94C4×C97—42no—
95C4×C97C2×#4112yes—
96Dic97—42no—
97C2×C97—41no—
98C2×C97#52511yescharacteristic
99C2×C97#57712yes—
100D97—82no—
101#3126—971nomaximal, Sylow
102C97#312611yescharacteristic, complemented, Sylow
103#540—971no—
104#519D9711yescharacteristic, complemented
105#482—972no—
106#479—972no—
107#460—972no—
108#448—972no—
109#440—972no—
110C4×SD16—972no—
111#433—971no—
112C22×Q8C2×D9711yescharacteristic
113C2×SD16—972no—
114C2×SD16—972no—
115#127—971no—
116#125—971no—
117#123—1942no—
118#121—972no—
119#121—1942no—
120C4×Q8—22no—
121C4×Q8C2×D9712yes—
122C4×Q8C2×D9712yes—
123C4×D4—972no—
124#117—22no—
125#117C2×D9711yescharacteristic
126#117C2×D9711yescharacteristic
127#107—972no—
128#106—972no—
129C4⋊C8—972no—
130#103—972no—
131#103—972no—
132#102—972no—
133#102—972no—
134C22⋊C8—972no—
135#97—971no—
136C4×C8—971no—
137C2×Q8—24no—
138C2×Q8—21no—
139C2×Q8C22×D9712yes—
140C2×D4—972no—
141C2×D4—1942no—
142C2×D4—1942no—
143C22×C4—21no—
144C22×C4—1942no—
145C22×C4C22×D9711yescharacteristic
146SD16—1944no—
147SD16—1944no—
148C2×C8—972no—
149C2×C8—972no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—972no—
153C4⋊C4—1942no—
154C4⋊C4C22⋊D9712yes—
155C4⋊C4C22×D9712yes—
156C22⋊C4—21no—
157C22⋊C4—1942no—
158C22⋊C4—1942no—
159C22⋊C4C22⋊D9712yes—
160C42—22no—
161C42—22no—
162C42—21no—
163C42C22×D9711yescharacteristic
164C42C22×D9711yescharacteristic
165C23—1942no—
166C23C2×C22⋊D9711yescharacteristic
167Q8—24no—
168Q8—22no—
169Q8#1309551114yes—
170D4—7762no—
171D4—7762no—
172D4—1942no—
173D4—1942no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—7762no—
179C2×C4—1942no—
180C2×C4C2×C22⋊D9712yes—
181C2×C4C2×C22⋊D9712yes—
182C2×C4C2×C22⋊D9711yescharacteristic
183C2×C4C23×D9711yescharacteristic
184C8—1942no—
185C8—1942no—
186C22—21no—
187C22—22no—
188C22—3882no—
189C22—3882no—
190C22—3882no—
191C22C22×C22⋊D9711yescharacteristic
192C4—24no—
193C4—21no—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—3882no—
198C4C2×#1309551112yes—
199C2—41no—
200C2—7762no—
201C2#2026124611yescharacteristic
202C2#2026154312yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#519×C97C2
socleC22×C97C22×D4
prime radical#20261340C2
hypercenter#519D97
hypocenterC97#3126
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. #20261059C2semidirect
3. #20261042C2semidirect
4. #20260935C2semidirect
5. #20260874C2semidirect
6. #20260833C2semidirect
7. C4×#13095498C2semidirect
8. #20260796C2semidirect
9. #519×C97C2semidirect
102. C97#3126semidirect
104. #519D97semidirect

Maximal subgroups

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

subgroupindex
#202613402
#202610592
#202610422
#202609352
#202608742
#202608332
C4×#130954982
#202607962
#519×C972
#312697