Finite Group Databasebeta

← #27140368

Subgroups — #27140368

Order 12416.

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

normal subgroups only

This group has 200 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#20261269C211yesmaximal, characteristic
2#20260792C211yesmaximal, characteristic, complemented
3#20260476C212yesmaximal, complemented
4#20260474C212yesmaximal, complemented
5#432×C97C211yesmaximal, characteristic, complemented
6#15690804—21no—
7#15690802C2211yescharacteristic
8#15690798C2211yescharacteristic
9#15690793—21no—
10#15690789C411yescharacteristic
11#15690784C411yescharacteristic
12C4×Dic194C411yescharacteristic
13C4×D388C411yescharacteristic
14C4×C22⋊D97—21no—
15#15690767—21no—
16C4⋊#11715895—21no—
17C22⋊#11715895—21no—
18#15690721C2212yes—
19C2×C42×C97C2211yescharacteristic
20C4⋊C8×C97—21no—
21C22⋊C8×C97—21no—
22#97×C97C2212yes—
23C2×Dic194C2×C411yescharacteristic
24C2×D388C2×C411yescharacteristic
25C2×C22⋊D97—21no—
26C2×C4×D97—21no—
27C2×#11715895—22no—
28C4⋊Dic97C2×C411yescharacteristic
29C4⋊Dic97C2×C411yescharacteristic
30#13095490—21no—
31#13095490—21no—
32#13095490—21no—
33#13095490—21no—
34C22⋊Dic97—21no—
35#13095488—21no—
36#13095488—21no—
37#13095488—21no—
38#13095488—21no—
39C4×Dic97—21no—
40C22×C4×C97C2×C411yescharacteristic
41C22×C4×C97D411yescharacteristic
42C22×C4×C97D411yescharacteristic
43C2×C8×C97—22no—
44C42×C97C2×C411yescharacteristic
45C42×C97D411yescharacteristic
46C42×C97D411yescharacteristic
47C42×C97C2311yescharacteristic
48C23×C97C22⋊C411yescharacteristic
49C22×D97—21no—
50C2×C4×C97—41no—
51C2×C4×C97—41no—
52C2×C4×C97—41no—
53C2×C4×C97C22⋊C411yescharacteristic
54C2×C4×C97C22⋊C411yescharacteristic
55C2×C4×C97C22⋊C411yescharacteristic
56C2×C4×C97C22×C411yescharacteristic
57C2×C4×C97C2×D411yescharacteristic
58C2×C4×C97C2×D411yescharacteristic
59C22⋊D97—42no—
60C2×Dic97—21no—
61C2×Dic97—21no—
62C2×Dic97—21no—
63D388—41no—
64C4×D97—42no—
65Dic194—41no—
66C8×C97—42no—
67#11715895—42no—
68C22×C97—21no—
69C22×C97—21no—
70C22×C97—21no—
71C22×C97C2×C22⋊C411yescharacteristic
72C2×D97—41no—
73C2×D97—41no—
74C2×D97—41no—
75C4×C97—21no—
76C4×C97—21no—
77C4×C97—21no—
78C4×C97—21no—
79C4×C97—21no—
80C4×C97—21no—
81Dic97—41no—
82Dic97—41no—
83Dic97—41no—
84C2×C97—41no—
85C2×C97#41311yescharacteristic
86C2×C97#42111yescharacteristic
87C2×C97#42111yescharacteristic
88D97—81no—
89#1535—971nomaximal, Sylow
90C97#153511yescharacteristic, complemented, Sylow
91#529—971no—
92#432—971no—
93#432D9711yescharacteristic, complemented
94#330—972no—
95#329—972no—
96#126—1941no—
97#125—971no—
98#124—971no—
99#123—1941no—
100C22⋊Q8—971no—
101#121—971no—
102C4×Q8—971no—
103C4×D4—971no—
104C4×D4—1941no—
105#117—1941no—
106C2×C42C2×D9711yescharacteristic
107C4⋊C8—21no—
108C4⋊C8—1941no—
109C22⋊C8—21no—
110C22⋊C8—1941no—
111#97—972no—
112#97C2×D9712yes—
113C2×Q8—971no—
114C2×D4—971no—
115C2×D4—1941no—
116C22×C4—1941no—
117C22×C4C4×D9711yescharacteristic
118C22×C4D38811yescharacteristic
119C22×C4C22⋊D9711yescharacteristic
120C2×C8—22no—
121C2×C8—1942no—
122C4⋊C4—971no—
123C4⋊C4—971no—
124C4⋊C4—1941no—
125C4⋊C4—1941no—
126C4⋊C4—1941no—
127C4⋊C4—1941no—
128C22⋊C4—1941no—
129C22⋊C4—1941no—
130C22⋊C4—1941no—
131C22⋊C4—1941no—
132C22⋊C4—1941no—
133C42—1941no—
134C42C4×D9711yescharacteristic
135C42D38811yescharacteristic
136C42C22⋊D9711yescharacteristic
137C42C22×D9711yescharacteristic
138C23—1941no—
139C23#1309548811yescharacteristic
140Q8—3881no—
141D4—3881no—
142D4—3882no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—3882no—
147C2×C4—1941no—
148C2×C4—1941no—
149C2×C4—1941no—
150C2×C4#1309548811yescharacteristic
151C2×C4#1309548811yescharacteristic
152C2×C4#1309548811yescharacteristic
153C2×C4C2×C4×D9711yescharacteristic
154C2×C4C2×C22⋊D9711yescharacteristic
155C2×C4C2×D38811yescharacteristic
156C8—42no—
157C8—3882no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—3881no—
162C22—3881no—
163C22—3881no—
164C22C2×#1309548811yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—3881no—
172C4—3881no—
173C4—3881no—
174C2—41no—
175C2—7761no—
176C2#2026071711yescharacteristic
177C2#2026075111yescharacteristic
178C2#2026075311yescharacteristic
subgroupG/H
centerC22C2×#13095488
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
Fitting#432×C97C2
socleC22×C97C2×C22⋊C4
prime radical#15690784C4
cosocleC42×C97C23
hypercenter#432D97
hypocenterC97#1535
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. #20260792C2semidirect
3. #20260476C2semidirect
4. #20260474C2semidirect
5. #432×C97C2semidirect
90. C97#1535semidirect
93. #432D97semidirect

Maximal subgroups

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

subgroupindex
#202612692
#202607922
#202604762
#202604742
#432×C972
#153597