Finite Group Databasebeta

← #27140906

Subgroups — #27140906

Order 12416.

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

normal subgroups only

This group has 194 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
2#20261085C211yesmaximal, characteristic, complemented
3#20261027C211yesmaximal, characteristic, complemented
4#20260482C211yesmaximal, characteristic, complemented
5#20260465C211yesmaximal, characteristic, complemented
6D4⋊#11715895C211yesmaximal, characteristic
7#476×C97C211yesmaximal, characteristic
8#15690810C2211yescharacteristic
9#15690803—22no—
10#15690797—21no—
11#15690789—21no—
12C4×Dic194C2211yescharacteristic
13#15690775—21no—
14#15690769—21no—
15#15690743—21no—
16C4⋊#11715895C2211yescharacteristic
17#15690734—22no—
18#15690733—21no—
19C4×#11715895C2211yescharacteristic
20C4⋊Q8×C97C2211yescharacteristic
21C4×D4×C97C2211yescharacteristic
22#106×C97—21no—
23C4⋊C8×C97C2211yescharacteristic
24#102×C97—21no—
25C2×Dic194—21no—
26C2×Dic194D411yescharacteristic
27C2×#11715895—21no—
28C2×#11715895—21no—
29C4⋊Dic97—21no—
30C4⋊Dic97D411yescharacteristic
31C4⋊Dic97D412yes—
32#13095490—22no—
33#13095490—21no—
34C22⋊Dic97—22no—
35C22⋊Dic97—22no—
36C4×Dic97—22no—
37C4×Dic97—21no—
38C2×Q8×C97—21no—
39C2×D4×C97D411yescharacteristic
40C22×C4×C97—21no—
41C2×C8×C97—21no—
42C4⋊C4×C97—21no—
43C4⋊C4×C97—21no—
44C4⋊C4×C97D411yescharacteristic
45C22⋊C4×C97—21no—
46C42×C97C2311yescharacteristic
47C23×C97—21no—
48Q8×C97—42no—
49D4×C97—21no—
50D4×C97—21no—
51C2×C4×C97—21no—
52C2×C4×C97—41no—
53C2×C4×C97—41no—
54C2×C4×C97—41no—
55C2×C4×C97C2×D411yescharacteristic
56C2×C4×C97C2×D411yescharacteristic
57C2×C4×C97C2×D411yescharacteristic
58C2×Dic97—21no—
59C2×Dic97—21no—
60C2×Dic97—22no—
61Dic194—21no—
62Dic194—21no—
63Dic194—41no—
64Dic194—41no—
65C8×C97—41no—
66#11715895—41no—
67#11715895—41no—
68C22×C97—41no—
69C22×C97—41no—
70C22×C97—41no—
71C22×C97#12011yescharacteristic
72C4×C97—21no—
73C4×C97—41no—
74C4×C97—81no—
75C4×C97#13511yescharacteristic
76C4×C97#13511yescharacteristic
77C4×C97#13711yescharacteristic
78C4×C97#13711yescharacteristic
79Dic97—41no—
80Dic97—42no—
81Dic97—41no—
82C2×C97—81no—
83C2×C97#45211yescharacteristic
84C2×C97#45211yescharacteristic
85C2×C97#45411yescharacteristic
86#1638—971nomaximal, Sylow
87C97#163811yescharacteristic, complemented, Sylow
88#556—971no—
89#487—971no—
90#477—971no—
91#476D9711yescharacteristic
92#332—971no—
93Q8⋊C8—971no—
94D4⋊C8—971no—
95C4⋊Q8—971no—
96C4⋊Q8C2×D9711yescharacteristic
97#126—1942no—
98#124—1941no—
99C22⋊Q8—1941no—
100C4×Q8—971no—
101C4×Q8—1941no—
102C4×D4C2×D9711yescharacteristic
103#117—1941no—
104#107—1941no—
105#106—21no—
106C4⋊C8—971no—
107C4⋊C8C2×D9711yescharacteristic
108#103—1941no—
109#103—1942no—
110#102—21no—
111C4×C8—971no—
112C2×Q8—21no—
113C2×Q8—971no—
114C2×Q8—1941no—
115C2×D4C22⋊D9711yescharacteristic
116C22×C4—21no—
117C2×C8—21no—
118C2×C8—1941no—
119C2×C8—1941no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—972no—
123C4⋊C4—971no—
124C4⋊C4—1942no—
125C4⋊C4—1941no—
126C4⋊C4—1941no—
127C4⋊C4C22⋊D9711yescharacteristic
128C22⋊C4—21no—
129C22⋊C4—1942no—
130C22⋊C4—1942no—
131C42—1942no—
132C42—1941no—
133C42C22×D9711yescharacteristic
134C23—21no—
135Q8—42no—
136Q8—3881no—
137Q8—3881no—
138Q8—1941no—
139Q8—1941no—
140D4—21no—
141D4—21no—
142C2×C4—21no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—1941no—
147C2×C4—1942no—
148C2×C4—1941no—
149C2×C4D4×D9711yescharacteristic
150C2×C4D4×D9711yescharacteristic
151C2×C4C2×C22⋊D9711yescharacteristic
152C8—41no—
153C8—3881no—
154C8—3881no—
155C22—41no—
156C22—41no—
157C22—41no—
158C22#1569077811yescharacteristic
159C4—21no—
160C4—41no—
161C4—81no—
162C4—3881no—
163C4—3882no—
164C4—3881no—
165C4#1569083011yescharacteristic
166C4#1569083111yescharacteristic
167C4#1569083911yescharacteristic
168C4#1569084111yescharacteristic
169C2—81no—
170C2#2026089411yescharacteristic
171C2#2026089511yescharacteristic
172C2#2026090611yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#476×C97C2
socleC22×C97#120
prime radicalC2×D4×C97D4
hypercenter#476D97
hypocenterC97#1638
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #20261085C2semidirect
3. #20261027C2semidirect
4. #20260482C2semidirect
5. #20260465C2semidirect
87. C97#1638semidirect

Maximal subgroups

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

subgroupindex
#202614552
#202610852
#202610272
#202604822
#202604652
D4⋊#117158952
#476×C972
#163897