Finite Group Databasebeta

← #27157891

Subgroups — #27157891

Order 12416.

292 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C2×#15690880C211yesmaximal, characteristic, complemented
2C2×#15690874C212yesmaximal, complemented
3#20261709C212yesmaximal, complemented
4C2×#15690864C211yesmaximal, characteristic, complemented
5C2×#15690829C211yesmaximal, characteristic
6C2×#13095521C2212yes—
7C2×#13095519C2211yescharacteristic
8C2×C4×#87604C2211yescharacteristic
9#15690880—22no—
10#15690874—24no—
11C22⋊#11715897—24no—
12#15690864—22no—
13C2×#13095510C2212yes—
14#15690829—22no—
15#15690829—22no—
16C2×Dic388C411yescharacteristic, complemented
17C2×#13095499C412yescomplemented
18C2×C8×D97C2211yescharacteristic
19C2×D8×C97C411yescharacteristic, complemented
20C22×#87604—21no—
21#13095521—22no—
22#13095521—22no—
23#13095519—21no—
24#13095519—21no—
25C2×#11715897—22no—
26C2×#11715897D412yes—
27C4×#87604—21no—
28C4×#87604D412yes—
29#13095510—22no—
30#13095510—22no—
31#13095510—22no—
32#13095510—22no—
33C2×Dic194C2×C412yes—
34C2×C22⋊D97—22no—
35C22×Dic97—22no—
36C2×C4×D97C2311yescharacteristic
37Dic388—22no—
38#13095499—24no—
39C2×#11715895C2×C411yescharacteristic
40C8×D97D412yes—
41C8×D97D412yes—
42C2×D4×C97C2×C412yes—
43D8×C97—22no—
44C2×C8×C97C2×C411yescharacteristic
45C23×C97—22no—
46C22×D97#4111yescharacteristic
47C2×#87604—22no—
48C2×#87604—42no—
49D4×C97—22no—
50D4×C97—22no—
51C2×C4×C97C22×C411yescharacteristic
52C22⋊D97—42no—
53C22⋊D97—42no—
54C2×Dic97—22no—
55C2×Dic97—42no—
56C2×Dic97—42no—
57C2×Dic97C2×D411yescharacteristic
58C4×D97C2×D411yescharacteristic
59C4×D97C2×D411yescharacteristic
60C4×D97C2×D411yescharacteristic
61C4×D97#4111yescharacteristic
62Dic194—22no—
63Dic194—22no—
64#11715897—22no—
65#11715897—42no—
66C8×C97C22⋊C412yes—
67#11715895C22⋊C412yes—
68C22×C97—42no—
69C22×C97—44no—
70C22×C97C4×D411yescharacteristic
71C2×D97—22no—
72C2×D97#12111yescharacteristic
73C2×D97#12411yescharacteristic
74#87604—42no—
75C4×C97C2×C22⋊C411yescharacteristic
76C4×C97C4×D411yescharacteristic
77Dic97—42no—
78Dic97#12111yescharacteristic
79Dic97#12711yescharacteristic
80C2×C97—82no—
81C2×C97#39011yescharacteristic
82C2×C97#44412yes—
83D97—21no—
84D97—21no—
85#1975—971nomaximal, Sylow
86C97#197511yescharacteristic, complemented, Sylow
87C2×#135—971no—
88C2×#108—971no—
89C2×#104—972no—
90#408—972no—
91C2×#97—971no—
92C2×#41—972no—
93#135—1942no—
94#135—1942no—
95C2×Q16—971no—
96C2×SD16—972no—
97C2×D8#8760411yescharacteristic, complemented
98C2×M16—972no—
99C22×C8—971no—
100C22×C8—971no—
101C2×C42—971no—
102#108—1942no—
103#104—1944no—
104C22⋊C8—1944no—
105#97—1942no—
106#41—1942no—
107#41—1942no—
108#41—1942no—
109#41—1942no—
110C2×Q8—972no—
111C2×D4—1942no—
112C2×D4C2×#8760412yes—
113C22×C4—971no—
114C22×C4—1942no—
115C22×C4—1941no—
116Q16—1942no—
117SD16—1944no—
118D8—22no—
119M16—1942no—
120M16—1942no—
121C2×C8—971no—
122C2×C8—972no—
123C2×C8—972no—
124C2×C8—972no—
125C2×C8—1942no—
126C2×C8—1941no—
127C2×C8—1941no—
128C2×C8C2×#8760411yescharacteristic
129C42—972no—
130C42—1941no—
131C23—22no—
132C23—971no—
133Q8—1942no—
134Q8—1942no—
135D4—22no—
136D4—22no—
137D4—3882no—
138D4—3882no—
139C2×C4—971no—
140C2×C4—971no—
141C2×C4—971no—
142C2×C4—971no—
143C2×C4—971no—
144C2×C4—3882no—
145C2×C4—3882no—
146C2×C4—3882no—
147C2×C4—1942no—
148C2×C4—1942no—
149C2×C4C22×#8760411yescharacteristic
150C8—972no—
151C8—3882no—
152C8—1942no—
153C8C22⋊#8760412yes—
154C22—42no—
155C22—44no—
156C22—971no—
157C22—971no—
158C22—1942no—
159C22D4×#8760411yescharacteristic
160C4—971no—
161C4—971no—
162C4—3882no—
163C4—3882no—
164C4C2×C22⋊#8760411yescharacteristic
165C4D4×#8760411yescharacteristic
166C2—82no—
167C2—1941no—
168C2—1941no—
169C2#2026168711yescharacteristic
170C2#2026178212yes—
subgroupG/H
centerC22D4×#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC2×D8×C97C4
socleC22×C97C4×D4
prime radicalC2×#15690829C2
cosocleC2×C4×D97C23
hypercenterC2×D8#87604
hypocenterC97#1975
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. C2×#15690880C2semidirect
2. C2×#15690874C2semidirect
3. #20261709C2semidirect
4. C2×#15690864C2semidirect
16. C2×Dic388C4semidirect
17. C2×#13095499C4semidirect
19. C2×D8×C97C4semidirect
86. C97#1975semidirect
97. C2×D8#87604semidirect

Maximal subgroups

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

subgroupindex
C2×#156908802
C2×#156908742
#202617092
C2×#156908642
C2×#156908292
#197597