Finite Group Databasebeta

← #27157645

Subgroups — #27157645

Order 12416.

262 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 62 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#20261721C211yesmaximal, characteristic
2#20261714C211yesmaximal, characteristic
3#20261606C212yesmaximal, complemented
4#20261602C212yesmaximal, complemented
5C4×#13095511C211yesmaximal, characteristic
6D97⋊M16C2212yes—
7#15690889C2212yes—
8#15690871—21no—
9#15690870—21no—
10#15690869—21no—
11#15690868—21no—
12C2×#13095511C2211yescharacteristic
13C4×Dic194—21no—
14C4×D388—21no—
15C4×C22⋊D97—21no—
16C4×C22⋊D97—21no—
17#15690769C411yescharacteristic, complemented
18#15690767C411yescharacteristic, complemented
19#15690767C2211yescharacteristic
20C42×D97C411yescharacteristic, complemented
21C42×D97C2211yescharacteristic
22C2×C42×C97C411yescharacteristic, complemented
23#13095521—22no—
24#13095520—22no—
25#13095519—22no—
26C4⋊#87604—22no—
27C22⋊#87604—22no—
28C4×#87604—22no—
29#13095511—22no—
30#13095511—22no—
31C2×Dic194D411yescharacteristic
32C2×D388D411yescharacteristic
33C2×C22⋊D97D411yescharacteristic
34C2×C22⋊D97D411yescharacteristic
35C2×C4×D97—21no—
36C2×C4×D97—21no—
37C2×C4×D97C2×C411yescharacteristic
38C2×C4×D97C2311yescharacteristic
39C4⋊Dic97—21no—
40#13095490—21no—
41#13095490—21no—
42C22⋊Dic97—21no—
43#13095488—21no—
44#13095488—21no—
45C4×Dic97—21no—
46C4×Dic97—21no—
47C4×Dic97C2×C411yescharacteristic
48C4×Dic97C2×C411yescharacteristic
49C22×C4×C97—21no—
50C22×C4×C97C2×C411yescharacteristic
51C42×C97—21no—
52C42×C97C2×C411yescharacteristic
53C42×C97C2×C411yescharacteristic
54C23×C97C22⋊C411yescharacteristic
55C22×D97C22⋊C411yescharacteristic
56C22×D97C2×D411yescharacteristic
57C2×#87604—42no—
58C2×C4×C97—21no—
59C2×C4×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97—21no—
62C2×C4×C97—42no—
63C2×C4×C97C22⋊C411yescharacteristic
64C2×C4×C97C22×C411yescharacteristic
65C22⋊D97—22no—
66C22⋊D97—22no—
67C2×Dic97—21no—
68C2×Dic97—21no—
69C2×Dic97C22⋊C411yescharacteristic
70C2×Dic97C2×D411yescharacteristic
71D388—22no—
72C4×D97—21no—
73C4×D97—21no—
74C4×D97—42no—
75C4×D97—42no—
76C4×D97#4112yes—
77C4×D97#4112yes—
78Dic194—22no—
79#11715897—42no—
80C22×C97—21no—
81C22×C97—41no—
82C22×C97C2×C22⋊C411yescharacteristic
83C2×D97—21no—
84C2×D97—41no—
85C2×D97—41no—
86C2×D97#12312yes—
87#87604—82no—
88C4×C97—22no—
89C4×C97—21no—
90C4×C97—41no—
91C4×C97#11711yescharacteristic
92C4×C97#11711yescharacteristic
93Dic97—21no—
94Dic97—41no—
95Dic97—41no—
96Dic97#12312yes—
97C2×C97—21no—
98C2×C97—41no—
99C2×C97#38111yescharacteristic
100D97—22no—
101D97—41no—
102#1827—971nomaximal, Sylow
103C97#182711yescharacteristic, complemented, Sylow
104C4×#41—971no—
105#413—971no—
106#410—971no—
107#339—972no—
108H(3,ℤ/4)—972no—
109C2×#41—971no—
110C2×M16—972no—
111C4×Q8—1941no—
112C4×D4—1941no—
113C4×D4—1941no—
114C4×D4—1941no—
115#117—971no—
116#117—971no—
117#117—971no—
118#117—972no—
119C2×C42—971no—
120C2×C42—971no—
121C2×C42#8760411yescharacteristic, complemented
122#101—1941no—
123#100—1941no—
124#99—1941no—
125#99—1941no—
126#41—1942no—
127#41—1942no—
128C2×Q8—971no—
129C2×D4—971no—
130C2×D4—971no—
131C2×D4—971no—
132C22×C4—21no—
133C22×C4—971no—
134C22×C4—971no—
135C22×C4—1941no—
136C22×C4—1941no—
137C22×C4C2×#8760411yescharacteristic
138M16—1942no—
139M16—1942no—
140C2×C8—1942no—
141C4⋊C4—1941no—
142C4⋊C4—1941no—
143C4⋊C4—1942no—
144C4⋊C4—1941no—
145C22⋊C4—1941no—
146C22⋊C4—1941no—
147C22⋊C4—1942no—
148C22⋊C4—1941no—
149C42—21no—
150C42—971no—
151C42—971no—
152C42—1941no—
153C42—1941no—
154C42—1942no—
155C42C2×#8760411yescharacteristic
156C42C2×#8760411yescharacteristic
157C23—971no—
158C23—971no—
159C23C22⋊#8760411yescharacteristic
160Q8—1942no—
161D4—1942no—
162D4—1942no—
163D4—1942no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—42no—
169C2×C4—971no—
170C2×C4—972no—
171C2×C4—972no—
172C2×C4—971no—
173C2×C4—3882no—
174C2×C4—3882no—
175C2×C4—3882no—
176C2×C4—1941no—
177C2×C4—1941no—
178C2×C4—1941no—
179C2×C4—1941no—
180C2×C4C22⋊#8760411yescharacteristic
181C2×C4C22×#8760411yescharacteristic
182C8—3882no—
183C22—21no—
184C22—41no—
185C22—972no—
186C22—3881no—
187C22—3881no—
188C22—1941no—
189C22C2×C22⋊#8760411yescharacteristic
190C4—22no—
191C4—21no—
192C4—41no—
193C4—7762no—
194C4—972no—
195C4—3881no—
196C4—3881no—
197C4—1941no—
198C4#1569088911yescharacteristic
199C4#1569088911yescharacteristic
200C2—21no—
201C2—41no—
202C2—3881no—
203C2—1942no—
204C2#2026167811yescharacteristic
subgroupG/H
centerC4#15690889
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC2×C42×C97C4
socleC2×C97#381
prime radicalC2×#13095511C22
cosocleC2×C4×D97C23
hypercenterC2×C42#87604
hypocenterC97#1827
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C42 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
3. #20261606C2semidirect
4. #20261602C2semidirect
17. #15690769C4semidirect
18. #15690767C4semidirect
20. C42×D97C4semidirect
22. C2×C42×C97C4semidirect
103. C97#1827semidirect
121. C2×C42#87604semidirect

Maximal subgroups

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

subgroupindex
#202617212
#202617142
#202616062
#202616022
C4×#130955112
#182797