Finite Group Databasebeta

← #27141838

Subgroups — #27141838

Order 12416.

262 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 73 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#20261203C211yesmaximal, characteristic
2#20260717C211yesmaximal, characteristic
3#20260708C211yesmaximal, characteristic
4#20260496C211yesmaximal, characteristic, complemented
5#20260494C211yesmaximal, characteristic, complemented
6#20260491C211yesmaximal, characteristic, complemented
7H(3,ℤ/4)×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2211yescharacteristic
9C2×#13095495C2211yescharacteristic
10#15690775—21no—
11#15690772—21no—
12#15690770—22no—
13#15690769C2211yescharacteristic
14#15690767C412yescomplemented
15C2×C4×Dic97C2211yescharacteristic
16C42×D97C412yescomplemented
17#15690728—21no—
18#15690726—21no—
19#15690724—22no—
20C2×M16×C97C2211yescharacteristic
21#117×C97C2211yescharacteristic
22#117×C97C2211yescharacteristic
23#13095511—21no—
24#13095511—21no—
25#13095511—21no—
26#13095511—21no—
27C2×Dic194D411yescharacteristic
28C2×D388D411yescharacteristic
29C2×C22⋊D97D412yes—
30C22×Dic97—21no—
31C2×C4×D97—22no—
32C2×C4×D97C2×C412yes—
33#13095495—21no—
34#13095495—21no—
35C2×#11715895—21no—
36C4⋊Dic97—21no—
37#13095490—22no—
38C22⋊Dic97—21no—
39#13095488—22no—
40C4×Dic97—21no—
41C4×Dic97—22no—
42C4×Dic97—21no—
43C4×Dic97C2×C411yescharacteristic
44C4×Dic97C2×C411yescharacteristic
45C22×C4×C97C2311yescharacteristic
46M16×C97—21no—
47M16×C97—21no—
48C2×C8×C97—21no—
49C4⋊C4×C97—21no—
50C4⋊C4×C97—21no—
51C22⋊C4×C97—21no—
52C22⋊C4×C97—21no—
53C42×C97—21no—
54C42×C97C2×C412yes—
55C23×C97C2×D411yescharacteristic
56C22×D97C22⋊C412yes—
57C2×C4×C97—22no—
58C2×C4×C97—41no—
59C2×C4×C97C22×C411yescharacteristic
60C2×C4×C97C2×D411yescharacteristic
61C2×C4×C97#4111yescharacteristic
62C2×C4×C97#4111yescharacteristic
63C2×C4×C97#4111yescharacteristic
64C2×C4×C97#4111yescharacteristic
65C22⋊D97—22no—
66C22⋊D97—22no—
67C2×Dic97—21no—
68C2×Dic97—21no—
69C2×Dic97—42no—
70C2×Dic97—41no—
71C2×Dic97C22⋊C412yes—
72D388—21no—
73D388—21no—
74C4×D97—22no—
75C4×D97—22no—
76C4×D97—44no—
77Dic194—21no—
78Dic194—21no—
79C8×C97—41no—
80#11715895—41no—
81C22×C97—41no—
82C22×C97C2×C22⋊C411yescharacteristic
83C22×C97#12311yescharacteristic
84C22×C97#12311yescharacteristic
85C2×D97—22no—
86C2×D97—42no—
87C4×C97—42no—
88C4×C97—81no—
89C4×C97#11711yescharacteristic
90C4×C97#11711yescharacteristic
91C4×C97#12311yescharacteristic
92C4×C97#12311yescharacteristic
93Dic97—22no—
94Dic97—22no—
95Dic97—41no—
96Dic97—81no—
97C2×C97—21no—
98C2×C97—21no—
99C2×C97—21no—
100C2×C97#38111yescharacteristic
101D97—42no—
102#1827—971nomaximal, Sylow
103C97#182711yescharacteristic, complemented, Sylow
104C4×#41—971no—
105#413—971no—
106#410—971no—
107#339—971no—
108#339—971no—
109H(3,ℤ/4)—971no—
110H(3,ℤ/4)D9711yescharacteristic, complemented
111C2×#41—971no—
112C2×M16—971no—
113C2×M16C2×D9711yescharacteristic
114C4×Q8—1941no—
115C4×D4—1942no—
116C4×D4—1941no—
117#117—972no—
118#117—971no—
119#117C2×D9711yescharacteristic
120#117C2×D9711yescharacteristic
121C2×C42—971no—
122C2×C42—972no—
123#101—1941no—
124#100—1941no—
125#99—1942no—
126#41—1941no—
127#41—1941no—
128#41—1941no—
129#41—1941no—
130C2×Q8—971no—
131C2×D4—972no—
132C2×D4—971no—
133C22×C4—972no—
134C22×C4—1942no—
135C22×C4—1941no—
136C22×C4C22×D9711yescharacteristic
137M16—21no—
138M16—21no—
139M16—1941no—
140M16—1941no—
141C2×C8—21no—
142C2×C8—1941no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—1942no—
146C4⋊C4—1941no—
147C22⋊C4—21no—
148C22⋊C4—21no—
149C22⋊C4—1942no—
150C22⋊C4—1941no—
151C42—21no—
152C42—971no—
153C42—971no—
154C42—1942no—
155C42—1941no—
156C42—1941no—
157C42C4×D9712yes—
158C23—972no—
159C23D4×D9711yescharacteristic
160Q8—1941no—
161Q8—1941no—
162D4—1942no—
163D4—1942no—
164D4—1941no—
165D4—1941no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—972no—
169C2×C4—3884no—
170C2×C4—3881no—
171C2×C4—3882no—
172C2×C4—1941no—
173C2×C4—1941no—
174C2×C4—1942no—
175C2×C4—1942no—
176C2×C4C2×C4×D9711yescharacteristic
177C2×C4D4×D9711yescharacteristic
178C2×C4#1309551011yescharacteristic
179C2×C4#1309551111yescharacteristic
180C2×C4#1309551111yescharacteristic
181C2×C4#1309551211yescharacteristic
182C8—41no—
183C8—3881no—
184C22—41no—
185C22—3882no—
186C22—1942no—
187C22C22⋊C4×D9711yescharacteristic
188C22#1569079211yescharacteristic
189C22#1569079511yescharacteristic
190C4—42no—
191C4—81no—
192C4—7761no—
193C4—3881no—
194C4—1942no—
195C4—1942no—
196C4#1569076711yescharacteristic
197C4#1569076811yescharacteristic
198C4#1569079211yescharacteristic
199C4#1569079511yescharacteristic
200C2—21no—
201C2—21no—
202C2—21no—
203C2—3882no—
204C2#2026059711yescharacteristic
subgroupG/H
centerC4#15690767
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingH(3,ℤ/4)×C97C2
socleC2×C97#381
prime radicalC2×#13095511C22
cosocleC22×C4×C97C23
hypercenterH(3,ℤ/4)D97
hypocenterC97#1827
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ #117 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
4. #20260496C2semidirect
5. #20260494C2semidirect
6. #20260491C2semidirect
7. H(3,ℤ/4)×C97C2semidirect
14. #15690767C4semidirect
16. C42×D97C4semidirect
103. C97#1827semidirect
110. H(3,ℤ/4)D97semidirect

Maximal subgroups

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

subgroupindex
#202612032
#202607172
#202607082
#202604962
#202604942
#202604912
H(3,ℤ/4)×C972
#182797