Finite Group Databasebeta

← #27151268

Subgroups — #27151268

Order 12416.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#13095501C211yesmaximal, characteristic, complemented
2#20261306C211yesmaximal, characteristic
3#20261213C211yesmaximal, characteristic, complemented
4#20260955C214yesmaximal, complemented
5#20260943C214yesmaximal, complemented
6#20260763C211yesmaximal, characteristic, complemented
7#20260745C211yesmaximal, characteristic, complemented
8#20260731C211yesmaximal, characteristic
9C2×#137×C97C211yesmaximal, characteristic
10C22×Dic194C2211yescharacteristic
11C2×#13095501—24no—
12C2×#13095501C2214yes—
13C22×#11715895C2211yescharacteristic
14#15690808C2212yes—
15#15690797C2212yes—
16#15690791—22no—
17#15690789—22no—
18Q8×Dic97—22no—
19Q8×Dic97C2214yes—
20#15690769—22no—
21#15690769C2211yescharacteristic
22#15690769C2211yescharacteristic
23#15690739C2214yes—
24#15690735C2214yes—
25#15690734C2212yes—
26#15690731C2212yes—
27C2×#41×C97C2211yescharacteristic
28C22×Q8×C97C2211yescharacteristic
29#137×C97—24no—
30C2×Q16×C97C2212yes—
31C2×SD16×C97C2212yes—
32C2×M16×C97C2211yescharacteristic
33C2×Dic194—22no—
34C2×Dic194—22no—
35C2×Dic194—21no—
36C2×Dic194C2312yes—
37C22×Dic97—21no—
38#13095501—28no—
39C2×#11715895—22no—
40C2×#11715895C2312yes—
41C4⋊Dic97—24no—
42C4⋊Dic97D412yes—
43C4⋊Dic97C2312yes—
44#13095490—24no—
45C22⋊Dic97—21no—
46C22⋊Dic97—24no—
47C22⋊Dic97D412yes—
48C4×Dic97—21no—
49C4×Dic97—24no—
50C4×Dic97C2312yes—
51#41×C97—22no—
52#41×C97—22no—
53C2×Q8×C97—21no—
54C2×Q8×C97—22no—
55C2×Q8×C97D414yes—
56C2×Q8×C97C2311yescharacteristic
57C2×Q8×C97C2312yes—
58C2×D4×C97—21no—
59C2×D4×C97C2311yescharacteristic
60C22×C4×C97—21no—
61C22×C4×C97—21no—
62C22×C4×C97C2311yescharacteristic
63Q16×C97—24no—
64SD16×C97—24no—
65M16×C97—22no—
66C2×C8×C97C2312yes—
67C23×C97—21no—
68C23×C97C2×D411yescharacteristic
69Q8×C97—24no—
70Q8×C97—22no—
71Q8×C97—21no—
72Q8×C97—21no—
73Q8×C97C2×D414yes—
74D4×C97—21no—
75D4×C97—21no—
76D4×C97—42no—
77C2×C4×C97—24no—
78C2×C4×C97—22no—
79C2×C4×C97—21no—
80C2×C4×C97—42no—
81C2×C4×C97C2×D411yescharacteristic
82C2×C4×C97C2×D412yes—
83C2×C4×C97#4112yes—
84C2×C4×C97C2411yescharacteristic
85C2×Dic97—22no—
86C2×Dic97—22no—
87C2×Dic97—42no—
88C2×Dic97C2×D414yes—
89Dic194—22no—
90Dic194—22no—
91Dic194—42no—
92C8×C97—22no—
93#11715895—22no—
94C22×C97—22no—
95C22×C97—41no—
96C22×C97—42no—
97C22×C97#12112yes—
98C22×C97C22×D411yescharacteristic
99C4×C97—24no—
100C4×C97—41no—
101C4×C97#12112yes—
102C4×C97C22×D411yescharacteristic
103C4×C97C2×#4111yescharacteristic
104Dic97—24no—
105Dic97—42no—
106Dic97—42no—
107C2×C97—22no—
108C2×C97—81no—
109C2×C97C2×#12111yescharacteristic
110C2×C97#57812yes—
111#3044—971nomaximal, Sylow
112C97#304411yescharacteristic, complemented, Sylow
113C2×#137D9711yescharacteristic
114C22×Q16—971no—
115#536—971no—
116#519—971no—
117#464—974no—
118#462—974no—
119#424—971no—
120#419—971no—
121#416—971no—
122C2×#41C2×D9711yescharacteristic
123C22×Q8—971no—
124C22×Q8C2×D9711yescharacteristic
125#137—24no—
126C2×Q16—974no—
127C2×Q16—1944no—
128C2×Q16C2×D9712yes—
129C2×SD16C2×D9712yes—
130C2×M16C2×D9711yescharacteristic
131C22×C8—971no—
132C4⋊Q8—972no—
133#124—972no—
134#123—1942no—
135C22⋊Q8—1942no—
136C4×Q8—974no—
137C4×Q8—1942no—
138#117—971no—
139#117—971no—
140#117—1942no—
141C4⋊C8—974no—
142#103—972no—
143#103—974no—
144#102—972no—
145#41—22no—
146#41—22no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—972no—
150C2×Q8—1942no—
151C2×Q8—1942no—
152C2×Q8—1941no—
153C2×Q8C22⋊D9714yes—
154C2×Q8C22×D9711yescharacteristic
155C2×Q8C22×D9712yes—
156C2×D4—21no—
157C2×D4C22×D9711yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—1941no—
161C22×C4C22×D9711yescharacteristic
162Q16—24no—
163Q16—1948no—
164SD16—24no—
165M16—22no—
166C2×C8—972no—
167C2×C8—1942no—
168C2×C8C22×D9712yes—
169C4⋊C4—972no—
170C4⋊C4—972no—
171C4⋊C4—1944no—
172C4⋊C4—1944no—
173C22⋊C4—972no—
174C22⋊C4—1944no—
175C22⋊C4—1941no—
176C42—972no—
177C42—1944no—
178C42—1941no—
179C23—21no—
180C23D4×D9711yescharacteristic
181Q8—24no—
182Q8—22no—
183Q8—21no—
184Q8—21no—
185Q8—3882no—
186Q8—1942no—
187Q8—1942no—
188Q8C2×C22⋊D9714yes—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—42no—
196C2×C4—974no—
197C2×C4—3882no—
198C2×C4—1942no—
199C2×C4—1942no—
200C2×C4D4×D9711yescharacteristic
201C2×C4C2×C22⋊D9712yes—
202C2×C4#1309551012yes—
203C2×C4C23×D9711yescharacteristic
204C8—22no—
205C8—1942no—
206C22—22no—
207C22—41no—
208C22—42no—
209C22#1569078212yes—
210C22C2×D4×D9711yescharacteristic
211C4—24no—
212C4—41no—
213C4—3882no—
214C4—3882no—
215C4—1944no—
216C4#1569078212yes—
217C4C22×C22⋊D9711yescharacteristic
218C4C2×#1309551011yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#1569078211yescharacteristic
222C2#2026155112yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#137×C97C2
socleC22×C97C22×D4
prime radicalC2×#41×C97C22
hypercenterC2×#137D97
hypocenterC97#3044
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#13095501C2semidirect
3. #20261213C2semidirect
4. #20260955C2semidirect
5. #20260943C2semidirect
6. #20260763C2semidirect
7. #20260745C2semidirect
112. C97#3044semidirect

Maximal subgroups

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

subgroupindex
C22×#130955012
#202613062
#202612132
#202609552
#202609432
#202607632
#202607452
#202607312
C2×#137×C972
#304497