Finite Group Databasebeta

← #27151394

Subgroups — #27151394

Order 12416.

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

normal subgroups only

This group has 484 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
1C2×#15690816C211yesmaximal, characteristic, complemented
2#20261313C212yesmaximal, complemented
3#20260990C212yesmaximal, complemented
4#20260985C212yesmaximal, complemented
5#20260961C214yesmaximal, complemented
6C2×#15690742C211yesmaximal, characteristic, complemented
7#20260736C212yesmaximal, complemented
8C2×#135×C97C211yesmaximal, characteristic, complemented
9C2×#13095511C2211yescharacteristic
10#15690816—24no—
11C2×#13095495C2211yescharacteristic
12C2×#13095494C2212yes—
13C2×C8×D97C2212yes—
14#15690789—22no—
15#15690786C2214yes—
16#15690784—22no—
17#15690782—24no—
18#15690781C2214yes—
19C2×C4⋊Dic97C2212yes—
20#15690742C2212yes—
21#15690742C2212yes—
22#15690734C2214yes—
23#15690731C2214yes—
24C2×#41×C97C2212yes—
25#135×C97—24no—
26C2×Q16×C97C2211yescharacteristic
27C2×SD16×C97C2212yes—
28C2×D8×C97C2211yescharacteristic
29C22×C8×C97C2211yescharacteristic
30#13095511—22no—
31#13095511—22no—
32C2×Dic194D411yescharacteristic
33C2×D388D411yescharacteristic
34C2×C22⋊D97—24no—
35C2×C22⋊D97D412yes—
36C22×Dic97—22no—
37C2×C4×D97C2312yes—
38#13095495—22no—
39#13095494—24no—
40C2×#11715895C2312yes—
41C8×D97—24no—
42C4⋊Dic97—22no—
43C4⋊Dic97C2314yes—
44#13095490—24no—
45C22⋊Dic97—24no—
46#13095488—24no—
47#41×C97—24no—
48#41×C97—24no—
49C2×Q8×C97C2312yes—
50C2×D4×C97—22no—
51C2×D4×C97C2312yes—
52C22×C4×C97—22no—
53C22×C4×C97C2311yescharacteristic
54Q16×C97—22no—
55SD16×C97—24no—
56D8×C97—22no—
57C2×C8×C97D414yescomplemented
58C2×C8×C97C2311yescharacteristic
59C2×C8×C97C2311yescharacteristic
60C23×C97—22no—
61C23×C97C2×D411yescharacteristic
62C22×D97C2×D412yes—
63Q8×C97—22no—
64Q8×C97—22no—
65D4×C97—22no—
66D4×C97—22no—
67D4×C97—44no—
68C2×C4×C97—22no—
69C2×C4×C97—44no—
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97C2×D412yes—
72C2×C4×C97#4112yes—
73C2×C4×C97C2411yescharacteristic
74C22⋊D97—24no—
75C22⋊D97—84no—
76C2×Dic97—24no—
77C2×Dic97—82no—
78C2×Dic97C2×D412yes—
79D388—22no—
80C4×D97—22no—
81C4×D97—22no—
82Dic194—22no—
83C8×C97C2×D412yes—
84C8×C97C2×D412yes—
85#11715895—22no—
86C22×C97—22no—
87C22×C97—42no—
88C22×C97—42no—
89C22×C97—42no—
90C22×C97#12112yes—
91C22×C97C22×D411yescharacteristic
92C2×D97—22no—
93C2×D97—24no—
94C4×C97—42no—
95C4×C97#12112yes—
96C4×C97C22×D411yescharacteristic
97C4×C97C2×#4111yescharacteristic
98Dic97—22no—
99Dic97—44no—
100C2×C97—22no—
101C2×C97—82no—
102C2×C97C2×#12111yescharacteristic
103C2×C97#57712yes—
104D97—42no—
105#3060—971nomaximal, Sylow
106C97#306011yescharacteristic, complemented, Sylow
107C2×#135D9711yescharacteristic, complemented
108C2×#131—971no—
109#537—972no—
110#470—972no—
111#469—972no—
112#465—974no—
113C2×#106—971no—
114#417—972no—
115C2×#41—971no—
116C2×#41C2×D9712yes—
117#135—24no—
118C2×Q16C2×D9711yescharacteristic
119C2×SD16C2×D9712yes—
120C2×D8C2×D9711yescharacteristic
121#131—1944no—
122C2×M16—971no—
123C2×M16—972no—
124C22×C8—972no—
125C22×C8C2×D9711yescharacteristic
126C22⋊Q8—974no—
127C22⋊Q8—1942no—
128#121—974no—
129#121—1942no—
130#121—1944no—
131C2×C4⋊C4—972no—
132#106—972no—
133#106—972no—
134#103—974no—
135#102—974no—
136#41—24no—
137#41—24no—
138#41—1942no—
139#41—1942no—
140C2×Q8—971no—
141C2×Q8C22×D9712yes—
142C2×D4—22no—
143C2×D4—972no—
144C2×D4—971no—
145C2×D4—1944no—
146C2×D4C22×D9712yes—
147C22×C4—22no—
148C22×C4—972no—
149C22×C4—1942no—
150C22×C4C22×D9711yescharacteristic
151Q16—22no—
152SD16—24no—
153D8—22no—
154M16—1942no—
155M16—1944no—
156C2×C8—972no—
157C2×C8—1944no—
158C2×C8C22⋊D9714yescomplemented
159C2×C8C22×D9711yescharacteristic
160C2×C8C22×D9711yescharacteristic
161C4⋊C4—974no—
162C4⋊C4—1944no—
163C4⋊C4—1942no—
164C22⋊C4—1944no—
165C22⋊C4—1944no—
166C23—22no—
167C23—972no—
168C23D4×D9711yescharacteristic
169Q8—22no—
170Q8—22no—
171Q8—1942no—
172D4—22no—
173D4—22no—
174D4—44no—
175D4—7764no—
176D4—1944no—
177D4—1942no—
178C2×C4—22no—
179C2×C4—44no—
180C2×C4—7762no—
181C2×C4—972no—
182C2×C4—1944no—
183C2×C4—1942no—
184C2×C4—1942no—
185C2×C4D4×D9711yescharacteristic
186C2×C4C2×C22⋊D9712yes—
187C2×C4#1309551012yes—
188C2×C4C23×D9711yescharacteristic
189C8—1942no—
190C8C2×C22⋊D9712yes—
191C8C2×C22⋊D9712yes—
192C22—22no—
193C22—42no—
194C22—42no—
195C22—42no—
196C22—1942no—
197C22—1944no—
198C22#1569078112yes—
199C22C2×D4×D9711yescharacteristic
200C4—42no—
201C4—3884no—
202C4—1942no—
203C4#1569078112yes—
204C4C22×C22⋊D9711yescharacteristic
205C4C2×#1309551011yescharacteristic
206C2—22no—
207C2—82no—
208C2—3882no—
209C2C2×#1569078111yescharacteristic
210C2#2026154212yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#135×C97C2
socleC22×C97C22×D4
hypercenterC2×#135D97
hypocenterC97#3060
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#15690816C2semidirect
2. #20261313C2semidirect
3. #20260990C2semidirect
4. #20260985C2semidirect
5. #20260961C2semidirect
6. C2×#15690742C2semidirect
7. #20260736C2semidirect
8. C2×#135×C97C2semidirect
57. C2×C8×C97D4semidirect
106. C97#3060semidirect
107. C2×#135D97semidirect
158. C2×C8C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908162
#202613132
#202609902
#202609852
#202609612
C2×#156907422
#202607362
C2×#135×C972
#306097