Finite Group Databasebeta

← #27142777

Subgroups — #27142777

Order 12416.

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

normal subgroups only

This group has 406 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
1C2×#15690836C212yesmaximal, complemented
2#20261316C211yesmaximal, characteristic, complemented
3C2×#15690738C212yesmaximal, complemented
4#20260504C211yesmaximal, characteristic, complemented
5#390×C97C211yesmaximal, characteristic, complemented
6C2×#13095511C2212yes—
7#15690836—24no—
8#15690836—24no—
9C2×#13095499—22no—
10C2×#13095496—22no—
11C2×#13095495C2212yes—
12#15690790—22no—
13#15690785—22no—
14#15690784—22no—
15#15690738—24no—
16C22×D4×C97C2211yescharacteristic
17C2×C4⋊C4×C97C2211yescharacteristic
18C2×C22⋊C4×C97—22no—
19C2×C42×C97C2211yescharacteristic
20#13095511—22no—
21#13095511—22no—
22#13095511—22no—
23#13095511—22no—
24C2×Dic194D412yes—
25C2×D388—22no—
26C2×D388D412yes—
27C2×C22⋊D97—22no—
28C2×C4×D97—22no—
29#13095499—28no—
30#13095496—28no—
31#13095495—22no—
32#13095495D414yescomplemented
33C2×#11715895—22no—
34C4⋊Dic97—22no—
35#13095488—24no—
36C24×C97—21no—
37C2×D4×C97—22no—
38C2×D4×C97—22no—
39C2×D4×C97—21no—
40C2×D4×C97D412yescomplemented
41C22×C4×C97—21no—
42C22×C4×C97D412yes—
43C22×C4×C97C2311yescharacteristic
44C4⋊C4×C97—22no—
45C22⋊C4×C97—44no—
46C42×C97—22no—
47C23×C97—22no—
48C23×C97—42no—
49C23×C97—42no—
50C23×C97—41no—
51C23×C97—41no—
52C23×C97C2×D411yescharacteristic
53C22×D97—22no—
54D4×C97—24no—
55D4×C97—42no—
56C2×C4×C97—22no—
57C2×C4×C97—22no—
58C2×C4×C97—41no—
59C2×C4×C97—41no—
60C2×C4×C97—42no—
61C2×C4×C97C2×D412yes—
62C2×C4×C97C2×D412yes—
63C2×C4×C97C2×D411yescharacteristic
64C2×C4×C97#4111yescharacteristic
65C22⋊D97—42no—
66C22⋊D97—42no—
67C2×Dic97—22no—
68D388—22no—
69D388—22no—
70D388—82no—
71C4×D97—42no—
72C4×D97—42no—
73Dic194—22no—
74Dic194—22no—
75#11715895—24no—
76C22×C97—21no—
77C22×C97—21no—
78C22×C97—44no—
79C22×C97—42no—
80C22×C97—82no—
81C22×C97—81no—
82C22×C97—81no—
83C22×C97#12011yescharacteristic
84C22×C97#12111yescharacteristic
85C22×C97#12711yescharacteristic
86C2×D97—42no—
87C2×D97—42no—
88C2×D97—42no—
89C4×C97—42no—
90C4×C97—42no—
91C4×C97#12012yes—
92C4×C97#12112yes—
93Dic97—42no—
94C2×C97—21no—
95C2×C97—21no—
96C2×C97—82no—
97C2×C97#39211yescharacteristic
98C2×C97#45311yescharacteristic
99C2×C97#45411yescharacteristic
100D97—82no—
101#2006—971nomaximal, Sylow
102C97#200611yescharacteristic, complemented, Sylow
103C2×AGL(1,ℤ/8)—972no—
104#537—971no—
105C2×#104—972no—
106#390D9711yescharacteristic, complemented
107#343—971no—
108C2×#41—972no—
109C22×D4C2×D9711yescharacteristic
110AGL(1,ℤ/8)—1944no—
111AGL(1,ℤ/8)—1944no—
112C2×SD16—1942no—
113C2×D8—1942no—
114C2×M16—972no—
115C22⋊Q8—1942no—
116#121—1942no—
117#121—1942no—
118C2×C4⋊C4C2×D9711yescharacteristic
119C2×C22⋊C4—22no—
120C2×C42C2×D9711yescharacteristic
121#104—1944no—
122C24—21no—
123#41—1942no—
124#41—1942no—
125#41—1942no—
126#41—1942no—
127C2×Q8—972no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—972no—
132C2×D4—1942no—
133C2×D4—1942no—
134C2×D4D38812yescomplemented
135C22×C4—21no—
136C22×C4—1942no—
137C22×C4C22⋊D9712yes—
138C22×C4C22×D9711yescharacteristic
139SD16—1948no—
140D8—1948no—
141M16—974no—
142M16—1942no—
143C2×C8—1942no—
144C4⋊C4—22no—
145C4⋊C4—1942no—
146C22⋊C4—44no—
147C22⋊C4—1944no—
148C42—22no—
149C23—22no—
150C23—42no—
151C23—42no—
152C23—41no—
153C23—41no—
154C23—1942no—
155C23C2×C22⋊D9711yescharacteristic
156Q8—1942no—
157Q8—1942no—
158D4—24no—
159D4—42no—
160D4—7762no—
161D4—3882no—
162D4—3882no—
163D4—1942no—
164D4—1942no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—3882no—
171C2×C4—3882no—
172C2×C4—1942no—
173C2×C4D4×D9712yes—
174C2×C4D4×D9712yes—
175C2×C4C2×D38811yescharacteristic
176C2×C4#1309551111yescharacteristic
177C8—1944no—
178C22—21no—
179C22—21no—
180C22—44no—
181C22—42no—
182C22—82no—
183C22—81no—
184C22—81no—
185C22—3882no—
186C22—3882no—
187C22—3882no—
188C22#1569077811yescharacteristic
189C22#1569078411yescharacteristic
190C22#1569080611yescharacteristic
191C4—42no—
192C4—42no—
193C4—3882no—
194C4#1569078012yes—
195C4#1569078312yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—7762no—
200C2#2026064411yescharacteristic
201C2#2026090211yescharacteristic
202C2#2026090711yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#390×C97C2
socleC22×C97#120
hypercenter#390D97
hypocenterC97#2006
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#15690836C2semidirect
2. #20261316C2semidirect
3. C2×#15690738C2semidirect
4. #20260504C2semidirect
5. #390×C97C2semidirect
32. #13095495D4semidirect
40. C2×D4×C97D4semidirect
102. C97#2006semidirect
106. #390D97semidirect
134. C2×D4D388semidirect

Maximal subgroups

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

subgroupindex
C2×#156908362
#202613162
C2×#156907382
#202605042
#390×C972
#200697