Finite Group Databasebeta

← #27171654

Subgroups — #27171654

Order 12928.

312 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 89 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#20265748C211yesmaximal, characteristic
2#20265187C212yesmaximal, complemented
3#20265151C211yesmaximal, characteristic
4#20264934C212yesmaximal, complemented
5#568×C101C211yesmaximal, characteristic
6C22×Dic202C2211yescharacteristic
7#15692253C412yescomplemented
8#15692242C412yescomplemented
9#15692236—22no—
10#15692234—22no—
11#15692214C2212yes—
12#15692214C2211yescharacteristic
13#15692180—24no—
14#15692170—22no—
15C2×#41×C101C2211yescharacteristic
16#131×C101—24no—
17C2×M16×C101—22no—
18C2×M16×C101C2212yes—
19C2×Dic202—22no—
20C2×Dic202—22no—
21C2×Dic202—21no—
22C2×Dic202C2×C412yes—
23C22×Dic101—21no—
24C4⋊Dic101—21no—
25C4⋊Dic101—22no—
26#13108053—24no—
27C22⋊Dic101—24no—
28C22⋊Dic101—21no—
29C22⋊Dic101—22no—
30C4×Dic101—22no—
31C4×Dic101C2×C412yes—
32#41×C101—22no—
33#41×C101—22no—
34C2×Q8×C101C2×C411yescharacteristic
35C2×D4×C101C2×C411yescharacteristic
36C2×D4×C101D412yes—
37C22×C4×C101D412yes—
38C22×C4×C101C2311yescharacteristic
39M16×C101—24no—
40M16×C101—22no—
41M16×C101—22no—
42C2×C8×C101—24no—
43C2×C8×C101D414yes—
44C23×C101C22⋊C412yes—
45C23×C101C2×D411yescharacteristic
46Q8×C101—22no—
47D4×C101—24no—
48D4×C101—22no—
49C2×C4×C101—22no—
50C2×C4×C101—22no—
51C2×C4×C101C22⋊C412yes—
52C2×C4×C101C22×C411yescharacteristic
53C2×C4×C101C2×D412yes—
54C2×C4×C101C2×D411yescharacteristic
55C2×C4×C101#4112yes—
56C2×Dic101—22no—
57C2×Dic101—22no—
58C2×Dic101—42no—
59C2×Dic101—42no—
60Dic202—24no—
61Dic202—41no—
62Dic202—41no—
63C8×C101—24no—
64C22×C101—22no—
65C22×C101—41no—
66C22×C101—42no—
67C22×C101C2×C22⋊C411yescharacteristic
68C22×C101#12112yes—
69C4×C101—22no—
70C4×C101C4×D412yes—
71C4×C101#12011yescharacteristic
72C4×C101#12311yescharacteristic
73Dic101—42no—
74Dic101—42no—
75Dic101—82no—
76C2×C101—21no—
77C2×C101—22no—
78C2×C101—42no—
79C2×C101#38611yescharacteristic
80#1909—1011nomaximal, Sylow
81C101#190911yescharacteristic, complemented, Sylow
82#568D10111yescharacteristic
83#536—1011no—
84#419—1012no—
85#410—1011no—
86H(3,ℤ/4)—1012no—
87C2×#41C2×D10111yescharacteristic
88C22×Q8—1011no—
89#131—24no—
90C2×M16—22no—
91C2×M16C2×D10112yes—
92C4⋊Q8—1012no—
93#124—1012no—
94#123—2022no—
95C22⋊Q8—2022no—
96#117—1011no—
97#117—1012no—
98#103—2024no—
99#99—2022no—
100#41—22no—
101#41—22no—
102C2×Q8—1012no—
103C2×Q8—2022no—
104C2×Q8—2022no—
105C2×Q8—2021no—
106C2×Q8C4×D10111yescharacteristic
107C2×D4C4×D10111yescharacteristic
108C2×D4D40412yes—
109C22×C4—2021no—
110C22×C4C22⋊D10112yes—
111C22×C4C22×D10111yescharacteristic
112M16—24no—
113M16—22no—
114M16—22no—
115C2×C8—24no—
116C2×C8C22⋊D10114yes—
117C4⋊C4—2022no—
118C4⋊C4—2024no—
119C4⋊C4—2021no—
120C22⋊C4—2022no—
121C22⋊C4—2024no—
122C22⋊C4—2021no—
123C42—1012no—
124C42—2022no—
125C23#1310805112yes—
126C23C2×D40411yescharacteristic
127Q8—22no—
128Q8—4041no—
129Q8—4041no—
130Q8—2024no—
131D4—24no—
132D4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—4042no—
136C2×C4—4042no—
137C2×C4—2022no—
138C2×C4—2022no—
139C2×C4#1310805112yes—
140C2×C4C2×C4×D10111yescharacteristic
141C2×C4C2×C22⋊D10112yes—
142C2×C4C2×C22⋊D10111yescharacteristic
143C2×C4#1310807412yes—
144C8—24no—
145C22—22no—
146C22—41no—
147C22—42no—
148C22C2×#1310805111yescharacteristic
149C22#1569222912yes—
150C4—22no—
151C4—8082no—
152C4—4042no—
153C4—4042no—
154C4C4×C22⋊D10112yes—
155C4#1569222411yescharacteristic
156C4#1569223811yescharacteristic
157C2—22no—
158C2—21no—
159C2—42no—
160C2#2026505611yescharacteristic
subgroupG/H
centerC2#20265056
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#568×C101C2
socleC2×C101#386
prime radicalC2×#41×C101C22
cosocleC22×C4×C101C23
hypercenter#568D101
hypocenterC101#1909
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #20265187C2semidirect
4. #20264934C2semidirect
7. #15692253C4semidirect
8. #15692242C4semidirect
81. C101#1909semidirect

Maximal subgroups

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

subgroupindex
#202657482
#202651872
#202651512
#202649342
#568×C1012
#1909101