Finite Group Databasebeta

← #27172173

Subgroups — #27172173

Order 12928.

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×#15692281C212yesmaximal, complemented
2#20265758C211yesmaximal, characteristic, complemented
3C2×#15692183C212yesmaximal, complemented
4#20264946C211yesmaximal, characteristic, complemented
5#390×C101C211yesmaximal, characteristic, complemented
6C2×#13108074C2212yes—
7#15692281—24no—
8#15692281—24no—
9C2×#13108062—22no—
10C2×#13108059—22no—
11C2×#13108058C2212yes—
12#15692235—22no—
13#15692230—22no—
14#15692229—22no—
15#15692183—24no—
16C22×D4×C101C2211yescharacteristic
17C2×C4⋊C4×C101C2211yescharacteristic
18C2×C22⋊C4×C101—22no—
19C2×C42×C101C2211yescharacteristic
20#13108074—22no—
21#13108074—22no—
22#13108074—22no—
23#13108074—22no—
24C2×Dic202D412yes—
25C2×D404—22no—
26C2×D404D412yes—
27C2×C22⋊D101—22no—
28C2×C4×D101—22no—
29#13108062—28no—
30#13108059—28no—
31#13108058—22no—
32#13108058D414yescomplemented
33C2×#11717579—22no—
34C4⋊Dic101—22no—
35#13108051—24no—
36C24×C101—21no—
37C2×D4×C101—22no—
38C2×D4×C101—22no—
39C2×D4×C101—21no—
40C2×D4×C101D412yescomplemented
41C22×C4×C101—21no—
42C22×C4×C101D412yes—
43C22×C4×C101C2311yescharacteristic
44C4⋊C4×C101—22no—
45C22⋊C4×C101—44no—
46C42×C101—22no—
47C23×C101—22no—
48C23×C101—42no—
49C23×C101—42no—
50C23×C101—41no—
51C23×C101—41no—
52C23×C101C2×D411yescharacteristic
53C22×D101—22no—
54D4×C101—24no—
55D4×C101—42no—
56C2×C4×C101—22no—
57C2×C4×C101—22no—
58C2×C4×C101—41no—
59C2×C4×C101—41no—
60C2×C4×C101—42no—
61C2×C4×C101C2×D412yes—
62C2×C4×C101C2×D412yes—
63C2×C4×C101C2×D411yescharacteristic
64C2×C4×C101#4111yescharacteristic
65C22⋊D101—42no—
66C22⋊D101—42no—
67C2×Dic101—22no—
68D404—22no—
69D404—22no—
70D404—82no—
71C4×D101—42no—
72C4×D101—42no—
73Dic202—22no—
74Dic202—22no—
75#11717579—24no—
76C22×C101—21no—
77C22×C101—21no—
78C22×C101—44no—
79C22×C101—42no—
80C22×C101—82no—
81C22×C101—81no—
82C22×C101—81no—
83C22×C101#12011yescharacteristic
84C22×C101#12111yescharacteristic
85C22×C101#12711yescharacteristic
86C2×D101—42no—
87C2×D101—42no—
88C2×D101—42no—
89C4×C101—42no—
90C4×C101—42no—
91C4×C101#12012yes—
92C4×C101#12112yes—
93Dic101—42no—
94C2×C101—21no—
95C2×C101—21no—
96C2×C101—82no—
97C2×C101#39211yescharacteristic
98C2×C101#45311yescharacteristic
99C2×C101#45411yescharacteristic
100D101—82no—
101#2006—1011nomaximal, Sylow
102C101#200611yescharacteristic, complemented, Sylow
103C2×AGL(1,ℤ/8)—1012no—
104#537—1011no—
105C2×#104—1012no—
106#390D10111yescharacteristic, complemented
107#343—1011no—
108C2×#41—1012no—
109C22×D4C2×D10111yescharacteristic
110AGL(1,ℤ/8)—2024no—
111AGL(1,ℤ/8)—2024no—
112C2×SD16—2022no—
113C2×D8—2022no—
114C2×M16—1012no—
115C22⋊Q8—2022no—
116#121—2022no—
117#121—2022no—
118C2×C4⋊C4C2×D10111yescharacteristic
119C2×C22⋊C4—22no—
120C2×C42C2×D10111yescharacteristic
121#104—2024no—
122C24—21no—
123#41—2022no—
124#41—2022no—
125#41—2022no—
126#41—2022no—
127C2×Q8—1012no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—1012no—
132C2×D4—2022no—
133C2×D4—2022no—
134C2×D4D40412yescomplemented
135C22×C4—21no—
136C22×C4—2022no—
137C22×C4C22⋊D10112yes—
138C22×C4C22×D10111yescharacteristic
139SD16—2028no—
140D8—2028no—
141M16—1014no—
142M16—2022no—
143C2×C8—2022no—
144C4⋊C4—22no—
145C4⋊C4—2022no—
146C22⋊C4—44no—
147C22⋊C4—2024no—
148C42—22no—
149C23—22no—
150C23—42no—
151C23—42no—
152C23—41no—
153C23—41no—
154C23—2022no—
155C23C2×C22⋊D10111yescharacteristic
156Q8—2022no—
157Q8—2022no—
158D4—24no—
159D4—42no—
160D4—8082no—
161D4—4042no—
162D4—4042no—
163D4—2022no—
164D4—2022no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—4042no—
171C2×C4—4042no—
172C2×C4—2022no—
173C2×C4D4×D10112yes—
174C2×C4D4×D10112yes—
175C2×C4C2×D40411yescharacteristic
176C2×C4#1310807411yescharacteristic
177C8—2024no—
178C22—21no—
179C22—21no—
180C22—44no—
181C22—42no—
182C22—82no—
183C22—81no—
184C22—81no—
185C22—4042no—
186C22—4042no—
187C22—4042no—
188C22#1569222311yescharacteristic
189C22#1569222911yescharacteristic
190C22#1569225111yescharacteristic
191C4—42no—
192C4—42no—
193C4—4042no—
194C4#1569222512yes—
195C4#1569222812yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—8082no—
200C2#2026508611yescharacteristic
201C2#2026534411yescharacteristic
202C2#2026534911yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#390×C101C2
socleC22×C101#120
hypercenter#390D101
hypocenterC101#2006
LCSG ▷ C22×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#15692281C2semidirect
2. #20265758C2semidirect
3. C2×#15692183C2semidirect
4. #20264946C2semidirect
5. #390×C101C2semidirect
32. #13108058D4semidirect
40. C2×D4×C101D4semidirect
102. C101#2006semidirect
106. #390D101semidirect
134. C2×D4D404semidirect

Maximal subgroups

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

subgroupindex
C2×#156922812
#202657582
C2×#156921832
#202649462
#390×C1012
#2006101