Finite Group Databasebeta

← #2849×C223 #43251864

Subgroups — #2849×C223 #43251864

Order 28544.

1102 conjugacy classes of subgroups; only the 264 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 264 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
1C24×C4×C223C211yesmaximal, characteristic, complemented
2C2×#123×C223C212yesmaximal, complemented
3C2×#120×C223C211yesmaximal, characteristic
4#386×C223C214yesmaximal, complemented
5#385×C223C214yesmaximal, complemented
6#381×C223C212yesmaximal, complemented
7#379×C223C211yesmaximal, characteristic, complemented
8C25×C223C2211yescharacteristic
9C22×D4×C223C2211yescharacteristic
10C22×D4×C223C2212yes—
11C23×C4×C223—24no—
12C23×C4×C223—28no—
13C23×C4×C223C2214yes—
14C23×C4×C223C2212yes—
15#123×C223—28no—
16#120×C223—24no—
17C2×C4⋊C4×C223C2214yescomplemented
18C2×C22⋊C4×C223C2211yescharacteristic
19C2×C22⋊C4×C223C2212yes—
20C2×C22⋊C4×C223C2212yes—
21C2×C22⋊C4×C223C2214yes—
22C2×C22⋊C4×C223C2214yescomplemented
23#95×C223C2218yescomplemented
24C24×C223—24no—
25C24×C223—28no—
26C24×C223D414yes—
27C24×C223C2311yescharacteristic
28C24×C223C2311yescharacteristic
29C24×C223C2312yes—
30C2×D4×C223—22no—
31C2×D4×C223—24no—
32C2×D4×C223—24no—
33C2×D4×C223—28no—
34C22×C4×C223—232no—
35C22×C4×C223—28no—
36C22×C4×C223—24no—
37C22×C4×C223—24no—
38C22×C4×C223—216no—
39C22×C4×C223D418yescomplemented
40C22×C4×C223C2312yes—
41C22×C4×C223C2314yes—
42C22×C4×C223C2311yescharacteristic
43C22×C4×C223C2314yes—
44C4⋊C4×C223—28no—
45C22⋊C4×C223—22no—
46C22⋊C4×C223—24no—
47C22⋊C4×C223—24no—
48C22⋊C4×C223—28no—
49C22⋊C4×C223—28no—
50C23×C223—232no—
51C23×C223—28no—
52C23×C223—216no—
53C23×C223—24no—
54C23×C223—24no—
55C23×C223—24no—
56C23×C223—22no—
57C23×C223—21no—
58C23×C223—24no—
59C23×C223C2×D412yes—
60C23×C223C2×D414yes—
61C23×C223C2×D414yes—
62C23×C223#4118yes—
63C23×C223C2411yescharacteristic
64D4×C223—44no—
65D4×C223—48no—
66C2×C4×C223—232no—
67C2×C4×C223—216no—
68C2×C4×C223—24no—
69C2×C4×C223—24no—
70C2×C4×C223—24no—
71C2×C4×C223—21no—
72C2×C4×C223—22no—
73C2×C4×C223—22no—
74C2×C4×C223—28no—
75C2×C4×C223—24no—
76C2×C4×C223C2×D418yes—
77C22×C223—232no—
78C22×C223—24no—
79C22×C223—24no—
80C22×C223—24no—
81C22×C223—28no—
82C22×C223—216no—
83C22×C223—41no—
84C22×C223—42no—
85C22×C223—44no—
86C22×C223#12014yes—
87C22×C223#12318yescomplemented
88C22×C223C22×D411yescharacteristic
89C22×C223C22×D412yes—
90C22×C223C2×#4114yes—
91C4×C223—28no—
92C4×C223—41no—
93C4×C223—42no—
94C4×C223—44no—
95C2×C223—24no—
96C2×C223—28no—
97C2×C223—81no—
98C2×C223C2×#12011yescharacteristic
99C2×C223C2×#12312yes—
100C2×C223#52514yes—
101C223#284911yescharacteristic, direct product factor, Sylow
102#2849C22311yesmaximal, characteristic, direct product factor, Sylow
103C24×C4C2×C22311yescharacteristic, complemented
104C2×#123C2×C22312yescomplemented
105C2×#120C2×C22311yescharacteristic
106#386C2×C22314yescomplemented
107#385C2×C22314yescomplemented
108#381C2×C22312yescomplemented
109#379C2×C22311yescharacteristic, complemented
110C25C22×C22311yescharacteristic
111C22×D4C22×C22311yescharacteristic
112C22×D4C22×C22312yes—
113C23×C4—24no—
114C23×C4—28no—
115C23×C4C22×C22314yes—
116C23×C4C22×C22312yes—
117#123—28no—
118#120—24no—
119C2×C4⋊C4C22×C22314yescomplemented
120C2×C22⋊C4C22×C22311yescharacteristic
121C2×C22⋊C4C22×C22312yes—
122C2×C22⋊C4C22×C22312yes—
123C2×C22⋊C4C22×C22314yes—
124C2×C22⋊C4C22×C22314yescomplemented
125#95C22×C22318yescomplemented
126C24—24no—
127C24—28no—
128C24D4×C22314yes—
129C24C23×C22311yescharacteristic
130C24C23×C22311yescharacteristic
131C24C23×C22312yes—
132C2×D4—22no—
133C2×D4—24no—
134C2×D4—24no—
135C2×D4—28no—
136C22×C4—232no—
137C22×C4—28no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—216no—
141C22×C4D4×C22318yescomplemented
142C22×C4C23×C22312yes—
143C22×C4C23×C22314yes—
144C22×C4C23×C22311yescharacteristic
145C22×C4C23×C22314yes—
146C4⋊C4—28no—
147C22⋊C4—22no—
148C22⋊C4—24no—
149C22⋊C4—24no—
150C22⋊C4—28no—
151C22⋊C4—28no—
152C23—232no—
153C23—28no—
154C23—216no—
155C23—24no—
156C23—24no—
157C23—24no—
158C23—22no—
159C23—21no—
160C23—24no—
161C23C2×D4×C22312yes—
162C23C2×D4×C22314yes—
163C23C2×D4×C22314yes—
164C23#41×C22318yes—
165C23C24×C22311yescharacteristic
166D4—44no—
167D4—48no—
168C2×C4—232no—
169C2×C4—216no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—24no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—28no—
177C2×C4—24no—
178C2×C4C2×D4×C22318yes—
179C22—232no—
180C22—24no—
181C22—24no—
182C22—24no—
183C22—28no—
184C22—216no—
185C22—41no—
186C22—42no—
187C22—44no—
188C22#120×C22314yes—
189C22#123×C22318yescomplemented
190C22C22×D4×C22311yescharacteristic
191C22C22×D4×C22312yes—
192C22C2×#41×C22314yes—
193C4—28no—
194C4—41no—
195C4—42no—
196C4—44no—
197C2—24no—
198C2—28no—
199C2—81no—
200C2C2×#120×C22311yescharacteristic
201C2C2×#123×C22312yes—
202C2#525×C22314yes—
subgroupG/H
center, socleC23×C223C24
commutator, Frattini, cosocleC23C24×C223
prime radicalC2×#120×C223C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C223 ◁ G

Complements

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

NH ≅ G/Ntype
1. C24×C4×C223C2semidirect
2. C2×#123×C223C2semidirect
4. #386×C223C2semidirect
5. #385×C223C2semidirect
6. #381×C223C2semidirect
7. #379×C223C2semidirect
17. C2×C4⋊C4×C223C22semidirect
22. C2×C22⋊C4×C223C22semidirect
23. #95×C223C22semidirect
39. C22×C4×C223D4semidirect
87. C22×C223#123semidirect
101. C223#2849direct (N × H)
102. #2849C223direct (N × H)
103. C24×C4C2×C223semidirect
104. C2×#123C2×C223semidirect
106. #386C2×C223semidirect
107. #385C2×C223semidirect
108. #381C2×C223semidirect
109. #379C2×C223semidirect
119. C2×C4⋊C4C22×C223semidirect
124. C2×C22⋊C4C22×C223semidirect
125. #95C22×C223semidirect
141. C22×C4D4×C223semidirect
189. C22#123×C223semidirect

Maximal subgroups

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

subgroupindex
C24×C4×C2232
C2×#123×C2232
C2×#120×C2232
#386×C2232
#385×C2232
#381×C2232
#379×C2232
#2849223