Finite Group Databasebeta

← #27170066

Subgroups — #27170066

Order 12928.

310 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 53 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#20265758C211yesmaximal, characteristic, complemented
2#20265311C212yesmaximal, complemented
3#20264910C212yesmaximal, complemented
4#20264903C211yesmaximal, characteristic, complemented
5#394×C101C211yesmaximal, characteristic, complemented
6C2×#13108074—21no—
7C2×#13108059—22no—
8#15692235—22no—
9#15692230—22no—
10#15692229—21no—
11#15692229C2212yes—
12#15692176—22no—
13C22⋊#11717579C2212yes—
14C22×D4×C101C2211yescharacteristic
15C2×C4⋊C4×C101C2211yescharacteristic
16C2×C22⋊C4×C101—21no—
17C2×C22⋊C4×C101—21no—
18#95×C101C2211yescharacteristic
19#13108074—42no—
20#13108074—42no—
21C2×Dic202—21no—
22C2×D404—21no—
23C2×D404—21no—
24C2×D404D412yes—
25C2×C22⋊D101—22no—
26C2×C4×D101—22no—
27#13108059—44no—
28C2×#11717579—22no—
29C4⋊Dic101—21no—
30C4⋊Dic101D412yes—
31#13108051—22no—
32#13108051—22no—
33C24×C101—21no—
34C2×D4×C101—22no—
35C2×D4×C101—42no—
36C22×C4×C101—21no—
37C22×C4×C101D411yescharacteristic
38C22×C4×C101D411yescharacteristic
39C22×C4×C101C2311yescharacteristic
40C4⋊C4×C101—22no—
41C22⋊C4×C101—42no—
42C22⋊C4×C101—42no—
43C23×C101—22no—
44C23×C101—41no—
45C23×C101—41no—
46C23×C101—41no—
47C23×C101—41no—
48C23×C101—41no—
49C23×C101—41no—
50C23×C101C2×D411yescharacteristic
51C22×D101—22no—
52D4×C101—24no—
53D4×C101—42no—
54C2×C4×C101—21no—
55C2×C4×C101—21no—
56C2×C4×C101—21no—
57C2×C4×C101—41no—
58C2×C4×C101—41no—
59C2×C4×C101—41no—
60C2×C4×C101—41no—
61C2×C4×C101—41no—
62C2×C4×C101D812yescomplemented
63C2×C4×C101C2×D412yes—
64C22⋊D101—42no—
65C22⋊D101—42no—
66C2×Dic101—22no—
67D404—42no—
68D404—42no—
69D404—81no—
70C4×D101—44no—
71Dic202—42no—
72#11717579—42no—
73C22×C101—21no—
74C22×C101—21no—
75C22×C101—42no—
76C22×C101—42no—
77C22×C101—42no—
78C22×C101—81no—
79C22×C101—81no—
80C22×C101—81no—
81C22×C101—81no—
82C22×C101#12011yescharacteristic
83C22×C101C2×D811yescharacteristic
84C22×C101AGL(1,ℤ/8)11yescharacteristic
85C2×D101—42no—
86C2×D101—42no—
87C2×D101—42no—
88C4×C101—22no—
89C4×C101—22no—
90C4×C101—41no—
91C4×C101—81no—
92Dic101—42no—
93C2×C101—21no—
94C2×C101—21no—
95C2×C101—82no—
96C2×C101#44711yescharacteristic
97C2×C101#45411yescharacteristic
98C2×C101B(4,𝔽2)11yescharacteristic
99D101—82no—
100#1598—1011nomaximal, Sylow
101C101#159811yescharacteristic, complemented, Sylow
102#537—1011no—
103#447—1012no—
104#394D10111yescharacteristic, complemented
105#327—1012no—
106#324—1011no—
107C2×#41—2021no—
108C22×D4C2×D10111yescharacteristic
109C2×D8—2022no—
110C22⋊Q8—2022no—
111#121—1012no—
112#121—2022no—
113#121—2021no—
114C2×C4⋊C4C2×D10111yescharacteristic
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117#102—2022no—
118C22⋊C8—1012no—
119#95C2×D10111yescharacteristic
120C24—21no—
121#41—4042no—
122#41—4042no—
123C2×Q8—2021no—
124C2×D4—22no—
125C2×D4—42no—
126C2×D4—1012no—
127C2×D4—2021no—
128C2×D4—2021no—
129C2×D4—2022no—
130C22×C4—21no—
131C22×C4—2022no—
132C22×C4C22⋊D10111yescharacteristic
133C22×C4C22⋊D10111yescharacteristic
134C22×C4C22×D10111yescharacteristic
135D8—4044no—
136C2×C8—2022no—
137C4⋊C4—22no—
138C4⋊C4—1012no—
139C4⋊C4—2021no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C22⋊C4—2022no—
143C22⋊C4—2022no—
144C23—22no—
145C23—41no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—2022no—
152C23C2×C22⋊D10111yescharacteristic
153Q8—4042no—
154D4—24no—
155D4—42no—
156D4—8081no—
157D4—4042no—
158D4—4042no—
159D4—4042no—
160D4—4042no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—4044no—
170C2×C4—2022no—
171C2×C4#1310805912yescomplemented
172C2×C4D4×D10112yes—
173C8—4042no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—81no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22—4042no—
184C22—4042no—
185C22—4042no—
186C22#1569222311yescharacteristic
187C22C2×#1310805911yescharacteristic
188C22#1569228011yescharacteristic
189C4—22no—
190C4—22no—
191C4—41no—
192C4—81no—
193C4—4042no—
194C2—21no—
195C2—21no—
196C2—82no—
197C2—8082no—
198C2#2026531011yescharacteristic
199C2#2026534911yescharacteristic
200C2#2026536311yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#394×C101C2
socleC22×C101#120
hypercenter#394D101
hypocenterC101#1598
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #20265758C2semidirect
2. #20265311C2semidirect
3. #20264910C2semidirect
4. #20264903C2semidirect
5. #394×C101C2semidirect
62. C2×C4×C101D8semidirect
101. C101#1598semidirect
104. #394D101semidirect
171. C2×C4#13108059semidirect

Maximal subgroups

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

subgroupindex
#202657582
#202653112
#202649102
#202649032
#394×C1012
#1598101