Finite Group Databasebeta

← #27170069

Subgroups — #27170069

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#20265313C212yesmaximal, complemented
3#20265094C211yesmaximal, characteristic, complemented
4#20264912C212yesmaximal, complemented
5#324×C101C211yesmaximal, characteristic, complemented
6C2×#13108074—21no—
7C22×D404C2211yescharacteristic
8C2×D808—22no—
9#15692235—22no—
10#15692230—22no—
11#15692229—21no—
12#15692229C2212yes—
13C2×#13108051—21no—
14C2×#13108051—21no—
15#15692177—22no—
16#15692162C2211yescharacteristic
17C2×C4⋊C4×C101C2211yescharacteristic
18C22⋊C8×C101C2212yes—
19C23×D101—21no—
20#13108074—42no—
21#13108074—42no—
22C2×Dic202—21no—
23C2×D404—22no—
24C2×D404—21no—
25C2×D404—21no—
26C2×D404—42no—
27C2×D404D412yes—
28C2×C22⋊D101—22no—
29C22×Dic101—21no—
30C2×C4×D101—22no—
31D808—44no—
32C4⋊Dic101—21no—
33C4⋊Dic101D412yes—
34#13108051—22no—
35#13108051—22no—
36#13108051—42no—
37#13108051—42no—
38C22×C4×C101D411yescharacteristic
39C22×C4×C101D411yescharacteristic
40C22×C4×C101C2311yescharacteristic
41C2×C8×C101—22no—
42C4⋊C4×C101—22no—
43C23×C101C2×D411yescharacteristic
44C22×D101—22no—
45C22×D101—22no—
46C22×D101—41no—
47C22×D101—41no—
48C22×D101—41no—
49C22×D101—41no—
50C22×D101—41no—
51C22×D101—41no—
52C2×C4×C101—21no—
53C2×C4×C101—21no—
54C2×C4×C101—21no—
55C2×C4×C101—41no—
56C2×C4×C101—41no—
57C2×C4×C101D812yescomplemented
58C2×C4×C101C2×D412yes—
59C22⋊D101—42no—
60C22⋊D101—42no—
61C2×Dic101—22no—
62C2×Dic101—41no—
63C2×Dic101—41no—
64C2×Dic101—41no—
65D404—24no—
66D404—42no—
67D404—42no—
68D404—42no—
69D404—81no—
70C4×D101—44no—
71Dic202—42no—
72C8×C101—42no—
73C22×C101—21no—
74C22×C101—21no—
75C22×C101#12011yescharacteristic
76C22×C101C2×D811yescharacteristic
77C22×C101AGL(1,ℤ/8)11yescharacteristic
78C2×D101—42no—
79C2×D101—42no—
80C2×D101—42no—
81C2×D101—42no—
82C2×D101—42no—
83C2×D101—42no—
84C2×D101—81no—
85C2×D101—81no—
86C2×D101—81no—
87C2×D101—81no—
88C4×C101—22no—
89C4×C101—22no—
90C4×C101—41no—
91Dic101—42no—
92Dic101—81no—
93C2×C101—21no—
94C2×C101—21no—
95C2×C101#44711yescharacteristic
96C2×C101#45411yescharacteristic
97C2×C101B(4,𝔽2)11yescharacteristic
98D101—82no—
99D101—82no—
100#1598—1011nomaximal, Sylow
101C101#159811yescharacteristic, complemented, Sylow
102#537—1011no—
103#447—1012no—
104#394—1011no—
105#327—1012no—
106#324D10111yescharacteristic, complemented
107C2×#41—2021no—
108C22×D4—1011no—
109C2×D8—2022no—
110C22⋊Q8—2022no—
111#121—1012no—
112#121—2022no—
113#121—2021no—
114C2×C4⋊C4C2×D10111yescharacteristic
115C2×C22⋊C4—2021no—
116C2×C22⋊C4—2021no—
117#102—2022no—
118C22⋊C8C2×D10112yes—
119#95—1011no—
120C24—2021no—
121#41—4042no—
122#41—4042no—
123C2×Q8—2021no—
124C2×D4—1012no—
125C2×D4—4042no—
126C2×D4—2021no—
127C2×D4—2021no—
128C2×D4—2022no—
129C2×D4—2022no—
130C22×C4—2022no—
131C22×C4—2021no—
132C22×C4D40411yescharacteristic
133C22×C4D40411yescharacteristic
134C22×C4C22×D10111yescharacteristic
135D8—4044no—
136C2×C8—22no—
137C4⋊C4—22no—
138C4⋊C4—1012no—
139C4⋊C4—2021no—
140C22⋊C4—4042no—
141C22⋊C4—4042no—
142C22⋊C4—2022no—
143C22⋊C4—2022no—
144C23—4041no—
145C23—4041no—
146C23—4041no—
147C23—4041no—
148C23—4041no—
149C23—4041no—
150C23—2022no—
151C23—2022no—
152C23C2×D40411yescharacteristic
153Q8—4042no—
154D4—8081no—
155D4—4042no—
156D4—4042no—
157D4—4042no—
158D4—4042no—
159D4—4042no—
160D4—2024no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—4044no—
167C2×C4—4041no—
168C2×C4—4041no—
169C2×C4—4041no—
170C2×C4—2022no—
171C2×C4D80812yescomplemented
172C2×C4D4×D10112yes—
173C8—42no—
174C22—21no—
175C22—21no—
176C22—8081no—
177C22—8081no—
178C22—8081no—
179C22—8081no—
180C22—4042no—
181C22—4042no—
182C22—4042no—
183C22—4042no—
184C22—4042no—
185C22—4042no—
186C22#1569222511yescharacteristic
187C22C2×D80811yescharacteristic
188C22#1569228311yescharacteristic
189C4—22no—
190C4—22no—
191C4—41no—
192C4—8081no—
193C4—4042no—
194C2—21no—
195C2—21no—
196C2—8082no—
197C2—8082no—
198C2#2026531311yescharacteristic
199C2#2026535211yescharacteristic
200C2#2026536511yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#324×C101C2
socleC22×C101#120
hypercenter#324D101
hypocenterC101#1598
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
1. #20265758C2semidirect
2. #20265313C2semidirect
3. #20265094C2semidirect
4. #20264912C2semidirect
5. #324×C101C2semidirect
57. C2×C4×C101D8semidirect
101. C101#1598semidirect
106. #324D101semidirect
171. C2×C4D808semidirect

Maximal subgroups

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

subgroupindex
#202657582
#202653132
#202650942
#202649122
#324×C1012
#1598101