Finite Group Databasebeta

← #27186469

Subgroups — #27186469

Order 12928.

266 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 66 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#20266180C211yesmaximal, characteristic, complemented
2C4×#13108083C211yesmaximal, characteristic, complemented
3#20266027C212yesmaximal, complemented
4#20266025C212yesmaximal, complemented
5#20265645C211yesmaximal, characteristic
6C2×#13108083—21no—
7C2×#13108083—21no—
8#15692322—21no—
9C4⋊#11717581C2212yes—
10#15692310—21no—
11C4×#11717581C2212yes—
12C2×#13108074C411yescharacteristic
13#15692220C411yescharacteristic
14#15692220C2211yescharacteristic
15#15692217C411yescharacteristic
16#15692217C2211yescharacteristic
17#15692215—21no—
18#15692215—21no—
19#15692212—21no—
20C2×C4×Dic101C2211yescharacteristic
21C42×D101—21no—
22#117×C101C411yescharacteristic
23#13108083—24no—
24#13108083—42no—
25C2×#11717581—22no—
26C2×#11717581—22no—
27#13108074—22no—
28#13108074—22no—
29C2×Dic202C2×C411yescharacteristic
30C2×D404C2×C411yescharacteristic
31C2×C22⋊D101—21no—
32C22×Dic101D411yescharacteristic
33C22×Dic101D411yescharacteristic
34C2×C4×D101—21no—
35C2×C4×D101—21no—
36C2×C4×D101—21no—
37#13108053—21no—
38#13108053—21no—
39#13108051—21no—
40#13108051—21no—
41C4×Dic101—21no—
42C4×Dic101—21no—
43C4×Dic101C2×C411yescharacteristic
44C4×Dic101D411yescharacteristic
45C4×Dic101D411yescharacteristic
46C4×Dic101C2311yescharacteristic
47C22×C4×C101C2×C411yescharacteristic
48C4⋊C4×C101C2×C411yescharacteristic
49C4⋊C4×C101C2×C411yescharacteristic
50C22⋊C4×C101—21no—
51C42×C101—21no—
52C23×C101C22⋊C411yescharacteristic
53C22×D101—21no—
54C2×C4×C101—21no—
55C2×C4×C101—21no—
56C2×C4×C101—21no—
57C2×C4×C101—21no—
58C2×C4×C101C22⋊C411yescharacteristic
59C2×C4×C101C22×C411yescharacteristic
60C22⋊D101—24no—
61C2×Dic101—21no—
62C2×Dic101—22no—
63C2×Dic101—21no—
64C2×Dic101—21no—
65C2×Dic101C22⋊C411yescharacteristic
66C2×Dic101C22⋊C411yescharacteristic
67C2×Dic101C2×D411yescharacteristic
68C2×Dic101C2×D411yescharacteristic
69D404—21no—
70D404M1612yescomplemented
71C4×D101—24no—
72C4×D101—42no—
73C4×D101—41no—
74C4×D101—41no—
75Dic202—21no—
76Dic202M1612yescomplemented
77#11717581—24no—
78#11717581—42no—
79C22×C101—21no—
80C22×C101—21no—
81C22×C101—21no—
82C22×C101C2×C22⋊C411yescharacteristic
83C2×D101—22no—
84C2×D101—41no—
85C2×D101—41no—
86C4×C101—21no—
87C4×C101—41no—
88C4×C101—41no—
89C4×C101C2×M1611yescharacteristic
90C4×C101C2×M1611yescharacteristic
91Dic101—22no—
92Dic101—21no—
93Dic101—21no—
94Dic101#10412yes—
95Dic101AGL(1,ℤ/8)11yescharacteristic
96Dic101#13711yescharacteristic
97C2×C101—41no—
98C2×C101#40711yescharacteristic
99C2×C101#41711yescharacteristic
100C2×C101C2×#10411yescharacteristic
101D101—42no—
102#1489—1011nomaximal, Sylow
103C101#148911yescharacteristic, complemented, Sylow
104C4×#41—1011no—
105#423—1011no—
106C4×M16—1011no—
107Q8⋊C8—1012no—
108D4⋊C8—1012no—
109C2×#41—1011no—
110C2×M16—2021no—
111C2×M16—2021no—
112C4×Q8—1011no—
113C4×Q8—1011no—
114C4×D4—1011no—
115C4×D4—1011no—
116C4×D4—2021no—
117C4×D4—2021no—
118#117—2021no—
119#117#8793811yescharacteristic
120C2×C42—1011no—
121C2×C42—2021no—
122C4⋊C8—1012no—
123C4⋊C8—2021no—
124#97—2021no—
125C4×C8—1012no—
126#41—2022no—
127#41—2022no—
128C2×Q8—1011no—
129C2×D4—1011no—
130C2×D4—2021no—
131C22×C4—1011no—
132C22×C4—1011no—
133C22×C4—2021no—
134C22×C4—2021no—
135C22×C4—2021no—
136C22×C4C2×#8793811yescharacteristic
137M16—4042no—
138M16—2024no—
139C2×C8—2022no—
140C2×C8—2022no—
141C4⋊C4—2021no—
142C4⋊C4—2021no—
143C4⋊C4C2×#8793811yescharacteristic
144C4⋊C4C2×#8793811yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—2021no—
147C22⋊C4—2021no—
148C42—21no—
149C42—1011no—
150C42—1011no—
151C42—1011no—
152C42—1011no—
153C42—2021no—
154C42—2021no—
155C23—2021no—
156C23C22⋊#8793811yescharacteristic
157Q8—1012no—
158Q8—2021no—
159D4—1012no—
160D4—2024no—
161D4—2021no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—1011no—
167C2×C4—1011no—
168C2×C4—1011no—
169C2×C4—1011no—
170C2×C4—4041no—
171C2×C4—4041no—
172C2×C4—4042no—
173C2×C4—2024no—
174C2×C4—2021no—
175C2×C4—2022no—
176C2×C4—2021no—
177C2×C4—2021no—
178C2×C4C22⋊#8793811yescharacteristic
179C2×C4C22×#8793811yescharacteristic
180C8—4042no—
181C8—2024no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—4041no—
186C22—4041no—
187C22—2022no—
188C22C2×C22⋊#8793811yescharacteristic
189C4—21no—
190C4—41no—
191C4—41no—
192C4—1011no—
193C4—1011no—
194C4—1012no—
195C4—2022no—
196C4—2021no—
197C4—2021no—
198C4D101⋊M1611yescharacteristic
199C4D101⋊M1611yescharacteristic
200C2—41no—
201C2—4042no—
202C2#2026614911yescharacteristic
203C2#2026616811yescharacteristic
204C2D101⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#117×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC2×#13108074C4
cosocleC4×Dic101C23
hypercenter#117#87938
hypocenterC101#1489
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #20266180C2semidirect
2. C4×#13108083C2semidirect
3. #20266027C2semidirect
4. #20266025C2semidirect
70. D404M16semidirect
76. Dic202M16semidirect
103. C101#1489semidirect

Maximal subgroups

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

subgroupindex
#202661802
C4×#131080832
#202660272
#202660252
#202656452
#1489101