Finite Group Databasebeta

← #27169761

Subgroups — #27169761

Order 12928.

200 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 65 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#20265711C211yesmaximal, characteristic
2#20265234C211yesmaximal, characteristic, complemented
3#20264918C212yesmaximal, complemented
4#20264916C212yesmaximal, complemented
5#432×C101C211yesmaximal, characteristic, complemented
6#15692249—21no—
7#15692247C2211yescharacteristic
8#15692243C2211yescharacteristic
9#15692238—21no—
10#15692234C411yescharacteristic
11#15692229C411yescharacteristic
12C4×Dic202C411yescharacteristic
13C4×D404C411yescharacteristic
14C4×C22⋊D101—21no—
15#15692212—21no—
16C4⋊#11717579—21no—
17C22⋊#11717579—21no—
18#15692166C2212yes—
19C2×C42×C101C2211yescharacteristic
20C4⋊C8×C101—21no—
21C22⋊C8×C101—21no—
22#97×C101C2212yes—
23C2×Dic202C2×C411yescharacteristic
24C2×D404C2×C411yescharacteristic
25C2×C22⋊D101—21no—
26C2×C4×D101—21no—
27C2×#11717579—22no—
28C4⋊Dic101C2×C411yescharacteristic
29C4⋊Dic101C2×C411yescharacteristic
30#13108053—21no—
31#13108053—21no—
32#13108053—21no—
33#13108053—21no—
34C22⋊Dic101—21no—
35#13108051—21no—
36#13108051—21no—
37#13108051—21no—
38#13108051—21no—
39C4×Dic101—21no—
40C22×C4×C101C2×C411yescharacteristic
41C22×C4×C101D411yescharacteristic
42C22×C4×C101D411yescharacteristic
43C2×C8×C101—22no—
44C42×C101C2×C411yescharacteristic
45C42×C101D411yescharacteristic
46C42×C101D411yescharacteristic
47C42×C101C2311yescharacteristic
48C23×C101C22⋊C411yescharacteristic
49C22×D101—21no—
50C2×C4×C101—41no—
51C2×C4×C101—41no—
52C2×C4×C101—41no—
53C2×C4×C101C22⋊C411yescharacteristic
54C2×C4×C101C22⋊C411yescharacteristic
55C2×C4×C101C22⋊C411yescharacteristic
56C2×C4×C101C22×C411yescharacteristic
57C2×C4×C101C2×D411yescharacteristic
58C2×C4×C101C2×D411yescharacteristic
59C22⋊D101—42no—
60C2×Dic101—21no—
61C2×Dic101—21no—
62C2×Dic101—21no—
63D404—41no—
64C4×D101—42no—
65Dic202—41no—
66C8×C101—42no—
67#11717579—42no—
68C22×C101—21no—
69C22×C101—21no—
70C22×C101—21no—
71C22×C101C2×C22⋊C411yescharacteristic
72C2×D101—41no—
73C2×D101—41no—
74C2×D101—41no—
75C4×C101—21no—
76C4×C101—21no—
77C4×C101—21no—
78C4×C101—21no—
79C4×C101—21no—
80C4×C101—21no—
81Dic101—41no—
82Dic101—41no—
83Dic101—41no—
84C2×C101—41no—
85C2×C101#41311yescharacteristic
86C2×C101#42111yescharacteristic
87C2×C101#42111yescharacteristic
88D101—81no—
89#1535—1011nomaximal, Sylow
90C101#153511yescharacteristic, complemented, Sylow
91#529—1011no—
92#432—1011no—
93#432D10111yescharacteristic, complemented
94#330—1012no—
95#329—1012no—
96#126—2021no—
97#125—1011no—
98#124—1011no—
99#123—2021no—
100C22⋊Q8—1011no—
101#121—1011no—
102C4×Q8—1011no—
103C4×D4—1011no—
104C4×D4—2021no—
105#117—2021no—
106C2×C42C2×D10111yescharacteristic
107C4⋊C8—21no—
108C4⋊C8—2021no—
109C22⋊C8—21no—
110C22⋊C8—2021no—
111#97—1012no—
112#97C2×D10112yes—
113C2×Q8—1011no—
114C2×D4—1011no—
115C2×D4—2021no—
116C22×C4—2021no—
117C22×C4C4×D10111yescharacteristic
118C22×C4D40411yescharacteristic
119C22×C4C22⋊D10111yescharacteristic
120C2×C8—22no—
121C2×C8—2022no—
122C4⋊C4—1011no—
123C4⋊C4—1011no—
124C4⋊C4—2021no—
125C4⋊C4—2021no—
126C4⋊C4—2021no—
127C4⋊C4—2021no—
128C22⋊C4—2021no—
129C22⋊C4—2021no—
130C22⋊C4—2021no—
131C22⋊C4—2021no—
132C22⋊C4—2021no—
133C42—2021no—
134C42C4×D10111yescharacteristic
135C42D40411yescharacteristic
136C42C22⋊D10111yescharacteristic
137C42C22×D10111yescharacteristic
138C23—2021no—
139C23#1310805111yescharacteristic
140Q8—4041no—
141D4—4041no—
142D4—4042no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—4042no—
147C2×C4—2021no—
148C2×C4—2021no—
149C2×C4—2021no—
150C2×C4#1310805111yescharacteristic
151C2×C4#1310805111yescharacteristic
152C2×C4#1310805111yescharacteristic
153C2×C4C2×C4×D10111yescharacteristic
154C2×C4C2×C22⋊D10111yescharacteristic
155C2×C4C2×D40411yescharacteristic
156C8—42no—
157C8—4042no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—4041no—
162C22—4041no—
163C22—4041no—
164C22C2×#1310805111yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—4041no—
172C4—4041no—
173C4—4041no—
174C2—41no—
175C2—8081no—
176C2#2026515911yescharacteristic
177C2#2026519311yescharacteristic
178C2#2026519511yescharacteristic
subgroupG/H
centerC22C2×#13108051
commutatorC2×C4×C101C22×C4
FrattiniC42C22×D101
Fitting#432×C101C2
socleC22×C101C2×C22⋊C4
prime radical#15692229C4
cosocleC42×C101C23
hypercenter#432D101
hypocenterC101#1535
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. #20265234C2semidirect
3. #20264918C2semidirect
4. #20264916C2semidirect
5. #432×C101C2semidirect
90. C101#1535semidirect
93. #432D101semidirect

Maximal subgroups

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

subgroupindex
#202657112
#202652342
#202649182
#202649162
#432×C1012
#1535101