Finite Group Databasebeta

← #27170201

Subgroups — #27170201

Order 12928.

236 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 55 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#20265888C211yesmaximal, characteristic, complemented
2#20265374C211yesmaximal, characteristic, complemented
3#20265369C211yesmaximal, characteristic
4#20264926C211yesmaximal, characteristic, complemented
5#20264909C211yesmaximal, characteristic, complemented
6#20264906C211yesmaximal, characteristic, complemented
7#492×C101C211yesmaximal, characteristic, complemented
8C2×#13108062—21no—
9C2×#13108059—21no—
10#15692246—21no—
11#15692235—21no—
12#15692231—22no—
13C4×Dic202C2211yescharacteristic
14C4×D404C2211yescharacteristic
15#15692217—21no—
16C4⋊C4×D101—21no—
17C4⋊#11717579C2211yescharacteristic
18C4⋊#11717579C2211yescharacteristic
19#15692176—21no—
20#15692176—21no—
21C2×SD16×C101—22no—
22C4⋊Q8×C101C2211yescharacteristic
23#127×C101C2211yescharacteristic
24C4×C8×C101C2211yescharacteristic
25C2×Dic202D411yescharacteristic
26C2×D404D411yescharacteristic
27C2×C4×D101—21no—
28C2×C4×D101—22no—
29#13108062—42no—
30#13108059—42no—
31C2×#11717579—21no—
32C2×#11717579—21no—
33C4⋊Dic101—22no—
34C4⋊Dic101D411yescharacteristic
35C4⋊Dic101D411yescharacteristic
36#13108053—22no—
37#13108053—21no—
38#13108051—22no—
39#13108051—21no—
40C4×Dic101—21no—
41C2×Q8×C101D412yes—
42C2×D4×C101—21no—
43C2×D4×C101—21no—
44C2×D4×C101—21no—
45SD16×C101—44no—
46C2×C8×C101—21no—
47C4⋊C4×C101—21no—
48C4⋊C4×C101—21no—
49C42×C101C2311yescharacteristic
50C23×C101—41no—
51C22×D101—21no—
52Q8×C101—42no—
53D4×C101—42no—
54D4×C101—41no—
55D4×C101—41no—
56D4×C101—41no—
57D4×C101—41no—
58C2×C4×C101—22no—
59C2×C4×C101C2×D411yescharacteristic
60C2×C4×C101C2×D411yescharacteristic
61C2×C4×C101C2×D411yescharacteristic
62C2×Dic101—21no—
63C2×Dic101—21no—
64C2×Dic101—21no—
65D404—21no—
66D404SD1612yescomplemented
67C4×D101—41no—
68C4×D101—41no—
69C4×D101—82no—
70Dic202—21no—
71Dic202—21no—
72C8×C101—22no—
73#11717579—41no—
74#11717579—41no—
75C22×C101—81no—
76C22×C101—81no—
77C22×C101—81no—
78C22×C101#12011yescharacteristic
79C2×D101—22no—
80C2×D101—41no—
81C2×D101—41no—
82C4×C101—21no—
83C4×C101—42no—
84C4×C101C2×SD1611yescharacteristic
85C4×C101#13511yescharacteristic
86C4×C101AGL(1,ℤ/8)11yescharacteristic
87C4×C101#13711yescharacteristic
88Dic101—41no—
89Dic101—41no—
90Dic101—41no—
91C2×C101—161no—
92C2×C101#44811yescharacteristic
93C2×C101#44911yescharacteristic
94C2×C101#45311yescharacteristic
95D101—42no—
96#1622—1011nomaximal, Sylow
97C101#162211yescharacteristic, complemented, Sylow
98#554—1011no—
99#492D10111yescharacteristic, complemented
100#460—1011no—
101#459—1011no—
102#333—1011no—
103Q8⋊C8—1011no—
104D4⋊C8—1011no—
105C2×SD16—22no—
106C2×SD16—2021no—
107C2×D8—2021no—
108C4⋊Q8C2×D10111yescharacteristic
109#127C2×D10111yescharacteristic
110#125—2021no—
111C22⋊Q8—2021no—
112C22⋊Q8—2022no—
113C4×Q8—1011no—
114C4×D4—1011no—
115C4×D4—2021no—
116C2×C4⋊C4—2021no—
117C4⋊C8—1011no—
118C4⋊C8—1011no—
119#102—2021no—
120#102—2021no—
121C4×C8C2×D10111yescharacteristic
122C2×Q8—1011no—
123C2×Q8C22⋊D10112yes—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—1011no—
128C22×C4—2022no—
129C22×C4—2021no—
130SD16—44no—
131SD16—4042no—
132D8—4042no—
133C2×C8—21no—
134C2×C8—2021no—
135C2×C8—2021no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—1011no—
139C4⋊C4—1011no—
140C4⋊C4—2022no—
141C4⋊C4—2022no—
142C4⋊C4—2021no—
143C22⋊C4—2022no—
144C22⋊C4—2021no—
145C42—2021no—
146C42C22×D10111yescharacteristic
147C23—41no—
148C23—2021no—
149Q8—42no—
150Q8—2021no—
151Q8—2021no—
152D4—42no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—1012no—
158D4—2021no—
159C2×C4—22no—
160C2×C4—8082no—
161C2×C4—4041no—
162C2×C4—4041no—
163C2×C4—2021no—
164C2×C4—2021no—
165C2×C4—2021no—
166C2×C4D4×D10111yescharacteristic
167C2×C4D4×D10111yescharacteristic
168C2×C4C2×C22⋊D10111yescharacteristic
169C8—22no—
170C8—4041no—
171C8—4041no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—4041no—
176C22—4041no—
177C22—2022no—
178C22#1569222311yescharacteristic
179C4—21no—
180C4—42no—
181C4—4041no—
182C4—4041no—
183C4—4041no—
184C4SD16×D10111yescharacteristic
185C4#1569227611yescharacteristic
186C4#1569228211yescharacteristic
187C4#1569228411yescharacteristic
188C2—161no—
189C2—4042no—
190C2#2026531411yescharacteristic
191C2#2026532111yescharacteristic
192C2#2026534411yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#492×C101C2
socleC22×C101#120
prime radical#20265369C2
hypercenter#492D101
hypocenterC101#1622
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #20265888C2semidirect
2. #20265374C2semidirect
4. #20264926C2semidirect
5. #20264909C2semidirect
6. #20264906C2semidirect
7. #492×C101C2semidirect
66. D404SD16semidirect
97. C101#1622semidirect
99. #492D101semidirect

Maximal subgroups

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

subgroupindex
#202658882
#202653742
#202653692
#202649262
#202649092
#202649062
#492×C1012
#1622101