Finite Group Databasebeta

← #27170468

Subgroups — #27170468

Order 12928.

184 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 45 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#20265931C211yesmaximal, characteristic
2#20265587C211yesmaximal, characteristic, complemented
3#20265546C211yesmaximal, characteristic, complemented
4#20265537C211yesmaximal, characteristic
5#20264918C211yesmaximal, characteristic, complemented
6#20264916C211yesmaximal, characteristic
7#330×C101C211yesmaximal, characteristic, complemented
8#15692255C2211yescharacteristic
9#15692254—21no—
10#15692250—21no—
11#15692250—21no—
12#15692247C2211yescharacteristic
13#15692243C2211yescharacteristic
14#15692240—21no—
15#15692235—21no—
16#15692233—21no—
17#15692188—21no—
18#15692186—21no—
19#15692180—21no—
20#15692180—21no—
21#15692180—21no—
22#15692177—21no—
23#15692166C2211yescharacteristic
24#125×C101C2211yescharacteristic
25#97×C101C2211yescharacteristic
26#97×C101C2211yescharacteristic
27C2×Dic202—21no—
28C2×Dic202—21no—
29C2×Dic202D411yescharacteristic
30C2×D404D411yescharacteristic
31C2×C4×D101—21no—
32C2×#11717579—21no—
33C4⋊Dic101—21no—
34C4⋊Dic101—21no—
35C4⋊Dic101—21no—
36C4⋊Dic101D411yescharacteristic
37C4⋊Dic101D411yescharacteristic
38#13108053—21no—
39#13108053—21no—
40#13108053—21no—
41#13108051—21no—
42#13108051—21no—
43#13108051—21no—
44#13108051—21no—
45#13108051—21no—
46C4×Dic101—21no—
47C2×C8×C101—21no—
48C2×C8×C101—21no—
49C4⋊C4×C101—21no—
50C4⋊C4×C101—21no—
51C4⋊C4×C101D411yescharacteristic
52C4⋊C4×C101D411yescharacteristic
53C42×C101C2311yescharacteristic
54C22×D101—21no—
55C2×C4×C101—21no—
56C2×C4×C101—21no—
57C2×C4×C101C2×D411yescharacteristic
58C2×C4×C101C2×D411yescharacteristic
59C2×C4×C101C2×D411yescharacteristic
60C2×Dic101—21no—
61C2×Dic101—21no—
62C2×Dic101—21no—
63C2×Dic101—41no—
64D404—41no—
65C4×D101—81no—
66Dic202—41no—
67Dic202—42no—
68Dic202—81no—
69C8×C101—41no—
70C8×C101—41no—
71#11717579—22no—
72C22×C101#12011yescharacteristic
73C2×D101—41no—
74C2×D101—41no—
75C2×D101—41no—
76C4×C101—21no—
77C4×C101—21no—
78C4×C101—21no—
79C4×C101—41no—
80C4×C101—41no—
81Dic101—41no—
82Dic101—41no—
83Dic101—41no—
84Dic101—81no—
85C2×C101#45411yescharacteristic
86C2×C101#45511yescharacteristic
87C2×C101#45611yescharacteristic
88D101—81no—
89#1664—1011nomaximal, Sylow
90C101#166411yescharacteristic, complemented, Sylow
91#563—1011no—
92#501—1011no—
93#491—1011no—
94#489—1011no—
95#330—1011no—
96#330D10111yescharacteristic, complemented
97#329—1011no—
98C4⋊Q8—1011no—
99C4⋊Q8—2021no—
100#126—2021no—
101#126—2021no—
102#125—1011no—
103#125C2×D10111yescharacteristic
104#124—1011no—
105#123—2021no—
106C22⋊Q8—2021no—
107C22⋊Q8—2021no—
108#107—2021no—
109#106—2021no—
110#103—2021no—
111#103—2021no—
112#103—2021no—
113#102—2021no—
114#97—1011no—
115#97C2×D10111yescharacteristic
116#97C2×D10111yescharacteristic
117C2×Q8—1011no—
118C2×Q8—2021no—
119C2×Q8—2021no—
120C2×D4—1011no—
121C22×C4—2021no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—2021no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—1011no—
128C4⋊C4—1011no—
129C4⋊C4—2021no—
130C4⋊C4—2021no—
131C4⋊C4—2021no—
132C4⋊C4—2021no—
133C4⋊C4—2021no—
134C4⋊C4—2021no—
135C4⋊C4D40411yescharacteristic
136C4⋊C4D40411yescharacteristic
137C22⋊C4—2021no—
138C22⋊C4—2021no—
139C22⋊C4—2021no—
140C22⋊C4—2021no—
141C22⋊C4—2021no—
142C42—2021no—
143C42C22×D10111yescharacteristic
144C23—2021no—
145Q8—8081no—
146Q8—4042no—
147Q8—4041no—
148D4—4041no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—8081no—
152C2×C4—4041no—
153C2×C4—2021no—
154C2×C4—2021no—
155C2×C4—2021no—
156C2×C4D4×D10111yescharacteristic
157C2×C4D4×D10111yescharacteristic
158C2×C4C2×D40411yescharacteristic
159C8—41no—
160C8—41no—
161C8—2022no—
162C22—4041no—
163C22—4041no—
164C22—4041no—
165C22#1569222511yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—8081no—
172C4—4041no—
173C4—4041no—
174C4—4041no—
175C2—8081no—
176C2#2026535111yescharacteristic
177C2#2026535611yescharacteristic
178C2#2026536211yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#330×C101C2
socleC22×C101#120
prime radicalC2×D404D4
hypercenter#330D101
hypocenterC101#1664
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #20265587C2semidirect
3. #20265546C2semidirect
5. #20264918C2semidirect
7. #330×C101C2semidirect
90. C101#1664semidirect
96. #330D101semidirect

Maximal subgroups

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

subgroupindex
#202659312
#202655872
#202655462
#202655372
#202649182
#202649162
#330×C1012
#1664101