Finite Group Databasebeta

← #27170294

Subgroups — #27170294

Order 12928.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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#20265842C211yesmaximal, characteristic
2#20265469C211yesmaximal, characteristic, complemented
3#20265464C211yesmaximal, characteristic
4#20264923C211yesmaximal, characteristic, complemented
5#20264909C211yesmaximal, characteristic, complemented
6#20264906C211yesmaximal, characteristic
7#487×C101C211yesmaximal, characteristic, complemented
8C2×Q8×D101—21no—
9#15692254—21no—
10#15692233—22no—
11#15692231—21no—
12C4×Dic202C2211yescharacteristic
13C4×D404C2211yescharacteristic
14#15692217—21no—
15#15692188—21no—
16#15692186—21no—
17C4⋊#11717579C2211yescharacteristic
18C4⋊#11717579C2211yescharacteristic
19#15692178—21no—
20#15692175—21no—
21C4⋊Q8×C101C2211yescharacteristic
22C4⋊Q8×C101C2211yescharacteristic
23#103×C101—22no—
24C4×C8×C101C2211yescharacteristic
25C2×Dic202—22no—
26C2×Dic202D411yescharacteristic
27Q8×D101—42no—
28Q8×D101—42no—
29C2×D404D411yescharacteristic
30C2×C4×D101—22no—
31C2×C4×D101—21no—
32C2×#11717579—21no—
33C2×#11717579—21no—
34C4⋊Dic101D411yescharacteristic
35C4⋊Dic101D411yescharacteristic
36#13108053—22no—
37#13108053—21no—
38#13108051—22no—
39#13108051—21no—
40C4×Dic101—21no—
41C2×Q8×C101—21no—
42C2×Q8×C101—21no—
43C2×C8×C101—21no—
44C4⋊C4×C101—21no—
45C4⋊C4×C101—21no—
46C4⋊C4×C101—21no—
47C4⋊C4×C101D412yes—
48C42×C101C2311yescharacteristic
49C22×D101—21no—
50Q8×C101—42no—
51Q8×C101—41no—
52Q8×C101—41no—
53C2×C4×C101—22no—
54C2×C4×C101—41no—
55C2×C4×C101C2×D411yescharacteristic
56C2×C4×C101C2×D411yescharacteristic
57C2×C4×C101C2×D411yescharacteristic
58C2×Dic101—21no—
59C2×Dic101—21no—
60C2×Dic101—21no—
61D404—21no—
62D404Q1612yes—
63C4×D101—44no—
64C4×D101—41no—
65C4×D101—41no—
66Dic202—21no—
67Dic202—44no—
68Dic202SD1612yes—
69C8×C101—41no—
70#11717579—41no—
71#11717579—41no—
72C22×C101#12011yescharacteristic
73C2×D101—22no—
74C2×D101—41no—
75C2×D101—41no—
76C4×C101—21no—
77C4×C101—42no—
78C4×C101—81no—
79C4×C101C2×SD1611yescharacteristic
80C4×C101C2×Q1611yescharacteristic
81C4×C101AGL(1,ℤ/8)11yescharacteristic
82C4×C101#13711yescharacteristic
83Dic101—22no—
84Dic101—41no—
85Dic101—41no—
86C2×C101#45011yescharacteristic
87C2×C101#45111yescharacteristic
88C2×C101#45411yescharacteristic
89D101—42no—
90#1637—1011nomaximal, Sylow
91C101#163711yescharacteristic, complemented, Sylow
92D4×Q8—1011no—
93#487D10111yescharacteristic, complemented
94#477—1011no—
95#476—1011no—
96#332—1011no—
97Q8⋊C8—1011no—
98D4⋊C8—1011no—
99C22×Q8—2021no—
100C4⋊Q8—2021no—
101C4⋊Q8C2×D10111yescharacteristic
102C4⋊Q8C2×D10111yescharacteristic
103C22⋊Q8—2021no—
104C22⋊Q8—2022no—
105C4×Q8—1011no—
106C4×D4—1011no—
107C4×D4—2021no—
108#107—2021no—
109#106—2021no—
110C4⋊C8—1011no—
111C4⋊C8—1011no—
112#103—22no—
113#103—2021no—
114#102—2021no—
115C4×C8C2×D10111yescharacteristic
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8—1011no—
119C2×Q8—4042no—
120C2×Q8—4042no—
121C2×Q8—2022no—
122C2×D4—1011no—
123C22×C4—2022no—
124C22×C4—2021no—
125C2×C8—21no—
126C2×C8—2021no—
127C2×C8—2021no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—1011no—
132C4⋊C4—1011no—
133C4⋊C4—2022no—
134C4⋊C4—2021no—
135C4⋊C4C22⋊D10112yes—
136C22⋊C4—2022no—
137C22⋊C4—2021no—
138C42—2021no—
139C42C22×D10111yescharacteristic
140C23—2021no—
141Q8—42no—
142Q8—41no—
143Q8—41no—
144Q8—1012no—
145Q8—4044no—
146Q8—2021no—
147D4—1012no—
148D4—2021no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—4044no—
152C2×C4—4041no—
153C2×C4—4041no—
154C2×C4—2021no—
155C2×C4—2021no—
156C2×C4—2021no—
157C2×C4D4×D10111yescharacteristic
158C2×C4D4×D10111yescharacteristic
159C2×C4C2×C22⋊D10111yescharacteristic
160C8—41no—
161C8—4041no—
162C8—4041no—
163C22—4041no—
164C22—4041no—
165C22—2022no—
166C22#1569222311yescharacteristic
167C4—21no—
168C4—42no—
169C4—81no—
170C4—4041no—
171C4—4041no—
172C4—2022no—
173C4SD16×D10111yescharacteristic
174C4Q16×D10111yescharacteristic
175C4#1569228211yescharacteristic
176C4#1569228711yescharacteristic
177C2—4042no—
178C2#2026532611yescharacteristic
179C2#2026533111yescharacteristic
180C2#2026534911yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#487×C101C2
socleC22×C101#120
prime radicalC2×D404D4
hypercenter#487D101
hypocenterC101#1637
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. #20265469C2semidirect
4. #20264923C2semidirect
5. #20264909C2semidirect
7. #487×C101C2semidirect
91. C101#1637semidirect
93. #487D101semidirect

Maximal subgroups

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

subgroupindex
#202658422
#202654692
#202654642
#202649232
#202649092
#202649062
#487×C1012
#1637101