Finite Group Databasebeta

← #27170315

Subgroups — #27170315

Order 12928.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 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#20265913C211yesmaximal, characteristic
2#20265459C211yesmaximal, characteristic, complemented
3#20265384C211yesmaximal, characteristic, complemented
4#20264921C211yesmaximal, characteristic
5#20264909C211yesmaximal, characteristic, complemented
6#20264909C211yesmaximal, characteristic, complemented
7#486×C101C211yesmaximal, characteristic
8C2×#13108064—21no—
9#15692242—21no—
10#15692242—21no—
11#15692236—22no—
12C4×Dic202C2211yescharacteristic
13C4×Dic202C2211yescharacteristic
14D4×Dic101—21no—
15D4×Dic101—21no—
16#15692186—21no—
17C4⋊#11717579C2211yescharacteristic
18C4⋊#11717579C2211yescharacteristic
19#15692179—21no—
20#15692178—21no—
21C4⋊Q8×C101C2211yescharacteristic
22#127×C101C2211yescharacteristic
23#102×C101—22no—
24C4×C8×C101C2211yescharacteristic
25C2×Dic202D411yescharacteristic
26C2×Dic202D411yescharacteristic
27C22×Dic101—22no—
28#13108064—42no—
29C2×#11717579—21no—
30C2×#11717579—21no—
31C4⋊Dic101D411yescharacteristic
32C4⋊Dic101D411yescharacteristic
33#13108053—21no—
34#13108053—21no—
35C22⋊Dic101—22no—
36C22⋊Dic101—22no—
37C22⋊Dic101—22no—
38C4×Dic101—21no—
39C4×Dic101—21no—
40C2×Q8×C101—21no—
41C2×D4×C101—21no—
42C2×D4×C101—21no—
43C2×D4×C101D412yes—
44C2×C8×C101—21no—
45C4⋊C4×C101—21no—
46C4⋊C4×C101—21no—
47C42×C101C2311yescharacteristic
48C23×C101—22no—
49Q8×C101—41no—
50Q8×C101—41no—
51D4×C101—42no—
52D4×C101—41no—
53D4×C101—41no—
54D4×C101—41no—
55D4×C101—41no—
56C2×C4×C101—41no—
57C2×C4×C101C2×D411yescharacteristic
58C2×C4×C101C2×D411yescharacteristic
59C2×C4×C101C2×D411yescharacteristic
60C2×Dic101—21no—
61C2×Dic101—21no—
62C2×Dic101—21no—
63C2×Dic101—21no—
64C2×Dic101—82no—
65Dic202—21no—
66Dic202—21no—
67Dic202—21no—
68Dic202—21no—
69C8×C101—41no—
70#11717579—41no—
71#11717579—41no—
72C22×C101—42no—
73C22×C101—42no—
74C22×C101—42no—
75C22×C101#12011yescharacteristic
76C4×C101—21no—
77C4×C101—81no—
78C4×C101#13511yescharacteristic
79C4×C101#13511yescharacteristic
80C4×C101AGL(1,ℤ/8)11yescharacteristic
81C4×C101#13711yescharacteristic
82Dic101—41no—
83Dic101—41no—
84Dic101—41no—
85Dic101—41no—
86C2×C101—82no—
87C2×C101#44911yescharacteristic
88C2×C101#45211yescharacteristic
89C2×C101#45511yescharacteristic
90#1640—1011nomaximal, Sylow
91C101#164011yescharacteristic, complemented, Sylow
92#559—1011no—
93#486D10111yescharacteristic
94#475—1011no—
95#462—1011no—
96#331—1011no—
97Q8⋊C8—1011no—
98Q8⋊C8—1011no—
99C2×Q16—2021no—
100C4⋊Q8C2×D10111yescharacteristic
101#127C2×D10111yescharacteristic
102#124—2021no—
103#124—2021no—
104#123—2022no—
105C4×Q8—1011no—
106C4×Q8—1011no—
107C4×D4—2021no—
108C4×D4—2021no—
109#106—2021no—
110C4⋊C8—1011no—
111C4⋊C8—1011no—
112#103—2021no—
113#103—2021no—
114#102—22no—
115C4×C8C2×D10111yescharacteristic
116C2×Q8—21no—
117C2×Q8—1011no—
118C2×Q8—1011no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4C22⋊D10112yes—
122C22×C4—2022no—
123Q16—4042no—
124C2×C8—21no—
125C2×C8—2021no—
126C2×C8—2021no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—1011no—
130C4⋊C4—1011no—
131C4⋊C4—2021no—
132C4⋊C4—2021no—
133C22⋊C4—2022no—
134C22⋊C4—2022no—
135C22⋊C4—2022no—
136C42—2021no—
137C42—2021no—
138C42C22×D10111yescharacteristic
139C23—22no—
140Q8—41no—
141Q8—41no—
142Q8—2021no—
143Q8—2021no—
144Q8—2021no—
145Q8—2021no—
146D4—42no—
147D4—41no—
148D4—41no—
149D4—41no—
150D4—41no—
151C2×C4—41no—
152C2×C4—8082no—
153C2×C4—2021no—
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—42no—
164C22—42no—
165C22—42no—
166C22#1569222311yescharacteristic
167C4—21no—
168C4—81no—
169C4—4041no—
170C4—4041no—
171C4—4041no—
172C4—4041no—
173C4#1569227411yescharacteristic
174C4#1569227611yescharacteristic
175C4#1569227911yescharacteristic
176C4#1569228411yescharacteristic
177C2—82no—
178C2#2026531911yescharacteristic
179C2#2026533611yescharacteristic
180C2#2026535411yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#486×C101C2
socleC22×C101#120
prime radical#127×C101C22
hypercenter#486D101
hypocenterC101#1640
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. #20265459C2semidirect
3. #20265384C2semidirect
5. #20264909C2semidirect
6. #20264909C2semidirect
91. C101#1640semidirect

Maximal subgroups

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

subgroupindex
#202659132
#202654592
#202653842
#202649212
#202649092
#486×C1012
#1640101