Finite Group Databasebeta

← #27170338

Subgroups — #27170338

Order 12928.

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

normal subgroups only

This group has 208 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#20265887C211yesmaximal, characteristic
2#20265529C211yesmaximal, characteristic, complemented
3#20265379C211yesmaximal, characteristic
4#20264925C211yesmaximal, characteristic, complemented
5Q8⋊#11717579C211yesmaximal, characteristic, complemented
6#20264904C211yesmaximal, characteristic
7#475×C101C211yesmaximal, characteristic, complemented
8C2×#13108061—21no—
9#15692255C2211yescharacteristic
10#15692247—21no—
11#15692233—22no—
12#15692231—21no—
13C4×D404—21no—
14C4×D404C2211yescharacteristic
15C4⋊C4×D101—21no—
16C4⋊#11717579C2211yescharacteristic
17#15692179—21no—
18#15692178—22no—
19C4×#11717579C2211yescharacteristic
20C4⋊Q8×C101C2211yescharacteristic
21C4×Q8×C101C2211yescharacteristic
22#106×C101—21no—
23C4⋊C8×C101C2211yescharacteristic
24#103×C101—21no—
25C2×Dic202D412yes—
26C2×D404D411yescharacteristic
27C2×C4×D101—21no—
28C2×C4×D101—22no—
29#13108061—42no—
30C2×#11717579—21no—
31C2×#11717579—21no—
32C4⋊Dic101—21no—
33C4⋊Dic101—21no—
34C4⋊Dic101D411yescharacteristic
35#13108053—22no—
36#13108053—22no—
37#13108051—22no—
38#13108051—21no—
39C2×Q8×C101—21no—
40C2×Q8×C101D411yescharacteristic
41C2×C8×C101—21no—
42C4⋊C4×C101—21no—
43C4⋊C4×C101—21no—
44C4⋊C4×C101—21no—
45C4⋊C4×C101D411yescharacteristic
46C42×C101—21no—
47C42×C101C2311yescharacteristic
48C22×D101—21no—
49Q8×C101—21no—
50Q8×C101—21no—
51Q8×C101—41no—
52Q8×C101—41no—
53C2×C4×C101—21no—
54C2×C4×C101—21no—
55C2×C4×C101—41no—
56C2×C4×C101C2×D411yescharacteristic
57C2×C4×C101C2×D411yescharacteristic
58C2×C4×C101C2×D411yescharacteristic
59C2×Dic101—21no—
60C2×Dic101—22no—
61D404—21no—
62D404SD1612yes—
63C4×D101—41no—
64C4×D101—41no—
65C4×D101—82no—
66Dic202—42no—
67C8×C101—41no—
68#11717579—41no—
69#11717579—41no—
70C22×C101#12011yescharacteristic
71C2×D101—22no—
72C2×D101—41no—
73C2×D101—41no—
74C4×C101—21no—
75C4×C101—41no—
76C4×C101—41no—
77C4×C101—81no—
78C4×C101C2×SD1611yescharacteristic
79C4×C101#13511yescharacteristic
80C4×C101#13711yescharacteristic
81C4×C101#13711yescharacteristic
82Dic101—41no—
83Dic101—42no—
84C2×C101#44811yescharacteristic
85C2×C101#45211yescharacteristic
86C2×C101#45511yescharacteristic
87D101—42no—
88#1643—1011nomaximal, Sylow
89C101#164311yescharacteristic, complemented, Sylow
90#554—1011no—
91#487—1011no—
92#475D10111yescharacteristic, complemented
93#461—1011no—
94#333—1011no—
95Q8⋊C8—1011no—
96D4⋊C8—1011no—
97C2×SD16—2021no—
98C4⋊Q8—1011no—
99C4⋊Q8C2×D10111yescharacteristic
100#125—2021no—
101C22⋊Q8—2022no—
102C22⋊Q8—2021no—
103C4×Q8C2×D10111yescharacteristic
104C4×D4—1011no—
105C4×D4—2021no—
106C2×C4⋊C4—2021no—
107#106—21no—
108C4⋊C8—1011no—
109C4⋊C8C2×D10111yescharacteristic
110#103—21no—
111#103—2022no—
112#103—2021no—
113C4×C8—1011no—
114C2×Q8—21no—
115C2×Q8—1012no—
116C2×Q8C22⋊D10111yescharacteristic
117C2×D4—1011no—
118C22×C4—2022no—
119C22×C4—2021no—
120SD16—4042no—
121C2×C8—21no—
122C2×C8—2021no—
123C2×C8—2021no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—1011no—
128C4⋊C4—2022no—
129C4⋊C4—2022no—
130C4⋊C4—2021no—
131C4⋊C4—2021no—
132C4⋊C4C22⋊D10111yescharacteristic
133C22⋊C4—2022no—
134C22⋊C4—2021no—
135C42—21no—
136C42C22×D10111yescharacteristic
137C23—2021no—
138Q8—21no—
139Q8—21no—
140Q8—41no—
141Q8—41no—
142Q8—4042no—
143D4—1012no—
144D4—2021no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—8082no—
149C2×C4—4041no—
150C2×C4—4041no—
151C2×C4—2021no—
152C2×C4—2022no—
153C2×C4D4×D10111yescharacteristic
154C2×C4D4×D10111yescharacteristic
155C2×C4C2×C22⋊D10111yescharacteristic
156C8—41no—
157C8—4041no—
158C8—4041no—
159C22—4041no—
160C22—4041no—
161C22—2022no—
162C22#1569222311yescharacteristic
163C4—21no—
164C4—41no—
165C4—41no—
166C4—81no—
167C4—4041no—
168C4—4042no—
169C4SD16×D10111yescharacteristic
170C4#1569227511yescharacteristic
171C4#1569228411yescharacteristic
172C4#1569228611yescharacteristic
173C2—4042no—
174C2#2026531411yescharacteristic
175C2#2026533711yescharacteristic
176C2#2026535311yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#475×C101C2
socleC22×C101#120
prime radicalC2×D404D4
hypercenter#475D101
hypocenterC101#1643
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #20265529C2semidirect
4. #20264925C2semidirect
5. Q8⋊#11717579C2semidirect
7. #475×C101C2semidirect
89. C101#1643semidirect
92. #475D101semidirect

Maximal subgroups

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

subgroupindex
#202658872
#202655292
#202653792
#202649252
Q8⋊#117175792
#202649042
#475×C1012
#1643101