Finite Group Databasebeta

← #27170355

Subgroups — #27170355

Order 12928.

200 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 59 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
1Q8×Dic202C211yesmaximal, characteristic
2#20265528C211yesmaximal, characteristic
3#20265463C211yesmaximal, characteristic
4#20265387C211yesmaximal, characteristic
5#20264925C211yesmaximal, characteristic
6#20264909C211yesmaximal, characteristic
7Q8⋊C8×C101C211yesmaximal, characteristic
8C2×#13108064—21no—
9#15692255—21no—
10#15692255C2211yescharacteristic
11#15692255C2211yescharacteristic
12#15692254—22no—
13#15692254—21no—
14Q8×Dic101—21no—
15C4×Dic202—21no—
16C4×Dic202C2211yescharacteristic
17#15692187—21no—
18C4⋊#11717579C2211yescharacteristic
19#15692180—22no—
20#15692180—21no—
21#15692178—21no—
22C4×Q8×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic
24C4×C8×C101C2211yescharacteristic
25C2×Dic202—21no—
26C2×Dic202—22no—
27C2×Dic202D411yescharacteristic
28C2×Dic202D412yes—
29#13108064—42no—
30C2×#11717579—21no—
31C4⋊Dic101—22no—
32C4⋊Dic101—21no—
33C4⋊Dic101—21no—
34C4⋊Dic101—21no—
35C4⋊Dic101—21no—
36C4⋊Dic101D411yescharacteristic
37#13108053—22no—
38#13108053—21no—
39C4×Dic101—22no—
40C4×Dic101—21no—
41C2×Q8×C101D411yescharacteristic
42C2×C8×C101—21no—
43C2×C8×C101—21no—
44C4⋊C4×C101—21no—
45C4⋊C4×C101D411yescharacteristic
46C42×C101—21no—
47C42×C101C2311yescharacteristic
48Q8×C101—21no—
49Q8×C101SD1612yes—
50C2×C4×C101—21no—
51C2×C4×C101—21no—
52C2×C4×C101C2×D411yescharacteristic
53C2×C4×C101C2×D411yescharacteristic
54C2×C4×C101C2×D411yescharacteristic
55C2×Dic101—21no—
56C2×Dic101—22no—
57C2×Dic101—21no—
58C2×Dic101—41no—
59Dic202—21no—
60Dic202—42no—
61Dic202—41no—
62Dic202—41no—
63Dic202—82no—
64Dic202Q1612yescomplemented
65C8×C101—41no—
66C8×C101—41no—
67#11717579—41no—
68C22×C101#12011yescharacteristic
69C4×C101—22no—
70C4×C101—21no—
71C4×C101—41no—
72C4×C101C2×SD1611yescharacteristic
73C4×C101C2×Q1611yescharacteristic
74C4×C101#13711yescharacteristic
75C4×C101#13711yescharacteristic
76Dic101—22no—
77Dic101—42no—
78Dic101—41no—
79Dic101—81no—
80C2×C101#44811yescharacteristic
81C2×C101#45111yescharacteristic
82C2×C101#45511yescharacteristic
83#1645—1011nomaximal, Sylow
84C101#164511yescharacteristic, complemented, Sylow
85Q82—1011no—
86#487—1011no—
87#475—1011no—
88#462—1011no—
89#333—1011no—
90Q8⋊C8—1011no—
91Q8⋊C8D10111yescharacteristic
92C2×Q16—2021no—
93C4⋊Q8—1011no—
94C4⋊Q8—1011no—
95C4⋊Q8—2021no—
96C4⋊Q8—2022no—
97C4⋊Q8—2021no—
98C4×Q8—1011no—
99C4×Q8—2021no—
100C4×Q8—2021no—
101C4×Q8C2×D10111yescharacteristic
102#106—2021no—
103C4⋊C8—1011no—
104C4⋊C8C2×D10111yescharacteristic
105#103—2021no—
106#103—2022no—
107#103—2021no—
108C4×C8C2×D10111yescharacteristic
109C2×Q8—1011no—
110C2×Q8—1012no—
111C2×Q8—2022no—
112C2×Q8—2021no—
113C2×Q8D40411yescharacteristic
114Q16—4042no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—2021no—
118C4⋊C4—21no—
119C4⋊C4—1011no—
120C4⋊C4—2022no—
121C4⋊C4—2022no—
122C4⋊C4—2021no—
123C4⋊C4—2021no—
124C4⋊C4—2021no—
125C4⋊C4—2021no—
126C4⋊C4—2021no—
127C4⋊C4D40411yescharacteristic
128C42—21no—
129C42—2022no—
130C42—2021no—
131C42C22×D10111yescharacteristic
132Q8—21no—
133Q8—8082no—
134Q8—1012no—
135Q8—4042no—
136Q8—4041no—
137Q8—4041no—
138Q8—2021no—
139Q8#1310806312yes—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—4041no—
143C2×C4—2021no—
144C2×C4—2022no—
145C2×C4—2021no—
146C2×C4D4×D10111yescharacteristic
147C2×C4D4×D10111yescharacteristic
148C2×C4C2×D40411yescharacteristic
149C8—41no—
150C8—41no—
151C8—4041no—
152C22#1569222511yescharacteristic
153C4—22no—
154C4—21no—
155C4—41no—
156C4—8081no—
157C4—4041no—
158C4—4042no—
159C4—2022no—
160C4C2×#1310806311yescharacteristic
161C4Q16×D10111yescharacteristic
162C4#1569228411yescharacteristic
163C4#1569228811yescharacteristic
164C2#2026531811yescharacteristic
165C2#2026533411yescharacteristic
166C2#2026535611yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
FittingQ8⋊C8×C101C2
socle, prime radicalC22×C101#120
hypercenterQ8⋊C8D101
hypocenterC101#1645
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
64. Dic202Q16semidirect
84. C101#1645semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic2022
#202655282
#202654632
#202653872
#202649252
#202649092
Q8⋊C8×C1012
#1645101