Finite Group Databasebeta

← #27172294

Subgroups — #27172294

Order 12928.

246 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 67 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#20265808C211yesmaximal, characteristic
2#20265187C212yesmaximal
3#20265082C211yesmaximal, characteristic
4#20264938C212yesmaximal
5#343×C101C211yesmaximal, characteristic
6C22×Dic202C2211yescharacteristic
7#15692254—22no—
8#15692246—22no—
9#15692234—22no—
10#15692214C2212yes—
11C2×C4×Dic101C2211yescharacteristic
12#15692180—22no—
13#15692180—22no—
14#15692162—22no—
15C2×M16×C101C2212yes—
16C2×C4⋊C4×C101C2211yescharacteristic
17C2×Dic202—22no—
18C2×Dic202—22no—
19C2×Dic202—21no—
20C2×Dic202—22no—
21C2×Dic202Q812yescomplemented
22C22×Dic101—21no—
23C22×Dic101—21no—
24C4⋊Dic101—22no—
25C4⋊Dic101D412yes—
26C4⋊Dic101D412yes—
27#13108053—24no—
28C22⋊Dic101—22no—
29C4×Dic101—22no—
30C4×Dic101—22no—
31C22×C4×C101D412yes—
32C22×C4×C101C2311yescharacteristic
33M16×C101—22no—
34M16×C101—22no—
35C2×C8×C101—22no—
36C4⋊C4×C101—22no—
37C23×C101C2×D411yescharacteristic
38C2×C4×C101—22no—
39C2×C4×C101—42no—
40C2×C4×C101C2×D412yes—
41C2×C4×C101C2×Q811yescharacteristic
42C2×C4×C101#4111yescharacteristic
43C2×C4×C101#4112yes—
44C2×Dic101—22no—
45C2×Dic101—22no—
46C2×Dic101—22no—
47C2×Dic101—22no—
48C2×Dic101—41no—
49C2×Dic101—41no—
50C2×Dic101—41no—
51C2×Dic101—41no—
52Dic202—24no—
53Dic202—42no—
54Dic202—82no—
55C8×C101—42no—
56C22×C101—21no—
57C22×C101—21no—
58C22×C101#12011yescharacteristic
59C22×C101C22⋊Q811yescharacteristic
60C22×C101#12411yescharacteristic
61C4×C101—42no—
62C4×C101C22⋊Q812yes—
63C4×C101#12412yes—
64Dic101—42no—
65Dic101—42no—
66Dic101—42no—
67Dic101—42no—
68C2×C101—21no—
69C2×C101—21no—
70C2×C101#39311yescharacteristic
71C2×C101#45511yescharacteristic
72C2×C101#45611yescharacteristic
73#2028—1011nomaximal, Sylow
74C101#202811yescharacteristic, complemented, Sylow
75#544—1011no—
76#419—1012no—
77#391—1011no—
78#343D10111yescharacteristic
79#339—1012no—
80C22×Q8—1011no—
81C2×M16C2×D10112yes—
82C4⋊Q8—2022no—
83#125—2022no—
84C22⋊Q8—2022no—
85#117—1012no—
86C2×C4⋊C4C2×D10111yescharacteristic
87C2×C42—1011no—
88#103—2022no—
89#103—2022no—
90#95—2022no—
91C2×Q8—1012no—
92C2×Q8—2022no—
93C2×Q8—2022no—
94C2×Q8—2022no—
95C2×Q8—2021no—
96C22×C4—2021no—
97C22×C4—2021no—
98C22×C4D40412yes—
99C22×C4C22×D10111yescharacteristic
100M16—22no—
101M16—22no—
102C2×C8—22no—
103C4⋊C4—22no—
104C4⋊C4—1012no—
105C4⋊C4—1012no—
106C4⋊C4—2024no—
107C4⋊C4—2022no—
108C22⋊C4—2022no—
109C42—2022no—
110C42—2022no—
111C23C2×D40411yescharacteristic
112Q8—8082no—
113Q8—4042no—
114Q8—2024no—
115C2×C4—22no—
116C2×C4—42no—
117C2×C4—4041no—
118C2×C4—4041no—
119C2×C4—4041no—
120C2×C4—4041no—
121C2×C4—2022no—
122C2×C4—2022no—
123C2×C4—2022no—
124C2×C4—2022no—
125C2×C4D4×D10112yes—
126C2×C4Q8×D10111yescharacteristic
127C2×C4#1310807311yescharacteristic
128C2×C4#1310807412yes—
129C8—42no—
130C22—21no—
131C22—21no—
132C22#1569222511yescharacteristic
133C22#1569223511yescharacteristic
134C22#1569224311yescharacteristic
135C4—42no—
136C4—4042no—
137C4—4042no—
138C4—4042no—
139C4—4042no—
140C4#1569223312yes—
141C4#1569224112yes—
142C2—21no—
143C2—21no—
144C2#2026509011yescharacteristic
145C2#2026535711yescharacteristic
146C2#2026536211yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#343×C101C2
socleC22×C101#120
prime radicalC23×C101C2×D4
hypercenter#343D101
hypocenterC101#2028
LCSG ▷ C22×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
21. C2×Dic202Q8semidirect
74. C101#2028semidirect

Maximal subgroups

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

subgroupindex
#202658082
#202651872
#202650822
#202649382
#343×C1012
#2028101