Finite Group Databasebeta

← #27173158

Subgroups — #27173158

Order 12928.

236 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 45 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#20265359C211yesmaximal, characteristic, complemented
2#20265349C212yesmaximal, complemented
3#20264976C211yesmaximal, characteristic, complemented
4#20264974C212yesmaximal, complemented
5#555×C101C211yesmaximal, characteristic, complemented
6#15692303C2211yescharacteristic
7#15692286—21no—
8#15692281—22no—
9#15692183—22no—
10#15692183—21no—
11#15692174C2211yescharacteristic
12#15692174C2212yes—
13C4⋊Q8×C101C2211yescharacteristic
14#124×C101—21no—
15#124×C101C2212yes—
16#121×C101—22no—
17C4×D4×C101—21no—
18#117×C101—22no—
19#13108075—41no—
20#13108074—22no—
21#13108074—21no—
22Q8×D101—41no—
23#13108064—41no—
24#13108062—42no—
25#13108061—41no—
26#13108059—42no—
27#13108058—22no—
28#13108058—21no—
29C2×Q8×C101D411yescharacteristic
30C2×Q8×C101C2311yescharacteristic
31C2×D4×C101—21no—
32C2×D4×C101—21no—
33C2×D4×C101D412yes—
34C22×C4×C101—22no—
35C4⋊C4×C101—21no—
36C4⋊C4×C101—22no—
37C22⋊C4×C101—22no—
38C22⋊C4×C101—22no—
39C22⋊C4×C101—22no—
40C42×C101—21no—
41C42×C101D412yes—
42C42×C101D411yescharacteristic
43C23×C101—22no—
44Q8×C101—41no—
45Q8×C101—41no—
46D4×C101—42no—
47D4×C101—42no—
48D4×C101—81no—
49C2×C4×C101—21no—
50C2×C4×C101—22no—
51C2×C4×C101—21no—
52C2×C4×C101—44no—
53C2×C4×C101C2×D411yescharacteristic
54C2×C4×C101C2×D412yes—
55C22⋊D101—41no—
56D404—22no—
57D404—21no—
58C4×D101—42no—
59C4×D101—41no—
60Dic202—22no—
61Dic202—21no—
62#11717579—42no—
63#11717579—41no—
64C22×C101—42no—
65C22×C101—42no—
66C22×C101—42no—
67C22×C101#12011yescharacteristic
68C2×D101—41no—
69C4×C101—22no—
70C4×C101—22no—
71C4×C101—21no—
72C4×C101—42no—
73C4×C101—41no—
74Dic101—41no—
75C2×C101—21no—
76C2×C101—82no—
77C2×C101B(4,𝔽2)11yescharacteristic
78D101—81no—
79#2204—1011nomaximal, Sylow
80C101#220411yescharacteristic, complemented, Sylow
81#555D10111yescharacteristic, complemented
82#456—1011no—
83#454—1012no—
84#356—1011no—
85#355—1012no—
862−1+4—1011no—
87#137—2021no—
88AGL(1,ℤ/8)—2022no—
89C4⋊Q8C2×D10111yescharacteristic
90#124—21no—
91#124C2×D10112yes—
92#121—22no—
93C4×D4—21no—
94#117—22no—
95#104—2022no—
96#104—2021no—
97#101—1011no—
98#101—1012no—
99#41—4041no—
100#41—2022no—
101#41—2021no—
102C2×Q8—4041no—
103C2×Q8C22⋊D10111yescharacteristic
104C2×Q8C22×D10111yescharacteristic
105C2×D4—21no—
106C2×D4—21no—
107C2×D4C22⋊D10112yes—
108C22×C4—22no—
109Q16—4041no—
110SD16—4042no—
111SD16—4041no—
112D8—4042no—
113M16—2022no—
114M16—2021no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C42—21no—
121C42C22⋊D10112yes—
122C42C22⋊D10111yescharacteristic
123C23—22no—
124Q8—41no—
125Q8—41no—
126Q8—2022no—
127Q8—2021no—
128D4—42no—
129D4—42no—
130D4—81no—
131D4—4041no—
132D4—2022no—
133D4—2021no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—44no—
138C2×C4—4042no—
139C2×C4—4041no—
140C2×C4C2×C22⋊D10111yescharacteristic
141C2×C4C2×C22⋊D10112yes—
142C8—4042no—
143C8—4041no—
144C22—42no—
145C22—42no—
146C22—42no—
147C22—4041no—
148C22#1569222411yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—42no—
153C4—41no—
154C4—4041no—
155C2—21no—
156C2—82no—
157C2—8081no—
158C2#2026536411yescharacteristic
subgroupG/H
centerC2#20265364
commutator, cosocleC2×Q8×C101C23
FrattiniC2×Q8C22×D101
Fitting#555×C101C2
socleC2×C101B(4,𝔽2)
hypercenter#555D101
hypocenterC101#2204
LCSG ▷ C2×Q8×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #20265359C2semidirect
2. #20265349C2semidirect
3. #20264976C2semidirect
4. #20264974C2semidirect
5. #555×C101C2semidirect
80. C101#2204semidirect
81. #555D101semidirect

Maximal subgroups

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

subgroupindex
#202653592
#202653492
#202649762
#202649742
#555×C1012
#2204101