Finite Group Databasebeta

← #27173085

Subgroups — #27173085

Order 12928.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#20265977C211yesmaximal, characteristic
2#20265620C212yesmaximal, complemented
3#20265435C211yesmaximal, characteristic, complemented
4#20264988C212yesmaximal, complemented
5#349×C101C211yesmaximal, characteristic, complemented
6C2×#13108074—21no—
7#15692288—22no—
8#15692283—22no—
9#15692277—21no—
10#15692277C2212yes—
11C2×Dic404C2211yescharacteristic, complemented
12#15692234—21no—
13Dic808—22no—
14#15692200—22no—
15#15692187C2211yescharacteristic, complemented
16#15692180—21no—
17C2×M16×C101C2211yescharacteristic
18M32×C101C2212yescomplemented
19#13108074—22no—
20#13108074—22no—
21#13108074—41no—
22#13108074—41no—
23C2×Dic202—21no—
24C2×Dic202—21no—
25C2×D404—21no—
26C2×C22⋊D101—41no—
27C2×C4×D101—41no—
28Dic404—21no—
29Dic404—22no—
30Dic404D412yes—
31#13108063—22no—
32#13108063—22no—
33D808—21no—
34D808D412yes—
35C4⋊Dic101—21no—
36#13108053—41no—
37C22⋊Dic101—41no—
38C22×C4×C101D411yescharacteristic
39M16×C101—22no—
40C2×C8×C101D411yescharacteristic
41C2×C8×C101C2311yescharacteristic
42C16×C101—22no—
43C23×C101D811yescharacteristic
44C22×D101—41no—
45C2×C4×C101—21no—
46C2×C4×C101—21no—
47C2×C4×C101D811yescharacteristic
48C2×C4×C101C2×D411yescharacteristic
49C22⋊D101—42no—
50C22⋊D101—42no—
51C2×Dic101—41no—
52C2×Dic101—41no—
53C2×Dic101—41no—
54D404—22no—
55D404—41no—
56C4×D101—42no—
57C4×D101—42no—
58Dic202—22no—
59Dic202—41no—
60Dic202—41no—
61Dic202—41no—
62C8×C101—21no—
63C8×C101C2×D412yes—
64C22×C101—21no—
65C22×C101—41no—
66C22×C101C2×D811yescharacteristic
67C2×D101—42no—
68C2×D101—82no—
69C4×C101—21no—
70C4×C101#12011yescharacteristic
71C4×C101AGL(1,ℤ/8)11yescharacteristic
72Dic101—42no—
73Dic101—81no—
74Dic101—81no—
75C2×C101—21no—
76C2×C101—41no—
77C2×C101#44711yescharacteristic
78D101—82no—
79#2191—1011nomaximal, Sylow
80C101#219111yescharacteristic, complemented, Sylow
81#575—1011no—
82#510—1012no—
83#470—1011no—
84#360—1012no—
85#349D10111yescharacteristic, complemented
86C2×#41—2021no—
87#137—2022no—
88AGL(1,ℤ/8)—2022no—
89#135—1012no—
90#135—2021no—
91C2×Q16—1011no—
92C2×M16C2×D10111yescharacteristic
93C22⋊Q8—2021no—
94Q32—2022no—
95SD32—2022no—
96M32C2×D10112yescomplemented
97#106—1011no—
98#103—2021no—
99#41—4041no—
100#41—4041no—
101#41—2022no—
102#41—2022no—
103C2×Q8—2021no—
104C2×Q8—2021no—
105C2×D4—4041no—
106C2×D4—2021no—
107C22×C4—4041no—
108C22×C4D40411yescharacteristic
109Q16—1012no—
110Q16—2021no—
111Q16—2022no—
112SD16—2022no—
113SD16—2022no—
114D8—1012no—
115D8—2021no—
116M16—22no—
117C2×C8D40411yescharacteristic
118C2×C8C22×D10111yescharacteristic
119C4⋊C4—4041no—
120C4⋊C4—2021no—
121C22⋊C4—4041no—
122C16—22no—
123C23—4041no—
124C23D80811yescharacteristic
125Q8—4041no—
126Q8—4041no—
127Q8—4041no—
128Q8—2022no—
129D4—4041no—
130D4—4042no—
131D4—4042no—
132D4—2022no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—4042no—
136C2×C4—4041no—
137C2×C4—4041no—
138C2×C4—4041no—
139C2×C4—4042no—
140C2×C4D80811yescharacteristic
141C2×C4C2×D40411yescharacteristic
142C8—21no—
143C8D4×D10112yes—
144C22—21no—
145C22—41no—
146C22—8082no—
147C22—4042no—
148C22C2×D80811yescharacteristic
149C4—21no—
150C4—8081no—
151C4—8081no—
152C4—4042no—
153C4#1569222511yescharacteristic
154C4#1569228311yescharacteristic
155C2—21no—
156C2—41no—
157C2—8082no—
158C2#2026531311yescharacteristic
subgroupG/H
centerC2#20265313
commutator, cosocleC2×C8×C101C23
FrattiniC2×C8C22×D101
Fitting#349×C101C2
socleC2×C101#447
prime radical#20265977C2
hypercenter#349D101
hypocenterC101#2191
LCSG ▷ C2×C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
2. #20265620C2semidirect
3. #20265435C2semidirect
4. #20264988C2semidirect
5. #349×C101C2semidirect
11. C2×Dic404C22semidirect
15. #15692187C22semidirect
18. M32×C101C22semidirect
80. C101#2191semidirect
85. #349D101semidirect
96. M32C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202659772
#202656202
#202654352
#202649882
#349×C1012
#2191101