Finite Group Databasebeta

← #27169765

Subgroups — #27169765

Order 12928.

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

normal subgroups only

This group has 238 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
1#20265726C211yesmaximal, characteristic
2#20265232C212yesmaximal, complemented
3#20264917C211yesmaximal, characteristic, complemented
4#20264916C212yesmaximal, complemented
5#329×C101C211yesmaximal, characteristic, complemented
6C2×#13108075C411yescharacteristic
7C2×#13108073C411yescharacteristic
8#15692253C411yescharacteristic
9#15692251C411yescharacteristic
10#15692243C2211yescharacteristic
11#15692228—22no—
12#15692215—22no—
13C42×D101C2211yescharacteristic
14#15692184—22no—
15#15692167—22no—
16#15692166C2212yes—
17#124×C101C2211yescharacteristic
18#97×C101C2212yes—
19#13108075—42no—
20#13108073—42no—
21C2×Dic202C2×C411yescharacteristic
22C2×D404—21no—
23C2×D404C2×C411yescharacteristic
24C2×C22⋊D101—21no—
25C2×C22⋊D101—22no—
26C22×Dic101—21no—
27C2×C4×D101—21no—
28C2×C4×D101C2×C411yescharacteristic
29C2×C4×D101D412yes—
30C2×#11717579—22no—
31#13108053—22no—
32#13108051—22no—
33C4×Dic101C2×C411yescharacteristic
34C4×Dic101D412yes—
35C2×Q8×C101C2×C411yescharacteristic
36C2×D4×C101C2×C411yescharacteristic
37C2×C8×C101—22no—
38C22⋊C4×C101—22no—
39C42×C101C2311yescharacteristic
40C23×C101—21no—
41C22×D101—21no—
42C22×D101C22⋊C411yescharacteristic
43Q8×C101—41no—
44D4×C101—41no—
45C2×C4×C101—21no—
46C2×C4×C101C22×C411yescharacteristic
47C2×C4×C101C2×D412yes—
48C22⋊D101—44no—
49C22⋊D101—42no—
50C2×Dic101—21no—
51C2×Dic101—42no—
52C2×Dic101C22⋊C412yes—
53C2×Dic101C22⋊C411yescharacteristic
54D404—41no—
55D404—42no—
56C4×D101—22no—
57C4×D101—42no—
58C4×D101—42no—
59Dic202—41no—
60C8×C101—42no—
61#11717579—42no—
62C22×C101—41no—
63C22×C101—42no—
64C22×C101C2×C22⋊C411yescharacteristic
65C2×D101—21no—
66C2×D101—22no—
67C2×D101—41no—
68C2×D101—42no—
69C4×C101—21no—
70C4×C101—22no—
71C4×C101—41no—
72Dic101—22no—
73Dic101—41no—
74Dic101#10012yes—
75C2×C101—81no—
76C2×C101C2×#10011yescharacteristic
77C2×C101#42112yes—
78D101—41no—
79D101—81no—
80#1536—1011nomaximal, Sylow
81C101#153611yescharacteristic, complemented, Sylow
82#532—1011no—
83#432—1012no—
84#329—1011no—
85#329—1012no—
86#329D10111yescharacteristic, complemented
87C2×#41—1011no—
88C2×#41—1011no—
89C4⋊Q8—1011no—
90#127—1011no—
91#124—1011no—
92#124C2×D10111yescharacteristic
93#121—2022no—
94C4×D4—2022no—
95C2×C42—1011no—
96C4⋊C8—2022no—
97C22⋊C8—2022no—
98#97—1012no—
99#97C2×D10112yes—
100#41—4042no—
101#41—4042no—
102C2×Q8—1011no—
103C2×Q8C4×D10111yescharacteristic
104C2×D4—1011no—
105C2×D4—2021no—
106C2×D4—2022no—
107C2×D4—2021no—
108C2×D4C4×D10111yescharacteristic
109C22×C4—1012no—
110C22×C4—1011no—
111C22×C4—2021no—
112C22×C4—2021no—
113C2×C8—22no—
114C2×C8—2022no—
115C4⋊C4—2022no—
116C22⋊C4—22no—
117C22⋊C4—2022no—
118C42—1011no—
119C42—1012no—
120C42C22×D10111yescharacteristic
121C23—21no—
122C23—1011no—
123C23—2021no—
124Q8—41no—
125Q8—4041no—
126D4—41no—
127D4—4041no—
128D4—4042no—
129D4—4042no—
130D4—4044no—
131C2×C4—21no—
132C2×C4—1011no—
133C2×C4—1012no—
134C2×C4—4042no—
135C2×C4—4042no—
136C2×C4—4042no—
137C2×C4—2022no—
138C2×C4—2021no—
139C2×C4C2×C4×D10111yescharacteristic
140C2×C4D4×D10112yes—
141C8—42no—
142C8—4042no—
143C22—41no—
144C22—42no—
145C22—4041no—
146C22—4042no—
147C22—2021no—
148C22—2022no—
149C22C22⋊C4×D10111yescharacteristic
150C4—21no—
151C4—22no—
152C4—41no—
153C4—1012no—
154C4—4041no—
155C4—2022no—
156C2—81no—
157C2—8081no—
158C2—4041no—
159C2#100×D10111yescharacteristic
160C2#2026519212yes—
subgroupG/H
centerC22C22⋊C4×D101
commutatorC2×C4×C101C22×C4
FrattiniC42C22×D101
Fitting#329×C101C2
socleC22×C101C2×C22⋊C4
prime radical#20265726C2
cosocleC42×C101C23
hypercenter#329D101
hypocenterC101#1536
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #20265232C2semidirect
3. #20264917C2semidirect
4. #20264916C2semidirect
5. #329×C101C2semidirect
81. C101#1536semidirect
86. #329D101semidirect

Maximal subgroups

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

subgroupindex
#202657262
#202652322
#202649172
#202649162
#329×C1012
#1536101