Finite Group Databasebeta

← #27140738

Subgroups — #27140738

Order 12416.

254 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261316C211yesmaximal, characteristic
2#20261048C212yesmaximal, complemented
3#20260468C212yesmaximal, complemented
4#20260461C211yesmaximal, characteristic, complemented
5#398×C97C211yesmaximal, characteristic, complemented
6C2×#13095511—21no—
7#15690790—22no—
8#15690785—22no—
9#15690784—21no—
10#15690784C2212yes—
11#15690743—22no—
12#15690730—22no—
13C22⋊#11715895C2212yes—
14C2×C4⋊C4×C97—21no—
15C2×C4⋊C4×C97C2211yescharacteristic
16C2×C4⋊C4×C97C2211yescharacteristic
17#95×C97—21no—
18#95×C97C2211yescharacteristic
19#13095511—42no—
20#13095511—42no—
21C2×Dic194—21no—
22C2×D388—21no—
23C2×D388—21no—
24C2×D388D412yes—
25C2×C22⋊D97—22no—
26C2×C4×D97—22no—
27C2×#11715895—22no—
28C4⋊Dic97—21no—
29C4⋊Dic97D412yes—
30#13095488—22no—
31#13095488—22no—
32C22×C4×C97—21no—
33C22×C4×C97—21no—
34C22×C4×C97D411yescharacteristic
35C22×C4×C97D411yescharacteristic
36C22×C4×C97C2311yescharacteristic
37C4⋊C4×C97—22no—
38C4⋊C4×C97—22no—
39C4⋊C4×C97—42no—
40C23×C97C2×D411yescharacteristic
41C22×D97—22no—
42C2×C4×C97—21no—
43C2×C4×C97—21no—
44C2×C4×C97—22no—
45C2×C4×C97—21no—
46C2×C4×C97—41no—
47C2×C4×C97—41no—
48C2×C4×C97—41no—
49C2×C4×C97—41no—
50C2×C4×C97—41no—
51C2×C4×C97—41no—
52C2×C4×C97—41no—
53C2×C4×C97D812yes—
54C2×C4×C97C2×D412yes—
55C22⋊D97—42no—
56C22⋊D97—42no—
57C2×Dic97—22no—
58D388—42no—
59D388—42no—
60D388—81no—
61C4×D97—44no—
62Dic194—42no—
63#11715895—42no—
64C22×C97—21no—
65C22×C97—21no—
66C22×C97#12011yescharacteristic
67C22×C97C2×D811yescharacteristic
68C22×C97AGL(1,ℤ/8)11yescharacteristic
69C2×D97—42no—
70C2×D97—42no—
71C2×D97—42no—
72C4×C97—22no—
73C4×C97—22no—
74C4×C97—42no—
75C4×C97—41no—
76C4×C97—81no—
77Dic97—42no—
78C2×C97—21no—
79C2×C97—21no—
80C2×C97#44711yescharacteristic
81C2×C97#45611yescharacteristic
82C2×C97#45811yescharacteristic
83D97—82no—
84#1610—971nomaximal, Sylow
85C97#161011yescharacteristic, complemented, Sylow
86#537—971no—
87#480—972no—
88#398D9711yescharacteristic, complemented
89#327—972no—
90#324—971no—
91C2×#41—1941no—
92C22⋊Q8—1942no—
93#121—972no—
94#121—1942no—
95#121—1941no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4C2×D9711yescharacteristic
98C2×C4⋊C4C2×D9711yescharacteristic
99#107—1942no—
100#102—1942no—
101C22⋊C8—972no—
102#95—21no—
103#95C2×D9711yescharacteristic
104#41—3882no—
105#41—3882no—
106C2×Q8—1941no—
107C2×D4—972no—
108C2×D4—1941no—
109C2×D4—1941no—
110C2×D4—1942no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—1942no—
114C22×C4C22⋊D9711yescharacteristic
115C22×C4C22⋊D9711yescharacteristic
116C22×C4C22×D9711yescharacteristic
117C2×C8—1942no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—42no—
121C4⋊C4—972no—
122C4⋊C4—1941no—
123C22⋊C4—1942no—
124C22⋊C4—1942no—
125C23—1942no—
126C23C2×C22⋊D9711yescharacteristic
127Q8—3882no—
128D4—7761no—
129D4—3882no—
130D4—3882no—
131D4—3882no—
132D4—3882no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—3884no—
145C2×C4—1942no—
146C2×C4#1309549612yes—
147C2×C4D4×D9712yes—
148C8—3882no—
149C22—21no—
150C22—21no—
151C22—3882no—
152C22—3882no—
153C22—3882no—
154C22#1569077811yescharacteristic
155C22C2×#1309549611yescharacteristic
156C22#1569083511yescharacteristic
157C4—22no—
158C4—22no—
159C4—42no—
160C4—41no—
161C4—81no—
162C4—3882no—
163C2—21no—
164C2—21no—
165C2—7762no—
166C2#2026086811yescharacteristic
167C2#2026091711yescharacteristic
168C2#2026092411yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#398×C97C2
socleC22×C97#120
prime radical#20261316C2
hypercenter#398D97
hypocenterC97#1610
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

Complements

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

NH ≅ G/Ntype
2. #20261048C2semidirect
3. #20260468C2semidirect
4. #20260461C2semidirect
5. #398×C97C2semidirect
85. C97#1610semidirect
88. #398D97semidirect

Maximal subgroups

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

subgroupindex
#202613162
#202610482
#202604682
#202604612
#398×C972
#161097