Finite Group Databasebeta

← #27140907

Subgroups — #27140907

Order 12416.

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

normal subgroups only

This group has 194 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#20261458C211yesmaximal, characteristic
2#20261085C211yesmaximal, characteristic, complemented
3#20261022C211yesmaximal, characteristic
4#20260480C211yesmaximal, characteristic, complemented
5Q8⋊#11715895C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic
7#477×C97C211yesmaximal, characteristic, complemented
8#15690810C2211yescharacteristic
9#15690805—22no—
10#15690798—21no—
11#15690786—21no—
12C4×Dic194—21no—
13C4×D388C2211yescharacteristic
14#15690768—21no—
15#15690741—21no—
16C4⋊#11715895C2211yescharacteristic
17#15690734—22no—
18#15690730—21no—
19C4×#11715895C2211yescharacteristic
20C4⋊Q8×C97C2211yescharacteristic
21C4×Q8×C97C2211yescharacteristic
22#107×C97—21no—
23C4⋊C8×C97C2211yescharacteristic
24#103×C97—21no—
25C2×Dic194—21no—
26C2×D388D411yescharacteristic
27C2×C4×D97—21no—
28C2×#11715895—21no—
29C2×#11715895—21no—
30C4⋊Dic97—21no—
31C4⋊Dic97D411yescharacteristic
32C4⋊Dic97D412yes—
33#13095490—22no—
34#13095488—22no—
35#13095488—22no—
36#13095488—21no—
37C4×Dic97—22no—
38C2×Q8×C97—21no—
39C2×Q8×C97D411yescharacteristic
40C2×C8×C97—21no—
41C4⋊C4×C97—21no—
42C4⋊C4×C97—21no—
43C4⋊C4×C97—21no—
44C4⋊C4×C97D411yescharacteristic
45C42×C97—21no—
46C42×C97C2311yescharacteristic
47C22×D97—21no—
48Q8×C97—21no—
49Q8×C97—21no—
50Q8×C97—42no—
51C2×C4×C97—21no—
52C2×C4×C97—21no—
53C2×C4×C97—41no—
54C2×C4×C97C2×D411yescharacteristic
55C2×C4×C97C2×D411yescharacteristic
56C2×C4×C97C2×D411yescharacteristic
57C2×Dic97—21no—
58C2×Dic97—22no—
59D388—21no—
60D388—21no—
61C4×D97—41no—
62C4×D97—41no—
63Dic194—41no—
64Dic194—41no—
65C8×C97—41no—
66#11715895—41no—
67#11715895—41no—
68C22×C97#12011yescharacteristic
69C2×D97—41no—
70C2×D97—41no—
71C2×D97—41no—
72C4×C97—21no—
73C4×C97—41no—
74C4×C97—41no—
75C4×C97—81no—
76C4×C97#13511yescharacteristic
77C4×C97#13511yescharacteristic
78C4×C97#13711yescharacteristic
79C4×C97#13711yescharacteristic
80Dic97—42no—
81Dic97—41no—
82C2×C97#45211yescharacteristic
83C2×C97#45211yescharacteristic
84C2×C97#45411yescharacteristic
85D97—81no—
86#1638—971nomaximal, Sylow
87C97#163811yescharacteristic, complemented, Sylow
88#556—971no—
89#487—971no—
90#477D9711yescharacteristic, complemented
91#476—971no—
92#332—971no—
93Q8⋊C8—971no—
94D4⋊C8—971no—
95C4⋊Q8—971no—
96C4⋊Q8C2×D9711yescharacteristic
97#126—1942no—
98#124—1941no—
99C22⋊Q8—1941no—
100C4×Q8—1941no—
101C4×Q8C2×D9711yescharacteristic
102C4×D4—971no—
103#117—1941no—
104#107—21no—
105#106—1941no—
106C4⋊C8—971no—
107C4⋊C8C2×D9711yescharacteristic
108#103—21no—
109#103—1942no—
110#102—1941no—
111C4×C8—971no—
112C2×Q8—21no—
113C2×Q8—1941no—
114C2×Q8C22⋊D9711yescharacteristic
115C2×D4—971no—
116C22×C4—1941no—
117C2×C8—21no—
118C2×C8—1941no—
119C2×C8—1941no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—972no—
124C4⋊C4—971no—
125C4⋊C4—1942no—
126C4⋊C4—1941no—
127C4⋊C4C22⋊D9711yescharacteristic
128C22⋊C4—1942no—
129C22⋊C4—1942no—
130C22⋊C4—1941no—
131C42—21no—
132C42—1942no—
133C42C22×D9711yescharacteristic
134C23—1941no—
135Q8—21no—
136Q8—21no—
137Q8—42no—
138Q8—3881no—
139Q8—3881no—
140D4—1941no—
141D4—1941no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—3881no—
146C2×C4—3881no—
147C2×C4—1941no—
148C2×C4—1942no—
149C2×C4D4×D9711yescharacteristic
150C2×C4D4×D9711yescharacteristic
151C2×C4C2×C22⋊D9711yescharacteristic
152C8—41no—
153C8—3881no—
154C8—3881no—
155C22—3881no—
156C22—3881no—
157C22—3881no—
158C22#1569077811yescharacteristic
159C4—21no—
160C4—41no—
161C4—41no—
162C4—81no—
163C4—3881no—
164C4—3882no—
165C4#1569083011yescharacteristic
166C4#1569083311yescharacteristic
167C4#1569084111yescharacteristic
168C4#1569084211yescharacteristic
169C2—7761no—
170C2#2026089511yescharacteristic
171C2#2026089611yescharacteristic
172C2#2026090611yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#477×C97C2
socleC22×C97#120
prime radicalC2×D388D4
hypercenter#477D97
hypocenterC97#1638
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #20261085C2semidirect
4. #20260480C2semidirect
5. Q8⋊#11715895C2semidirect
7. #477×C97C2semidirect
87. C97#1638semidirect
90. #477D97semidirect

Maximal subgroups

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

subgroupindex
#202614582
#202610852
#202610222
#202604802
Q8⋊#117158952
#202604622
#477×C972
#163897