Finite Group Databasebeta

← #27140861

Subgroups — #27140861

Order 12416.

256 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 57 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#20261407C211yesmaximal, characteristic
2#20261109C211yesmaximal, characteristic, complemented
3#20261022C211yesmaximal, characteristic, complemented
4#20260478C211yesmaximal, characteristic, complemented
5Q8⋊#11715895C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic, complemented
7#477×C97C211yesmaximal, characteristic, complemented
8C2×#13095512—21no—
9C2×#13095498—22no—
10#15690807—21no—
11#15690807C2211yescharacteristic
12#15690785—22no—
13#15690785—21no—
14C4×D388—21no—
15C4×D388C2211yescharacteristic
16#15690741—21no—
17C4⋊#11715895C2211yescharacteristic
18#15690730—21no—
19C4×#11715895C2211yescharacteristic
20C4⋊Q8×C97C2211yescharacteristic
21C4×Q8×C97C2211yescharacteristic
22#107×C97—21no—
23C4⋊C8×C97C2211yescharacteristic
24#103×C97—21no—
25#13095512—42no—
26#13095512—42no—
27C2×D388—22no—
28C2×D388—21no—
29C2×D388—21no—
30C2×D388—22no—
31C2×D388D411yescharacteristic
32C2×D388D412yes—
33C2×C4×D97—22no—
34C2×C4×D97—21no—
35#13095498—44no—
36C2×#11715895—21no—
37C2×#11715895—21no—
38C4⋊Dic97D411yescharacteristic
39#13095488—22no—
40#13095488—21no—
41C2×Q8×C97—21no—
42C2×Q8×C97D411yescharacteristic
43C2×C8×C97—21no—
44C4⋊C4×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97—21no—
47C4⋊C4×C97D411yescharacteristic
48C42×C97—21no—
49C42×C97C2311yescharacteristic
50C22×D97—21no—
51C22×D97—22no—
52Q8×C97—21no—
53Q8×C97—42no—
54Q8×C97SD1612yescomplemented
55C2×C4×C97—21no—
56C2×C4×C97—21no—
57C2×C4×C97—41no—
58C2×C4×C97C2×D411yescharacteristic
59C2×C4×C97C2×D411yescharacteristic
60C2×C4×C97C2×D411yescharacteristic
61C2×Dic97—21no—
62D388—21no—
63D388—21no—
64D388—42no—
65D388—44no—
66D388—41no—
67D388—41no—
68D388—41no—
69D388—41no—
70D388—82no—
71C4×D97—44no—
72C4×D97—41no—
73C4×D97—41no—
74C8×C97—41no—
75#11715895—22no—
76#11715895—41no—
77C22×C97#12011yescharacteristic
78C2×D97—42no—
79C2×D97—41no—
80C2×D97—41no—
81C2×D97—42no—
82C2×D97—42no—
83C2×D97—41no—
84C4×C97—22no—
85C4×C97—21no—
86C4×C97—41no—
87C4×C97—81no—
88C4×C97C2×SD1611yescharacteristic
89C4×C97#13511yescharacteristic
90C4×C97AGL(1,ℤ/8)11yescharacteristic
91C4×C97#13711yescharacteristic
92Dic97—41no—
93C2×C97#45011yescharacteristic
94C2×C97#45211yescharacteristic
95C2×C97#45311yescharacteristic
96D97—81no—
97D97—82no—
98#1631—971nomaximal, Sylow
99C97#163111yescharacteristic, complemented, Sylow
100#550—971no—
101#492—971no—
102#477D9711yescharacteristic, complemented
103#476—971no—
104#331—971no—
105Q8⋊C8—971no—
106D4⋊C8—971no—
107C2×#41—1941no—
108C2×SD16—1942no—
109C4⋊Q8C2×D9711yescharacteristic
110#127—971no—
111#127—1941no—
112#121—1942no—
113#121—1941no—
114C4×Q8C2×D9711yescharacteristic
115C4×D4—971no—
116C4×D4—1941no—
117#107—21no—
118#106—1941no—
119C4⋊C8—971no—
120C4⋊C8C2×D9711yescharacteristic
121#103—21no—
122#102—1941no—
123C4×C8—971no—
124#41—3882no—
125#41—3882no—
126C2×Q8—21no—
127C2×Q8C22⋊D9711yescharacteristic
128C2×D4—972no—
129C2×D4—971no—
130C2×D4—1942no—
131C2×D4—1942no—
132C2×D4—1941no—
133C2×D4—1941no—
134C22×C4—1942no—
135C22×C4—1941no—
136SD16—3884no—
137C2×C8—21no—
138C2×C8—1941no—
139C2×C8—1941no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—971no—
144C4⋊C4C22⋊D9711yescharacteristic
145C22⋊C4—1942no—
146C22⋊C4—1941no—
147C42—21no—
148C42C22×D9711yescharacteristic
149C23—1942no—
150C23—1941no—
151Q8—21no—
152Q8—42no—
153Q8#1309549812yescomplemented
154D4—7762no—
155D4—3882no—
156D4—3884no—
157D4—3881no—
158D4—3881no—
159D4—3881no—
160D4—3881no—
161D4—1941no—
162D4—1941no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—3884no—
167C2×C4—3881no—
168C2×C4—3881no—
169C2×C4—1941no—
170C2×C4D4×D9711yescharacteristic
171C2×C4D4×D9711yescharacteristic
172C2×C4C2×C22⋊D9711yescharacteristic
173C8—41no—
174C8—3881no—
175C8—1942no—
176C22—3882no—
177C22—3881no—
178C22—3882no—
179C22—3881no—
180C22—3881no—
181C22—3882no—
182C22#1569077811yescharacteristic
183C4—22no—
184C4—21no—
185C4—41no—
186C4—81no—
187C4—3881no—
188C4C2×#1309549811yescharacteristic
189C4#1569083311yescharacteristic
190C4#1569083511yescharacteristic
191C4#1569084211yescharacteristic
192C2—7762no—
193C2—7761no—
194C2#2026088511yescharacteristic
195C2#2026089611yescharacteristic
196C2#2026090111yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#477×C97C2
socleC22×C97#120
prime radical#20261407C2
hypercenter#477D97
hypocenterC97#1631
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. #20261109C2semidirect
3. #20261022C2semidirect
4. #20260478C2semidirect
5. Q8⋊#11715895C2semidirect
6. #20260462C2semidirect
7. #477×C97C2semidirect
54. Q8×C97SD16semidirect
99. C97#1631semidirect
102. #477D97semidirect
153. Q8#13095498semidirect

Maximal subgroups

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

subgroupindex
#202614072
#202611092
#202610222
#202604782
Q8⋊#117158952
#202604622
#477×C972
#163197