Finite Group Databasebeta

← #27140928

Subgroups — #27140928

Order 12416.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 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#20261406C211yesmaximal, characteristic
2#20261083C211yesmaximal, characteristic, complemented
3#20260942C211yesmaximal, characteristic, complemented
4#20260478C211yesmaximal, characteristic, complemented
5#20260465C211yesmaximal, characteristic, complemented
6D4⋊#11715895C211yesmaximal, characteristic, complemented
7#474×C97C211yesmaximal, characteristic, complemented
8C2×#13095511—21no—
9C2×#13095501—21no—
10#15690807C2211yescharacteristic
11#15690806—21no—
12#15690784—21no—
13#15690783—22no—
14C4×Dic194C2211yescharacteristic
15#15690772—21no—
16C4⋊#11715895C2211yescharacteristic
17#15690734—21no—
18#15690730—22no—
19C4×#11715895C2211yescharacteristic
20C4⋊Q8×C97C2211yescharacteristic
21C4×D4×C97C2211yescharacteristic
22#107×C97—21no—
23C4⋊C8×C97C2211yescharacteristic
24#102×C97—21no—
25#13095511—42no—
26#13095511—42no—
27C2×Dic194D411yescharacteristic
28C2×D388—21no—
29C2×D388—21no—
30C2×D388D412yes—
31C2×C22⋊D97—22no—
32C2×C22⋊D97—22no—
33C2×C4×D97—22no—
34#13095501—42no—
35C2×#11715895—21no—
36C2×#11715895—21no—
37C4⋊Dic97D411yescharacteristic
38#13095490—21no—
39#13095488—22no—
40C4×Dic97—21no—
41C2×Q8×C97—21no—
42C2×D4×C97D411yescharacteristic
43C22×C4×C97—21no—
44C2×C8×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97—21no—
47C4⋊C4×C97D411yescharacteristic
48C22⋊C4×C97—21no—
49C42×C97C2311yescharacteristic
50C23×C97—21no—
51C22×D97—22no—
52Q8×C97—41no—
53Q8×C97—41no—
54D4×C97—21no—
55D4×C97—21no—
56C2×C4×C97—21no—
57C2×C4×C97—41no—
58C2×C4×C97—41no—
59C2×C4×C97—41no—
60C2×C4×C97C2×D411yescharacteristic
61C2×C4×C97C2×D411yescharacteristic
62C2×C4×C97C2×D411yescharacteristic
63C22⋊D97—44no—
64C22⋊D97—82no—
65C2×Dic97—21no—
66C2×Dic97—21no—
67D388—42no—
68D388—41no—
69D388—41no—
70D388—41no—
71D388—41no—
72C4×D97—44no—
73Dic194—21no—
74Dic194D812yes—
75C8×C97—41no—
76#11715895—41no—
77#11715895—41no—
78C22×C97—41no—
79C22×C97—41no—
80C22×C97—41no—
81C22×C97#12011yescharacteristic
82C2×D97—42no—
83C2×D97—42no—
84C2×D97—42no—
85C4×C97—21no—
86C4×C97—41no—
87C4×C97—81no—
88C4×C97C2×D811yescharacteristic
89C4×C97#13511yescharacteristic
90C4×C97AGL(1,ℤ/8)11yescharacteristic
91C4×C97#13711yescharacteristic
92Dic97—22no—
93Dic97—41no—
94C2×C97—81no—
95C2×C97#44711yescharacteristic
96C2×C97#45211yescharacteristic
97C2×C97#45511yescharacteristic
98D97—82no—
99#1641—971nomaximal, Sylow
100C97#164111yescharacteristic, complemented, Sylow
101#550—971no—
102#486—971no—
103#474D9711yescharacteristic, complemented
104#462—971no—
105#331—971no—
106Q8⋊C8—971no—
107D4⋊C8—971no—
108C2×#41—1941no—
109C2×Q16—1941no—
110C4⋊Q8C2×D9711yescharacteristic
111#127—971no—
112#127—1941no—
113#121—1942no—
114#121—1941no—
115C4×Q8—971no—
116C4×D4—1941no—
117C4×D4C2×D9711yescharacteristic
118#107—21no—
119C4⋊C8—971no—
120C4⋊C8C2×D9711yescharacteristic
121#103—1941no—
122#102—21no—
123#102—1942no—
124C4×C8—971no—
125#41—3882no—
126#41—3882no—
127C2×Q8—21no—
128C2×Q8—971no—
129C2×D4—972no—
130C2×D4—1942no—
131C2×D4—1942no—
132C2×D4—1941no—
133C2×D4—1941no—
134C2×D4C22⋊D9711yescharacteristic
135C22×C4—21no—
136C22×C4—1942no—
137Q16—3882no—
138C2×C8—21no—
139C2×C8—1941no—
140C2×C8—1941no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—971no—
144C4⋊C4—1941no—
145C4⋊C4C22⋊D9711yescharacteristic
146C22⋊C4—21no—
147C22⋊C4—1942no—
148C42—1941no—
149C42C22×D9711yescharacteristic
150C23—21no—
151C23—1942no—
152Q8—41no—
153Q8—41no—
154Q8—972no—
155Q8—1941no—
156D4—21no—
157D4—21no—
158D4—7762no—
159D4—3882no—
160D4—3884no—
161D4—3881no—
162D4—3881no—
163D4—3881no—
164D4—3881no—
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—3884no—
170C2×C4—1941no—
171C2×C4—1941no—
172C2×C4D4×D9711yescharacteristic
173C2×C4D4×D9711yescharacteristic
174C2×C4C2×C22⋊D9711yescharacteristic
175C8—41no—
176C8—3881no—
177C8—3881no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—3882no—
182C22—3882no—
183C22—3882no—
184C22#1569077811yescharacteristic
185C4—21no—
186C4—41no—
187C4—81no—
188C4—3881no—
189C4—1942no—
190C4D8×D9711yescharacteristic
191C4#1569083011yescharacteristic
192C4#1569083411yescharacteristic
193C4#1569084111yescharacteristic
194C2—81no—
195C2—7762no—
196C2#2026086711yescharacteristic
197C2#2026089511yescharacteristic
198C2#2026091111yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#474×C97C2
socleC22×C97#120
prime radical#20261406C2
hypercenter#474D97
hypocenterC97#1641
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #20261083C2semidirect
3. #20260942C2semidirect
4. #20260478C2semidirect
5. #20260465C2semidirect
6. D4⋊#11715895C2semidirect
7. #474×C97C2semidirect
100. C97#1641semidirect
103. #474D97semidirect

Maximal subgroups

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

subgroupindex
#202614062
#202610832
#202609422
#202604782
#202604652
D4⋊#117158952
#474×C972
#164197