Finite Group Databasebeta

← #27140962

Subgroups — #27140962

Order 12416.

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

normal subgroups only

This group has 212 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#20261450C211yesmaximal, characteristic
2#20261087C211yesmaximal, characteristic, complemented
3#20261012C211yesmaximal, characteristic, complemented
4#20260483C211yesmaximal, characteristic, complemented
5D4⋊#11715895C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic, complemented
7#461×C97C211yesmaximal, characteristic, complemented
8#15690810C2211yescharacteristic
9#15690796—22no—
10#15690784—21no—
11#15690781—21no—
12C4×D388C2211yescharacteristic
13#15690769—21no—
14#15690768—21no—
15#15690743—21no—
16C4⋊#11715895C2211yescharacteristic
17#15690733—22no—
18#15690730—21no—
19C4×#11715895C2211yescharacteristic
20C2×SD16×C97—21no—
21C4⋊Q8×C97C2211yescharacteristic
22C4×D4×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24#103×C97—21no—
25C2×Dic194D412yes—
26C2×D388D411yescharacteristic
27C2×C22⋊D97—22no—
28C2×C4×D97—21no—
29C2×#11715895—21no—
30C2×#11715895—21no—
31C4⋊Dic97—21no—
32C4⋊Dic97—21no—
33C4⋊Dic97D411yescharacteristic
34C22⋊Dic97—22no—
35#13095488—22no—
36#13095488—21no—
37C4×Dic97—22no—
38C2×Q8×C97—21no—
39C2×D4×C97D411yescharacteristic
40C22×C4×C97—21no—
41SD16×C97—42no—
42C2×C8×C97—21no—
43C4⋊C4×C97—21no—
44C4⋊C4×C97—21no—
45C4⋊C4×C97D411yescharacteristic
46C22⋊C4×C97—21no—
47C42×C97C2311yescharacteristic
48C23×C97—21no—
49C22×D97—21no—
50Q8×C97—41no—
51Q8×C97—41no—
52D4×C97—21no—
53D4×C97—21no—
54C2×C4×C97—21no—
55C2×C4×C97—41no—
56C2×C4×C97—41no—
57C2×C4×C97—41no—
58C2×C4×C97C2×D411yescharacteristic
59C2×C4×C97C2×D411yescharacteristic
60C2×C4×C97C2×D411yescharacteristic
61C22⋊D97—82no—
62C2×Dic97—21no—
63C2×Dic97—22no—
64D388—21no—
65D388—21no—
66C4×D97—41no—
67C4×D97—41no—
68Dic194—42no—
69C8×C97—41no—
70#11715895—41no—
71#11715895—41no—
72C22×C97—41no—
73C22×C97—41no—
74C22×C97—41no—
75C22×C97#12011yescharacteristic
76C2×D97—41no—
77C2×D97—41no—
78C2×D97—41no—
79C4×C97—21no—
80C4×C97—41no—
81C4×C97—81no—
82C4×C97#13511yescharacteristic
83C4×C97#13511yescharacteristic
84C4×C97AGL(1,ℤ/8)11yescharacteristic
85C4×C97#13711yescharacteristic
86Dic97—41no—
87Dic97—42no—
88C2×C97—81no—
89C2×C97#44911yescharacteristic
90C2×C97#45211yescharacteristic
91C2×C97#45511yescharacteristic
92D97—81no—
93#1646—971nomaximal, Sylow
94C97#164611yescharacteristic, complemented, Sylow
95#555—971no—
96#487—971no—
97#474—971no—
98#461D9711yescharacteristic, complemented
99#333—971no—
100D4⋊C8—971no—
101D4⋊C8—971no—
102C2×SD16—21no—
103C4⋊Q8—971no—
104C4⋊Q8C2×D9711yescharacteristic
105#124—1942no—
106#121—1941no—
107#121—1941no—
108C4×D4—971no—
109C4×D4C2×D9711yescharacteristic
110#117—1941no—
111#117—1941no—
112#107—1941no—
113C4⋊C8—971no—
114C4⋊C8C2×D9711yescharacteristic
115#103—21no—
116#103—1942no—
117#102—1941no—
118C4×C8—971no—
119C2×Q8—21no—
120C2×Q8—972no—
121C2×D4—971no—
122C2×D4—1942no—
123C2×D4C22⋊D9711yescharacteristic
124C22×C4—21no—
125C22×C4—1941no—
126SD16—42no—
127C2×C8—21no—
128C2×C8—1941no—
129C2×C8—1941no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—971no—
133C4⋊C4—1941no—
134C4⋊C4—1941no—
135C4⋊C4C22⋊D9711yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—1942no—
138C22⋊C4—1942no—
139C22⋊C4—1941no—
140C42—1942no—
141C42C22×D9711yescharacteristic
142C23—21no—
143C23—1941no—
144Q8—41no—
145Q8—41no—
146Q8—3882no—
147D4—21no—
148D4—21no—
149D4—7762no—
150D4—1941no—
151D4—1941no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—3881no—
157C2×C4—3881no—
158C2×C4—1941no—
159C2×C4—1942no—
160C2×C4D4×D9711yescharacteristic
161C2×C4D4×D9711yescharacteristic
162C2×C4C2×C22⋊D9711yescharacteristic
163C8—41no—
164C8—3881no—
165C8—3881no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—3881no—
170C22—3881no—
171C22—3881no—
172C22#1569077811yescharacteristic
173C4—21no—
174C4—41no—
175C4—81no—
176C4—3881no—
177C4—3882no—
178C4#1569083011yescharacteristic
179C4#1569083111yescharacteristic
180C4#1569083611yescharacteristic
181C4#1569083911yescharacteristic
182C2—81no—
183C2—7761no—
184C2#2026087811yescharacteristic
185C2#2026089411yescharacteristic
186C2#2026091111yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#461×C97C2
socleC22×C97#120
prime radical#20261450C2
hypercenter#461D97
hypocenterC97#1646
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #20261087C2semidirect
3. #20261012C2semidirect
4. #20260483C2semidirect
5. D4⋊#11715895C2semidirect
6. #20260462C2semidirect
7. #461×C97C2semidirect
94. C97#1646semidirect
98. #461D97semidirect

Maximal subgroups

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

subgroupindex
#202614502
#202610872
#202610122
#202604832
D4⋊#117158952
#202604622
#461×C972
#164697