Finite Group Databasebeta

← #27140846

Subgroups — #27140846

Order 12416.

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

normal subgroups only

This group has 216 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#20261083C211yesmaximal, characteristic, complemented
2#20260935C211yesmaximal, characteristic, complemented
3#20260930C211yesmaximal, characteristic
4#20260481C211yesmaximal, characteristic, complemented
5Q8⋊#11715895C211yesmaximal, characteristic, complemented
6D4⋊#11715895C211yesmaximal, characteristic, complemented
7#556×C97C211yesmaximal, characteristic, complemented
8C2×#13095498—21no—
9C2×#13095496—21no—
10#15690807C2211yescharacteristic
11C4⋊#11715895C2211yescharacteristic
12C4⋊#11715895C2211yescharacteristic
13#15690730—21no—
14#15690730—21no—
15#15690730—22no—
16C4×#11715895C2211yescharacteristic
17C4⋊Q8×C97C2211yescharacteristic
18#126×C97—22no—
19#124×C97—21no—
20C22⋊Q8×C97—21no—
21C4×Q8×C97—21no—
22C4×Q8×C97C2211yescharacteristic
23C4×D4×C97C2211yescharacteristic
24#117×C97—21no—
25C2×D388—21no—
26C2×D388—21no—
27C2×D388—21no—
28#13095498—42no—
29#13095496—42no—
30C2×#11715895—21no—
31C2×#11715895—21no—
32C2×#11715895—21no—
33C2×Q8×C97—21no—
34C2×Q8×C97D411yescharacteristic
35C2×D4×C97D411yescharacteristic
36C22×C4×C97—21no—
37C4⋊C4×C97—22no—
38C4⋊C4×C97—21no—
39C4⋊C4×C97—21no—
40C4⋊C4×C97D411yescharacteristic
41C4⋊C4×C97D411yescharacteristic
42C4⋊C4×C97D412yes—
43C22⋊C4×C97—22no—
44C22⋊C4×C97—22no—
45C22⋊C4×C97—21no—
46C42×C97—22no—
47C42×C97—21no—
48C42×C97C2311yescharacteristic
49C23×C97—21no—
50C22×D97—41no—
51Q8×C97—21no—
52Q8×C97—21no—
53Q8×C97—41no—
54Q8×C97—41no—
55D4×C97—21no—
56D4×C97—21no—
57C2×C4×C97—21no—
58C2×C4×C97—21no—
59C2×C4×C97—21no—
60C2×C4×C97—22no—
61C2×C4×C97—41no—
62C2×C4×C97—41no—
63C2×C4×C97C2×D411yescharacteristic
64C2×C4×C97C2×D411yescharacteristic
65C2×C4×C97C2×D411yescharacteristic
66D388—42no—
67D388—41no—
68D388—41no—
69D388—41no—
70D388—41no—
71#11715895—41no—
72#11715895—41no—
73#11715895—41no—
74C22×C97—41no—
75C22×C97—41no—
76C22×C97—41no—
77C22×C97#12011yescharacteristic
78C2×D97—81no—
79C2×D97—81no—
80C2×D97—81no—
81C4×C97—21no—
82C4×C97—41no—
83C4×C97—41no—
84C4×C97—42no—
85C4×C97—41no—
86C4×C97#13511yescharacteristic
87C4×C97#13511yescharacteristic
88C4×C97AGL(1,ℤ/8)11yescharacteristic
89C4×C97AGL(1,ℤ/8)11yescharacteristic
90C2×C97—81no—
91C2×C97#44911yescharacteristic
92C2×C97#44911yescharacteristic
93C2×C97#45411yescharacteristic
94D97—161no—
95#1628—971nomaximal, Sylow
96C97#162811yescharacteristic, complemented, Sylow
97#556D9711yescharacteristic, complemented
98#486—971no—
99#460—971no—
100#459—971no—
101#332—971no—
102Q8⋊C8—971no—
103D4⋊C8—971no—
104C2×SD16—1941no—
105C2×D8—1941no—
106C4⋊Q8C2×D9711yescharacteristic
107#127—971no—
108#126—22no—
109#124—21no—
110C22⋊Q8—21no—
111C4×Q8—21no—
112C4×Q8C2×D9711yescharacteristic
113C4×D4C2×D9711yescharacteristic
114#117—21no—
115C4⋊C8—971no—
116C4⋊C8—971no—
117#102—1941no—
118#102—1941no—
119#102—1942no—
120C4×C8—971no—
121C2×Q8—21no—
122C2×Q8C22⋊D9711yescharacteristic
123C2×D4—1941no—
124C2×D4—1941no—
125C2×D4—1941no—
126C2×D4C22⋊D9711yescharacteristic
127C22×C4—21no—
128SD16—3882no—
129D8—3882no—
130C2×C8—1941no—
131C2×C8—1941no—
132C2×C8—1941no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4C22⋊D9711yescharacteristic
137C4⋊C4C22⋊D9711yescharacteristic
138C4⋊C4C22⋊D9712yes—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—21no—
142C42—22no—
143C42—21no—
144C42C22×D9711yescharacteristic
145C23—21no—
146C23—3881no—
147Q8—21no—
148Q8—21no—
149Q8—41no—
150Q8—41no—
151D4—21no—
152D4—21no—
153D4—3882no—
154D4—3881no—
155D4—3881no—
156D4—3881no—
157D4—3881no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4C2×C22⋊D9711yescharacteristic
165C2×C4C2×C22⋊D9711yescharacteristic
166C2×C4C2×C22⋊D9711yescharacteristic
167C8—3881no—
168C8—3881no—
169C8—3881no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—7761no—
174C22—7761no—
175C22—7761no—
176C22#1569077911yescharacteristic
177C4—21no—
178C4—41no—
179C4—41no—
180C4—42no—
181C4—41no—
182C4#1569083011yescharacteristic
183C4#1569083011yescharacteristic
184C4#1569083511yescharacteristic
185C4#1569083511yescharacteristic
186C2—81no—
187C2—15521no—
188C2#2026088011yescharacteristic
189C2#2026088011yescharacteristic
190C2#2026090811yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#556×C97C2
socleC22×C97#120
prime radical#20260930C2
hypercenter#556D97
hypocenterC97#1628
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #20261083C2semidirect
2. #20260935C2semidirect
4. #20260481C2semidirect
5. Q8⋊#11715895C2semidirect
6. D4⋊#11715895C2semidirect
7. #556×C97C2semidirect
96. C97#1628semidirect
97. #556D97semidirect

Maximal subgroups

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

subgroupindex
#202610832
#202609352
#202609302
#202604812
Q8⋊#117158952
D4⋊#117158952
#556×C972
#162897