Finite Group Databasebeta

← #27143662

Subgroups — #27143662

Order 12416.

230 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 51 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
1C2×#15690833C211yesmaximal, characteristic, complemented
2C2×#15690753C211yesmaximal, characteristic, complemented
3C2×D1552C211yesmaximal, characteristic, complemented
4#20260972C211yesmaximal, characteristic, complemented
5#20260535C211yesmaximal, characteristic, complemented
6#20260514C211yesmaximal, characteristic, complemented
7#358×C97C211yesmaximal, characteristic, complemented
8C2×#13095512—21no—
9#15690833—22no—
10#15690833—22no—
11C2×#13095498—21no—
12C2×D776C2211yescharacteristic
13C2×D776C2211yescharacteristic
14C2×C8×D97C2211yescharacteristic
15#15690785—21no—
16#15690753—22no—
17D1552—22no—
18C2×#13095486C2211yescharacteristic
19#15690730—21no—
20C2×Q16×C97C2211yescharacteristic
21C2×C16×C97C2211yescharacteristic
22#107×C97C2211yescharacteristic
23#13095512—24no—
24#13095512—42no—
25C2×D388—21no—
26C2×D388—21no—
27C2×D388—41no—
28C2×D388—41no—
29C2×C4×D97—41no—
30C2×C4×D97D411yescharacteristic
31#13095498—24no—
32D776—21no—
33D776—21no—
34D776—21no—
35D776D412yescomplemented
36C2×#11715895D411yescharacteristic
37C8×D97—21no—
38C8×D97—21no—
39#13095488—41no—
40#13095486—21no—
41C2×Q8×C97—21no—
42Q16×C97—21no—
43Q16×C97D412yescomplemented
44C2×C8×C97C2311yescharacteristic
45C4⋊C4×C97—21no—
46C16×C97—21no—
47C22×D97—41no—
48C22×D97—41no—
49C22×D97D811yescharacteristic
50Q8×C97—22no—
51Q8×C97—41no—
52C2×C4×C97—41no—
53C2×C4×C97—41no—
54C2×C4×C97C2×D411yescharacteristic
55C2×Dic97D811yescharacteristic
56D388—22no—
57D388—44no—
58D388—41no—
59D388—41no—
60D388—41no—
61D388—82no—
62C4×D97—21no—
63C4×D97—21no—
64C4×D97—44no—
65C8×C97C2×D411yescharacteristic
66C8×C97C2×D411yescharacteristic
67#11715895—21no—
68C22×C97C2×D811yescharacteristic
69C2×D97—21no—
70C2×D97—21no—
71C2×D97—21no—
72C2×D97—42no—
73C2×D97—81no—
74C2×D97—81no—
75C2×D97—81no—
76C2×D97—81no—
77C2×D97—81no—
78C4×C97—42no—
79C4×C97—81no—
80C4×C97#12011yescharacteristic
81C4×C97AGL(1,ℤ/8)11yescharacteristic
82Dic97—21no—
83C2×C97#44711yescharacteristic
84C2×C97#50811yescharacteristic
85C2×C97#50911yescharacteristic
86D97—41no—
87D97—82no—
88D97—161no—
89#2186—971nomaximal, Sylow
90C97#218611yescharacteristic, complemented, Sylow
91C2×#135—971no—
92C2×SD32—971no—
93C2×D16—971no—
94#466—971no—
95#358D9711yescharacteristic, complemented
96#357—971no—
97C22⋊C16—971no—
98C2×#41—1941no—
99#135—1942no—
100#135—1942no—
101C2×Q16C2×D9711yescharacteristic
102C2×SD16—1941no—
103C2×D8—971no—
104C2×D8—971no—
105C22×C8—971no—
106#121—1941no—
107SD32—1942no—
108D16—1942no—
109C2×C16—971no—
110C2×C16C2×D9711yescharacteristic
111#107C2×D9711yescharacteristic
112#102—1941no—
113#41—3882no—
114#41—1944no—
115C2×Q8—21no—
116C2×D4—3881no—
117C2×D4—3881no—
118C2×D4—1941no—
119C2×D4—1941no—
120C22×C4—971no—
121C22×C4—3881no—
122Q16—21no—
123Q16D38812yescomplemented
124SD16—1944no—
125D8—972no—
126D8—1941no—
127D8—1941no—
128D8—1941no—
129C2×C8—971no—
130C2×C8—1941no—
131C2×C8—1941no—
132C2×C8C22×D9711yescharacteristic
133C4⋊C4—21no—
134C22⋊C4—3881no—
135C16—21no—
136C16—1941no—
137C23—971no—
138C23—3881no—
139C23—3881no—
140Q8—22no—
141Q8—41no—
142D4—7762no—
143D4—3884no—
144D4—3881no—
145D4—3881no—
146D4—3881no—
147D4—1942no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—971no—
151C2×C4—3884no—
152C2×C4—1941no—
153C2×C4—1941no—
154C2×C4D4×D9711yescharacteristic
155C8—1941no—
156C8D4×D9711yescharacteristic
157C8C2×D38811yescharacteristic
158C22—7761no—
159C22—7761no—
160C22—7761no—
161C22—7761no—
162C22—7761no—
163C22—3882no—
164C22—1941no—
165C22—1941no—
166C22—1941no—
167C22D8×D9711yescharacteristic
168C4—42no—
169C4—81no—
170C4—1941no—
171C4#1569078011yescharacteristic
172C4#1569083711yescharacteristic
173C2—7762no—
174C2—15521no—
175C2—3881no—
176C2#2026087011yescharacteristic
177C2#2026116811yescharacteristic
178C2#2026117211yescharacteristic
subgroupG/H
centerC22D8×D97
commutator, cosocleC2×C8×C97C23
FrattiniC2×C8C22×D97
Fitting#358×C97C2
socleC22×C97C2×D8
hypercenter#358D97
hypocenterC97#2186
LCSG ▷ C2×C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ #358

Complements

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

NH ≅ G/Ntype
1. C2×#15690833C2semidirect
2. C2×#15690753C2semidirect
3. C2×D1552C2semidirect
4. #20260972C2semidirect
5. #20260535C2semidirect
6. #20260514C2semidirect
7. #358×C97C2semidirect
35. D776D4semidirect
43. Q16×C97D4semidirect
90. C97#2186semidirect
95. #358D97semidirect
123. Q16D388semidirect

Maximal subgroups

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

subgroupindex
C2×#156908332
C2×#156907532
C2×D15522
#202609722
#202605352
#202605142
#358×C972
#218697