Finite Group Databasebeta

← #3497⋊C3 #85609

Subgroups — #3497⋊C3 #85609

Order 384.

372 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 62 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
1C2×2−1+4⋊C3C212yesmaximal, complemented
2C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
3D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#548⋊C3C212yesmaximal, complemented
5C2×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#3497C311yesmaximal, characteristic, complemented, Sylow
72−1+4⋊C3—24no—
8C2×#41⋊C3C2212yes—
9C22×SL(2,𝔽3)C2212yes—
10C22×SL(2,𝔽3)C2211yescharacteristic
11C3×#121—41nomaximal
12C4×SL(2,𝔽3)C2211yescharacteristic
13C4×SL(2,𝔽3)C2211yescharacteristic
14C2×2−1+4C2×C312yescomplemented
15#550—31no—
16#550C2×C311yescharacteristic, complemented
17D4×Q8—31no—
18D4×Q8C2×C311yescharacteristic, complemented
19#548—32no—
20#548C2×C312yescomplemented
21#537—32no—
22#532—31no—
23#532—31no—
24C2×C4×Q8C2×C311yescharacteristic, complemented
25C2×C3×D4—42no—
26C2×C3×D4—41no—
27C22×C3×C4—41no—
28#41⋊C3—22no—
29C2×SL(2,𝔽3)—22no—
30C2×SL(2,𝔽3)D412yescomplemented
31C2×SL(2,𝔽3)C2311yescharacteristic
32C3×C4⋊C4—41no—
33C3×C22⋊C4—42no—
342−1+4—24no—
352−1+4—64no—
36C2×#41—32no—
37C2×#41—32no—
38C2×#41—32no—
39C2×#41C22×C312yes—
40C22×Q8—32no—
41C22×Q8C22×C312yes—
42C22×Q8C22×C311yescharacteristic
43C4⋊Q8—31no—
44C4⋊Q8—31no—
45#127—31no—
46#127—31no—
47#124—32no—
48#124—32no—
49C22⋊Q8—34no—
50C22⋊Q8—32no—
51C22⋊Q8—32no—
52#121—32no—
53#121—32no—
54#121—34no—
55#121—32no—
56#121—31no—
57#121—32no—
58#121A411yescharacteristic
59C4×Q8—31no—
60C4×Q8—31no—
61C4×Q8C22×C311yescharacteristic
62C4×Q8C22×C311yescharacteristic
63C4×D4—32no—
64C4×D4—32no—
65C4×D4—31no—
66C4×D4—31no—
67C4×D4—32no—
68C2×C4⋊C4—31no—
69C2×C42—31no—
70C23×C3—41no—
71C23×C3—42no—
72C3×D4—82no—
73C3×D4—84no—
74C2×C3×C4—41no—
75C2×C3×C4—41no—
76C2×C3×C4—42no—
77C2×C3×C4—82no—
78SL(2,𝔽3)C2×D411yescharacteristic
79#41—22no—
80#41—64no—
81#41—64no—
82#41—64no—
83#41—64no—
84#41—64no—
85#41—64no—
86#41—62no—
87C2×Q8—22no—
88C2×Q8—32no—
89C2×Q8—31no—
90C2×Q8—32no—
91C2×Q8—32no—
92C2×Q8—64no—
93C2×Q8—64no—
94C2×Q8—62no—
95C2×Q8C3×D412yescomplemented
96C2×Q8C23×C311yescharacteristic
97C2×D4—32no—
98C2×D4—32no—
99C2×D4—32no—
100C2×D4—32no—
101C2×D4—32no—
102C2×D4—31no—
103C2×D4C2×A411yescharacteristic
104C2×D4C2×A412yes—
105C22×C4—31no—
106C22×C4—34no—
107C22×C4—32no—
108C22×C4—31no—
109C22×C4—32no—
110C22×C4C2×A411yescharacteristic
111C4⋊C4—31no—
112C4⋊C4—31no—
113C4⋊C4—32no—
114C4⋊C4—32no—
115C4⋊C4—31no—
116C4⋊C4—32no—
117C4⋊C4C2×A411yescharacteristic
118C22⋊C4—34no—
119C22⋊C4—32no—
120C22⋊C4—32no—
121C22⋊C4—32no—
122C22⋊C4C2×A412yes—
123C42—31no—
124C42—31no—
125C42—31no—
126C42—31no—
127C22×C3—42no—
128C22×C3—41no—
129C22×C3—81no—
130C22×C3—81no—
131C22×C3—82no—
132C22×C3—82no—
133C22×C3—82no—
134C3×C4—42no—
135C3×C4—81no—
136C3×C4—82no—
137C23—32no—
138C23C22×A412yes—
139C23C22×A411yescharacteristic
140Q8—32no—
141Q8—31no—
142Q8—32no—
143Q8—64no—
144Q8—62no—
145Q8—62no—
146Q8C2×C3×D411yescharacteristic
147D4—24no—
148D4—22no—
149D4—64no—
150D4—62no—
151D4—62no—
152D4—64no—
153D4—62no—
154D4—122no—
155D4—122no—
156C2×C4—22no—
157C2×C4—32no—
158C2×C4—32no—
159C2×C4—31no—
160C2×C4—31no—
161C2×C4—31no—
162C2×C4—31no—
163C2×C4—32no—
164C2×C4—62no—
165C2×C4—62no—
166C2×C4—62no—
167C2×C4—62no—
168C2×C4—68no—
169C2×C4—121no—
170C2×C4C22×A411yescharacteristic
171C2×C4C22×A412yes—
172C2×C4C22×A411yescharacteristic
173C2×C3—41no—
174C2×C3—41no—
175C2×C3—41no—
176C2×C3—82no—
177C2×C3—162no—
178C22—21no—
179C22—21no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—62no—
184C22—62no—
185C22—62no—
186C22D4×A412yes—
187C22C23×A411yescharacteristic
188C4—22no—
189C4—21no—
190C4—32no—
191C4—31no—
192C4—31no—
193C4—62no—
194C4—61no—
195C4—61no—
196C42−1+4⋊C312yescomplemented
197C3—41noSylow
198C2—22no—
199C2—42no—
200C2—122no—
201C2C2×D4×A411yescharacteristic
202C2C2×2−1+4⋊C311yescharacteristic
203C2#594511yescharacteristic
subgroupG/H
center, Frattini, socleC22C23×A4
commutator, cosocleC2×Q8C23×C3
Fitting#3497C3
hypercenter#121A4
hypocenterQ8C2×C3×D4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×2−1+4⋊C3C2semidirect
2. C4⋊(#41⋊C3)C2semidirect
3. D4×SL(2,𝔽3)C2semidirect
4. #548⋊C3C2semidirect
5. C2×C4×SL(2,𝔽3)C2semidirect
6. #3497C3semidirect
14. C2×2−1+4C2×C3semidirect
16. #550C2×C3semidirect
18. D4×Q8C2×C3semidirect
20. #548C2×C3semidirect
24. C2×C4×Q8C2×C3semidirect
30. C2×SL(2,𝔽3)D4semidirect
95. C2×Q8C3×D4semidirect
196. C42−1+4⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×2−1+4⋊C32
C4⋊(#41⋊C3)2
D4×SL(2,𝔽3)2
#548⋊C32
C2×C4×SL(2,𝔽3)2
#34973
C3×#1214