Finite Group Databasebeta

← C4×2−1+4⋊C3 #85581

Subgroups — C4×2−1+4⋊C3 #85581

Order 384.

372 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 86 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⋊C3C211yesmaximal, characteristic
2#520⋊C3C211yesmaximal, characteristic, complemented
3#519⋊C3C212yesmaximal, complemented
4C4×#41⋊C3C211yesmaximal, characteristic, complemented
5C2×C4×SL(2,𝔽3)C212yesmaximal, complemented
6C4×2−1+4C311yesmaximal, characteristic, complemented, Sylow
72−1+4⋊C3C414yesdirect product factor
8C2×#41⋊C3C2211yescharacteristic
9C22×SL(2,𝔽3)C2212yes—
10C3×C4×D4—41nomaximal
11C4×SL(2,𝔽3)C2211yescharacteristic
12C4×SL(2,𝔽3)C2212yes—
13C4×SL(2,𝔽3)C2211yescharacteristic
14C2×2−1+4C2×C311yescharacteristic
15#520—31no—
16#520—32no—
17#520C2×C311yescharacteristic, complemented
18#519—31no—
19#519C2×C312yescomplemented
20C4×#41—31no—
21C4×#41—32no—
22C4×#41C2×C311yescharacteristic, complemented
23C2×C4×Q8—31no—
24C2×C4×Q8C2×C312yescomplemented
25C2×C3×D4—41no—
26C22×C3×C4—42no—
27#41⋊C3C2×C412yescomplemented
28C2×SL(2,𝔽3)C2×C414yescomplemented
29C2×SL(2,𝔽3)C2311yescharacteristic
30C3×C4⋊C4—41no—
31C3×C22⋊C4—42no—
32C3×C42—41no—
332−1+4—34no—
342−1+4C3×C414yescomplemented
35C2×#41—31no—
36C2×#41—32no—
37C2×#41C22×C311yescharacteristic
38C22×Q8—31no—
39C22×Q8C22×C312yes—
40C4×Q8—32no—
41C4×Q8—31no—
42C4×Q8—32no—
43C4×Q8—32no—
44C4×Q8—31no—
45C4×Q8—32no—
46C4×Q8—32no—
47C4×Q8C22×C311yescharacteristic
48C4×Q8C22×C312yes—
49C4×Q8C22×C311yescharacteristic
50C4×D4—34no—
51C4×D4—31no—
52C4×D4—32no—
53C4×D4—31no—
54C4×D4—31no—
55C4×D4—32no—
56C4×D4—32no—
57C4×D4A411yescharacteristic
58#117—34no—
59#117—31no—
60#117—32no—
61#117—32no—
62#117—31no—
63C2×C4⋊C4—32no—
64C2×C4⋊C4—32no—
65C2×C4⋊C4—31no—
66C2×C42—32no—
67C2×C42—32no—
68C2×C42—31no—
69C23×C3—42no—
70C3×D4—44no—
71C2×C3×C4—41no—
72C2×C3×C4—41no—
73C2×C3×C4—42no—
74C2×C3×C4—41no—
75C2×C3×C4—84no—
76SL(2,𝔽3)C22×C411yescharacteristic
77#41—38no—
78#41—34no—
79#41—34no—
80#41—38no—
81#41—32no—
82#41C2×C3×C412yescomplemented
83C2×Q8—34no—
84C2×Q8—34no—
85C2×Q8—31no—
86C2×Q8—32no—
87C2×Q8—34no—
88C2×Q8C2×C3×C414yescomplemented
89C2×Q8C23×C311yescharacteristic
90C2×D4—31no—
91C2×D4—32no—
92C2×D4C2×A411yescharacteristic
93C22×C4—32no—
94C22×C4—32no—
95C22×C4—31no—
96C22×C4—32no—
97C22×C4—32no—
98C22×C4—31no—
99C22×C4—31no—
100C22×C4C2×A412yes—
101C4⋊C4—31no—
102C4⋊C4—32no—
103C4⋊C4—31no—
104C4⋊C4—34no—
105C4⋊C4—32no—
106C4⋊C4—31no—
107C4⋊C4—32no—
108C4⋊C4C2×A411yescharacteristic
109C22⋊C4—32no—
110C22⋊C4—32no—
111C22⋊C4—32no—
112C22⋊C4C2×A412yes—
113C42—34no—
114C42—32no—
115C42—31no—
116C42—32no—
117C42—31no—
118C42—32no—
119C42—31no—
120C42C2×A411yescharacteristic
121C22×C3—44no—
122C22×C3—41no—
123C22×C3—82no—
124C22×C3—82no—
125C3×C4—42no—
126C3×C4—42no—
127C3×C4—81no—
128C3×C4—82no—
129C23—31no—
130C23C22×A412yes—
131Q8—34no—
132Q8—31no—
133Q8—34no—
134Q8—34no—
135Q8C22×C3×C411yescharacteristic
136D4—32no—
137D4—32no—
138D4—38no—
139D4C4×A414yes—
140C2×C4—24no—
141C2×C4—38no—
142C2×C4—34no—
143C2×C4—34no—
144C2×C4—32no—
145C2×C4—31no—
146C2×C4—32no—
147C2×C4—32no—
148C2×C4—31no—
149C2×C4—32no—
150C2×C4—31no—
151C2×C4—62no—
152C2×C4—121no—
153C2×C4—122no—
154C2×C4—122no—
155C2×C4C22×A412yes—
156C2×C4C22×A411yescharacteristic
157C2×C4C22×A411yescharacteristic
158C2×C4C22×A411yescharacteristic
159C2×C3—41no—
160C2×C3—41no—
161C2×C3—41no—
162C2×C3—84no—
163C22—22no—
164C22—22no—
165C22—32no—
166C22—61no—
167C22—61no—
168C22C2×C4×A414yes—
169C22C23×A411yescharacteristic
170C4—22no—
171C4—21no—
172C4—34no—
173C4—31no—
174C4—31no—
175C4—61no—
176C4—61no—
177C4—62no—
178C4C2×C4×A412yes—
179C42−1+4⋊C312yesdirect product factor
180C3—41noSylow
181C2—24no—
182C2—62no—
183C2C22×C4×A411yescharacteristic
184C2C2×2−1+4⋊C311yescharacteristic
185C2#594511yescharacteristic
subgroupG/H
centerC2×C4C22×A4
commutator, hypocenterQ8C22×C3×C4
Frattini, socleC22C23×A4
FittingC4×2−1+4C3
prime radicalC2×2−1+4⋊C3C2
cosocleC2×Q8C23×C3
hypercenterC4×D4A4
LCSG ▷ Q8
UCSC1 ◁ C2×C4 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. #520⋊C3C2semidirect
3. #519⋊C3C2semidirect
4. C4×#41⋊C3C2semidirect
5. C2×C4×SL(2,𝔽3)C2semidirect
6. C4×2−1+4C3semidirect
7. 2−1+4⋊C3C4direct (N × H)
17. #520C2×C3semidirect
19. #519C2×C3semidirect
22. C4×#41C2×C3semidirect
24. C2×C4×Q8C2×C3semidirect
27. #41⋊C3C2×C4semidirect
28. C2×SL(2,𝔽3)C2×C4semidirect
34. 2−1+4C3×C4semidirect
82. #41C2×C3×C4semidirect
88. C2×Q8C2×C3×C4semidirect
179. C42−1+4⋊C3direct (N × H)

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
#520⋊C32
#519⋊C32
C4×#41⋊C32
C2×C4×SL(2,𝔽3)2
C4×2−1+43
C3×C4×D44