Finite Group Databasebeta

← #3486⋊C3 #85605

Subgroups — #3486⋊C3 #85605

Order 384.

402 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 64 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, complemented
2C23×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3D4×SL(2,𝔽3)C212yesmaximal, complemented
4#548⋊C3C212yesmaximal, complemented
5#519⋊C3C211yesmaximal, characteristic, complemented
6#3486C311yesmaximal, characteristic, complemented, Sylow
72−1+4⋊C3—22no—
8C2×#41⋊C3C2211yescharacteristic, complemented
9C22×SL(2,𝔽3)—22no—
10C22×SL(2,𝔽3)C2212yes—
11C22×SL(2,𝔽3)C2211yescharacteristic
12C22×SL(2,𝔽3)C2211yescharacteristic, complemented
13C3×#120—41nomaximal
14C4×SL(2,𝔽3)C2212yescomplemented
15C2×2−1+4C2×C311yescharacteristic, complemented
16C23×Q8C2×C311yescharacteristic, complemented
17D4×Q8—32no—
18D4×Q8C2×C312yescomplemented
19#548—32no—
20#548C2×C312yescomplemented
21#536—31no—
22#536—31no—
23#525—31no—
24#525—31no—
25#519C2×C311yescharacteristic, complemented
26C24×C3—41no—
27C2×C3×D4—41no—
28C2×C3×D4—42no—
29#41⋊C3—21no—
30C2×SL(2,𝔽3)—22no—
31C2×SL(2,𝔽3)—21no—
32C2×SL(2,𝔽3)D412yescomplemented
33C2×SL(2,𝔽3)C2311yescharacteristic
34C3×C22⋊C4—41no—
35C3×C22⋊C4—42no—
362−1+4—22no—
372−1+4—62no—
38C2×#41—31no—
39C2×#41—31no—
40C2×#41—31no—
41C2×#41C22×C311yescharacteristic, complemented
42C22×Q8—22no—
43C22×Q8—31no—
44C22×Q8—31no—
45C22×Q8—32no—
46C22×Q8—62no—
47C22×Q8C22×C312yes—
48C22×Q8C22×C311yescharacteristic
49C22×Q8C22×C311yescharacteristic, complemented
50C23×C4—31no—
51C4⋊Q8—32no—
52C4⋊Q8—32no—
53#124—32no—
54#124—32no—
55#123—32no—
56#123—31no—
57#123—31no—
58C22⋊Q8—34no—
59C22⋊Q8—32no—
60C22⋊Q8—32no—
61C22⋊Q8—32no—
62C22⋊Q8—32no—
63#121—32no—
64#121—32no—
65#120—31no—
66#120A411yescharacteristic
67C4×Q8—32no—
68C4×Q8C22×C312yescomplemented
69C4×D4—34no—
70C4×D4—32no—
71C4×D4—32no—
72#117—31no—
73#117—31no—
74C23×C3—41no—
75C23×C3—42no—
76C23×C3—41no—
77C23×C3—82no—
78C23×C3—84no—
79C3×D4—82no—
80C3×D4—84no—
81C2×C3×C4—42no—
82C2×C3×C4—41no—
83SL(2,𝔽3)C2×D411yescharacteristic
84C24C2×A411yescharacteristic
85#41—21no—
86#41—62no—
87#41—62no—
88#41—62no—
89#41—62no—
90#41—62no—
91#41—62no—
92#41—61no—
93C2×Q8—22no—
94C2×Q8—21no—
95C2×Q8—32no—
96C2×Q8—31no—
97C2×Q8—31no—
98C2×Q8—32no—
99C2×Q8—32no—
100C2×Q8—31no—
101C2×Q8—62no—
102C2×Q8—62no—
103C2×Q8—62no—
104C2×Q8—64no—
105C2×Q8—62no—
106C2×Q8—62no—
107C2×Q8—61no—
108C2×Q8—64no—
109C2×Q8C3×D412yescomplemented
110C2×Q8C23×C311yescharacteristic
111C2×D4—31no—
112C2×D4—32no—
113C2×D4—31no—
114C2×D4—31no—
115C2×D4C2×A412yes—
116C2×D4C2×A411yescharacteristic
117C22×C4—31no—
118C22×C4—32no—
119C22×C4—32no—
120C22×C4—32no—
121C22×C4—31no—
122C22×C4—31no—
123C22×C4—31no—
124C22×C4—62no—
125C22×C4—62no—
126C4⋊C4—32no—
127C4⋊C4—34no—
128C4⋊C4—32no—
129C4⋊C4—32no—
130C4⋊C4—32no—
131C22⋊C4—34no—
132C22⋊C4—31no—
133C22⋊C4—32no—
134C22⋊C4—32no—
135C22⋊C4—32no—
136C22⋊C4C2×A411yescharacteristic
137C22⋊C4C2×A412yes—
138C42—32no—
139C42—32no—
140C22×C3—42no—
141C22×C3—44no—
142C22×C3—41no—
143C22×C3—82no—
144C22×C3—82no—
145C22×C3—82no—
146C22×C3—82no—
147C22×C3—81no—
148C22×C3—88no—
149C3×C4—82no—
150C3×C4—81no—
151C23—22no—
152C23—24no—
153C23—31no—
154C23C22×A411yescharacteristic
155C23C22×A411yescharacteristic
156C23C22×A412yes—
157Q8—32no—
158Q8—31no—
159Q8—32no—
160Q8—62no—
161Q8—62no—
162Q8—61no—
163Q8—62no—
164Q8—61no—
165Q8—64no—
166Q8C2×C3×D411yescharacteristic
167D4—22no—
168D4—24no—
169D4—62no—
170D4—61no—
171D4—61no—
172D4—62no—
173D4—64no—
174C2×C4—32no—
175C2×C4—32no—
176C2×C4—31no—
177C2×C4—31no—
178C2×C4—32no—
179C2×C4—32no—
180C2×C4—31no—
181C2×C4—61no—
182C2×C4—61no—
183C2×C4—62no—
184C2×C4—64no—
185C2×C4—62no—
186C2×C4—61no—
187C2×C4—61no—
188C2×C4—64no—
189C2×C4—62no—
190C2×C4C22×A412yes—
191C2×C4C22×A411yescharacteristic
192C2×C3—41no—
193C2×C3—42no—
194C2×C3—82no—
195C2×C3—84no—
196C2×C3—161no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—21no—
202C22—22no—
203C22—61no—
204C22—62no—
205C22D4×A412yes—
206C222−1+4⋊C314yescomplemented
207C22C23×A411yescharacteristic
208C4—21no—
209C4—22no—
210C4—32no—
211C4—31no—
212C4—31no—
213C4—61no—
214C4—62no—
215C4—62no—
216C3—41noSylow
217C2—22no—
218C2—24no—
219C2—41no—
220C2—121no—
221C2C2×D4×A411yescharacteristic
222C2C2×2−1+4⋊C312yes—
subgroupG/H
center, Frattini, socleC22C23×A4
commutator, cosocleC2×Q8C23×C3
Fitting#3486C3
hypercenter#120A4
hypocenterQ8C2×C3×D4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×2−1+4⋊C3C2semidirect
2. C23×SL(2,𝔽3)C2semidirect
3. D4×SL(2,𝔽3)C2semidirect
4. #548⋊C3C2semidirect
5. #519⋊C3C2semidirect
6. #3486C3semidirect
8. C2×#41⋊C3C22semidirect
12. C22×SL(2,𝔽3)C22semidirect
14. C4×SL(2,𝔽3)C22semidirect
15. C2×2−1+4C2×C3semidirect
16. C23×Q8C2×C3semidirect
18. D4×Q8C2×C3semidirect
20. #548C2×C3semidirect
25. #519C2×C3semidirect
32. C2×SL(2,𝔽3)D4semidirect
41. C2×#41C22×C3semidirect
49. C22×Q8C22×C3semidirect
68. C4×Q8C22×C3semidirect
109. C2×Q8C3×D4semidirect
206. C222−1+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
C23×SL(2,𝔽3)2
D4×SL(2,𝔽3)2
#548⋊C32
#519⋊C32
#34863
C3×#1204