Finite Group Databasebeta

← #3489⋊C3 #85608

Subgroups — #3489⋊C3 #85608

Order 384.

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

normal subgroups only

This group has 377 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⋊C3C211yesmaximal, characteristic, complemented
2C22×#41⋊C3C211yesmaximal, characteristic, complemented
3D4×SL(2,𝔽3)C212yesmaximal, complemented
4#548⋊C3C212yesmaximal, complemented
5#520⋊C3C211yesmaximal, characteristic, complemented
6#3489C311yesmaximal, characteristic, complemented, Sylow
72−1+4⋊C3—22no—
8C2×#41⋊C3—22no—
9C2×#41⋊C3C2211yescharacteristic
10C2×#41⋊C3C2211yescharacteristic
11C22×SL(2,𝔽3)C2211yescharacteristic
12C22×SL(2,𝔽3)C2212yescomplemented
13C3×#121—41nomaximal
14C4×SL(2,𝔽3)C2212yescomplemented
15C2×2−1+4C2×C311yescharacteristic, complemented
16C22×#41C2×C311yescharacteristic, complemented
17D4×Q8C2×C312yescomplemented
18#548C2×C312yescomplemented
19#547—32no—
20#546—32no—
21#537—31no—
22#536—31no—
23#525—31no—
24C2×C22⋊Q8—31no—
25#520C2×C311yescharacteristic, complemented
26C2×C3×D4—41no—
27C2×C3×D4—42no—
28C22×C3×C4—41no—
29#41⋊C3—21no—
30#41⋊C3D412yescomplemented
31C2×SL(2,𝔽3)—21no—
32C2×SL(2,𝔽3)—22no—
33C2×SL(2,𝔽3)C2311yescharacteristic
34C3×C4⋊C4—41no—
35C3×C22⋊C4—42no—
362−1+4—22no—
372−1+4—62no—
38C2×#41—22no—
39C2×#41—31no—
40C2×#41—32no—
41C2×#41—31no—
42C2×#41—31no—
43C2×#41—62no—
44C2×#41C22×C311yescharacteristic
45C2×#41C22×C311yescharacteristic
46C22×Q8—31no—
47C22×Q8C22×C312yes—
48C22×Q8C22×C311yescharacteristic
49C22×D4—31no—
50C23×C4—31no—
51C4⋊Q8—32no—
52#124—32no—
53#123—34no—
54#123—32no—
55C22⋊Q8—32no—
56C22⋊Q8—31no—
57C22⋊Q8—31no—
58C22⋊Q8—32no—
59C22⋊Q8—32no—
60C22⋊Q8—32no—
61#121—32no—
62#121—34no—
63#121—31no—
64#121A411yescharacteristic
65#120—32no—
66C4×Q8C22×C312yes—
67C4×D4—32no—
68C4×D4—32no—
69C4×D4—32no—
70#117—31no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4—31no—
73C2×C22⋊C4—32no—
74C23×C3—41no—
75C23×C3—42no—
76C3×D4—82no—
77C3×D4—84no—
78C2×C3×C4—41no—
79C2×C3×C4—42no—
80C2×C3×C4—41no—
81C2×C3×C4—82no—
82SL(2,𝔽3)C2×D411yescharacteristic
83C24—31no—
84#41—21no—
85#41—32no—
86#41—62no—
87#41—62no—
88#41—62no—
89#41—62no—
90#41—62no—
91#41—64no—
92#41—64no—
93#41—62no—
94#41—61no—
95#41C3×D412yescomplemented
96C2×Q8—21no—
97C2×Q8—22no—
98C2×Q8—31no—
99C2×Q8—32no—
100C2×Q8—31no—
101C2×Q8—62no—
102C2×Q8—62no—
103C2×Q8—61no—
104C2×Q8—62no—
105C2×Q8C23×C311yescharacteristic
106C2×D4—31no—
107C2×D4—31no—
108C2×D4—32no—
109C2×D4—32no—
110C2×D4—32no—
111C2×D4—62no—
112C2×D4—62no—
113C2×D4C2×A412yes—
114C2×D4C2×A411yescharacteristic
115C22×C4—31no—
116C22×C4—32no—
117C22×C4—32no—
118C22×C4—31no—
119C22×C4—32no—
120C22×C4—31no—
121C22×C4—31no—
122C22×C4—31no—
123C22×C4—62no—
124C22×C4—62no—
125C22×C4C2×A411yescharacteristic
126C4⋊C4—32no—
127C4⋊C4—34no—
128C4⋊C4—32no—
129C4⋊C4—31no—
130C4⋊C4—32no—
131C4⋊C4C2×A411yescharacteristic
132C22⋊C4—34no—
133C22⋊C4—32no—
134C22⋊C4—32no—
135C22⋊C4—32no—
136C22⋊C4—32no—
137C22⋊C4C2×A412yes—
138C42—32no—
139C22×C3—42no—
140C22×C3—41no—
141C22×C3—81no—
142C22×C3—81no—
143C22×C3—82no—
144C22×C3—82no—
145C22×C3—82no—
146C3×C4—42no—
147C3×C4—81no—
148C3×C4—82no—
149C23—31no—
150C23—31no—
151C23—62no—
152C23—62no—
153C23—121no—
154C23C22×A412yes—
155C23C22×A411yescharacteristic
156Q8—31no—
157Q8—62no—
158Q8—61no—
159Q8—62no—
160Q8—61no—
161Q8—62no—
162Q8C2×C3×D411yescharacteristic
163D4—24no—
164D4—22no—
165D4—32no—
166D4—32no—
167D4—61no—
168D4—61no—
169D4—64no—
170D4—64no—
171D4—122no—
172C2×C4—22no—
173C2×C4—31no—
174C2×C4—32no—
175C2×C4—32no—
176C2×C4—32no—
177C2×C4—32no—
178C2×C4—31no—
179C2×C4—61no—
180C2×C4—61no—
181C2×C4—62no—
182C2×C4—61no—
183C2×C4—61no—
184C2×C4—62no—
185C2×C4—62no—
186C2×C4—64no—
187C2×C4—62no—
188C2×C4—122no—
189C2×C4C22×A412yes—
190C2×C4C22×A411yescharacteristic
191C2×C4C22×A411yescharacteristic
192C2×C3—41no—
193C2×C3—41no—
194C2×C3—41no—
195C2×C3—82no—
196C2×C3—162no—
197C22—21no—
198C22—21no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—32no—
203C22—61no—
204C22—61no—
205C22—61no—
206C22—61no—
207C22—61no—
208C22—124no—
209C222−1+4⋊C312yescomplemented
210C22C23×A411yescharacteristic
211C4—21no—
212C4—22no—
213C4—31no—
214C4—31no—
215C4—61no—
216C4—62no—
217C4—62no—
218C4D4×A412yes—
219C3—41noSylow
220C2—22no—
221C2—42no—
222C2—62no—
223C2—121no—
224C2C2×D4×A411yescharacteristic
225C2C2×2−1+4⋊C311yescharacteristic
226C2#594511yescharacteristic
subgroupG/H
center, Frattini, socleC22C23×A4
commutator, cosocleC2×Q8C23×C3
Fitting#3489C3
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. C22×#41⋊C3C2semidirect
3. D4×SL(2,𝔽3)C2semidirect
4. #548⋊C3C2semidirect
5. #520⋊C3C2semidirect
6. #3489C3semidirect
12. C22×SL(2,𝔽3)C22semidirect
14. C4×SL(2,𝔽3)C22semidirect
15. C2×2−1+4C2×C3semidirect
16. C22×#41C2×C3semidirect
17. D4×Q8C2×C3semidirect
18. #548C2×C3semidirect
25. #520C2×C3semidirect
30. #41⋊C3D4semidirect
95. #41C3×D4semidirect
209. 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
C22×#41⋊C32
D4×SL(2,𝔽3)2
#548⋊C32
#520⋊C32
#34893
C3×#1214