Finite Group Databasebeta

← C3×#73088 #11920866

Subgroups — C3×#73088 #11920866

Order 1152.

313 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 44 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
1C3×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C3×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic, complemented
4#73088C311yesmaximal, characteristic, direct product factor
5C3×#1627—31nomaximal
6C22×C3×SL(2,𝔽3)—21no—
7C2×C3×GL(2,𝔽3)—21no—
8C2×C3×#213—41nomaximal
9C3×C4×SL(2,𝔽3)C2211yescharacteristic
10C3×D4×Q8D311yescharacteristic, complemented
11D4×SL(2,𝔽3)—21no—
12D4×SL(2,𝔽3)C2×C311yescharacteristic, complemented
13C4⋊GL(2,𝔽3)C2×C311yescharacteristic, complemented
14C3×#486—31no—
15C3×#460—31no—
16C3×#459—31no—
17SL(2,𝔽3)⋊C8C2×C311yescharacteristic, complemented
18C3×#332—31no—
19C3×Q8⋊C8—31no—
20C3×D4⋊C8—31no—
21C2×C32×D4—41no—
22C2×C3×D12—41no—
23C2×C3×SL(2,𝔽3)—41no—
24C2×C3×SL(2,𝔽3)D411yescharacteristic
25C3×GL(2,𝔽3)—41no—
26C3×#213—44no—
27C2×C3×C3⋊C8—41no—
28#1627—31noSylow
29C22×C3×Q8—21no—
30C22×SL(2,𝔽3)—22no—
31C22×SL(2,𝔽3)—21no—
32C2×GL(2,𝔽3)—21no—
33C2×C3×SD16—61no—
34C2×C3×D8—61no—
35C3×C4⋊Q8—31no—
36C3×#127—31no—
37C3×C22⋊Q8—61no—
38C3×C4×Q8C2×D311yescharacteristic
39C3×C4×D4—31no—
40C2×#213—41no—
41C4×SL(2,𝔽3)—21no—
42C4×SL(2,𝔽3)C22×C311yescharacteristic
43C3×C4⋊C8—31no—
44C3×C4⋊C8—31no—
45C3×#102—61no—
46C3×#102—61no—
47C3×#102—62no—
48C3×C4×C8—31no—
49C23×C32—81no—
50C22×C3×D3—81no—
51C32×D4—42no—
52C32×D4—81no—
53C2×C32×C4—41no—
54C3×D12—42no—
55C3×D12—81no—
56C3×SL(2,𝔽3)D811yescharacteristic, complemented
57C3×C3⋊C8—42no—
58D4×Q8C3×D311yescharacteristic, complemented
59#486—31no—
60#460—31no—
61#459—31no—
62#332—31no—
63Q8⋊C8—31no—
64D4⋊C8—31no—
65C2×C3×Q8—41no—
66C2×C3×Q8—61no—
67C2×C3×Q8—121no—
68C2×C3×Q8#6711yescharacteristic
69C2×C3×D4—41no—
70C2×C3×D4—61no—
71C2×C3×D4—61no—
72C2×C3×D4—61no—
73C2×C3×D4—81no—
74C2×C3×D4S411yescharacteristic
75C22×C3×C4—61no—
76C2×D12—41no—
77C2×SL(2,𝔽3)—21no—
78C2×SL(2,𝔽3)—42no—
79C2×SL(2,𝔽3)—41no—
80C2×SL(2,𝔽3)C3×D411yescharacteristic
81GL(2,𝔽3)—41no—
82C3×SD16—121no—
83C3×SD16—121no—
84C3×D8—122no—
85C2×C3×C8—61no—
86C2×C3×C8—61no—
87C2×C3×C8—61no—
88C3×C4⋊C4—31no—
89C3×C4⋊C4—32no—
90C3×C4⋊C4—31no—
91C3×C22⋊C4—61no—
92C3×C42—31no—
93#213—44no—
94C2×C3⋊C8—41no—
95C22×C32—41no—
96C22×C32—82no—
97C22×C32—161no—
98C22×C32—161no—
99C2×C3×D3—82no—
100C2×C3×D3—161no—
101C2×C3×D3—161no—
102C32×C4—41no—
103C32×C4—41no—
104C22×Q8—21no—
105C2×SD16—61no—
106C2×D8—61no—
107C4⋊Q8—31no—
108#127—31no—
109C22⋊Q8—61no—
110C4×Q8C2×C3×D311yescharacteristic
111C4×D4—31no—
112C4⋊C8—31no—
113C4⋊C8—31no—
114#102—61no—
115#102—61no—
116#102—62no—
117C4×C8—31no—
118C23×C3—21no—
119C23×C3—81no—
120C23×C3—82no—
121C23×C3—121no—
122C22×D3—81no—
123C3×Q8—31no—
124C3×Q8—121no—
125C3×Q8—121no—
126C3×Q8#21311yescharacteristic, complemented
127C3×D4—21no—
128C3×D4—42no—
129C3×D4—81no—
130C3×D4—82no—
131C3×D4—82no—
132C3×D4—121no—
133C3×D4—121no—
134C3×D4—122no—
135C3×D4—121no—
136C3×D4—121no—
137C3×D4GL(2,𝔽3)12yescomplemented
138C2×C3×C4—31no—
139C2×C3×C4—31no—
140C2×C3×C4—41no—
141C2×C3×C4—61no—
142C2×C3×C4—81no—
143C2×C3×C4—121no—
144C2×C3×C4—121no—
145C2×C3×C4C2×S411yescharacteristic
146D12—42no—
147D12—81no—
148SL(2,𝔽3)—21no—
149SL(2,𝔽3)C3×D811yescharacteristic, complemented
150C3×C8—121no—
151C3×C8—121no—
152C3×C8—121no—
153C3⋊C8—42no—
154C2×C32—41no—
155C2×C32—41no—
156C2×C32—41no—
157C2×C32—162no—
158C3×D3—162no—
159C2×Q8—41no—
160C2×Q8—61no—
161C2×Q8—121no—
162C2×Q8C3×#6711yescharacteristic
163C2×D4—61no—
164C2×D4—61no—
165C2×D4—61no—
166C2×D4C3×S411yescharacteristic
167C22×C4—61no—
168SD16—121no—
169SD16—121no—
170D8—122no—
171C2×C8—61no—
172C2×C8—61no—
173C2×C8—61no—
174C4⋊C4—31no—
175C4⋊C4—32no—
176C4⋊C4—31no—
177C22⋊C4—61no—
178C42—31no—
179C22×C3—22no—
180C22×C3—41no—
181C22×C3—41no—
182C22×C3—41no—
183C22×C3—82no—
184C22×C3—84no—
185C22×C3—81no—
186C22×C3—161no—
187C22×C3—161no—
188C22×C3—162no—
189C22×C3—162no—
190C22×C3—241no—
191C22×C3—241no—
192C22×C3—241no—
193C22×C3GL(2,ℤ/4)11yescharacteristic
194C2×D3—82no—
195C2×D3—161no—
196C2×D3—161no—
197C3×C4—31no—
198C3×C4—31no—
199C3×C4—41no—
200C3×C4—41no—
201C3×C4—61no—
202C3×C4—81no—
203C3×C4—81no—
204C3×C4—121no—
205C3×C4C2×GL(2,𝔽3)11yescharacteristic
206C3×C4#98311yescharacteristic
207C32—41noSylow
208C23—21no—
209C23—121no—
210Q8—31no—
211Q8—121no—
212Q8—121no—
213Q8C3×#21311yescharacteristic, complemented
214D4—21no—
215D4—121no—
216D4—121no—
217D4—122no—
218D4—121no—
219D4—121no—
220D4C3×GL(2,𝔽3)12yescomplemented
221C2×C4—31no—
222C2×C4—31no—
223C2×C4—61no—
224C2×C4—121no—
225C2×C4—121no—
226C2×C4C2×C3×S411yescharacteristic
227C8—121no—
228C8—121no—
229C8—121no—
230C2×C3—42no—
231C2×C3—41no—
232C2×C3—41no—
233C2×C3—41no—
234C2×C3—81no—
235C2×C3—81no—
236C2×C3—81no—
237C2×C3—162no—
238C2×C3—164no—
239C2×C3—481no—
240C2×C3#541511yescharacteristic
241C2×C3#542111yescharacteristic
242C2×C3#543111yescharacteristic
243D3—162no—
244C22—22no—
245C22—41no—
246C22—41no—
247C22—241no—
248C22—241no—
249C22—241no—
250C22C3×GL(2,ℤ/4)11yescharacteristic
251C4—31no—
252C4—31no—
253C4—61no—
254C4—121no—
255C4C2×C3×GL(2,𝔽3)11yescharacteristic
256C4C3×#98311yescharacteristic
257C3—41no—
258C3—81no—
259C3#7308811yescharacteristic, direct product factor
260C2—42no—
261C2—481no—
262C2C3×#541511yescharacteristic
263C2C3×#542111yescharacteristic
264C2C3×#543111yescharacteristic
subgroupG/H
center, socleC22×C3GL(2,ℤ/4)
commutator, cosocleC4×SL(2,𝔽3)C22×C3
FrattiniC2×C4C2×C3×S4
FittingC3×D4×Q8D3
hypercenterC2×C3×D4S4
hypocenterSL(2,𝔽3)C3×D8
LCSG ▷ C4×SL(2,𝔽3) ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C3×D4×SL(2,𝔽3)C2semidirect
2. C3×C4⋊GL(2,𝔽3)C2semidirect
3. C3×SL(2,𝔽3)⋊C8C2semidirect
4. #73088C3direct (N × H)
10. C3×D4×Q8D3semidirect
12. D4×SL(2,𝔽3)C2×C3semidirect
13. C4⋊GL(2,𝔽3)C2×C3semidirect
17. SL(2,𝔽3)⋊C8C2×C3semidirect
56. C3×SL(2,𝔽3)D8semidirect
58. D4×Q8C3×D3semidirect
126. C3×Q8#213semidirect
137. C3×D4GL(2,𝔽3)semidirect
149. SL(2,𝔽3)C3×D8semidirect
213. Q8C3×#213semidirect
220. D4C3×GL(2,𝔽3)semidirect
259. C3#73088direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D4×SL(2,𝔽3)2
C3×C4⋊GL(2,𝔽3)2
C3×SL(2,𝔽3)⋊C82
#730883
C3×#16273
C2×C3×#2134