Finite Group Databasebeta

← C3×#73156 #11920934

Subgroups — C3×#73156 #11920934

Order 1152.

429 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 56 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×C3×2−1+4⋊C3C211yesmaximal, characteristic
2C3×#5420C212yesmaximal, complemented
3#73156C311yesmaximal, characteristic, direct product factor
4C3×#1796—31nomaximal
5C3×2−1+4⋊C3—22no—
6C2×C3×#41⋊C3C411yescharacteristic, complemented
7C22×C3×SL(2,𝔽3)C411yescharacteristic, complemented
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C3×#859—22no—
10C3×#834—41nomaximal
11C2×C3×2−1+4D311yescharacteristic
12C2×2−1+4⋊C3—21no—
13C2×2−1+4⋊C3C2×C311yescharacteristic
14#5420C2×C312yescomplemented
15C3×#421—32no—
16C3×#419—32no—
17C3×#413—31no—
18C3×#410—31no—
19C2×C32×D4—41no—
20C3×#41⋊C3—21no—
21C2×C3×SL(2,𝔽3)—21no—
22C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
23C2×C3×SL(2,𝔽3)D412yes—
24C3×#217—42no—
25#1796—31noSylow
26C3×2−1+4—22no—
27C3×2−1+4—62no—
28C2×C3×#41—31no—
29C2×C3×#41—31no—
30C2×C3×#41—31no—
31C2×C3×#41Dic311yescharacteristic, complemented
32C22×C3×Q8—31no—
33C22×C3×Q8Dic311yescharacteristic, complemented
34C22×C3×Q8C2×D311yescharacteristic
352−1+4⋊C3—24no—
362−1+4⋊C3—22no—
37C2×#41⋊C3—21no—
38C2×#41⋊C3C3×C411yescharacteristic, complemented
39C22×SL(2,𝔽3)—21no—
40C22×SL(2,𝔽3)—21no—
41C22×SL(2,𝔽3)C3×C411yescharacteristic, complemented
42C22×SL(2,𝔽3)C22×C311yescharacteristic
43C2×C3×M16—32no—
44C3×#117—32no—
45#859—22no—
46C3×#104—64no—
47C3×#103—62no—
48C3×#103—62no—
49C3×#101—61no—
50C3×#100—61no—
51C3×#99—61no—
52C3×#99—61no—
53#834—41no—
54C23×C32—41no—
55C23×C32—41no—
56C32×D4—82no—
57C2×C32×C4—41no—
58C2×C3×Dic3—82no—
59C3×SL(2,𝔽3)C22⋊C411yescharacteristic
60C2×2−1+4C3×D311yescharacteristic
61#421—32no—
62#419—32no—
63#413—31no—
64#410—31no—
65C3×#41—21no—
66C3×#41—34no—
67C3×#41—62no—
68C3×#41—62no—
69C3×#41—62no—
70C3×#41—62no—
71C3×#41—62no—
72C3×#41—61no—
73C2×C3×Q8—21no—
74C2×C3×Q8—31no—
75C2×C3×Q8—32no—
76C2×C3×Q8—31no—
77C2×C3×Q8—31no—
78C2×C3×Q8—62no—
79C2×C3×Q8—62no—
80C2×C3×Q8—61no—
81C2×C3×Q8C2×Dic311yescharacteristic
82C2×C3×Q8#6712yes—
83C2×C3×D4—31no—
84C2×C3×D4—31no—
85C2×C3×D4—31no—
86C2×C3×D4—41no—
87C2×C3×D4—81no—
88C2×C3×D4S411yescharacteristic
89C22×C3×C4—31no—
90C22×C3×C4—31no—
91C22×C3×C4—31no—
92C22×C3×C4—61no—
93#41⋊C3—22no—
94#41⋊C3—21no—
95C2×SL(2,𝔽3)—22no—
96C2×SL(2,𝔽3)—21no—
97C2×SL(2,𝔽3)—21no—
98C2×SL(2,𝔽3)—22no—
99C2×SL(2,𝔽3)C2×C3×C411yescharacteristic
100C2×SL(2,𝔽3)C3×D412yes—
101C3×M16—62no—
102C3×M16—62no—
103C2×C3×C8—62no—
104C3×C4⋊C4—62no—
105C3×C22⋊C4—62no—
106C3×C42—62no—
107#217—42no—
108C22×C32—41no—
109C22×C32—42no—
110C22×C32—81no—
111C22×C32—161no—
112C22×C32—161no—
113C32×C4—81no—
114C3×Dic3—162no—
1152−1+4—22no—
1162−1+4—62no—
117C2×#41—31no—
118C2×#41—31no—
119C2×#41—31no—
120C2×#41C3×Dic311yescharacteristic, complemented
121C22×Q8—31no—
122C22×Q8C3×Dic311yescharacteristic, complemented
123C22×Q8C2×C3×D311yescharacteristic
124C2×M16—32no—
125#117—32no—
126#104—64no—
127#103—62no—
128#103—62no—
129#101—61no—
130#100—61no—
131#99—61no—
132#99—61no—
133C23×C3—31no—
134C23×C3—41no—
135C23×C3—41no—
136C23×C3—81no—
137C23×C3—81no—
138C23×C3SL(2,ℤ/4)11yescharacteristic
139C23×C3C2×S411yescharacteristic
140C3×Q8—31no—
141C3×Q8—32no—
142C3×Q8—32no—
143C3×Q8—62no—
144C3×Q8—61no—
145C3×Q8—61no—
146C3×Q8#21711yescharacteristic
147C3×D4—22no—
148C3×D4—32no—
149C3×D4—62no—
150C3×D4—61no—
151C3×D4—62no—
152C3×D4—82no—
153C3×D4—84no—
154C2×C3×C4—31no—
155C2×C3×C4—31no—
156C2×C3×C4—32no—
157C2×C3×C4—32no—
158C2×C3×C4—31no—
159C2×C3×C4—41no—
160C2×C3×C4—61no—
161C2×C3×C4—61no—
162C2×C3×C4—61no—
163C2×C3×C4—61no—
164C2×C3×C4—64no—
165C2×C3×C4—81no—
166C2×C3×C4—122no—
167C2×C3×C4SL(2,ℤ/4)11yescharacteristic
168C2×Dic3—82no—
169SL(2,𝔽3)—21no—
170SL(2,𝔽3)C3×C22⋊C411yescharacteristic
171C3×C8—122no—
172C2×C32—41no—
173C2×C32—81no—
174C2×C32—82no—
175C2×C32—161no—
176#41—21no—
177#41—34no—
178#41—62no—
179#41—62no—
180#41—62no—
181#41—62no—
182#41—62no—
183#41—61no—
184C2×Q8—21no—
185C2×Q8—31no—
186C2×Q8—32no—
187C2×Q8—31no—
188C2×Q8—31no—
189C2×Q8—62no—
190C2×Q8—62no—
191C2×Q8—61no—
192C2×Q8C2×C3×Dic311yescharacteristic
193C2×Q8C3×#6712yes—
194C2×D4—31no—
195C2×D4—31no—
196C2×D4—31no—
197C2×D4C3×S411yescharacteristic
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—61no—
202M16—62no—
203M16—62no—
204C2×C8—62no—
205C4⋊C4—62no—
206C22⋊C4—62no—
207C42—62no—
208C22×C3—21no—
209C22×C3—41no—
210C22×C3—41no—
211C22×C3—41no—
212C22×C3—42no—
213C22×C3—61no—
214C22×C3—81no—
215C22×C3—82no—
216C22×C3—81no—
217C22×C3—82no—
218C22×C3—121no—
219C22×C3—161no—
220C22×C3—161no—
221C22×C3—162no—
222C22×C3—162no—
223C22×C3C2×SL(2,ℤ/4)11yescharacteristic
224C22×C3GL(2,ℤ/4)12yes—
225C3×C4—21no—
226C3×C4—32no—
227C3×C4—31no—
228C3×C4—31no—
229C3×C4—61no—
230C3×C4—81no—
231C3×C4—82no—
232C3×C4—242no—
233Dic3—162no—
234C32—41noSylow
235C23—31no—
236C23C3×SL(2,ℤ/4)11yescharacteristic
237C23C2×C3×S411yescharacteristic
238Q8—31no—
239Q8—32no—
240Q8—32no—
241Q8—62no—
242Q8—61no—
243Q8—61no—
244Q8C3×#21711yescharacteristic
245D4—22no—
246D4—32no—
247D4—62no—
248D4—61no—
249D4—62no—
250C2×C4—31no—
251C2×C4—31no—
252C2×C4—32no—
253C2×C4—32no—
254C2×C4—31no—
255C2×C4—61no—
256C2×C4—61no—
257C2×C4—61no—
258C2×C4—61no—
259C2×C4—64no—
260C2×C4—122no—
261C2×C4C3×SL(2,ℤ/4)11yescharacteristic
262C8—122no—
263C2×C3—22no—
264C2×C3—21no—
265C2×C3—41no—
266C2×C3—41no—
267C2×C3—82no—
268C2×C3—81no—
269C2×C3—82no—
270C2×C3—81no—
271C2×C3—121no—
272C2×C3—162no—
273C2×C3—161no—
274C2×C3—162no—
275C2×C3#543211yescharacteristic
276C22—21no—
277C22—41no—
278C22—41no—
279C22—61no—
280C22—121no—
281C22C2×C3×SL(2,ℤ/4)11yescharacteristic
282C22C3×GL(2,ℤ/4)12yes—
283C4—21no—
284C4—32no—
285C4—31no—
286C4—31no—
287C4—61no—
288C4—242no—
289C3—41no—
290C3—81no—
291C3#7315611yescharacteristic, direct product factor
292C2—22no—
293C2—21no—
294C2—41no—
295C2—121no—
296C2C3×#543211yescharacteristic
subgroupG/H
center, socleC2×C3#5432
commutatorC2×SL(2,𝔽3)C2×C3×C4
FrattiniC23C2×C3×S4
FittingC2×C3×2−1+4D3
prime radicalC2×C3×2−1+4⋊C3C2
cosocleC22×SL(2,𝔽3)C22×C3
hypercenterC2×C3×D4S4
hypocenterSL(2,𝔽3)C3×C22⋊C4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C3×#5420C2semidirect
3. #73156C3direct (N × H)
6. C2×C3×#41⋊C3C4semidirect
7. C22×C3×SL(2,𝔽3)C4semidirect
14. #5420C2×C3semidirect
31. C2×C3×#41Dic3semidirect
33. C22×C3×Q8Dic3semidirect
38. C2×#41⋊C3C3×C4semidirect
41. C22×SL(2,𝔽3)C3×C4semidirect
120. C2×#41C3×Dic3semidirect
122. C22×Q8C3×Dic3semidirect
291. C3#73156direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
C3×#54202
#731563
C3×#17963
C3×#8344