Finite Group Databasebeta

← C3×#73149 #11920927

Subgroups — C3×#73149 #11920927

Order 1152.

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

normal subgroups only

This group has 477 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
1C22×C3×#41⋊C3C211yesmaximal, characteristic
2C2×C3×GU(2,𝔽3)C212yesmaximal, complemented
3#73149C311yesmaximal, characteristic, direct product factor
4C3×#1789—31nomaximal
5C2×C3×#41⋊C3—22no—
6C2×C3×#41⋊C3C411yescharacteristic, complemented
7C2×C3×#41⋊C3C2211yescharacteristic
8C22×C3×SL(2,𝔽3)C411yescharacteristic, complemented
9C3×GU(2,𝔽3)—22no—
10C3×#830—41nomaximal
11C22×C3×#41D311yescharacteristic
12C22×#41⋊C3—21no—
13C22×#41⋊C3C2×C311yescharacteristic
14C2×GU(2,𝔽3)C2×C312yescomplemented
15C2×C3×#104—32no—
16C2×C3×#104—32no—
17C3×#407—31no—
18C3×C4×C22⋊C4—31no—
19C22×C32×C4—41no—
20C3×#41⋊C3—21no—
21C3×#41⋊C3D412yes—
22C2×C3×SL(2,𝔽3)—21no—
23C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
24C2×C3×C3⋊C8—42no—
25#1789—31noSylow
26C2×C3×#41—22no—
27C2×C3×#41—31no—
28C2×C3×#41—31no—
29C2×C3×#41—62no—
30C2×C3×#41Dic311yescharacteristic
31C2×C3×#41C2×D311yescharacteristic
32C22×C3×Q8Dic311yescharacteristic
33C22×C3×D4—31no—
34C23×C3×C4—31no—
35C2×#41⋊C3—24no—
36C2×#41⋊C3—22no—
37C2×#41⋊C3—21no—
38C2×#41⋊C3—21no—
39C2×#41⋊C3C3×C411yescharacteristic, complemented
40C2×#41⋊C3C22×C311yescharacteristic
41C22×SL(2,𝔽3)—21no—
42C22×SL(2,𝔽3)C3×C411yescharacteristic, complemented
43C2×C3×M16—32no—
44C2×C3×C22⋊C4—61no—
45C2×C3×C42—32no—
46GU(2,𝔽3)—22no—
47C3×#104—62no—
48C3×#104—64no—
49C3×#104—62no—
50C3×C22⋊C8—61no—
51C3×C22⋊C8—61no—
52C3×#95—61no—
53#830—41no—
54C23×C32—41no—
55C2×C32×C4—41no—
56C2×C32×C4—41no—
57C2×C32×C4—82no—
58C3×SL(2,𝔽3)C22⋊C411yescharacteristic, complemented
59C3×C3⋊C8—82no—
60C22×#41C3×D311yescharacteristic
61C2×#104—32no—
62C2×#104—32no—
63#407—31no—
64C4×C22⋊C4—31no—
65C24×C3—31no—
66C3×#41—21no—
67C3×#41—32no—
68C3×#41—34no—
69C3×#41—62no—
70C3×#41—62no—
71C3×#41—61no—
72C3×#41—62no—
73C3×#41#6712yes—
74C2×C3×Q8—21no—
75C2×C3×Q8—31no—
76C2×C3×Q8—61no—
77C2×C3×Q8C2×Dic311yescharacteristic
78C2×C3×D4—31no—
79C2×C3×D4—31no—
80C2×C3×D4—61no—
81C2×C3×D4—62no—
82C2×C3×D4—62no—
83C22×C3×C4—31no—
84C22×C3×C4—31no—
85C22×C3×C4—31no—
86C22×C3×C4—31no—
87C22×C3×C4—31no—
88C22×C3×C4—41no—
89C22×C3×C4—62no—
90C22×C3×C4—62no—
91C22×C3×C4—62no—
92C22×C3×C4—81no—
93C22×C3×C4S411yescharacteristic
94#41⋊C3—22no—
95#41⋊C3—21no—
96#41⋊C3—22no—
97#41⋊C3C3×D412yes—
98C2×SL(2,𝔽3)—22no—
99C2×SL(2,𝔽3)—21no—
100C2×SL(2,𝔽3)—21no—
101C2×SL(2,𝔽3)C2×C3×C411yescharacteristic
102C3×M16—62no—
103C3×M16—62no—
104C2×C3×C8—62no—
105C3×C22⋊C4—64no—
106C3×C42—62no—
107C3×C42—62no—
108C2×C3⋊C8—42no—
109C22×C32—42no—
110C22×C32—41no—
111C22×C32—81no—
112C22×C32—81no—
113C32×C4—42no—
114C32×C4—81no—
115C2×#41—22no—
116C2×#41—31no—
117C2×#41—31no—
118C2×#41—62no—
119C2×#41C3×Dic311yescharacteristic
120C2×#41C2×C3×D311yescharacteristic
121C22×Q8C3×Dic311yescharacteristic
122C22×D4—31no—
123C23×C4—31no—
124C2×M16—32no—
125C2×C22⋊C4—61no—
126C2×C42—32no—
127#104—62no—
128#104—64no—
129#104—62no—
130C22⋊C8—61no—
131C22⋊C8—61no—
132#95—61no—
133C23×C3—31no—
134C23×C3—31no—
135C23×C3—41no—
136C23×C3—62no—
137C23×C3—62no—
138C23×C3—81no—
139C23×C3—121no—
140C23×C3SL(2,ℤ/4)11yescharacteristic
141C3×Q8—31no—
142C3×Q8—32no—
143C3×Q8—61no—
144C3×Q8#21711yescharacteristic
145C3×D4—32no—
146C3×D4—32no—
147C3×D4—61no—
148C3×D4—61no—
149C3×D4—64no—
150C2×C3×C4—22no—
151C2×C3×C4—31no—
152C2×C3×C4—32no—
153C2×C3×C4—32no—
154C2×C3×C4—31no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4—61no—
158C2×C3×C4—61no—
159C2×C3×C4—62no—
160C2×C3×C4—61no—
161C2×C3×C4—61no—
162C2×C3×C4—62no—
163C2×C3×C4—64no—
164C2×C3×C4—82no—
165C2×C3×C4—84no—
166C2×C3×C4—81no—
167C2×C3×C4—81no—
168C2×C3×C4—122no—
169C2×C3×C4—122no—
170C2×C3×C4SL(2,ℤ/4)11yescharacteristic
171C2×C3×C4C2×S411yescharacteristic
172SL(2,𝔽3)—21no—
173SL(2,𝔽3)C3×C22⋊C411yescharacteristic, complemented
174C3×C8—122no—
175C3⋊C8—82no—
176C2×C32—41no—
177C2×C32—41no—
178C2×C32—41no—
179C2×C32—82no—
180C24—31no—
181#41—21no—
182#41—32no—
183#41—34no—
184#41—62no—
185#41—62no—
186#41—61no—
187#41—62no—
188#41C3×#6712yes—
189C2×Q8—21no—
190C2×Q8—31no—
191C2×Q8—61no—
192C2×Q8C2×C3×Dic311yescharacteristic
193C2×D4—31no—
194C2×D4—31no—
195C2×D4—61no—
196C2×D4—62no—
197C2×D4—62no—
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—31no—
202C22×C4—31no—
203C22×C4—62no—
204C22×C4—62no—
205C22×C4—62no—
206C22×C4C3×S411yescharacteristic
207M16—62no—
208M16—62no—
209C2×C8—62no—
210C22⋊C4—64no—
211C42—62no—
212C42—62no—
213C22×C3—21no—
214C22×C3—21no—
215C22×C3—32no—
216C22×C3—41no—
217C22×C3—42no—
218C22×C3—61no—
219C22×C3—61no—
220C22×C3—61no—
221C22×C3—61no—
222C22×C3—61no—
223C22×C3—81no—
224C22×C3—81no—
225C22×C3—82no—
226C22×C3—82no—
227C22×C3—81no—
228C22×C3—82no—
229C22×C3—124no—
230C22×C3GU(2,𝔽3)12yescomplemented
231C22×C3C2×SL(2,ℤ/4)11yescharacteristic
232C3×C4—21no—
233C3×C4—31no—
234C3×C4—31no—
235C3×C4—42no—
236C3×C4—61no—
237C3×C4—81no—
238C3×C4—82no—
239C3×C4—82no—
240C3×C4—124no—
241C3×C4GL(2,ℤ/4)12yes—
242C32—41noSylow
243C23—31no—
244C23—31no—
245C23—62no—
246C23—62no—
247C23—121no—
248C23C3×SL(2,ℤ/4)11yescharacteristic
249Q8—31no—
250Q8—32no—
251Q8—61no—
252Q8C3×#21711yescharacteristic
253D4—32no—
254D4—32no—
255D4—61no—
256D4—61no—
257D4—64no—
258C2×C4—22no—
259C2×C4—31no—
260C2×C4—32no—
261C2×C4—32no—
262C2×C4—31no—
263C2×C4—61no—
264C2×C4—61no—
265C2×C4—62no—
266C2×C4—61no—
267C2×C4—61no—
268C2×C4—62no—
269C2×C4—64no—
270C2×C4—122no—
271C2×C4—122no—
272C2×C4C3×SL(2,ℤ/4)11yescharacteristic
273C2×C4C2×C3×S411yescharacteristic
274C8—122no—
275C2×C3—22no—
276C2×C3—41no—
277C2×C3—41no—
278C2×C3—41no—
279C2×C3—62no—
280C2×C3—82no—
281C2×C3—84no—
282C2×C3—81no—
283C2×C3—81no—
284C2×C3—81no—
285C2×C3—121no—
286C2×C3C2×GU(2,𝔽3)11yescharacteristic
287C2×C3#542311yescharacteristic
288C2×C3#543211yescharacteristic
289C22—21no—
290C22—21no—
291C22—32no—
292C22—61no—
293C22—61no—
294C22—61no—
295C22—61no—
296C22—61no—
297C22—124no—
298C22C3×GU(2,𝔽3)12yescomplemented
299C22C2×C3×SL(2,ℤ/4)11yescharacteristic
300C4—21no—
301C4—31no—
302C4—31no—
303C4—61no—
304C4—124no—
305C4C3×GL(2,ℤ/4)12yes—
306C3—41no—
307C3—81no—
308C3#7314911yescharacteristic, direct product factor
309C2—22no—
310C2—62no—
311C2—121no—
312C2C2×C3×GU(2,𝔽3)11yescharacteristic
313C2C3×#542311yescharacteristic
314C2C3×#543211yescharacteristic
subgroupG/H
centerC2×C3×C4C2×S4
commutatorC2×SL(2,𝔽3)C2×C3×C4
FrattiniC2×C4C2×C3×S4
FittingC22×C3×#41D3
socleC22×C3C2×SL(2,ℤ/4)
prime radicalC22×C3×#41⋊C3C2
cosocleC2×#41⋊C3C22×C3
hypercenterC22×C3×C4S4
hypocenterSL(2,𝔽3)C3×C22⋊C4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×GU(2,𝔽3)C2semidirect
3. #73149C3direct (N × H)
6. C2×C3×#41⋊C3C4semidirect
8. C22×C3×SL(2,𝔽3)C4semidirect
14. C2×GU(2,𝔽3)C2×C3semidirect
39. C2×#41⋊C3C3×C4semidirect
42. C22×SL(2,𝔽3)C3×C4semidirect
58. C3×SL(2,𝔽3)C22⋊C4semidirect
173. SL(2,𝔽3)C3×C22⋊C4semidirect
230. C22×C3GU(2,𝔽3)semidirect
298. C22C3×GU(2,𝔽3)semidirect
308. C3#73149direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C2×C3×GU(2,𝔽3)2
#731493
C3×#17893
C3×#8304