Finite Group Databasebeta

← C3×#73155 #11920933

Subgroups — C3×#73155 #11920933

Order 1152.

443 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 443 conjugacy classes of subgroups — too many to draw. Only its 60 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×#5426C211yesmaximal, characteristic, complemented
3C2×C3×GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73155C311yesmaximal, characteristic, direct product factor
5C3×#1793—31nomaximal
6C3×2−1+4⋊C3—21no—
7C3×2−1+4⋊C3C412yescomplemented
8C2×C3×#41⋊C3C2211yescharacteristic
9C22×C3×SL(2,𝔽3)—21no—
10C3×GU(2,𝔽3)—21no—
11C3×#859—21no—
12C3×#832—41nomaximal
13C2×C3×2−1+4D311yescharacteristic
14C2×2−1+4⋊C3—21no—
15C2×2−1+4⋊C3C2×C311yescharacteristic
16#5426C2×C311yescharacteristic, complemented
17C2×GU(2,𝔽3)C2×C311yescharacteristic, complemented
18C2×C3×#104—31no—
19C2×C3×#104—31no—
20C3×#417—31no—
21C3×#417—31no—
22C2×C3×#101—31no—
23C3×#391—31no—
24C2×C32×D4—41no—
25C3×#41⋊C3C2×C411yescharacteristic
26C3×#41⋊C3D411yescharacteristic
27C2×C3×SL(2,𝔽3)—22no—
28C2×C3×SL(2,𝔽3)D411yescharacteristic
29C3×C4⋊Dic3—41no—
30C2×C3×C3⋊C8—41no—
31#1793—31noSylow
32C3×2−1+4—21no—
33C3×2−1+4—32no—
34C3×2−1+4—61no—
35C3×2−1+4Dic312yescomplemented
36C2×C3×#41—31no—
37C2×C3×#41—61no—
38C2×C3×#41C2×D311yescharacteristic
39C22×C3×Q8—21no—
40C22×C3×Q8—31no—
412−1+4⋊C3—22no—
422−1+4⋊C3—21no—
432−1+4⋊C3—22no—
442−1+4⋊C3C3×C412yescomplemented
45C2×#41⋊C3—21no—
46C2×#41⋊C3C22×C311yescharacteristic
47C22×SL(2,𝔽3)—22no—
48C22×SL(2,𝔽3)—21no—
49C2×C3×M16—31no—
50C2×C3×M16—31no—
51C2×C3×C4⋊C4—31no—
52C2×C3×C42—31no—
53GU(2,𝔽3)—21no—
54#859—21no—
55C3×#104—61no—
56C3×#104—62no—
57C3×#104—61no—
58C3×#103—61no—
59C3×#103—61no—
60C3×#102—61no—
61C3×#102—61no—
62C3×#101—62no—
63C3×#95—62no—
64#832—41no—
65C23×C32—81no—
66C32×D4—42no—
67C32×D4—81no—
68C2×C32×C4—41no—
69C2×C3×Dic3—81no—
70C3×SL(2,𝔽3)C22⋊C411yescharacteristic
71C3×C3⋊C8—81no—
72C2×2−1+4C3×D311yescharacteristic
73C2×#104—31no—
74C2×#104—31no—
75#417—31no—
76#417—31no—
77C2×#101—31no—
78#391—31no—
79C3×#41—31no—
80C3×#41—31no—
81C3×#41—32no—
82C3×#41—32no—
83C3×#41—62no—
84C3×#41—61no—
85C3×#41—61no—
86C3×#41—64no—
87C3×#41—62no—
88C3×#41C2×Dic311yescharacteristic
89C3×#41#6711yescharacteristic
90C2×C3×Q8—22no—
91C2×C3×Q8—32no—
92C2×C3×Q8—32no—
93C2×C3×Q8—31no—
94C2×C3×Q8—61no—
95C2×C3×Q8—61no—
96C2×C3×Q8—61no—
97C2×C3×Q8—62no—
98C2×C3×Q8#6711yescharacteristic
99C2×C3×D4—31no—
100C2×C3×D4—41no—
101C2×C3×D4—61no—
102C2×C3×D4—81no—
103C2×C3×D4S411yescharacteristic
104C22×C3×C4—31no—
105C22×C3×C4—32no—
106C22×C3×C4—61no—
107C22×C3×C4—61no—
108C22×C3×C4—61no—
109#41⋊C3—21no—
110#41⋊C3—21no—
111#41⋊C3C2×C3×C411yescharacteristic
112#41⋊C3C3×D411yescharacteristic
113C2×SL(2,𝔽3)—24no—
114C2×SL(2,𝔽3)—22no—
115C2×SL(2,𝔽3)—21no—
116C2×SL(2,𝔽3)C3×D411yescharacteristic
117C3×M16—61no—
118C3×M16—61no—
119C3×M16—61no—
120C3×M16—61no—
121C2×C3×C8—61no—
122C2×C3×C8—61no—
123C3×C4⋊C4—61no—
124C3×C4⋊C4—61no—
125C3×C42—61no—
126C3×C42—61no—
127C4⋊Dic3—41no—
128C2×C3⋊C8—41no—
129C22×C32—41no—
130C22×C32—82no—
131C22×C32—161no—
132C22×C32—161no—
133C32×C4—41no—
134C32×C4—41no—
135C3×Dic3—161no—
1362−1+4—21no—
1372−1+4—32no—
1382−1+4—61no—
1392−1+4C3×Dic312yescomplemented
140C2×#41—31no—
141C2×#41—61no—
142C2×#41C2×C3×D311yescharacteristic
143C22×Q8—21no—
144C22×Q8—31no—
145C2×M16—31no—
146C2×M16—31no—
147C2×C4⋊C4—31no—
148C2×C42—31no—
149#104—61no—
150#104—62no—
151#104—61no—
152#103—61no—
153#103—61no—
154#102—61no—
155#102—61no—
156#101—62no—
157#95—62no—
158C23×C3—21no—
159C23×C3—31no—
160C23×C3—81no—
161C23×C3—82no—
162C3×Q8—31no—
163C3×Q8—32no—
164C3×Q8—62no—
165C3×Q8—62no—
166C3×Q8—61no—
167C3×Q8#21711yescharacteristic
168C3×D4—21no—
169C3×D4—31no—
170C3×D4—31no—
171C3×D4—42no—
172C3×D4—61no—
173C3×D4—62no—
174C3×D4—62no—
175C3×D4—81no—
176C3×D4—82no—
177C3×D4—82no—
178C3×D4SL(2,ℤ/4)12yes—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—34no—
185C2×C3×C4—41no—
186C2×C3×C4—62no—
187C2×C3×C4—62no—
188C2×C3×C4—62no—
189C2×C3×C4—61no—
190C2×C3×C4—62no—
191C2×C3×C4—81no—
192C2×C3×C4—121no—
193C2×C3×C4—121no—
194C2×C3×C4—122no—
195C2×C3×C4—121no—
196C2×C3×C4C2×S411yescharacteristic
197C2×Dic3—81no—
198SL(2,𝔽3)—21no—
199SL(2,𝔽3)C3×C22⋊C411yescharacteristic
200C3×C8—121no—
201C3×C8—121no—
202C3⋊C8—81no—
203C2×C32—41no—
204C2×C32—41no—
205C2×C32—41no—
206C2×C32—162no—
207#41—31no—
208#41—31no—
209#41—32no—
210#41—32no—
211#41—62no—
212#41—61no—
213#41—61no—
214#41—64no—
215#41—62no—
216#41C2×C3×Dic311yescharacteristic
217#41C3×#6711yescharacteristic
218C2×Q8—22no—
219C2×Q8—32no—
220C2×Q8—32no—
221C2×Q8—31no—
222C2×Q8—61no—
223C2×Q8—61no—
224C2×Q8—61no—
225C2×Q8—62no—
226C2×Q8C3×#6711yescharacteristic
227C2×D4—31no—
228C2×D4—61no—
229C2×D4C3×S411yescharacteristic
230C22×C4—31no—
231C22×C4—32no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235M16—61no—
236M16—61no—
237M16—61no—
238M16—61no—
239C2×C8—61no—
240C2×C8—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C42—61no—
244C42—61no—
245C22×C3—22no—
246C22×C3—31no—
247C22×C3—31no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3—41no—
251C22×C3—61no—
252C22×C3—61no—
253C22×C3—82no—
254C22×C3—84no—
255C22×C3—81no—
256C22×C3—161no—
257C22×C3—161no—
258C22×C3—162no—
259C22×C3—162no—
260C22×C3GL(2,ℤ/4)11yescharacteristic
261C3×C4—31no—
262C3×C4—31no—
263C3×C4—41no—
264C3×C4—41no—
265C3×C4—62no—
266C3×C4—81no—
267C3×C4—81no—
268C3×C4—122no—
269C3×C4—241no—
270C3×C4C2×SL(2,ℤ/4)11yescharacteristic
271C3×C4GL(2,ℤ/4)11yescharacteristic
272Dic3—161no—
273C32—41noSylow
274C23—21no—
275C23—31no—
276Q8—31no—
277Q8—32no—
278Q8—62no—
279Q8—62no—
280Q8—61no—
281Q8C3×#21711yescharacteristic
282D4—21no—
283D4—31no—
284D4—31no—
285D4—61no—
286D4—62no—
287D4—62no—
288D4C3×SL(2,ℤ/4)12yes—
289C2×C4—31no—
290C2×C4—31no—
291C2×C4—31no—
292C2×C4—31no—
293C2×C4—31no—
294C2×C4—34no—
295C2×C4—62no—
296C2×C4—62no—
297C2×C4—62no—
298C2×C4—61no—
299C2×C4—62no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—122no—
303C2×C4—121no—
304C2×C4C2×C3×S411yescharacteristic
305C8—121no—
306C8—121no—
307C2×C3—42no—
308C2×C3—41no—
309C2×C3—41no—
310C2×C3—41no—
311C2×C3—61no—
312C2×C3—61no—
313C2×C3—81no—
314C2×C3—81no—
315C2×C3—81no—
316C2×C3—162no—
317C2×C3—164no—
318C2×C3#543011yescharacteristic
319C2×C3#543111yescharacteristic
320C2×C3#543211yescharacteristic
321C22—22no—
322C22—31no—
323C22—31no—
324C22—41no—
325C22—41no—
326C22—61no—
327C22—61no—
328C22C3×GL(2,ℤ/4)11yescharacteristic
329C4—31no—
330C4—31no—
331C4—62no—
332C4—122no—
333C4—241no—
334C4C2×C3×SL(2,ℤ/4)11yescharacteristic
335C4C3×GL(2,ℤ/4)11yescharacteristic
336C3—41no—
337C3—81no—
338C3#7315511yescharacteristic, direct product factor
339C2—42no—
340C2—61no—
341C2—61no—
342C2C3×#543011yescharacteristic
343C2C3×#543111yescharacteristic
344C2C3×#543211yescharacteristic
subgroupG/H
center, socleC22×C3GL(2,ℤ/4)
commutator#41⋊C3C2×C3×C4
FrattiniC2×C4C2×C3×S4
FittingC2×C3×2−1+4D3
prime radicalC2×C3×2−1+4⋊C3C2
cosocleC2×#41⋊C3C22×C3
hypercenterC2×C3×D4S4
hypocenterSL(2,𝔽3)C3×C22⋊C4
LCSG ▷ #41⋊C3 ▷ 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
2. C3×#5426C2semidirect
3. C2×C3×GU(2,𝔽3)C2semidirect
4. #73155C3direct (N × H)
7. C3×2−1+4⋊C3C4semidirect
16. #5426C2×C3semidirect
17. C2×GU(2,𝔽3)C2×C3semidirect
35. C3×2−1+4Dic3semidirect
44. 2−1+4⋊C3C3×C4semidirect
139. 2−1+4C3×Dic3semidirect
338. C3#73155direct (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×#54262
C2×C3×GU(2,𝔽3)2
#731553
C3×#17933
C3×#8324