Finite Group Databasebeta

← C3×Dic3×GL(2,𝔽3) #13165889

Subgroups — C3×Dic3×GL(2,𝔽3) #13165889

Order 1728.

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

normal subgroups only

This group has 461 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×C32×GL(2,𝔽3)C211yesmaximal, characteristic
2C3×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×C3⋊#859C211yesmaximal, characteristic, complemented
4C3×C4×GL(2,𝔽3)—31nomaximal
5Dic3×GL(2,𝔽3)C311yesmaximal, characteristic, direct product factor
6C3×Dic3×SD16—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×C3×D3×Dic3—41nomaximal
9C32×GL(2,𝔽3)C412yescomplemented
10C2×C3×GL(2,𝔽3)—21no—
11C2×C3×GL(2,𝔽3)D311yescharacteristic
12C2×C3×GL(2,𝔽3)C2×C311yescharacteristic
13C2×C32×SD16—31no—
14C3×Q8×Dic3D311yescharacteristic, complemented
15C3×D4×Dic3—31no—
16C3×C4×SL(2,𝔽3)—31no—
17Dic3×SL(2,𝔽3)—21no—
18Dic3×SL(2,𝔽3)C2×C311yescharacteristic, complemented
19C3⋊#859C2×C311yescharacteristic, complemented
20C3×#859—31no—
21C3×#859—61no—
22C3×#835—31no—
23C3×#832—31no—
24C3×#817—31no—
25C3×C8×Dic3—31no—
26C22×C32×D3—41no—
27C2×C3×C3⋊Dic3—41no—
28C2×C32×Dic3—41no—
29C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
30C3×D3×Dic3—44no—
31C4×GL(2,𝔽3)—31no—
32C3×C4×SD16—91no—
33Dic3×SD16—31no—
34C2×C32×Q8C2×D311yescharacteristic
35C2×C32×D4—31no—
36C22×C3×Dic3—61no—
37C2×C3×C4×D3—121no—
38C2×C3×SL(2,𝔽3)—21no—
39C2×C3×SL(2,𝔽3)—21no—
40C2×C3×SL(2,𝔽3)—21no—
41C2×C3×SL(2,𝔽3)—41no—
42C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
43C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
44C2×D3×Dic3—41no—
45C3×GL(2,𝔽3)—22no—
46C3×GL(2,𝔽3)Dic312yesdirect product factor
47C3×GL(2,𝔽3)C3×C412yescomplemented
48C32×SD16—32no—
49C32×SD16—32no—
50C2×C32×C8—31no—
51C3×#217—61no—
52C3×C4⋊Dic3—31no—
53C3×C4⋊Dic3—31no—
54C3×C4×Dic3—31no—
55C2×C3×C3⋊C8—31no—
56C22×C33—41no—
57C2×C32×D3—42no—
58C2×C32×D3—42no—
59C2×C32×D3—42no—
60C3×C3⋊Dic3—42no—
61C32×Dic3—42no—
62C2×GL(2,𝔽3)C3×D311yescharacteristic
63C2×C3×SD16—31no—
64C2×C3×SD16—31no—
65C2×C3×SD16—61no—
66C3×C4×Q8—31no—
67C3×C4×D4—91no—
68Q8×Dic3C3×D311yescharacteristic, complemented
69D4×Dic3—31no—
70C4×SL(2,𝔽3)—31no—
71C4×SL(2,𝔽3)—61no—
72#859—31no—
73#859—61no—
74C3×#106—91no—
75C3×#103—91no—
76C3×#102—91no—
77C3×C4×C8—91no—
78#835—31no—
79#832—31no—
80#817—31no—
81C8×Dic3—31no—
82C23×C32—61no—
83C22×C3×D3—41no—
84C22×C3×D3—41no—
85C22×C3×D3—81no—
86C32×Q8—31no—
87C32×Q8C4×D311yescharacteristic, complemented
88C32×D4—32no—
89C32×D4—61no—
90C2×C32×C4—31no—
91C2×C32×C4—121no—
92C2×C3⋊Dic3—41no—
93C2×C3×Dic3—31no—
94C2×C3×Dic3—41no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—81no—
97C2×C3×Dic3—121no—
98C2×C3×Dic3—122no—
99C2×C3×Dic3—241no—
100C2×C3×Dic3S411yescharacteristic
101C3×C4×D3—124no—
102C3×SL(2,𝔽3)—21no—
103C3×SL(2,𝔽3)—21no—
104C3×SL(2,𝔽3)—21no—
105C3×SL(2,𝔽3)—41no—
106C3×SL(2,𝔽3)C2×Dic311yescharacteristic, complemented
107C3×SL(2,𝔽3)C2×C3×C411yescharacteristic, complemented
108D3×Dic3—44no—
109C32×C8—32no—
110C3×C3⋊C8—61no—
111C4×SD16—91noSylow
112C2×C33—41no—
113C2×C33—41no—
114C2×C33—41no—
115C32×D3—44no—
116C2×C3×Q8—21no—
117C2×C3×Q8D3211yescharacteristic
118C2×C3×Q8C2×C3×D311yescharacteristic
119C2×C3×D4—31no—
120C2×C3×D4—31no—
121C2×C3×D4—61no—
122C22×C3×C4—181no—
123C22×Dic3—61no—
124C2×C4×D3—121no—
125C2×SL(2,𝔽3)—21no—
126C2×SL(2,𝔽3)—21no—
127C2×SL(2,𝔽3)—41no—
128C2×SL(2,𝔽3)C2×C3×D311yescharacteristic
129GL(2,𝔽3)C3×Dic312yesdirect product factor
130C3×SD16—32no—
131C3×SD16—32no—
132C3×SD16—32no—
133C3×SD16—32no—
134C3×SD16—62no—
135C3×SD16—62no—
136C2×C3×C8—31no—
137C2×C3×C8—31no—
138C2×C3×C8—61no—
139C2×C3×C8—91no—
140C3×C4⋊C4—91no—
141C3×C4⋊C4—91no—
142C3×C22⋊C4—181no—
143C3×C42—91no—
144#217—61no—
145C4⋊Dic3—31no—
146C4⋊Dic3—31no—
147C4×Dic3—31no—
148C2×C3⋊C8—31no—
149C22×C32—41no—
150C22×C32—41no—
151C22×C32—62no—
152C22×C32—81no—
153C22×C32—81no—
154C22×C32—81no—
155C22×C32—121no—
156C22×C32—121no—
157C22×C32—161no—
158C22×C32C2×S411yescharacteristic
159C2×C3×D3—42no—
160C2×C3×D3—42no—
161C2×C3×D3—42no—
162C2×C3×D3—42no—
163C2×C3×D3—42no—
164C2×C3×D3—42no—
165C2×C3×D3—82no—
166C2×C3×D3—82no—
167C2×C3×D3—82no—
168C32×C4—31no—
169C32×C4—31no—
170C32×C4—122no—
171C3⋊Dic3—42no—
172C3×Dic3—42no—
173C3×Dic3—61no—
174C3×Dic3—82no—
175C3×Dic3—122no—
176C3×Dic3—121no—
177C3×Dic3—242no—
178C3×Dic3GL(2,𝔽3)12yesdirect product factor
179C2×SD16—31no—
180C4×Q8—31no—
181C4×D4—91no—
182#106—91no—
183#103—91no—
184#102—91no—
185C4×C8—91no—
186C33—41noSylow
187C23×C3—61no—
188C23×C3—61no—
189C23×C3—121no—
190C22×D3—41no—
191C3×Q8—21no—
192C3×Q8—31no—
193C3×Q8—31no—
194C3×Q8—61no—
195C3×Q8D3×Dic311yescharacteristic, complemented
196C3×Q8C3×C4×D311yescharacteristic, complemented
197C3×D4—32no—
198C3×D4—32no—
199C3×D4—62no—
200C3×D4—61no—
201C3×D4—61no—
202C3×D4—121no—
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—31no—
206C2×C3×C4—61no—
207C2×C3×C4—91no—
208C2×C3×C4—121no—
209C2×C3×C4—181no—
210C2×C3×C4—241no—
211C2×C3×C4—362no—
212C2×Dic3—31no—
213C2×Dic3—61no—
214C2×Dic3—121no—
215C2×Dic3—122no—
216C2×Dic3—241no—
217C2×Dic3C3×S411yescharacteristic
218C4×D3—124no—
219SL(2,𝔽3)—21no—
220SL(2,𝔽3)—21no—
221SL(2,𝔽3)—41no—
222SL(2,𝔽3)C2×C3×Dic311yescharacteristic, complemented
223C3×C8—32no—
224C3×C8—32no—
225C3×C8—62no—
226C3×C8—181no—
227C3⋊C8—61no—
228C2×C32—41no—
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32—81no—
235C2×C32—81no—
236C2×C32—81no—
237C2×C32—81no—
238C2×C32—81no—
239C2×C32—81no—
240C2×C32—81no—
241C2×C32—81no—
242C2×C32—81no—
243C2×C32—122no—
244C2×C32—161no—
245C2×C32—161no—
246C2×C32—161no—
247C2×C32C4×S411yescharacteristic
248C2×C32C2×GL(2,𝔽3)11yescharacteristic
249C2×C32#98511yescharacteristic
250C3×D3—44no—
251C3×D3—44no—
252C3×D3—84no—
253C2×Q8C3×D3211yescharacteristic
254C2×D4—31no—
255C22×C4—181no—
256SD16—32no—
257SD16—32no—
258C2×C8—31no—
259C2×C8—91no—
260C4⋊C4—91no—
261C4⋊C4—91no—
262C22⋊C4—181no—
263C42—91no—
264C22×C3—21no—
265C22×C3—41no—
266C22×C3—62no—
267C22×C3—62no—
268C22×C3—81no—
269C22×C3—81no—
270C22×C3—122no—
271C22×C3—121no—
272C22×C3—121no—
273C22×C3—121no—
274C22×C3—121no—
275C22×C3—161no—
276C22×C3—241no—
277C22×C3—241no—
278C22×C3D3×S411yescharacteristic
279C22×C3C2×C3×S411yescharacteristic
280C2×D3—42no—
281C2×D3—42no—
282C2×D3—42no—
283C3×C4—32no—
284C3×C4—31no—
285C3×C4—31no—
286C3×C4—31no—
287C3×C4—31no—
288C3×C4—61no—
289C3×C4—61no—
290C3×C4—122no—
291C3×C4—181no—
292C3×C4—242no—
293C3×C4—361no—
294Dic3—61no—
295Dic3—122no—
296Dic3—121no—
297Dic3—242no—
298Dic3C3×GL(2,𝔽3)12yesdirect product factor
299C32—41no—
300C32—41no—
301C32—81no—
302C32—81no—
303C32—81no—
304C32—161no—
305C32C4×GL(2,𝔽3)11yescharacteristic, complemented
306C23—61no—
307Q8—31no—
308Q8C3×D3×Dic311yescharacteristic, complemented
309D4—32no—
310D4—61no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—91no—
314C2×C4—181no—
315C2×C4—362no—
316C8—32no—
317C8—181no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—41no—
322C2×C3—41no—
323C2×C3—41no—
324C2×C3—81no—
325C2×C3—81no—
326C2×C3—81no—
327C2×C3—81no—
328C2×C3—81no—
329C2×C3—81no—
330C2×C3—122no—
331C2×C3—122no—
332C2×C3—161no—
333C2×C3—161no—
334C2×C3—161no—
335C2×C3—242no—
336C2×C3#6419711yescharacteristic
337C2×C3D3×GL(2,𝔽3)11yescharacteristic
338C2×C3Dic3×S411yescharacteristic
339C2×C3C3×C4×S411yescharacteristic
340C2×C3C2×C3×GL(2,𝔽3)11yescharacteristic
341C2×C3C3×#98511yescharacteristic
342D3—44no—
343C22—62no—
344C22—121no—
345C22—121no—
346C22C3×D3×S411yescharacteristic
347C4—32no—
348C4—31no—
349C4—31no—
350C4—181no—
351C4—361no—
352C3—21no—
353C3—41no—
354C3—81no—
355C3—81no—
356C3—161no—
357C3Dic3×GL(2,𝔽3)11yescharacteristic, direct product factor
358C3C3×C4×GL(2,𝔽3)11yescharacteristic, complemented
359C2—122no—
360C2C3×#6419711yescharacteristic
361C2C3×D3×GL(2,𝔽3)11yescharacteristic
362C2C3×Dic3×S411yescharacteristic
subgroupG/H
center, hypercenterC22×C3D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C2×C3×C4
FrattiniC22C3×D3×S4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C32×GL(2,𝔽3)C2
cosocleC2×C3×SL(2,𝔽3)C22×C3
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C3×Dic3×SL(2,𝔽3)C2semidirect
3. C3×C3⋊#859C2semidirect
5. Dic3×GL(2,𝔽3)C3direct (N × H)
9. C32×GL(2,𝔽3)C4semidirect
14. C3×Q8×Dic3D3semidirect
18. Dic3×SL(2,𝔽3)C2×C3semidirect
19. C3⋊#859C2×C3semidirect
29. C32×SL(2,𝔽3)C2×C4semidirect
46. C3×GL(2,𝔽3)Dic3direct (N × H)
47. C3×GL(2,𝔽3)C3×C4semidirect
68. Q8×Dic3C3×D3semidirect
87. C32×Q8C4×D3semidirect
106. C3×SL(2,𝔽3)C2×Dic3semidirect
107. C3×SL(2,𝔽3)C2×C3×C4semidirect
129. GL(2,𝔽3)C3×Dic3direct (N × H)
178. C3×Dic3GL(2,𝔽3)direct (N × H)
195. C3×Q8D3×Dic3semidirect
196. C3×Q8C3×C4×D3semidirect
222. SL(2,𝔽3)C2×C3×Dic3semidirect
298. Dic3C3×GL(2,𝔽3)direct (N × H)
305. C32C4×GL(2,𝔽3)semidirect
308. Q8C3×D3×Dic3semidirect
357. C3Dic3×GL(2,𝔽3)direct (N × H)
358. C3C3×C4×GL(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
C2×C32×GL(2,𝔽3)2
C3×Dic3×SL(2,𝔽3)2
C3×C3⋊#8592
C3×C4×GL(2,𝔽3)3
Dic3×GL(2,𝔽3)3
C3×Dic3×SD163
C2×C3×D3×Dic34