Finite Group Databasebeta

← C3×#10593208 #13165884

Subgroups — C3×#10593208 #13165884

Order 1728.

409 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 409 conjugacy classes of subgroups — too many to draw. Only its 50 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×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic
2C2×C32×#226C211yesmaximal, characteristic, complemented
3C3×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C3×#5395—31nomaximal
5#10593208C311yesmaximal, characteristic, direct product factor
6C3×#5185—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×C3⋊GL(2,𝔽3)—21no—
9C32×#226—21no—
10C3×#3722—41nomaximal
11C2×C3⋊GL(2,𝔽3)C2×C311yescharacteristic
12C2×C3×GL(2,𝔽3)—31no—
13C2×C3×GL(2,𝔽3)—61no—
14C2×C3×#226—21no—
15C2×C3×#226D311yescharacteristic, complemented
16C2×C3×#226C2×C311yescharacteristic, complemented
17C2×C32×Q16—31no—
18C3×#947—31no—
19C3×Q8×Dic3D311yescharacteristic, complemented
20C2×C3×#215—31no—
21C3×C4×SL(2,𝔽3)—31no—
22Dic3×SL(2,𝔽3)—21no—
23Dic3×SL(2,𝔽3)C2×C311yescharacteristic, complemented
24C3×#835—31no—
25C3×#821—31no—
26C3×Dic3⋊C8—31no—
27C22×C3×C3⋊D3—41no—
28C2×C32×Dic3—41no—
29C2×C32×Dic3—41no—
30C32×SL(2,𝔽3)D411yescharacteristic
31#5395—31no—
32C3×#464—91no—
33#5185—31no—
34C2×C32×Q8—31no—
35C2×C32×Q8C2×D311yescharacteristic
36C2×C3×D12—31no—
37C2×C3×SL(2,𝔽3)—21no—
38C2×C3×SL(2,𝔽3)—21no—
39C2×C3×SL(2,𝔽3)—21no—
40C2×C3×SL(2,𝔽3)—41no—
41C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
42C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
43C3⋊GL(2,𝔽3)—21no—
44C3×GL(2,𝔽3)—62no—
45C3×GL(2,𝔽3)—61no—
46C3×#226—22no—
47C3×#226—21no—
48C3×#226—21no—
49C32×Q16—61no—
50C32×Q16—61no—
51C2×C32×C8—31no—
52C3×#215—61no—
53C3×#215—61no—
54C3×(C2×D3)⋊C4—61no—
55C3×(C2×D3)⋊C4—61no—
56C3×(C2×D3)⋊C4—121no—
57C3×C4⋊Dic3—31no—
58C3×C4×Dic3—31no—
59C2×C3×C3⋊C8—31no—
60#3722—41no—
61C22×C33—41no—
62C2×C3×C3⋊D3—42no—
63C2×C3×C3⋊D3—81no—
64C2×C3×C3⋊D3—81no—
65C32×Dic3—81no—
66C32×Dic3—81no—
67C2×GL(2,𝔽3)—31no—
68C2×GL(2,𝔽3)—61no—
69C2×#226C3×D311yescharacteristic, complemented
70C2×C3×Q16—31no—
71C2×C3×Q16—31no—
72C2×C3×Q16—61no—
73C2×C3×SD16—91no—
74C3×#124—91no—
75C3×C4×Q8—31no—
76#947—31no—
77Q8×Dic3C3×D311yescharacteristic, complemented
78C2×#215—31no—
79C4×SL(2,𝔽3)—31no—
80C4×SL(2,𝔽3)—61no—
81C3×C4⋊C8—91no—
82C3×#103—91no—
83C3×#102—91no—
84#835—31no—
85#821—31no—
86Dic3⋊C8—31no—
87C22×C3⋊D3—41no—
88C22×C3×D3—61no—
89C22×C3×D3—121no—
90C22×C3×D3—241no—
91C32×Q8—31no—
92C32×Q8—61no—
93C32×Q8—61no—
94C32×Q8D1211yescharacteristic
95C2×C32×C4—31no—
96C2×C32×C4—61no—
97C2×C32×C4—121no—
98C2×C3×Dic3—31no—
99C2×C3×Dic3—41no—
100C2×C3×Dic3—41no—
101C2×C3×Dic3—41no—
102C2×C3×Dic3—81no—
103C2×C3×Dic3—81no—
104C2×C3×Dic3S411yescharacteristic
105C3×D12—61no—
106C3×D12—61no—
107C3×SL(2,𝔽3)—21no—
108C3×SL(2,𝔽3)—21no—
109C3×SL(2,𝔽3)—21no—
110C3×SL(2,𝔽3)—41no—
111C3×SL(2,𝔽3)#6711yescharacteristic
112C3×SL(2,𝔽3)C3×D411yescharacteristic
113C32×C8—61no—
114C3×C3⋊C8—61no—
115#464—91noSylow
116C2×C33—41no—
117C2×C33—41no—
118C2×C33—41no—
119C3×C3⋊D3—82no—
120C2×C3×Q8—21no—
121C2×C3×Q8—31no—
122C2×C3×Q8—31no—
123C2×C3×Q8—61no—
124C2×C3×Q8D3211yescharacteristic
125C2×C3×Q8C2×C3×D311yescharacteristic
126C2×C3×D4—91no—
127C2×D12—31no—
128C2×SL(2,𝔽3)—21no—
129C2×SL(2,𝔽3)—21no—
130C2×SL(2,𝔽3)—41no—
131C2×SL(2,𝔽3)C2×C3×D311yescharacteristic
132GL(2,𝔽3)—62no—
133GL(2,𝔽3)—61no—
134#226—21no—
135C3×Q16—62no—
136C3×Q16—61no—
137C3×Q16—61no—
138C3×Q16—62no—
139C3×Q16—61no—
140C3×Q16—61no—
141C3×SD16—181no—
142C3×SD16—181no—
143C2×C3×C8—31no—
144C2×C3×C8—31no—
145C2×C3×C8—61no—
146C2×C3×C8—91no—
147C3×C4⋊C4—91no—
148C3×C22⋊C4—181no—
149C3×C22⋊C4—181no—
150C3×C42—91no—
151#215—61no—
152#215—61no—
153(C2×D3)⋊C4—61no—
154(C2×D3)⋊C4—61no—
155(C2×D3)⋊C4—121no—
156C4⋊Dic3—31no—
157C4×Dic3—31no—
158C2×C3⋊C8—31no—
159C22×C32—41no—
160C22×C32—41no—
161C22×C32—81no—
162C22×C32—81no—
163C22×C32—81no—
164C22×C32—161no—
165C22×C32C2×S411yescharacteristic
166C2×C3⋊D3—42no—
167C2×C3⋊D3—81no—
168C2×C3⋊D3—81no—
169C2×C3×D3—122no—
170C2×C3×D3—121no—
171C2×C3×D3—121no—
172C2×C3×D3—121no—
173C2×C3×D3—241no—
174C2×C3×D3—241no—
175C2×C3×D3—242no—
176C2×C3×D3—242no—
177C2×C3×D3—242no—
178C32×C4—31no—
179C32×C4—31no—
180C32×C4—121no—
181C32×C4—241no—
182C3×Dic3—21no—
183C3×Dic3—61no—
184C3×Dic3—81no—
185C3×Dic3—82no—
186C3×Dic3—81no—
187C3×Dic3—81no—
188C3×Dic3—82no—
189C2×Q16—31no—
190C2×SD16—91no—
191#124—91no—
192C4×Q8—31no—
193C4⋊C8—91no—
194#103—91no—
195#102—91no—
196C33—41noSylow
197C23×C3—181no—
198C22×D3—61no—
199C22×D3—121no—
200C22×D3—241no—
201C3×Q8—21no—
202C3×Q8—31no—
203C3×Q8—31no—
204C3×Q8—61no—
205C3×Q8—61no—
206C3×Q8—61no—
207C3×Q8—61no—
208C3×Q8—61no—
209C3×Q8—62no—
210C3×Q8—121no—
211C3×Q8#62611yescharacteristic
212C3×Q8C3×D1211yescharacteristic
213C3×D4—181no—
214C3×D4—181no—
215C2×C3×C4—31no—
216C2×C3×C4—31no—
217C2×C3×C4—31no—
218C2×C3×C4—61no—
219C2×C3×C4—61no—
220C2×C3×C4—61no—
221C2×C3×C4—91no—
222C2×C3×C4—121no—
223C2×C3×C4—121no—
224C2×C3×C4—241no—
225C2×Dic3—31no—
226C2×Dic3—41no—
227C2×Dic3C3×S411yescharacteristic
228D12—61no—
229D12—61no—
230SL(2,𝔽3)—21no—
231SL(2,𝔽3)—21no—
232SL(2,𝔽3)—41no—
233SL(2,𝔽3)C3×#6711yescharacteristic
234C3×C8—61no—
235C3×C8—61no—
236C3×C8—62no—
237C3×C8—181no—
238C3⋊C8—61no—
239C2×C32—41no—
240C2×C32—41no—
241C2×C32—41no—
242C2×C32—41no—
243C2×C32—41no—
244C2×C32—41no—
245C2×C32—81no—
246C2×C32—81no—
247C2×C32—81no—
248C2×C32—81no—
249C2×C32—81no—
250C2×C32—81no—
251C2×C32—81no—
252C2×C32—81no—
253C2×C32—81no—
254C2×C32—161no—
255C2×C32—161no—
256C2×C32—161no—
257C2×C32C4⋊S411yescharacteristic
258C2×C32#98311yescharacteristic
259C2×C32#98511yescharacteristic
260C3⋊D3—82no—
261C3×D3—242no—
262C3×D3—244no—
263C3×D3—241no—
264C2×Q8—31no—
265C2×Q8C3×D3211yescharacteristic
266C2×D4—91no—
267Q16—61no—
268Q16—61no—
269SD16—181no—
270SD16—181no—
271C2×C8—31no—
272C2×C8—91no—
273C4⋊C4—91no—
274C22⋊C4—181no—
275C22⋊C4—181no—
276C42—91no—
277C22×C3—21no—
278C22×C3—41no—
279C22×C3—81no—
280C22×C3—81no—
281C22×C3—161no—
282C22×C3—361no—
283C22×C3—361no—
284C22×C3—361no—
285C22×C3D3×S411yescharacteristic
286C22×C3C2×C3×S411yescharacteristic
287C2×D3—122no—
288C2×D3—121no—
289C2×D3—121no—
290C2×D3—121no—
291C2×D3—241no—
292C2×D3—241no—
293C2×D3—242no—
294C2×D3—242no—
295C2×D3—242no—
296C3×C4—31no—
297C3×C4—31no—
298C3×C4—31no—
299C3×C4—31no—
300C3×C4—61no—
301C3×C4—61no—
302C3×C4—61no—
303C3×C4—122no—
304C3×C4—121no—
305C3×C4—121no—
306C3×C4—181no—
307C3×C4—241no—
308C3×C4—242no—
309Dic3—21no—
310Dic3—61no—
311Dic3—81no—
312C32—41no—
313C32—41no—
314C32—81no—
315C32—81no—
316C32—81no—
317C32—161no—
318C32#539511yescharacteristic, complemented
319C23—181no—
320Q8—31no—
321Q8—61no—
322Q8—61no—
323Q8C3×#62611yescharacteristic
324D4—181no—
325D4—181no—
326C2×C4—31no—
327C2×C4—31no—
328C2×C4—61no—
329C2×C4—91no—
330C8—61no—
331C8—181no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—21no—
335C2×C3—41no—
336C2×C3—41no—
337C2×C3—41no—
338C2×C3—81no—
339C2×C3—81no—
340C2×C3—81no—
341C2×C3—81no—
342C2×C3—81no—
343C2×C3—81no—
344C2×C3—161no—
345C2×C3—161no—
346C2×C3—161no—
347C2×C3—721no—
348C2×C3#6419811yescharacteristic
349C2×C3#6420211yescharacteristic
350C2×C3#6420711yescharacteristic
351C2×C3C3×C4⋊S411yescharacteristic
352C2×C3C3×#98311yescharacteristic
353C2×C3C3×#98511yescharacteristic
354D3—242no—
355D3—244no—
356D3—241no—
357C22—361no—
358C22—361no—
359C22—361no—
360C22C3×D3×S411yescharacteristic
361C4—31no—
362C4—31no—
363C4—61no—
364C4—121no—
365C4—181no—
366C3—21no—
367C3—41no—
368C3—81no—
369C3—81no—
370C3—161no—
371C3#1059320811yescharacteristic, direct product factor
372C3C3×#539511yescharacteristic, complemented
373C2—721no—
374C2C3×#6419811yescharacteristic
375C2C3×#6420211yescharacteristic
376C2C3×#6420711yescharacteristic
subgroupG/H
center, hypercenterC22×C3D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C22×C3
FrattiniC22C3×D3×S4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3×C3⋊GL(2,𝔽3)C2
hypocenterC3×SL(2,𝔽3)C3×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C2×C32×#226C2semidirect
3. C3×Dic3×SL(2,𝔽3)C2semidirect
5. #10593208C3direct (N × H)
15. C2×C3×#226D3semidirect
16. C2×C3×#226C2×C3semidirect
19. C3×Q8×Dic3D3semidirect
23. Dic3×SL(2,𝔽3)C2×C3semidirect
69. C2×#226C3×D3semidirect
77. Q8×Dic3C3×D3semidirect
318. C32#5395semidirect
371. C3#10593208direct (N × H)
372. C3C3×#5395semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊GL(2,𝔽3)2
C2×C32×#2262
C3×Dic3×SL(2,𝔽3)2
C3×#53953
#105932083
C3×#51853
C3×#37224