Finite Group Databasebeta

← C3×#10593661 #13173029

Subgroups — C3×#10593661 #13173029

Order 1728.

559 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 66 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×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×C3×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×C3⋊#859C211yesmaximal, characteristic, complemented
4#10593661C311yesmaximal, characteristic, direct product factor
5C3×#5427—33nomaximal
6C3×#5238—31nomaximal
7C32×#41⋊C3—21no—
8C2×C32×SL(2,𝔽3)C2211yescharacteristic
9C3×C3⋊GL(2,𝔽3)—21no—
10C3×#3752—41nomaximal
11C2×C32×#41D311yescharacteristic, complemented
12C2×C3×#41⋊C3—21no—
13C2×C3×#41⋊C3—23no—
14C2×C3×#41⋊C3D313yescomplemented
15C2×C3×#41⋊C3C2×C311yescharacteristic, complemented
16C2×C3⋊GL(2,𝔽3)C2×C311yescharacteristic, complemented
17C2×C3×GL(2,𝔽3)—33no—
18C2×C3×#215—31no—
19C2×C3×#213—31no—
20C3×#930—31no—
21C3⋊#859C2×C311yescharacteristic, complemented
22C3×#859—33no—
23C3×#835—31no—
24C3×#832—31no—
25C3×#830—31no—
26C22×C3×C3⋊D3—41no—
27C2×C33×C4—41no—
28C2×C3×C3⋊Dic3—41no—
29C32×SL(2,𝔽3)D411yescharacteristic, complemented
30#5427—33no—
31C3×#449—91no—
32#5238—31no—
33C32×#41—21no—
34C32×#41—61no—
35C2×C32×Q8C2×D311yescharacteristic
36C2×C32×D4—31no—
37C22×C32×C4—31no—
38C2×C3×#67—61no—
39C2×C3×D12—31no—
40C3×#41⋊C3—22no—
41C3×#41⋊C3—26no—
42C3×#41⋊C3—23no—
43C3×#41⋊C3—21no—
44C2×C3×SL(2,𝔽3)—21no—
45C2×C3×SL(2,𝔽3)—23no—
46C2×C3×SL(2,𝔽3)C2×D313yes—
47C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
48C3⋊GL(2,𝔽3)—21no—
49C3×GL(2,𝔽3)—63no—
50#3752—41no—
51C3×#215—61no—
52C3×#215—61no—
53C3×#213—62no—
54C3×(C2×D3)⋊C4—61no—
55C3×(C2×D3)⋊C4—123no—
56C3×C4⋊Dic3—31no—
57C2×C3×C3⋊C8—31no—
58C2×C3×C3⋊C8—31no—
59C22×C33—41no—
60C2×C3×C3⋊D3—42no—
61C2×C3×C3⋊D3—81no—
62C2×C3×C3⋊D3—81no—
63C33×C4—81no—
64C3×C3⋊Dic3—81no—
65C2×C3×#41—21no—
66C2×C3×#41C3×D311yescharacteristic, complemented
67C2×C3×#41C3⋊D311yescharacteristic, complemented
68C2×#41⋊C3—23no—
69C2×#41⋊C3C3×D313yescomplemented
70C2×GL(2,𝔽3)—33no—
71C2×C3×SD16—91no—
72C2×C3×D8—91no—
73C3×#121—91no—
74C2×#215—31no—
75C2×#213—31no—
76#930—31no—
77#859—33no—
78C3×#103—91no—
79C3×#102—91no—
80C3×C22⋊C8—91no—
81#835—31no—
82#832—31no—
83#830—31no—
84C23×C32—31no—
85C22×C3⋊D3—41no—
86C22×C3×D3—61no—
87C22×C3×D3—123no—
88C32×Q8—31no—
89C32×Q8#6711yescharacteristic
90C32×D4—32no—
91C32×D4—61no—
92C2×C32×C4—31no—
93C2×C32×C4—41no—
94C2×C32×C4—43no—
95C2×C32×C4—61no—
96C2×C32×C4—61no—
97C2×C32×C4—83no—
98C2×C32×C4—81no—
99C2×C32×C4S411yescharacteristic
100C2×C3⋊Dic3—41no—
101C3×#67—122no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3—123no—
104C3×D12—61no—
105C3×D12—61no—
106C3×SL(2,𝔽3)—21no—
107C3×SL(2,𝔽3)—23no—
108C3×SL(2,𝔽3)#6713yescomplemented
109C3×SL(2,𝔽3)C3×D411yescharacteristic, complemented
110C3×C3⋊C8—61no—
111C3×C3⋊C8—61no—
112#449—91noSylow
113C2×C33—41no—
114C2×C33—41no—
115C2×C33—41no—
116C3×C3⋊D3—82no—
117C3×#41—22no—
118C3×#41—21no—
119C3×#41—21no—
120C3×#41—61no—
121C3×#41—61no—
122C3×#41—62no—
123C2×C3×Q8—21no—
124C2×C3×Q8C2×C3×D311yescharacteristic
125C2×C3×Q8C2×C3⋊D311yescharacteristic
126C2×C3×D4—31no—
127C2×C3×D4—31no—
128C2×C3×D4—61no—
129C2×C3×D4—91no—
130C2×C3×D4—181no—
131C22×C3×C4—31no—
132C22×C3×C4—31no—
133C22×C3×C4—61no—
134C2×#67—61no—
135C2×D12—31no—
136#41⋊C3—26no—
137#41⋊C3—23no—
138C2×SL(2,𝔽3)—23no—
139C2×SL(2,𝔽3)C2×C3×D313yes—
140GL(2,𝔽3)—63no—
141C3×SD16—181no—
142C3×SD16—181no—
143C3×D8—182no—
144C2×C3×C8—91no—
145C2×C3×C8—91no—
146C3×C4⋊C4—91no—
147C3×C22⋊C4—181no—
148#215—61no—
149#215—61no—
150#213—62no—
151(C2×D3)⋊C4—61no—
152(C2×D3)⋊C4—123no—
153C4⋊Dic3—31no—
154C2×C3⋊C8—31no—
155C2×C3⋊C8—31no—
156C22×C32—41no—
157C22×C32—43no—
158C22×C32—61no—
159C22×C32—61no—
160C22×C32—61no—
161C22×C32—83no—
162C22×C32—81no—
163C22×C32C2×S411yescharacteristic
164C2×C3⋊D3—42no—
165C2×C3⋊D3—81no—
166C2×C3⋊D3—81no—
167C2×C3×D3—126no—
168C2×C3×D3—121no—
169C2×C3×D3—121no—
170C2×C3×D3—121no—
171C2×C3×D3—243no—
172C2×C3×D3—243no—
173C32×C4—21no—
174C32×C4—31no—
175C32×C4—31no—
176C32×C4—81no—
177C32×C4—83no—
178C32×C4—86no—
179C32×C4—82no—
180C3⋊Dic3—81no—
181C3×Dic3—121no—
182C3×Dic3—243no—
183C2×#41C3×C3⋊D311yescharacteristic, complemented
184C2×SD16—91no—
185C2×D8—91no—
186#121—91no—
187#103—91no—
188#102—91no—
189C22⋊C8—91no—
190C33—41noSylow
191C23×C3—31no—
192C23×C3—31no—
193C23×C3—61no—
194C23×C3—181no—
195C22×D3—61no—
196C22×D3—123no—
197C3×Q8—21no—
198C3×Q8—31no—
199C3×Q8—31no—
200C3×Q8—61no—
201C3×Q8C3×#6711yescharacteristic
202C3×Q8#63811yescharacteristic
203C3×D4—32no—
204C3×D4—32no—
205C3×D4—62no—
206C3×D4—61no—
207C3×D4—61no—
208C3×D4—62no—
209C3×D4—181no—
210C3×D4—181no—
211C3×D4—362no—
212C2×C3×C4—21no—
213C2×C3×C4—31no—
214C2×C3×C4—31no—
215C2×C3×C4—43no—
216C2×C3×C4—61no—
217C2×C3×C4—61no—
218C2×C3×C4—61no—
219C2×C3×C4—61no—
220C2×C3×C4—61no—
221C2×C3×C4—62no—
222C2×C3×C4—62no—
223C2×C3×C4—83no—
224C2×C3×C4—181no—
225C2×C3×C4C3×S411yescharacteristic
226C2×C3×C4C3⋊S411yescharacteristic
227#67—122no—
228C2×Dic3—61no—
229C2×Dic3—123no—
230D12—61no—
231D12—61no—
232SL(2,𝔽3)—23no—
233SL(2,𝔽3)C3×#6713yescomplemented
234C3×C8—181no—
235C3×C8—181no—
236C3⋊C8—61no—
237C3⋊C8—61no—
238C2×C32—41no—
239C2×C32—41no—
240C2×C32—41no—
241C2×C32—43no—
242C2×C32—43no—
243C2×C32—43no—
244C2×C32—83no—
245C2×C32—83no—
246C2×C32—83no—
247C2×C32—81no—
248C2×C32—81no—
249C2×C32—81no—
250C2×C32—121no—
251C2×C32#98511yescharacteristic
252C2×C32#98611yescharacteristic
253C2×C32GL(2,ℤ/4)11yescharacteristic
254C3⋊D3—82no—
255C3×D3—246no—
256C3×D3—241no—
257#41—21no—
258#41—61no—
259C2×Q8C2×C3×C3⋊D311yescharacteristic
260C2×D4—31no—
261C2×D4—91no—
262C2×D4—181no—
263C22×C4—31no—
264SD16—181no—
265SD16—181no—
266D8—182no—
267C2×C8—91no—
268C2×C8—91no—
269C4⋊C4—91no—
270C22⋊C4—181no—
271C22×C3—21no—
272C22×C3—43no—
273C22×C3—62no—
274C22×C3—61no—
275C22×C3—61no—
276C22×C3—61no—
277C22×C3—61no—
278C22×C3—61no—
279C22×C3—61no—
280C22×C3—83no—
281C22×C3—121no—
282C22×C3—121no—
283C22×C3—361no—
284C22×C3—361no—
285C22×C3—361no—
286C22×C3C2×C3×S411yescharacteristic
287C22×C3C2×C3⋊S411yescharacteristic
288C2×D3—126no—
289C2×D3—121no—
290C2×D3—121no—
291C2×D3—121no—
292C2×D3—243no—
293C2×D3—243no—
294C3×C4—22no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—31no—
298C3×C4—31no—
299C3×C4—31no—
300C3×C4—31no—
301C3×C4—61no—
302C3×C4—61no—
303C3×C4—83no—
304C3×C4—86no—
305C3×C4—361no—
306Dic3—121no—
307Dic3—243no—
308C32—41no—
309C32—43no—
310C32—83no—
311C32—81no—
312C32#542711yescharacteristic, complemented
313C23—31no—
314C23—181no—
315Q8—31no—
316Q8C3×#63811yescharacteristic
317D4—32no—
318D4—61no—
319D4—181no—
320D4—181no—
321D4—362no—
322C2×C4—31no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—181no—
326C2×C4C3×C3⋊S411yescharacteristic
327C8—181no—
328C8—181no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—43no—
333C2×C3—43no—
334C2×C3—43no—
335C2×C3—83no—
336C2×C3—83no—
337C2×C3—83no—
338C2×C3—122no—
339C2×C3—121no—
340C2×C3—121no—
341C2×C3—721no—
342C2×C3C3×#98511yescharacteristic
343C2×C3C3×#98611yescharacteristic
344C2×C3C3×GL(2,ℤ/4)11yescharacteristic
345C2×C3#6426611yescharacteristic
346C2×C3#6426711yescharacteristic
347C2×C3#6426911yescharacteristic
348D3—246no—
349D3—241no—
350C22—61no—
351C22—61no—
352C22—61no—
353C22—361no—
354C22—361no—
355C22—361no—
356C22C2×C3×C3⋊S411yescharacteristic
357C4—21no—
358C4—31no—
359C4—31no—
360C4—361no—
361C3—21no—
362C3—43no—
363C3—83no—
364C3C3×#542711yescharacteristic, complemented
365C3#1059366111yescharacteristic, direct product factor
366C2—121no—
367C2—721no—
368C2C3×#6426611yescharacteristic
369C2C3×#6426711yescharacteristic
370C2C3×#6426911yescharacteristic
subgroupG/H
centerC22×C3C2×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C22×C3
FrattiniC22C2×C3×C3⋊S4
FittingC2×C32×#41D3
socleC22×C32C2×S4
hypercenterC2×C3×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C3×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#41⋊C3C2semidirect
2. C2×C3×C3⋊GL(2,𝔽3)C2semidirect
3. C3×C3⋊#859C2semidirect
4. #10593661C3direct (N × H)
11. C2×C32×#41D3semidirect
14. C2×C3×#41⋊C3D3semidirect
15. C2×C3×#41⋊C3C2×C3semidirect
16. C2×C3⋊GL(2,𝔽3)C2×C3semidirect
21. C3⋊#859C2×C3semidirect
29. C32×SL(2,𝔽3)D4semidirect
66. C2×C3×#41C3×D3semidirect
67. C2×C3×#41C3⋊D3semidirect
69. C2×#41⋊C3C3×D3semidirect
108. C3×SL(2,𝔽3)#67semidirect
109. C3×SL(2,𝔽3)C3×D4semidirect
183. C2×#41C3×C3⋊D3semidirect
233. SL(2,𝔽3)C3×#67semidirect
312. C32#5427semidirect
364. C3C3×#5427semidirect
365. C3#10593661direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
C2×C3×C3⋊GL(2,𝔽3)2
C3×C3⋊#8592
#105936613
C3×#54273
C3×#52383
C3×#37524