Finite Group Databasebeta

← C3×#10593663 #13173031

Subgroups — C3×#10593663 #13173031

Order 1728.

496 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 58 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
1C32×#994C211yesmaximal, characteristic, complemented
2C3×#64267C211yesmaximal, characteristic, complemented
3C3×C3⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#10593663C311yesmaximal, characteristic, direct product factor
5C3×#5429—33nomaximal
6C3×#5241—31nomaximal
7C32×#41⋊C3—21no—
8C32×#41⋊C3C2211yescharacteristic
9C3×C3⋊GL(2,𝔽3)—21no—
10C3×#3755—41nomaximal
11C32×H(5,𝔽2)D311yescharacteristic, complemented
12C3×#994—21no—
13C3×#994—23no—
14C3×#994D313yescomplemented
15C3×#994C2×C311yescharacteristic, complemented
16#64267C2×C311yescharacteristic, complemented
17C3×#986—33no—
18C3×#949—31no—
19C3×#949—31no—
20C3×#940—31no—
21C3⋊GU(2,𝔽3)C2×C311yescharacteristic, complemented
22C3×GU(2,𝔽3)—33no—
23C3×#837—31no—
24C3×#837—31no—
25C3×#833—31no—
26C33×Q8—41no—
27C3×C4⋊(C3⋊D3)—41no—
28C32×SL(2,𝔽3)D411yescharacteristic, complemented
29C3×C3⋊(C3⋊C8)—41no—
30#5429—33no—
31C3×#453—91no—
32#5241—31no—
33C32×#41—21no—
34C32×#41—31no—
35C32×#41C2×D311yescharacteristic
36C2×C32×D4—31no—
37C2×C32×D4—61no—
38C2×C3×#67—62no—
39C2×C3×D12—31no—
40C3×#41⋊C3—22no—
41C3×#41⋊C3—26no—
42C3×#41⋊C3—21no—
43C3×#41⋊C3—23no—
44C3×#41⋊C3—23no—
45C3×#41⋊C3—21no—
46C3×#41⋊C3C2×D313yes—
47C3×#41⋊C3C22×C311yescharacteristic
48C3⋊GL(2,𝔽3)—21no—
49C3×GL(2,𝔽3)—63no—
50#3755—41no—
51C3×#215—61no—
52C3×#215—61no—
53C3×#215—123no—
54C3×#213—61no—
55C3×#213—61no—
56C3×C4×Dic3—31no—
57C3×#208—31no—
58C3×#208—31no—
59C2×C3×C3⋊D3—81no—
60C33×C4—41no—
61C33×C4—81no—
62C3×H(5,𝔽2)—21no—
63C3×H(5,𝔽2)C3×D311yescharacteristic, complemented
64C3×H(5,𝔽2)C3⋊D311yescharacteristic, complemented
65#994—23no—
66#994C3×D313yescomplemented
67#986—33no—
68C3×AGL(1,ℤ/8)—91no—
69C3×AGL(1,ℤ/8)—91no—
70C3×#127—91no—
71#949—31no—
72#949—31no—
73#940—31no—
74GU(2,𝔽3)—33no—
75C3×#104—91no—
76C3×#104—91no—
77C3×#100—91no—
78#837—31no—
79#837—31no—
80#833—31no—
81C23×C32—32no—
82C22×C3×D3—61no—
83C32×Q8—41no—
84C32×Q8—43no—
85C32×Q8—83no—
86C32×Q8—81no—
87C32×Q8#6711yescharacteristic
88C32×Q8S411yescharacteristic
89C32×D4—31no—
90C32×D4—31no—
91C32×D4—61no—
92C32×D4—61no—
93C2×C32×C4—31no—
94C2×C32×C4—61no—
95C4⋊(C3⋊D3)—41no—
96C3×#67—124no—
97C2×C3×Dic3—61no—
98C3×D12—61no—
99C3×D12—61no—
100C3×D12—123no—
101C3×SL(2,𝔽3)—21no—
102C3×SL(2,𝔽3)—23no—
103C3×SL(2,𝔽3)#6713yescomplemented
104C3×SL(2,𝔽3)C3×D411yescharacteristic, complemented
105C3⋊(C3⋊C8)—41no—
106C3×C3⋊C8—61no—
107C3×C3⋊C8—61no—
108C3×C3⋊C8—123no—
109#453—91noSylow
110C2×C33—41no—
111C3×C3⋊D3—161no—
112C3×#41—22no—
113C3×#41—21no—
114C3×#41—21no—
115C3×#41—21no—
116C3×#41—31no—
117C3×#41—31no—
118C3×#41—61no—
119C3×#41C2×C3×D311yescharacteristic
120C3×#41C2×C3⋊D311yescharacteristic
121C2×C3×D4—31no—
122C2×C3×D4—31no—
123C2×C3×D4—61no—
124C2×C3×D4—61no—
125C2×C3×D4—61no—
126C2×C3×D4—62no—
127C2×C3×D4—91no—
128C2×C3×D4—182no—
129C2×#67—62no—
130C2×D12—31no—
131#41⋊C3—26no—
132#41⋊C3—23no—
133#41⋊C3—23no—
134#41⋊C3C2×C3×D313yes—
135GL(2,𝔽3)—63no—
136C3×SD16—181no—
137C3×SD16—181no—
138C3×D8—181no—
139C3×D8—181no—
140C3×M16—91no—
141C3×M16—91no—
142C3×C42—91no—
143#215—61no—
144#215—61no—
145#215—123no—
146#213—61no—
147#213—61no—
148C4×Dic3—31no—
149#208—31no—
150#208—31no—
151C22×C32—31no—
152C22×C32—61no—
153C22×C32—122no—
154C2×C3⋊D3—81no—
155C2×C3×D3—122no—
156C2×C3×D3—121no—
157C2×C3×D3—243no—
158C32×C4—21no—
159C32×C4—31no—
160C32×C4—41no—
161C32×C4—43no—
162C32×C4—83no—
163C32×C4—81no—
164C32×C4—83no—
165C32×C4—86no—
166C32×C4—82no—
167C32×C4—81no—
168C32×C4C2×S411yescharacteristic
169C3×Dic3—62no—
170H(5,𝔽2)C3×C3⋊D311yescharacteristic, complemented
171AGL(1,ℤ/8)—91no—
172AGL(1,ℤ/8)—91no—
173#127—91no—
174#104—91no—
175#104—91no—
176#100—91no—
177C33—41noSylow
178C23×C3—32no—
179C23×C3—32no—
180C23×C3—62no—
181C23×C3—181no—
182C22×D3—61no—
183C3×Q8—21no—
184C3×Q8—21no—
185C3×Q8—43no—
186C3×Q8—83no—
187C3×Q8C3×#6711yescharacteristic
188C3×Q8#63811yescharacteristic
189C3×Q8C3×S411yescharacteristic
190C3×Q8C3⋊S411yescharacteristic
191C3×D4—31no—
192C3×D4—31no—
193C3×D4—31no—
194C3×D4—31no—
195C3×D4—61no—
196C3×D4—61no—
197C3×D4—61no—
198C3×D4—61no—
199C3×D4—62no—
200C3×D4—61no—
201C3×D4—61no—
202C3×D4—62no—
203C3×D4—181no—
204C3×D4—181no—
205C3×D4—364no—
206C2×C3×C4—31no—
207C2×C3×C4—31no—
208C2×C3×C4—61no—
209C2×C3×C4—61no—
210C2×C3×C4—61no—
211C2×C3×C4—62no—
212C2×C3×C4—181no—
213#67—124no—
214C2×Dic3—61no—
215D12—61no—
216D12—61no—
217D12—123no—
218SL(2,𝔽3)—23no—
219SL(2,𝔽3)C3×#6713yescomplemented
220C3×C8—181no—
221C3×C8—181no—
222C3⋊C8—61no—
223C3⋊C8—61no—
224C3⋊C8—123no—
225C2×C32—41no—
226C2×C32—43no—
227C2×C32—61no—
228C2×C32—83no—
229C2×C32—81no—
230C2×C32—121no—
231C2×C32GL(2,ℤ/4)11yescharacteristic
232C3⋊D3—161no—
233C3×D3—241no—
234C3×D3—483no—
235#41—21no—
236#41—31no—
237#41C2×C3×C3⋊D311yescharacteristic
238C2×D4—31no—
239C2×D4—61no—
240C2×D4—91no—
241C2×D4—182no—
242SD16—181no—
243SD16—181no—
244D8—181no—
245D8—181no—
246M16—91no—
247M16—91no—
248C42—91no—
249C22×C3—31no—
250C22×C3—31no—
251C22×C3—61no—
252C22×C3—62no—
253C22×C3—61no—
254C22×C3—61no—
255C22×C3—122no—
256C22×C3—122no—
257C22×C3—242no—
258C22×C3—362no—
259C22×C3—361no—
260C2×D3—122no—
261C2×D3—121no—
262C2×D3—243no—
263C3×C4—22no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—31no—
268C3×C4—31no—
269C3×C4—43no—
270C3×C4—61no—
271C3×C4—83no—
272C3×C4—86no—
273C3×C4—83no—
274C3×C4—182no—
275C3×C4C2×C3×S411yescharacteristic
276C3×C4C2×C3⋊S411yescharacteristic
277Dic3—62no—
278C32—41no—
279C32—43no—
280C32—83no—
281C32—81no—
282C32#542911yescharacteristic, complemented
283C23—32no—
284C23—181no—
285Q8C3×#63811yescharacteristic
286Q8C3×C3⋊S411yescharacteristic
287D4—31no—
288D4—31no—
289D4—61no—
290D4—61no—
291D4—181no—
292D4—181no—
293D4—364no—
294C2×C4—31no—
295C2×C4—61no—
296C2×C4—181no—
297C8—181no—
298C8—181no—
299C2×C3—21no—
300C2×C3—43no—
301C2×C3—61no—
302C2×C3—61no—
303C2×C3—83no—
304C2×C3—121no—
305C2×C3—122no—
306C2×C3—121no—
307C2×C3—121no—
308C2×C3—721no—
309C2×C3C3×GL(2,ℤ/4)11yescharacteristic
310C2×C3#6426911yescharacteristic
311D3—241no—
312D3—483no—
313C22—31no—
314C22—61no—
315C22—122no—
316C22—362no—
317C22—361no—
318C4—21no—
319C4—31no—
320C4—182no—
321C4C2×C3×C3⋊S411yescharacteristic
322C3—21no—
323C3—43no—
324C3—83no—
325C3C3×#542911yescharacteristic, complemented
326C3#1059366311yescharacteristic, direct product factor
327C2—61no—
328C2—121no—
329C2—721no—
330C2C3×#6426911yescharacteristic
subgroupG/H
centerC2×C3#64269
commutator, cosocleC3×#41⋊C3C22×C3
FrattiniC4C2×C3×C3⋊S4
FittingC32×H(5,𝔽2)D3
socleC2×C32GL(2,ℤ/4)
hypercenterC3×Q8C3⋊S4
hypocenterC3×SL(2,𝔽3)C3×D4
LCSG ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×Q8

Complements

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

NH ≅ G/Ntype
1. C32×#994C2semidirect
2. C3×#64267C2semidirect
3. C3×C3⋊GU(2,𝔽3)C2semidirect
4. #10593663C3direct (N × H)
11. C32×H(5,𝔽2)D3semidirect
14. C3×#994D3semidirect
15. C3×#994C2×C3semidirect
16. #64267C2×C3semidirect
21. C3⋊GU(2,𝔽3)C2×C3semidirect
28. C32×SL(2,𝔽3)D4semidirect
63. C3×H(5,𝔽2)C3×D3semidirect
64. C3×H(5,𝔽2)C3⋊D3semidirect
66. #994C3×D3semidirect
103. C3×SL(2,𝔽3)#67semidirect
104. C3×SL(2,𝔽3)C3×D4semidirect
170. H(5,𝔽2)C3×C3⋊D3semidirect
219. SL(2,𝔽3)C3×#67semidirect
282. C32#5429semidirect
325. C3C3×#5429semidirect
326. C3#10593663direct (N × H)

Maximal subgroups

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

subgroupindex
C32×#9942
C3×#642672
C3×C3⋊GU(2,𝔽3)2
#105936633
C3×#54293
C3×#52413
C3×#37554