Finite Group Databasebeta

← C2×C3×C3⋊#994 #13181446

Subgroups — C2×C3×C3⋊#994 #13181446

Order 1728.

1099 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 1099 conjugacy classes of subgroups — too many to draw. Only its 180 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
2C3×C3⋊#994C214yesmaximal, direct product factor
3C2×C3×C3⋊(#41⋊C3)C212yesmaximal, complemented
4C2×C3×#1009C311yesmaximal, characteristic, complemented
5C2×C3×#994—31nomaximal
6C2×C3⋊#994C313yesmaximal, direct product factor
7C2×C32×Dic6—41nomaximal
8C32×#41⋊C3C2212yescomplemented
9C2×C32×SL(2,𝔽3)C2211yescharacteristic, complemented
10C3×C3⋊(#41⋊C3)C2214yescomplemented
11C2×C32×#41C2×C311yescharacteristic
12C3×#1009—34no—
13C3×#1009C2×C314yescomplemented
14C2×C3×#239C2×C312yes—
15C2×C3×D3×D4—32no—
16C2×C3×#235—31no—
17C22×C3×D12—31no—
18C3×#994—34no—
19C2×C3×#41⋊C3—21no—
20C2×C3×#41⋊C3—23no—
21C2×C3×#41⋊C3—32no—
22C2×C3×#41⋊C3D311yescharacteristic, complemented
23C2×C3×#41⋊C3C2×C313yescomplemented
24C3⋊#994C2×C3112yesdirect product factor
25C2×C3⋊(#41⋊C3)C2×C316yescomplemented
26C2×C33×C4—41no—
27C2×C32×Dic3—42no—
28C32×Dic6—44no—
29C32×SL(2,𝔽3)C2311yescharacteristic, complemented
30C2×C3×H(5,𝔽2)—31no—
31C2×#1009C3211yescharacteristic, complemented
32C2×#994—33no—
33C23×C3×D3—32no—
34C32×#41—32no—
35C32×#41C22×C312yes—
36C2×C32×Q8—121no—
37C2×C32×Q8C22×C311yescharacteristic
38C2×C32×D4—31no—
39C22×C32×C4—31no—
40C2×C3×#67—32no—
41C3×#239—34no—
42C3×#239C22×C314yes—
43C3×D3×D4—38no—
44C3×D3×D4—38no—
45C3×#235—34no—
46C3×#235—34no—
47C2×C3×D12—34no—
48C2×C3×D12—32no—
49C2×C3×D12—31no—
50C2×C3×D12—34no—
51C2×C3×C4×D3—32no—
52C2×C3×Dic6—43no—
53C2×C3×Dic6A411yescharacteristic
54C3×#41⋊C3—22no—
55C3×#41⋊C3—26no—
56C3×#41⋊C3—34no—
57C3×#41⋊C3C2×D312yescomplemented
58C3×#41⋊C3C22×C316yescomplemented
59C2×C3×SL(2,𝔽3)—21no—
60C2×C3×SL(2,𝔽3)—23no—
61C2×C3×SL(2,𝔽3)C2×D311yescharacteristic, complemented
62C2×C3×SL(2,𝔽3)C22×C313yescomplemented
63C3⋊(#41⋊C3)C22×C3112yescomplemented
64C22×C33—41no—
65C33×C4—42no—
66C32×Dic3—44no—
67C3×H(5,𝔽2)—34no—
68C3×H(5,𝔽2)—94no—
69C2×C3×#41—21no—
70C2×C3×#41—32no—
71C2×C3×#41—91no—
72C2×C3×#41C3×D311yescharacteristic
73C2×C3×#41C2×C3211yescharacteristic
74C22×C3×D4—91no—
75C22×C3×D4—92no—
76#1009—34no—
77#1009C2×C3214yescomplemented
78C2×#239C2×C3212yes—
79C2×D3×D4—32no—
80C2×#235—31no—
81C22×D12—31no—
82#994—312no—
83C2×#41⋊C3—23no—
84C2×#41⋊C3—36no—
85C2×#41⋊C3C3×D313yescomplemented
86C23×C32—31no—
87C22×C3×D3—32no—
88C22×C3×D3—38no—
89C22×C3×D3—124no—
90C32×Q8—31no—
91C32×Q8—124no—
92C32×Q8C23×C311yescharacteristic
93C32×D4—32no—
94C32×D4—32no—
95C2×C32×C4—31no—
96C2×C32×C4—32no—
97C2×C32×C4—32no—
98C2×C32×C4—41no—
99C2×C32×C4—43no—
100C2×C32×C4—83no—
101C2×C32×C4—81no—
102C2×C32×C4—122no—
103C2×C32×C4C2×A411yescharacteristic
104C3×#67—38no—
105C2×C3×Dic3—46no—
106C2×C3×Dic3C2×A412yes—
107C3×D12—34no—
108C3×D12—34no—
109C3×D12—34no—
110C3×C4×D3—34no—
111C3×C4×D3—34no—
112C3×Dic6—412no—
113C3×Dic6C2×A414yes—
114C3×SL(2,𝔽3)—21no—
115C3×SL(2,𝔽3)—23no—
116C3×SL(2,𝔽3)C22×D311yescharacteristic, complemented
117C3×SL(2,𝔽3)C23×C313yescomplemented
118C2×H(5,𝔽2)—31noSylow
119C2×C33—42no—
120C2×C33—41no—
121C24×C3—92no—
122C23×D3—32no—
123C3×#41—22no—
124C3×#41—32no—
125C3×#41—32no—
126C3×#41—34no—
127C3×#41—62no—
128C3×#41—94no—
129C3×#41—94no—
130C3×#41—94no—
131C3×#41C2×C3×D312yes—
132C3×#41C22×C3212yes—
133C2×C3×Q8—21no—
134C2×C3×Q8—31no—
135C2×C3×Q8—123no—
136C2×C3×Q8C2×C3×D311yescharacteristic
137C2×C3×Q8C22×C3211yescharacteristic
138C2×C3×D4—31no—
139C2×C3×D4—31no—
140C2×C3×D4—61no—
141C2×C3×D4—92no—
142C2×C3×D4—92no—
143C2×C3×D4—94no—
144C2×C3×D4—94no—
145C2×C3×D4—98no—
146C2×C3×D4—98no—
147C2×C3×D4—91no—
148C22×C3×C4—31no—
149C22×C3×C4—31no—
150C22×C3×C4—61no—
151C22×C3×C4—92no—
152C2×#67—32no—
153#239—34no—
154#239C22×C3214yes—
155D3×D4—38no—
156D3×D4—38no—
157#235—34no—
158#235—34no—
159C2×D12—32no—
160C2×D12—34no—
161C2×D12—31no—
162C2×D12—34no—
163C2×C4×D3—32no—
164C2×Dic6C3×A411yescharacteristic
165#41⋊C3—26no—
166#41⋊C3—312no—
167#41⋊C3C2×C3×D316yescomplemented
168C2×SL(2,𝔽3)—23no—
169C2×SL(2,𝔽3)C2×C3×D313yescomplemented
170C22×C32—32no—
171C22×C32—41no—
172C22×C32—43no—
173C22×C32—62no—
174C22×C32—83no—
175C22×C32—81no—
176C22×C32C22×A411yescharacteristic
177C2×C3×D3—34no—
178C2×C3×D3—64no—
179C2×C3×D3—128no—
180C32×C4—31no—
181C32×C4—31no—
182C32×C4—42no—
183C32×C4—46no—
184C32×C4—86no—
185C32×C4—82no—
186C32×C4—124no—
187C32×C4C22×A412yes—
188C3×Dic3—412no—
189C3×Dic3C22×A414yes—
190H(5,𝔽2)—34no—
191H(5,𝔽2)—94no—
192C2×#41—32no—
193C2×#41—91no—
194C2×#41C32×D311yescharacteristic
195C22×D4—91no—
196C22×D4—92no—
197C33—41noSylow
198C23×C3—31no—
199C23×C3—31no—
200C23×C3—61no—
201C23×C3—98no—
202C23×C3—92no—
203C23×C3—364no—
204C22×D3—32no—
205C22×D3—38no—
206C22×D3—124no—
207C3×Q8—21no—
208C3×Q8—31no—
209C3×Q8—31no—
210C3×Q8—34no—
211C3×Q8—61no—
212C3×Q8—1212no—
213C3×Q8C22×C3×D311yescharacteristic
214C3×Q8C23×C3211yescharacteristic
215C3×D4—32no—
216C3×D4—32no—
217C3×D4—32no—
218C3×D4—32no—
219C3×D4—62no—
220C3×D4—62no—
221C3×D4—98no—
222C3×D4—94no—
223C3×D4—94no—
224C3×D4—94no—
225C2×C3×C4—21no—
226C2×C3×C4—32no—
227C2×C3×C4—31no—
228C2×C3×C4—31no—
229C2×C3×C4—32no—
230C2×C3×C4—32no—
231C2×C3×C4—32no—
232C2×C3×C4—32no—
233C2×C3×C4—43no—
234C2×C3×C4—62no—
235C2×C3×C4—62no—
236C2×C3×C4—61no—
237C2×C3×C4—83no—
238C2×C3×C4—94no—
239C2×C3×C4—94no—
240C2×C3×C4—126no—
241C2×C3×C4D3×A411yescharacteristic
242C2×C3×C4C2×C3×A411yescharacteristic
243#67—38no—
244C2×Dic3C2×C3×A412yes—
245D12—34no—
246D12—34no—
247D12—34no—
248C4×D3—34no—
249C4×D3—34no—
250Dic6C2×C3×A414yes—
251SL(2,𝔽3)—23no—
252SL(2,𝔽3)C22×C3×D313yescomplemented
253C2×C32—42no—
254C2×C32—41no—
255C2×C32—46no—
256C2×C32—43no—
257C2×C32—62no—
258C2×C32—83no—
259C2×C32—86no—
260C2×C32—82no—
261C2×C32—81no—
262C2×C32#99412yescomplemented
263C2×C32C23×A411yescharacteristic
264C3×D3—64no—
265C24—92no—
266#41—32no—
267#41—34no—
268#41—94no—
269#41—94no—
270#41—94no—
271#41C2×C32×D312yes—
272C2×Q8—31no—
273C2×Q8C2×C32×D311yescharacteristic
274C2×D4—31no—
275C2×D4—92no—
276C2×D4—92no—
277C2×D4—94no—
278C2×D4—94no—
279C2×D4—98no—
280C2×D4—98no—
281C2×D4—91no—
282C22×C4—31no—
283C22×C4—92no—
284C22×C3—21no—
285C22×C3—32no—
286C22×C3—32no—
287C22×C3—43no—
288C22×C3—62no—
289C22×C3—62no—
290C22×C3—62no—
291C22×C3—83no—
292C22×C3—94no—
293C22×C3—122no—
294C22×C3—184no—
295C22×C3—368no—
296C22×C3C2×D3×A411yescharacteristic
297C22×C3C22×C3×A411yescharacteristic
298C2×D3—34no—
299C2×D3—64no—
300C2×D3—128no—
301C3×C4—22no—
302C3×C4—31no—
303C3×C4—31no—
304C3×C4—34no—
305C3×C4—31no—
306C3×C4—31no—
307C3×C4—46no—
308C3×C4—61no—
309C3×C4—61no—
310C3×C4—86no—
311C3×C4—1212no—
312C3×C4C2×D3×A412yes—
313C3×C4C22×C3×A412yes—
314Dic3C22×C3×A414yes—
315C32—41no—
316C32—43no—
317C32—83no—
318C32—81no—
319C32C2×#99411yescharacteristic, complemented
320C23—31no—
321C23—98no—
322C23—92no—
323C23—364no—
324Q8—31no—
325Q8—34no—
326Q8C22×C32×D311yescharacteristic
327D4—32no—
328D4—32no—
329D4—98no—
330D4—94no—
331D4—94no—
332D4—94no—
333C2×C4—31no—
334C2×C4—32no—
335C2×C4—32no—
336C2×C4—32no—
337C2×C4—94no—
338C2×C4—94no—
339C2×C4C3×D3×A411yescharacteristic
340C2×C3—22no—
341C2×C3—21no—
342C2×C3—46no—
343C2×C3—43no—
344C2×C3—62no—
345C2×C3—62no—
346C2×C3—86no—
347C2×C3—83no—
348C2×C3—122no—
349C2×C3—184no—
350C2×C3C3⋊#99412yesdirect product factor
351C2×C3C3×#99412yescomplemented
352C2×C3C22×D3×A411yescharacteristic
353C2×C3C23×C3×A411yescharacteristic
354D3—64no—
355C22—32no—
356C22—62no—
357C22—94no—
358C22—184no—
359C22—368no—
360C22C2×C3×D3×A411yescharacteristic
361C4—31no—
362C4—31no—
363C4—34no—
364C4C2×C3×D3×A412yes—
365C3—21no—
366C3—43no—
367C3—83no—
368C3C2×C3⋊#99411yescharacteristic, direct product factor
369C3C2×C3×#99411yescharacteristic, complemented
370C2—62no—
371C2—184no—
372C2C3×C3⋊#99412yesdirect product factor
373C2C22×C3×D3×A411yescharacteristic
subgroupG/H
centerC22×C3C2×D3×A4
commutator, cosocle, hypocenterC3×Q8C23×C32
FrattiniC2C22×C3×D3×A4
FittingC2×C32×#41C2×C3
socleC22×C32C22×A4
hypercenterC2×C3×C4D3×A4
LCSG ▷ C3×Q8
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. C3×C3⋊#994C2direct (N × H)
3. C2×C3×C3⋊(#41⋊C3)C2semidirect
4. C2×C3×#1009C3semidirect
6. C2×C3⋊#994C3direct (N × H)
8. C32×#41⋊C3C22semidirect
9. C2×C32×SL(2,𝔽3)C22semidirect
10. C3×C3⋊(#41⋊C3)C22semidirect
13. C3×#1009C2×C3semidirect
22. C2×C3×#41⋊C3D3semidirect
23. C2×C3×#41⋊C3C2×C3semidirect
24. C3⋊#994C2×C3direct (N × H)
25. C2×C3⋊(#41⋊C3)C2×C3semidirect
29. C32×SL(2,𝔽3)C23semidirect
31. C2×#1009C32semidirect
57. C3×#41⋊C3C2×D3semidirect
58. C3×#41⋊C3C22×C3semidirect
61. C2×C3×SL(2,𝔽3)C2×D3semidirect
62. C2×C3×SL(2,𝔽3)C22×C3semidirect
63. C3⋊(#41⋊C3)C22×C3semidirect
77. #1009C2×C32semidirect
85. C2×#41⋊C3C3×D3semidirect
116. C3×SL(2,𝔽3)C22×D3semidirect
117. C3×SL(2,𝔽3)C23×C3semidirect
167. #41⋊C3C2×C3×D3semidirect
169. C2×SL(2,𝔽3)C2×C3×D3semidirect
252. SL(2,𝔽3)C22×C3×D3semidirect
262. C2×C32#994semidirect
319. C32C2×#994semidirect
350. C2×C3C3⋊#994direct (N × H)
351. C2×C3C3×#994semidirect
368. C3C2×C3⋊#994direct (N × H)
369. C3C2×C3×#994semidirect
372. C2C3×C3⋊#994direct (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
C3×C3⋊#9942
C2×C3×C3⋊(#41⋊C3)2
C2×C3×#10093
C2×C3×#9943
C2×C3⋊#9943
C2×C32×Dic64