Finite Group Databasebeta

← C3×#6291⋊C3 #14518626

Subgroups — C3×#6291⋊C3 #14518626

Order 1944.

783 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 783 conjugacy classes of subgroups — too many to draw. Only its 94 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⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
2C2×C3×D3×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C32×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4C3×#6291C311yesmaximal, characteristic, complemented
5C32×#626—31nomaximal
6C3×#638⋊C3—31nomaximal
7#6291⋊C3C313yesmaximal, direct product factor
8C2×C32×H(3,𝔽3)C2211yescharacteristic
9C3×C3⋊((C3⋊D3)⋊C3)—21no—
10C3×D3×H(3,𝔽3)—21no—
11C2×C3×C3⋊(C3⋊D3)C2×C311yescharacteristic, complemented
12C2×C32×C3⋊D3—31no—
13C2×C33×D3—21no—
14C2×C33×D3D311yescharacteristic, complemented
15C2×C33×D3C2×C311yescharacteristic, complemented
16C32×C3⋊Dic3C2×C311yescharacteristic, complemented
17C33×Dic3—31no—
18C22×C3×H(3,𝔽3)—31no—
19C2×C3⋊((C3⋊D3)⋊C3)C2×C313yescomplemented
20C2×D3×H(3,𝔽3)—23no—
21C2×D3×H(3,𝔽3)C2×C313yescomplemented
22C2×C3×(C3⋊D3)⋊C3—31no—
23C2×C3×(C3⋊D3)⋊C3—61no—
24C3×(C3⋊Dic3)⋊C3—21no—
25C3×(C3⋊Dic3)⋊C3—23no—
26C3×(C3⋊Dic3)⋊C3D311yescharacteristic, complemented
27C3×(C3⋊Dic3)⋊C3C2×C313yescomplemented
28C32×H(3,𝔽3)D411yescharacteristic, complemented, Sylow
29C3×#638—31no—
30C32×#67—91no—
31C32×D12—91no—
32#6291C3211yescharacteristic, complemented
33C3×#626—31no—
34C3×#626—33no—
35C3×#626—91no—
36#638⋊C3—33no—
37C2×C34—21no—
38C2×C34C2×D311yescharacteristic
39C2×C34C22×C311yescharacteristic
40C3×C3⋊(C3⋊D3)—21no—
41C32×C3⋊D3—61no—
42C33×D3—22no—
43C33×D3—21no—
44C33×D3—21no—
45C2×C3×H(3,𝔽3)—21no—
46C2×C3×H(3,𝔽3)—21no—
47C2×C3×H(3,𝔽3)—23no—
48C2×C3×H(3,𝔽3)—23no—
49C2×C3×H(3,𝔽3)—43no—
50C2×C3×H(3,𝔽3)—41no—
51C2×C3×H(3,𝔽3)—61no—
52C2×C3×H(3,𝔽3)C2×D311yescharacteristic
53C2×C3×H(3,𝔽3)C22×C313yes—
54C3⋊((C3⋊D3)⋊C3)—23no—
55D3×H(3,𝔽3)—26no—
56D3×H(3,𝔽3)—23no—
57C3×(C3⋊D3)⋊C3—62no—
58C3×(C3⋊D3)⋊C3—61no—
59C22×C33—31no—
60C22×C33—31no—
61C22×C33—61no—
62C2×C3⋊(C3⋊D3)C2×C3211yescharacteristic, complemented
63C2×C3×C3⋊D3—33no—
64C2×C3×C3⋊D3—31no—
65C2×C3×C3⋊D3—31no—
66C2×C3×C3⋊D3—61no—
67C2×C3×C3⋊D3—91no—
68C2×C3×C3⋊D3—181no—
69C2×C32×D3—21no—
70C2×C32×D3—23no—
71C2×C32×D3—31no—
72C2×C32×D3—31no—
73C2×C32×D3—61no—
74C2×C32×D3—61no—
75C2×C32×D3—62no—
76C2×C32×D3—61no—
77C2×C32×D3—91no—
78C2×C32×D3—91no—
79C2×C32×D3—181no—
80C2×C32×D3C3×D311yescharacteristic
81C2×C32×D3C3×D313yescomplemented
82C2×C32×D3C2×C3211yescharacteristic, complemented
83C33×C4—91no—
84C3×C3⋊Dic3—21no—
85C3×C3⋊Dic3C3×D311yescharacteristic, complemented
86C3×C3⋊Dic3C2×C3211yescharacteristic, complemented
87C32×Dic3—31no—
88C32×Dic3—33no—
89C32×Dic3—31no—
90C32×Dic3—63no—
91C32×Dic3—61no—
92C32×Dic3—91no—
93C22×H(3,𝔽3)—33no—
94C22×H(3,𝔽3)—63no—
95C2×(C3⋊D3)⋊C3—33no—
96C2×(C3⋊D3)⋊C3—63no—
97(C3⋊Dic3)⋊C3—23no—
98(C3⋊Dic3)⋊C3C3×D313yescomplemented
99C34—21no—
100C34#6711yescharacteristic, complemented
101C34C3×D411yescharacteristic, complemented
102C3×H(3,𝔽3)—21no—
103C3×H(3,𝔽3)—21no—
104C3×H(3,𝔽3)—23no—
105C3×H(3,𝔽3)—23no—
106C3×H(3,𝔽3)—43no—
107C3×H(3,𝔽3)—41no—
108C3×H(3,𝔽3)D1211yescharacteristic, complemented
109C3×H(3,𝔽3)C3×D413yescomplemented
110C32×D4—271no—
111#638—31no—
112C3×#67—93no—
113C3×#67—91no—
114C3×#67—271no—
115C3×D12—93no—
116C3×D12—91no—
117#626—31no—
118#626—91no—
119C2×C33—21no—
120C2×C33—23no—
121C2×C33—21no—
122C2×C33—21no—
123C2×C33—21no—
124C2×C33—21no—
125C2×C33—23no—
126C2×C33—31no—
127C2×C33—31no—
128C2×C33—41no—
129C2×C33—43no—
130C2×C33—41no—
131C2×C33—61no—
132C2×C33—61no—
133C2×C33—61no—
134C2×C33—61no—
135C2×C33—62no—
136C2×C33—61no—
137C2×C33—62no—
138C2×C33—61no—
139C2×C33—61no—
140C2×C33—121no—
141C2×C33—121no—
142C2×C33—121no—
143C2×C33—122no—
144C2×C33D3211yescharacteristic
145C2×C33C2×C3×D311yescharacteristic
146C2×C33C2×C3×D311yescharacteristic
147C2×C33C2×C3×D313yes—
148C2×C33C22×C3211yescharacteristic
149C3⋊(C3⋊D3)—21no—
150C3×C3⋊D3—61no—
151C3×C3⋊D3—63no—
152C3×C3⋊D3—62no—
153C3×C3⋊D3—61no—
154C3×C3⋊D3—181no—
155C3×C3⋊D3—182no—
156C32×D3—21no—
157C32×D3—22no—
158C32×D3—26no—
159C32×D3—23no—
160C32×D3—21no—
161C32×D3—61no—
162C32×D3—61no—
163C32×D3—62no—
164C32×D3—62no—
165C32×D3—64no—
166C32×D3—62no—
167C32×D3—181no—
168C32×D3—182no—
169C32×D3—181no—
170C2×H(3,𝔽3)—23no—
171C2×H(3,𝔽3)—23no—
172C2×H(3,𝔽3)—23no—
173C2×H(3,𝔽3)—43no—
174C2×H(3,𝔽3)—43no—
175C2×H(3,𝔽3)—46no—
176C2×H(3,𝔽3)—43no—
177C2×H(3,𝔽3)—66no—
178C2×H(3,𝔽3)—63no—
179C2×H(3,𝔽3)C2×C3×D313yes—
180(C3⋊D3)⋊C3—66no—
181(C3⋊D3)⋊C3—63no—
182C22×C32—31no—
183C22×C32—33no—
184C22×C32—31no—
185C22×C32—63no—
186C22×C32—61no—
187C22×C32—91no—
188C22×C32—91no—
189C22×C32—181no—
190C22×C32—181no—
191C22×C32—181no—
192C22×C32—182no—
193C22×C32—271no—
194C2×C3⋊D3—31no—
195C2×C3⋊D3—31no—
196C2×C3⋊D3—61no—
197C2×C3⋊D3—91no—
198C2×C3⋊D3—181no—
199C2×C3×D3—21no—
200C2×C3×D3—33no—
201C2×C3×D3—31no—
202C2×C3×D3—61no—
203C2×C3×D3—63no—
204C2×C3×D3—93no—
205C2×C3×D3—91no—
206C2×C3×D3—93no—
207C2×C3×D3—91no—
208C2×C3×D3—181no—
209C2×C3×D3—183no—
210C2×C3×D3—271no—
211C2×C3×D3—541no—
212C2×C3×D3(C3⋊D3)⋊C311yescharacteristic, complemented
213C2×C3×D3C32×D311yescharacteristic
214C32×C4—91no—
215C32×C4—93no—
216C32×C4—91no—
217C32×C4—183no—
218C32×C4—181no—
219C3⋊Dic3C32×D311yescharacteristic, complemented
220C3×Dic3—33no—
221C3×Dic3—31no—
222C3×Dic3—31no—
223C3×Dic3—63no—
224C3×Dic3—61no—
225C3×Dic3—91no—
226C3×Dic3—91no—
227C3×Dic3—181no—
228C33—21no—
229C33—21no—
230C33—23no—
231C33—21no—
232C33—21no—
233C33—21no—
234C33—23no—
235C33—31no—
236C33—31no—
237C33—41no—
238C33—43no—
239C33—41no—
240C33—61no—
241C33—61no—
242C33—61no—
243C33—61no—
244C33—62no—
245C33—61no—
246C33—121no—
247C33—121no—
248C33—121no—
249C33—122no—
250C33#62611yescharacteristic, complemented
251C33C3×D1211yescharacteristic, complemented
252C33C3×#6711yescharacteristic
253C33C3×#6713yescomplemented
254C33C32×D411yescharacteristic, complemented
255H(3,𝔽3)—23no—
256H(3,𝔽3)—23no—
257H(3,𝔽3)—23no—
258H(3,𝔽3)—43no—
259H(3,𝔽3)—43no—
260H(3,𝔽3)—46no—
261H(3,𝔽3)—43no—
262H(3,𝔽3)C3×D1213yescomplemented
263C3×D4—271no—
264C3×D4—273no—
265#67—91no—
266#67—271no—
267D12—91no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32—21no—
273C2×C32—23no—
274C2×C32—23no—
275C2×C32—31no—
276C2×C32—31no—
277C2×C32—33no—
278C2×C32—31no—
279C2×C32—41no—
280C2×C32—43no—
281C2×C32—41no—
282C2×C32—61no—
283C2×C32—63no—
284C2×C32—63no—
285C2×C32—61no—
286C2×C32—61no—
287C2×C32—61no—
288C2×C32—62no—
289C2×C32—61no—
290C2×C32—61no—
291C2×C32—61no—
292C2×C32—61no—
293C2×C32—61no—
294C2×C32—63no—
295C2×C32—62no—
296C2×C32—66no—
297C2×C32—63no—
298C2×C32—61no—
299C2×C32—121no—
300C2×C32—121no—
301C2×C32—123no—
302C2×C32—121no—
303C2×C32—121no—
304C2×C32—123no—
305C2×C32—122no—
306C2×C32—123no—
307C2×C32—121no—
308C2×C32—122no—
309C2×C32—126no—
310C2×C32—121no—
311C2×C32—181no—
312C2×C32—181no—
313C2×C32—182no—
314C2×C32—182no—
315C2×C32—184no—
316C2×C32—182no—
317C2×C32—541no—
318C2×C32C2×(C3⋊D3)⋊C311yescharacteristic
319C2×C32C3×D3211yescharacteristic
320C2×C32C3×D3213yes—
321C2×C32C2×C32×D311yescharacteristic
322C2×C32C2×C32×D311yescharacteristic
323C3⋊D3—61no—
324C3⋊D3—62no—
325C3⋊D3—61no—
326C3⋊D3—181no—
327C3⋊D3—182no—
328C3×D3—22no—
329C3×D3—21no—
330C3×D3—21no—
331C3×D3—63no—
332C3×D3—61no—
333C3×D3—62no—
334C3×D3—66no—
335C3×D3—182no—
336C3×D3—183no—
337C3×D3—186no—
338C3×D3—181no—
339C3×D3—181no—
340C3×D3—183no—
341C3×D3—541no—
342C3×D3—542no—
343C22×C3—31no—
344C22×C3—31no—
345C22×C3—61no—
346C22×C3—91no—
347C22×C3—93no—
348C22×C3—181no—
349C22×C3—183no—
350C22×C3—273no—
351C22×C3—271no—
352C2×D3—91no—
353C2×D3—91no—
354C2×D3—181no—
355C2×D3—271no—
356C2×D3—541no—
357C2×D3C3×(C3⋊D3)⋊C311yescharacteristic, complemented
358C3×C4—93no—
359C3×C4—91no—
360C3×C4—91no—
361C3×C4—181no—
362C3×C4—183no—
363Dic3—31no—
364Dic3—91no—
365C32—21no—
366C32—21no—
367C32—21no—
368C32—21no—
369C32—21no—
370C32—23no—
371C32—23no—
372C32—31no—
373C32—33no—
374C32—31no—
375C32—31no—
376C32—41no—
377C32—43no—
378C32—41no—
379C32—61no—
380C32—63no—
381C32—61no—
382C32—61no—
383C32—63no—
384C32—61no—
385C32—62no—
386C32—61no—
387C32—61no—
388C32—61no—
389C32—63no—
390C32—61no—
391C32—121no—
392C32—121no—
393C32—121no—
394C32—121no—
395C32—123no—
396C32—123no—
397C32—122no—
398C32—123no—
399C32—121no—
400C32—122no—
401C32—126no—
402C32—121no—
403C32#638⋊C311yescharacteristic, complemented
404C32C3×#62611yescharacteristic
405C32C3×#62613yescomplemented
406C32C32×D1211yescharacteristic, complemented
407C32C32×#6711yescharacteristic
408D4—271noSylow
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—31no—
413C2×C3—33no—
414C2×C3—41no—
415C2×C3—62no—
416C2×C3—61no—
417C2×C3—61no—
418C2×C3—63no—
419C2×C3—61no—
420C2×C3—61no—
421C2×C3—63no—
422C2×C3—121no—
423C2×C3—123no—
424C2×C3—181no—
425C2×C3—183no—
426C2×C3—182no—
427C2×C3—186no—
428C2×C3—543no—
429C2×C3—541no—
430C2×C3D3×(C3⋊D3)⋊C311yescharacteristic
431C2×C3C2×C3×(C3⋊D3)⋊C311yescharacteristic
432C2×C3C32×D3211yescharacteristic
433D3—21no—
434D3—182no—
435D3—181no—
436D3—181no—
437D3—541no—
438D3—542no—
439C22—31no—
440C22—271no—
441C4—91no—
442C3—21no—
443C3—21no—
444C3—21no—
445C3—31no—
446C3—33no—
447C3—41no—
448C3—61no—
449C3—61no—
450C3—63no—
451C3—63no—
452C3—121no—
453C3—123no—
454C3#6291⋊C311yescharacteristic, direct product factor
455C3C3×#638⋊C311yescharacteristic, complemented
456C3C32×#62611yescharacteristic
457C2—61no—
458C2—541no—
459C2C3×D3×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
center, hypercenterC2×C3D3×(C3⋊D3)⋊C3
commutator, cosocleC2×C33C22×C32
FrattiniC2×C3C32×D32
FittingC2×C32×H(3,𝔽3)C22
socleC2×C33C2×C3×D3
hypocenterC33C32×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×C3⋊((C3⋊D3)⋊C3)C2semidirect
2. C2×C3×D3×H(3,𝔽3)C2semidirect
3. C32×(C3⋊Dic3)⋊C3C2semidirect
4. C3×#6291C3semidirect
7. #6291⋊C3C3direct (N × H)
11. C2×C3×C3⋊(C3⋊D3)C2×C3semidirect
14. C2×C33×D3D3semidirect
15. C2×C33×D3C2×C3semidirect
16. C32×C3⋊Dic3C2×C3semidirect
19. C2×C3⋊((C3⋊D3)⋊C3)C2×C3semidirect
21. C2×D3×H(3,𝔽3)C2×C3semidirect
26. C3×(C3⋊Dic3)⋊C3D3semidirect
27. C3×(C3⋊Dic3)⋊C3C2×C3semidirect
28. C32×H(3,𝔽3)D4semidirect
32. #6291C32semidirect
62. C2×C3⋊(C3⋊D3)C2×C32semidirect
81. C2×C32×D3C3×D3semidirect
82. C2×C32×D3C2×C32semidirect
85. C3×C3⋊Dic3C3×D3semidirect
86. C3×C3⋊Dic3C2×C32semidirect
98. (C3⋊Dic3)⋊C3C3×D3semidirect
100. C34#67semidirect
101. C34C3×D4semidirect
108. C3×H(3,𝔽3)D12semidirect
109. C3×H(3,𝔽3)C3×D4semidirect
212. C2×C3×D3(C3⋊D3)⋊C3semidirect
219. C3⋊Dic3C32×D3semidirect
250. C33#626semidirect
251. C33C3×D12semidirect
253. C33C3×#67semidirect
254. C33C32×D4semidirect
262. H(3,𝔽3)C3×D12semidirect
357. C2×D3C3×(C3⋊D3)⋊C3semidirect
403. C32#638⋊C3semidirect
405. C32C3×#626semidirect
406. C32C32×D12semidirect
454. C3#6291⋊C3direct (N × H)
455. C3C3×#638⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊((C3⋊D3)⋊C3)2
C2×C3×D3×H(3,𝔽3)2
C32×(C3⋊Dic3)⋊C32
C3×#62913
C32×#6263
C3×#638⋊C33
#6291⋊C33