Finite Group Databasebeta

← C3×#6292⋊C3 #14518620

Subgroups — C3×#6292⋊C3 #14518620

Order 1944.

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

normal subgroups only

This group has 761 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×C32×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3C3×Dic3×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4C3×#6292C311yesmaximal, characteristic, complemented
5C32×#626—31nomaximal
6C3×C4⋊((C3⋊D3)⋊C3)—31nomaximal
7#6292⋊C3C313yesmaximal, direct product factor
8C2×C32×H(3,𝔽3)C2211yescharacteristic
9C3×C3⋊((C3⋊D3)⋊C3)—21no—
10C32×(C3⋊D3)⋊C3—21no—
11C2×C3×C3⋊(C3⋊D3)C2×C311yescharacteristic, complemented
12C2×C32×C3⋊D3—31no—
13C2×C32×C3⋊D3C2×C311yescharacteristic, complemented
14C2×C33×D3—31no—
15C33×Dic3—21no—
16C33×Dic3D311yescharacteristic, complemented
17C33×Dic3C2×C311yescharacteristic, complemented
18C2×C3⋊((C3⋊D3)⋊C3)C2×C313yescomplemented
19C2×C3×(C3⋊D3)⋊C3—21no—
20C2×C3×(C3⋊D3)⋊C3—23no—
21C2×C3×(C3⋊D3)⋊C3—31no—
22C2×C3×(C3⋊D3)⋊C3—61no—
23C2×C3×(C3⋊D3)⋊C3D311yescharacteristic, complemented
24C2×C3×(C3⋊D3)⋊C3C2×C313yescomplemented
25C3×C4×H(3,𝔽3)—31no—
26Dic3×H(3,𝔽3)—23no—
27Dic3×H(3,𝔽3)C2×C313yescomplemented
28C32×H(3,𝔽3)D411yescharacteristic, complemented, Sylow
29C3×C4⋊(C3⋊D3)—31no—
30C32×#67—91no—
31C32×D12—91no—
32#6292C3211yescharacteristic, complemented
33C3×#626—31no—
34C3×#626—33no—
35C3×#626—91no—
36C4⋊((C3⋊D3)⋊C3)—33no—
37C2×C34—21no—
38C2×C34C2×D311yescharacteristic
39C2×C34C22×C311yescharacteristic
40C3×C3⋊(C3⋊D3)—21no—
41C32×C3⋊D3—21no—
42C32×C3⋊D3—61no—
43C33×D3—61no—
44C2×C3×H(3,𝔽3)—21no—
45C2×C3×H(3,𝔽3)—21no—
46C2×C3×H(3,𝔽3)—23no—
47C2×C3×H(3,𝔽3)—23no—
48C2×C3×H(3,𝔽3)—43no—
49C2×C3×H(3,𝔽3)—41no—
50C2×C3×H(3,𝔽3)C2×D311yescharacteristic
51C2×C3×H(3,𝔽3)C22×C313yes—
52C3⋊((C3⋊D3)⋊C3)—23no—
53C3×(C3⋊D3)⋊C3—22no—
54C3×(C3⋊D3)⋊C3—26no—
55C3×(C3⋊D3)⋊C3—21no—
56C3×(C3⋊D3)⋊C3—23no—
57C3×(C3⋊D3)⋊C3—62no—
58C3×(C3⋊D3)⋊C3—61no—
59C22×C33—91no—
60C2×C3⋊(C3⋊D3)C2×C3211yescharacteristic, complemented
61C2×C3×C3⋊D3—21no—
62C2×C3×C3⋊D3—33no—
63C2×C3×C3⋊D3—31no—
64C2×C3×C3⋊D3—31no—
65C2×C3×C3⋊D3—61no—
66C2×C3×C3⋊D3—91no—
67C2×C3×C3⋊D3—181no—
68C2×C3×C3⋊D3C3×D311yescharacteristic, complemented
69C2×C3×C3⋊D3C2×C3211yescharacteristic, complemented
70C2×C32×D3—31no—
71C2×C32×D3—33no—
72C2×C32×D3—31no—
73C2×C32×D3—63no—
74C2×C32×D3—61no—
75C2×C32×D3—91no—
76C2×C32×D3—91no—
77C2×C32×D3—91no—
78C2×C32×D3—181no—
79C33×C4—31no—
80C33×C4—31no—
81C33×C4—61no—
82C32×Dic3—21no—
83C32×Dic3—23no—
84C32×Dic3—31no—
85C32×Dic3—31no—
86C32×Dic3—61no—
87C32×Dic3—61no—
88C32×Dic3—62no—
89C32×Dic3—61no—
90C32×Dic3C3×D311yescharacteristic
91C32×Dic3C3×D313yescomplemented
92C32×Dic3C2×C3211yescharacteristic, complemented
93C2×(C3⋊D3)⋊C3—23no—
94C2×(C3⋊D3)⋊C3—33no—
95C2×(C3⋊D3)⋊C3—63no—
96C2×(C3⋊D3)⋊C3C3×D313yescomplemented
97C4×H(3,𝔽3)—33no—
98C4×H(3,𝔽3)—63no—
99C34—21no—
100C34D1211yescharacteristic, 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)#6711yescharacteristic, complemented
109C3×H(3,𝔽3)C3×D413yescomplemented
110C32×D4—271no—
111C4⋊(C3⋊D3)—31no—
112C3×#67—93no—
113C3×#67—91no—
114C3×D12—93no—
115C3×D12—91no—
116C3×D12—271no—
117#626—31no—
118#626—91no—
119C2×C33—21no—
120C2×C33—23no—
121C2×C33—21no—
122C2×C33—21no—
123C2×C33—21no—
124C2×C33—23no—
125C2×C33—21no—
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—121no—
138C2×C33—121no—
139C2×C33—121no—
140C2×C33—122no—
141C2×C33—181no—
142C2×C33D3211yescharacteristic
143C2×C33C2×C3×D311yescharacteristic
144C2×C33C2×C3×D313yes—
145C2×C33C2×C3×D311yescharacteristic
146C2×C33C22×C3211yescharacteristic
147C3⋊(C3⋊D3)—21no—
148C3×C3⋊D3—22no—
149C3×C3⋊D3—21no—
150C3×C3⋊D3—21no—
151C3×C3⋊D3—61no—
152C3×C3⋊D3—63no—
153C3×C3⋊D3—62no—
154C3×C3⋊D3—61no—
155C3×C3⋊D3—181no—
156C3×C3⋊D3—182no—
157C32×D3—62no—
158C32×D3—66no—
159C32×D3—61no—
160C32×D3—61no—
161C32×D3—63no—
162C32×D3—181no—
163C32×D3—181no—
164C32×D3—182no—
165C32×D3—181no—
166C2×H(3,𝔽3)—23no—
167C2×H(3,𝔽3)—23no—
168C2×H(3,𝔽3)—23no—
169C2×H(3,𝔽3)—43no—
170C2×H(3,𝔽3)—43no—
171C2×H(3,𝔽3)—46no—
172C2×H(3,𝔽3)—43no—
173C2×H(3,𝔽3)C2×C3×D313yes—
174(C3⋊D3)⋊C3—26no—
175(C3⋊D3)⋊C3—23no—
176(C3⋊D3)⋊C3—66no—
177(C3⋊D3)⋊C3—63no—
178C22×C32—91no—
179C22×C32—93no—
180C22×C32—91no—
181C22×C32—183no—
182C22×C32—181no—
183C22×C32—271no—
184C2×C3⋊D3—31no—
185C2×C3⋊D3—31no—
186C2×C3⋊D3—61no—
187C2×C3⋊D3—91no—
188C2×C3⋊D3—181no—
189C2×C3⋊D3C32×D311yescharacteristic, complemented
190C2×C3×D3—33no—
191C2×C3×D3—31no—
192C2×C3×D3—31no—
193C2×C3×D3—63no—
194C2×C3×D3—61no—
195C2×C3×D3—91no—
196C2×C3×D3—91no—
197C2×C3×D3—93no—
198C2×C3×D3—91no—
199C2×C3×D3—93no—
200C2×C3×D3—91no—
201C2×C3×D3—181no—
202C2×C3×D3—181no—
203C2×C3×D3—183no—
204C2×C3×D3—271no—
205C2×C3×D3—541no—
206C32×C4—31no—
207C32×C4—33no—
208C32×C4—31no—
209C32×C4—63no—
210C32×C4—61no—
211C32×C4—91no—
212C32×C4—91no—
213C32×C4—181no—
214C32×C4—181no—
215C32×C4—181no—
216C32×C4—182no—
217C3×Dic3—21no—
218C3×Dic3—33no—
219C3×Dic3—31no—
220C3×Dic3—61no—
221C3×Dic3—63no—
222C3×Dic3(C3⋊D3)⋊C311yescharacteristic, complemented
223C3×Dic3C32×D311yescharacteristic
224C33—21no—
225C33—21no—
226C33—23no—
227C33—21no—
228C33—21no—
229C33—23no—
230C33—21no—
231C33—31no—
232C33—31no—
233C33—41no—
234C33—43no—
235C33—41no—
236C33—61no—
237C33—61no—
238C33—61no—
239C33—61no—
240C33—62no—
241C33—61no—
242C33—121no—
243C33—121no—
244C33—121no—
245C33—122no—
246C33#62611yescharacteristic, complemented
247C33C3×D1211yescharacteristic
248C33C3×D1213yescomplemented
249C33C3×#6711yescharacteristic, complemented
250C33C32×D411yescharacteristic, complemented
251H(3,𝔽3)—23no—
252H(3,𝔽3)—23no—
253H(3,𝔽3)—23no—
254H(3,𝔽3)—43no—
255H(3,𝔽3)—43no—
256H(3,𝔽3)—46no—
257H(3,𝔽3)—43no—
258H(3,𝔽3)C3×#6713yescomplemented
259C3×D4—271no—
260C3×D4—273no—
261#67—91no—
262D12—91no—
263D12—271no—
264C2×C32—21no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—23no—
270C2×C32—23no—
271C2×C32—31no—
272C2×C32—31no—
273C2×C32—33no—
274C2×C32—31no—
275C2×C32—41no—
276C2×C32—43no—
277C2×C32—41no—
278C2×C32—61no—
279C2×C32—63no—
280C2×C32—61no—
281C2×C32—63no—
282C2×C32—61no—
283C2×C32—61no—
284C2×C32—62no—
285C2×C32—61no—
286C2×C32—61no—
287C2×C32—61no—
288C2×C32—61no—
289C2×C32—63no—
290C2×C32—121no—
291C2×C32—121no—
292C2×C32—123no—
293C2×C32—121no—
294C2×C32—121no—
295C2×C32—123no—
296C2×C32—122no—
297C2×C32—123no—
298C2×C32—121no—
299C2×C32—122no—
300C2×C32—126no—
301C2×C32—121no—
302C2×C32—181no—
303C2×C32—183no—
304C2×C32—186no—
305C2×C32—182no—
306C2×C32—181no—
307C2×C32—541no—
308C2×C32C2×(C3⋊D3)⋊C311yescharacteristic
309C2×C32C3×D3211yescharacteristic
310C2×C32C3×D3213yes—
311C2×C32C2×C32×D311yescharacteristic
312C2×C32C2×C32×D311yescharacteristic
313C3⋊D3—21no—
314C3⋊D3—61no—
315C3⋊D3—62no—
316C3⋊D3—61no—
317C3⋊D3—181no—
318C3⋊D3—182no—
319C3×D3—66no—
320C3×D3—62no—
321C3×D3—61no—
322C3×D3—63no—
323C3×D3—61no—
324C3×D3—181no—
325C3×D3—181no—
326C3×D3—182no—
327C3×D3—182no—
328C3×D3—183no—
329C3×D3—186no—
330C3×D3—181no—
331C3×D3—181no—
332C3×D3—183no—
333C3×D3—541no—
334C3×D3—542no—
335C22×C3—93no—
336C22×C3—91no—
337C22×C3—91no—
338C22×C3—181no—
339C22×C3—183no—
340C22×C3—273no—
341C22×C3—271no—
342C2×D3—31no—
343C2×D3—91no—
344C2×D3—91no—
345C2×D3—91no—
346C2×D3—181no—
347C2×D3—271no—
348C2×D3—541no—
349C3×C4—31no—
350C3×C4—31no—
351C3×C4—61no—
352C3×C4—91no—
353C3×C4—93no—
354C3×C4—181no—
355C3×C4—183no—
356Dic3C3×(C3⋊D3)⋊C311yescharacteristic, complemented
357C32—21no—
358C32—21no—
359C32—21no—
360C32—23no—
361C32—21no—
362C32—21no—
363C32—23no—
364C32—31no—
365C32—31no—
366C32—33no—
367C32—31no—
368C32—41no—
369C32—43no—
370C32—41no—
371C32—61no—
372C32—63no—
373C32—61no—
374C32—61no—
375C32—63no—
376C32—61no—
377C32—62no—
378C32—61no—
379C32—61no—
380C32—61no—
381C32—63no—
382C32—61no—
383C32—121no—
384C32—121no—
385C32—121no—
386C32—121no—
387C32—123no—
388C32—123no—
389C32—122no—
390C32—123no—
391C32—121no—
392C32—122no—
393C32—126no—
394C32—121no—
395C32C4⋊((C3⋊D3)⋊C3)11yescharacteristic, complemented
396C32C3×#62611yescharacteristic
397C32C3×#62613yescomplemented
398C32C32×D1211yescharacteristic
399C32C32×#6711yescharacteristic, complemented
400D4—271noSylow
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—31no—
405C2×C3—33no—
406C2×C3—41no—
407C2×C3—61no—
408C2×C3—63no—
409C2×C3—61no—
410C2×C3—63no—
411C2×C3—121no—
412C2×C3—123no—
413C2×C3—182no—
414C2×C3—183no—
415C2×C3—186no—
416C2×C3—181no—
417C2×C3—181no—
418C2×C3—543no—
419C2×C3—541no—
420C2×C3D3×(C3⋊D3)⋊C311yescharacteristic
421C2×C3C2×C3×(C3⋊D3)⋊C311yescharacteristic
422C2×C3C32×D3211yescharacteristic
423D3—61no—
424D3—181no—
425D3—182no—
426D3—181no—
427D3—181no—
428D3—541no—
429D3—542no—
430C22—91no—
431C22—271no—
432C4—31no—
433C3—21no—
434C3—21no—
435C3—21no—
436C3—31no—
437C3—33no—
438C3—41no—
439C3—61no—
440C3—63no—
441C3—61no—
442C3—63no—
443C3—121no—
444C3—123no—
445C3#6292⋊C311yescharacteristic, direct product factor
446C3C3×C4⋊((C3⋊D3)⋊C3)11yescharacteristic, complemented
447C3C32×#62611yescharacteristic
448C2—181no—
449C2—541no—
450C2C3×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×C32×(C3⋊D3)⋊C3C2semidirect
3. C3×Dic3×H(3,𝔽3)C2semidirect
4. C3×#6292C3semidirect
7. #6292⋊C3C3direct (N × H)
11. C2×C3×C3⋊(C3⋊D3)C2×C3semidirect
13. C2×C32×C3⋊D3C2×C3semidirect
16. C33×Dic3D3semidirect
17. C33×Dic3C2×C3semidirect
18. C2×C3⋊((C3⋊D3)⋊C3)C2×C3semidirect
23. C2×C3×(C3⋊D3)⋊C3D3semidirect
24. C2×C3×(C3⋊D3)⋊C3C2×C3semidirect
27. Dic3×H(3,𝔽3)C2×C3semidirect
28. C32×H(3,𝔽3)D4semidirect
32. #6292C32semidirect
60. C2×C3⋊(C3⋊D3)C2×C32semidirect
68. C2×C3×C3⋊D3C3×D3semidirect
69. C2×C3×C3⋊D3C2×C32semidirect
91. C32×Dic3C3×D3semidirect
92. C32×Dic3C2×C32semidirect
96. C2×(C3⋊D3)⋊C3C3×D3semidirect
100. C34D12semidirect
101. C34C3×D4semidirect
108. C3×H(3,𝔽3)#67semidirect
109. C3×H(3,𝔽3)C3×D4semidirect
189. C2×C3⋊D3C32×D3semidirect
222. C3×Dic3(C3⋊D3)⋊C3semidirect
246. C33#626semidirect
248. C33C3×D12semidirect
249. C33C3×#67semidirect
250. C33C32×D4semidirect
258. H(3,𝔽3)C3×#67semidirect
356. Dic3C3×(C3⋊D3)⋊C3semidirect
395. C32C4⋊((C3⋊D3)⋊C3)semidirect
397. C32C3×#626semidirect
399. C32C32×#67semidirect
445. C3#6292⋊C3direct (N × H)
446. C3C3×C4⋊((C3⋊D3)⋊C3)semidirect

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×C32×(C3⋊D3)⋊C32
C3×Dic3×H(3,𝔽3)2
C3×#62923
C32×#6263
C3×C4⋊((C3⋊D3)⋊C3)3
#6292⋊C33