Finite Group Databasebeta

← C3×#11722396 #72276546

Subgroups — C3×#11722396 #72276546

Order 2592.

982 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 982 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C22×C32×C22⋊C9C211yesmaximal, characteristic, complemented
2C2×C3×C3⋊((C22⋊C9)⋊C2)C211yesmaximal, characteristic, complemented
3C3×C3⋊((C22⋊C9)⋊C4)C211yesmaximal, characteristic, complemented
4C3×#64162—31nomaximal
5#11722396C311yesmaximal, characteristic, direct product factor
6C3×#63694—33nomaximal
7C2×C32×C22⋊C9—21no—
8C2×C32×C22⋊C9C2211yescharacteristic
9C3×C3⋊((C22⋊C9)⋊C2)—21no—
10C3×#6233—41nomaximal
11C24×C33D311yescharacteristic
12C2×C3×#638—31no—
13C2×C3×#638—31no—
14C2×C3×#638—31no—
15C22×C3×C22⋊C9—23no—
16C22×C3×C22⋊C9—21no—
17C22×C3×C22⋊C9D313yescomplemented
18C22×C3×C22⋊C9C2×C311yescharacteristic, complemented
19C2×C3⋊((C22⋊C9)⋊C2)C2×C311yescharacteristic, complemented
20C2×C3×(C22⋊C9)⋊C2—33no—
21C3×#3757—31no—
22C3×#3757—31no—
23C3×#3757—31no—
24C3⋊((C22⋊C9)⋊C4)C2×C311yescharacteristic, complemented
25C3×(C22⋊C9)⋊C4—33no—
26C22×C32×C9—41no—
27C2×C3×C3⋊D9—41no—
28C32×C22⋊C9D411yescharacteristic, complemented
29C3×C3⋊Dic9—41no—
30#64162—31no—
31C3×#953—93no—
32C3×#953—91no—
33#63694—33no—
34C23×C33—21no—
35C23×C33—31no—
36C23×C33—31no—
37C23×C33—61no—
38C23×C33C2×D311yescharacteristic
39C22×C3×C3⋊D3—31no—
40C3×#638—61no—
41C3×#638—62no—
42C3×#638—61no—
43C3×#638—61no—
44C3×#638—61no—
45C2×C3×C3⋊Dic3—31no—
46C2×C3×C3⋊Dic3—31no—
47C2×C3×C3⋊Dic3—31no—
48C2×C3×C22⋊C9—22no—
49C2×C3×C22⋊C9—26no—
50C2×C3×C22⋊C9—21no—
51C2×C3×C22⋊C9—23no—
52C2×C3×C22⋊C9—23no—
53C2×C3×C22⋊C9—21no—
54C2×C3×C22⋊C9C2×D313yes—
55C2×C3×C22⋊C9C22×C311yescharacteristic
56C3⋊((C22⋊C9)⋊C2)—21no—
57C3×(C22⋊C9)⋊C2—63no—
58#6233—41no—
59C3×#611—123no—
60C2×C32×C9—41no—
61C2×C32×C9—81no—
62C3×C3⋊D9—81no—
63C24×C32—23no—
64C24×C32—21no—
65C24×C32D913yescomplemented
66C24×C32C3×D311yescharacteristic
67C24×C32C3⋊D311yescharacteristic
68C2×#638—31no—
69C2×#638—31no—
70C2×#638—31no—
71C2×C3×#67—93no—
72C2×C3×#67—91no—
73C2×C3×#67—93no—
74C2×C3×#67—91no—
75C2×C3×#67—93no—
76C2×C3×#67—91no—
77C22×C22⋊C9—23no—
78C22×C22⋊C9C3×D313yescomplemented
79C2×(C22⋊C9)⋊C2—33no—
80#3757—31no—
81#3757—31no—
82#3757—31no—
83C3×#217—93no—
84C3×#217—91no—
85C3×#217—93no—
86C3×#217—91no—
87C3×#217—93no—
88C3×#217—91no—
89(C22⋊C9)⋊C4—33no—
90C22×C33—31no—
91C22×C33—31no—
92C22×C33—32no—
93C22×C33—31no—
94C22×C33—61no—
95C22×C33—61no—
96C22×C33—61no—
97C22×C33#6711yescharacteristic
98C22×C33S411yescharacteristic
99C2×C3×C3⋊D3—61no—
100C2×C3×C3⋊D3—61no—
101C2×C3×C3⋊D3—61no—
102C3×C3⋊Dic3—61no—
103C3×C3⋊Dic3—61no—
104C3×C3⋊Dic3—61no—
105C22×C3×C9—41no—
106C22×C3×C9—43no—
107C22×C3×C9—83no—
108C22×C3×C9—81no—
109C2×C3⋊D9—41no—
110C2×C3×D9—123no—
111C3×C22⋊C9—21no—
112C3×C22⋊C9—23no—
113C3×C22⋊C9#6713yescomplemented
114C3×C22⋊C9C3×D411yescharacteristic, complemented
115C3⋊Dic9—41no—
116C3×Dic9—123no—
117C3×#120—271no—
118#953—93no—
119#953—91no—
120C32×C9—41noSylow
121C23×C32—22no—
122C23×C32—26no—
123C23×C32—23no—
124C23×C32—21no—
125C23×C32—21no—
126C23×C32—23no—
127C23×C32—21no—
128C23×C32—31no—
129C23×C32—31no—
130C23×C32—33no—
131C23×C32—31no—
132C23×C32—33no—
133C23×C32—31no—
134C23×C32—63no—
135C23×C32—61no—
136C23×C32—63no—
137C23×C32—61no—
138C23×C32—63no—
139C23×C32—61no—
140C23×C32—61no—
141C23×C32—66no—
142C23×C32—62no—
143C23×C32C2×D913yes—
144C23×C32C2×C3×D311yescharacteristic
145C23×C32C2×C3⋊D311yescharacteristic
146C22×C3⋊D3—31no—
147C22×C3×D3—91no—
148C22×C3×D3—93no—
149#638—61no—
150#638—62no—
151#638—61no—
152#638—61no—
153#638—61no—
154C2×C3⋊Dic3—31no—
155C2×C3⋊Dic3—31no—
156C2×C3⋊Dic3—31no—
157C3×#67—183no—
158C3×#67—181no—
159C3×#67—182no—
160C3×#67—186no—
161C3×#67—181no—
162C3×#67—183no—
163C3×#67—183no—
164C3×#67—183no—
165C3×#67—181no—
166C3×#67—181no—
167C2×C3×Dic3—91no—
168C2×C3×Dic3—91no—
169C2×C3×Dic3—93no—
170C2×C3×Dic3—93no—
171C2×C3×Dic3—91no—
172C2×C3×Dic3—93no—
173C2×C22⋊C9—26no—
174C2×C22⋊C9—23no—
175C2×C22⋊C9—23no—
176C2×C22⋊C9C2×C3×D313yes—
177(C22⋊C9)⋊C2—63no—
178#611—123no—
179C2×C33—21no—
180C2×C33—31no—
181C2×C33—31no—
182C2×C33—61no—
183C2×C33C2×S411yescharacteristic
184C3×C3⋊D3—121no—
185C2×C3×C9—41no—
186C2×C3×C9—43no—
187C2×C3×C9—83no—
188C2×C3×C9—81no—
189C2×C3×C9—83no—
190C2×C3×C9—86no—
191C2×C3×C9—82no—
192C2×C3×C9—81no—
193C3⋊D9—81no—
194C3×D9—243no—
195C24×C3—23no—
196C24×C3—21no—
197C24×C3C3×D913yescomplemented
198C24×C3C3⋊D911yescharacteristic, complemented
199C24×C3C3×C3⋊D311yescharacteristic
200C2×C3×D4—271no—
201C2×C3×D4—271no—
202C2×C3×D4—271no—
203C2×#67—93no—
204C2×#67—91no—
205C2×#67—93no—
206C2×#67—91no—
207C2×#67—93no—
208C2×#67—91no—
209C3×C22⋊C4—271no—
210C3×C22⋊C4—271no—
211C3×C22⋊C4—271no—
212#217—93no—
213#217—91no—
214#217—93no—
215#217—91no—
216#217—93no—
217#217—91no—
218C22×C32—23no—
219C22×C32—23no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—31no—
223C22×C32—31no—
224C22×C32—32no—
225C22×C32—31no—
226C22×C32—33no—
227C22×C32—31no—
228C22×C32—31no—
229C22×C32—33no—
230C22×C32—32no—
231C22×C32—36no—
232C22×C32—31no—
233C22×C32—33no—
234C22×C32—61no—
235C22×C32—63no—
236C22×C32—61no—
237C22×C32—63no—
238C22×C32—63no—
239C22×C32—61no—
240C22×C32—66no—
241C22×C32—62no—
242C22×C32—61no—
243C22×C32—61no—
244C22×C32—63no—
245C22×C32—61no—
246C22×C32—61no—
247C22×C32—63no—
248C22×C32—62no—
249C22×C32—66no—
250C22×C32—62no—
251C22×C32—66no—
252C22×C32—61no—
253C22×C32—61no—
254C22×C32—63no—
255C22×C32—62no—
256C22×C32—66no—
257C22×C32#61113yescomplemented
258C22×C32(C22⋊C9)⋊C213yescomplemented
259C22×C32C3×#6711yescharacteristic
260C22×C32#63811yescharacteristic
261C22×C32C3×S411yescharacteristic
262C22×C32C3⋊S411yescharacteristic
263C2×C3⋊D3—61no—
264C2×C3⋊D3—61no—
265C2×C3⋊D3—61no—
266C2×C3×D3—181no—
267C2×C3×D3—181no—
268C2×C3×D3—183no—
269C2×C3×D3—183no—
270C2×C3×D3—181no—
271C2×C3×D3—183no—
272C3⋊Dic3—61no—
273C3⋊Dic3—61no—
274C3⋊Dic3—61no—
275C3×Dic3—181no—
276C3×Dic3—181no—
277C3×Dic3—183no—
278C3×Dic3—183no—
279C3×Dic3—181no—
280C3×Dic3—183no—
281C22×C9—43no—
282C22×C9—83no—
283C2×D9—123no—
284C22⋊C9—23no—
285C22⋊C9C3×#6713yescomplemented
286Dic9—123no—
287#120—271noSylow
288C33GL(2,ℤ/4)11yescharacteristic
289C3×C9—41no—
290C3×C9—43no—
291C3×C9—83no—
292C3×C9—81no—
293C23×C3—26no—
294C23×C3—22no—
295C23×C3—23no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—23no—
299C23×C3—21no—
300C23×C3—31no—
301C23×C3—33no—
302C23×C3—31no—
303C23×C3—33no—
304C23×C3—31no—
305C23×C3—31no—
306C23×C3—63no—
307C23×C3—61no—
308C23×C3—63no—
309C23×C3—61no—
310C23×C3—63no—
311C23×C3—61no—
312C23×C3—61no—
313C23×C3—66no—
314C23×C3—62no—
315C23×C3—271no—
316C23×C3C2×C3×D913yes—
317C23×C3C2×C3⋊D911yescharacteristic
318C23×C3C2×C3×C3⋊D311yescharacteristic
319C22×D3—91no—
320C22×D3—93no—
321C3×D4—541no—
322C3×D4—542no—
323C3×D4—541no—
324C3×D4—541no—
325C3×D4—541no—
326C2×C3×C4—271no—
327C2×C3×C4—271no—
328C2×C3×C4—271no—
329#67—183no—
330#67—181no—
331#67—186no—
332#67—182no—
333#67—181no—
334#67—183no—
335#67—183no—
336#67—183no—
337#67—181no—
338#67—181no—
339C2×Dic3—91no—
340C2×Dic3—91no—
341C2×Dic3—93no—
342C2×Dic3—93no—
343C2×Dic3—91no—
344C2×Dic3—93no—
345C2×C32—22no—
346C2×C32—26no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—23no—
350C2×C32—21no—
351C2×C32—23no—
352C2×C32—31no—
353C2×C32—31no—
354C2×C32—31no—
355C2×C32—31no—
356C2×C32—33no—
357C2×C32—33no—
358C2×C32—63no—
359C2×C32—61no—
360C2×C32—63no—
361C2×C32—61no—
362C2×C32—66no—
363C2×C32—62no—
364C2×C32—61no—
365C2×C32—61no—
366C2×C32—63no—
367C2×C32C2×(C22⋊C9)⋊C213yes—
368C2×C32C2×C3×S411yescharacteristic
369C2×C32C2×C3⋊S411yescharacteristic
370C3⋊D3—121no—
371C3×D3—361no—
372C3×D3—363no—
373C2×C9—43no—
374C2×C9—83no—
375C2×C9—86no—
376C2×C9—83no—
377D9—243no—
378C24C3×C3⋊D911yescharacteristic, complemented
379C2×D4—271no—
380C2×D4—271no—
381C2×D4—271no—
382C22⋊C4—271no—
383C22⋊C4—271no—
384C22⋊C4—271no—
385C22×C3—23no—
386C22×C3—21no—
387C22×C3—23no—
388C22×C3—21no—
389C22×C3—33no—
390C22×C3—31no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—33no—
394C22×C3—31no—
395C22×C3—36no—
396C22×C3—32no—
397C22×C3—32no—
398C22×C3—31no—
399C22×C3—33no—
400C22×C3—31no—
401C22×C3—61no—
402C22×C3—63no—
403C22×C3—61no—
404C22×C3—63no—
405C22×C3—63no—
406C22×C3—61no—
407C22×C3—62no—
408C22×C3—66no—
409C22×C3—63no—
410C22×C3—61no—
411C22×C3—61no—
412C22×C3—63no—
413C22×C3—61no—
414C22×C3—61no—
415C22×C3—62no—
416C22×C3—66no—
417C22×C3—62no—
418C22×C3—66no—
419C22×C3—63no—
420C22×C3—61no—
421C22×C3—61no—
422C22×C3—62no—
423C22×C3—66no—
424C22×C3—541no—
425C22×C3—541no—
426C22×C3—541no—
427C22×C3C3×#61113yescomplemented
428C22×C3#623311yescharacteristic, complemented
429C22×C3C3×(C22⋊C9)⋊C213yescomplemented
430C22×C3C3⋊((C22⋊C9)⋊C2)11yescharacteristic, complemented
431C22×C3C3×#63811yescharacteristic
432C22×C3C3×C3⋊S411yescharacteristic
433C2×D3—183no—
434C2×D3—183no—
435C2×D3—181no—
436C2×D3—181no—
437C2×D3—181no—
438C2×D3—183no—
439C3×C4—541no—
440C3×C4—541no—
441C3×C4—541no—
442Dic3—183no—
443Dic3—183no—
444Dic3—181no—
445Dic3—181no—
446Dic3—181no—
447Dic3—183no—
448C32—23no—
449C32—21no—
450C32#6369413yescomplemented
451C32C3×GL(2,ℤ/4)11yescharacteristic
452C32#6426911yescharacteristic
453C9—43no—
454C9—83no—
455C23—21no—
456C23—31no—
457C23—31no—
458C23—61no—
459C23—271no—
460C23C2×C3×C3⋊D911yescharacteristic
461D4—541no—
462D4—542no—
463D4—541no—
464D4—541no—
465D4—541no—
466C2×C4—271no—
467C2×C4—271no—
468C2×C4—271no—
469C2×C3—22no—
470C2×C3—26no—
471C2×C3—23no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—23no—
476C2×C3—33no—
477C2×C3—33no—
478C2×C3—31no—
479C2×C3—31no—
480C2×C3—31no—
481C2×C3—31no—
482C2×C3—61no—
483C2×C3—63no—
484C2×C3—61no—
485C2×C3—63no—
486C2×C3—62no—
487C2×C3—66no—
488C2×C3—63no—
489C2×C3—61no—
490C2×C3—61no—
491C2×C3—1081no—
492C2×C3C2×C3×(C22⋊C9)⋊C213yes—
493C2×C3C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
494C2×C3C2×C3×C3⋊S411yescharacteristic
495D3—363no—
496D3—361no—
497C22—31no—
498C22—31no—
499C22—32no—
500C22—31no—
501C22—61no—
502C22—61no—
503C22—61no—
504C22—541no—
505C22—541no—
506C22—541no—
507C22C3×#623311yescharacteristic, complemented
508C22C3×C3⋊((C22⋊C9)⋊C2)11yescharacteristic, complemented
509C4—541no—
510C4—541no—
511C4—541no—
512C3—21no—
513C3—23no—
514C3C3×#6369413yescomplemented
515C3#1172239611yescharacteristic, direct product factor
516C3C3×#6426911yescharacteristic
517C2—21no—
518C2—31no—
519C2—31no—
520C2—61no—
521C2—1081no—
522C2C2×C3×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
subgroupG/H
centerC2×C3C2×C3⋊((C22⋊C9)⋊C2)
commutator, cosocleC2×C3×C22⋊C9C22×C3
FrattiniC2×C3C2×C3×C3⋊S4
FittingC24×C33D3
socleC23×C33C2×D3
hypercenterC22×C3C3⋊((C22⋊C9)⋊C2)
hypocenterC3×C22⋊C9C3×D4
LCSG ▷ C2×C3×C22⋊C9 ▷ C3×C22⋊C9
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C22×C32×C22⋊C9C2semidirect
2. C2×C3×C3⋊((C22⋊C9)⋊C2)C2semidirect
3. C3×C3⋊((C22⋊C9)⋊C4)C2semidirect
5. #11722396C3direct (N × H)
17. C22×C3×C22⋊C9D3semidirect
18. C22×C3×C22⋊C9C2×C3semidirect
19. C2×C3⋊((C22⋊C9)⋊C2)C2×C3semidirect
24. C3⋊((C22⋊C9)⋊C4)C2×C3semidirect
28. C32×C22⋊C9D4semidirect
65. C24×C32D9semidirect
78. C22×C22⋊C9C3×D3semidirect
113. C3×C22⋊C9#67semidirect
114. C3×C22⋊C9C3×D4semidirect
197. C24×C3C3×D9semidirect
198. C24×C3C3⋊D9semidirect
257. C22×C32#611semidirect
258. C22×C32(C22⋊C9)⋊C2semidirect
285. C22⋊C9C3×#67semidirect
378. C24C3×C3⋊D9semidirect
427. C22×C3C3×#611semidirect
428. C22×C3#6233semidirect
429. C22×C3C3×(C22⋊C9)⋊C2semidirect
430. C22×C3C3⋊((C22⋊C9)⋊C2)semidirect
450. C32#63694semidirect
507. C22C3×#6233semidirect
508. C22C3×C3⋊((C22⋊C9)⋊C2)semidirect
514. C3C3×#63694semidirect
515. C3#11722396direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×C22⋊C92
C2×C3×C3⋊((C22⋊C9)⋊C2)2
C3×C3⋊((C22⋊C9)⋊C4)2
C3×#641623
#117223963
C3×#636943
C3×#62334