Finite Group Databasebeta

← C3×#72022 #11788371

Subgroups — C3×#72022 #11788371

Order 1152.

737 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#568C211yesmaximal, characteristic
2C2×C3×#951C211yesmaximal, characteristic
3C3×#5236C211yesmaximal, characteristic
4C2×C3×#824C211yesmaximal, characteristic, complemented
5C3×#5120C211yesmaximal, characteristic, complemented
6C3×C3⋊#338C211yesmaximal, characteristic, complemented
7C3×#4559C211yesmaximal, characteristic, complemented
8C3×#1902—31nomaximal
9#72022C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C32×#131—22no—
13C32×#131—22no—
14C2×C32×M16—22no—
15C2×C32×M16C2211yescharacteristic
16C2×C32×M16C2211yescharacteristic
17C3×#951—22no—
18C3×#951—22no—
19C2×C3×#216C412yescomplemented
20C2×C3×#214C412yescomplemented
21C3×#926C2211yescharacteristic
22C2×C3×#208C2211yescharacteristic
23C2×C3×#208C2211yescharacteristic
24C3×#837—22no—
25C3×#824—22no—
26C3×#816—22no—
27C2×C3×#137—31no—
28C3×#568—21no—
29C3×#568D311yescharacteristic
30C3×#568C2×C311yescharacteristic
31C2×#951C2×C311yescharacteristic
32C3×#421—31no—
33C3×#419—31no—
34C2×C3×#101—31no—
35#5236C2×C311yescharacteristic
36C2×#824C2×C311yescharacteristic, complemented
37#5120C2×C311yescharacteristic, complemented
38C3×#344—31no—
39C3×#338—31no—
40C3⋊#338C2×C311yescharacteristic, complemented
41#4559C2×C311yescharacteristic, complemented
42C32×#41—21no—
43C32×#41—21no—
44C32×#41D412yes—
45C32×#41D412yes—
46C2×C32×Q8C2×C411yescharacteristic
47C2×C32×D4—21no—
48C2×C32×D4C2×C411yescharacteristic
49C22×C32×C4—21no—
50C22×C32×C4C2311yescharacteristic
51C22×C3×Dic3—21no—
52C2×C3×Dic6—21no—
53C2×C3×Dic6—21no—
54C2×C3×Dic6—21no—
55C2×C3×Dic6—21no—
56C2×C3×Dic6—21no—
57C2×C3×Dic6C2×C412yes—
58C32×M16—24no—
59C32×M16—21no—
60C32×M16—21no—
61C32×M16—21no—
62C32×M16D412yes—
63C2×C32×C8—24no—
64C2×C32×C8—21no—
65C2×C32×C8D412yes—
66C3×#217—21no—
67C3×#216—24no—
68C3×#214—24no—
69C3×C4⋊Dic3—21no—
70C3×C4×Dic3—21no—
71C3×#208—21no—
72C3×#208—21no—
73C3×#208—21no—
74C3×#208—21no—
75C2×C3×C3⋊C8—21no—
76C2×C3×C3⋊C8C2×C412yes—
77#1902—31noSylow
78C2×C3×#41—21no—
79C2×C3×#41C2×D311yescharacteristic
80C2×C3×#41C22×C311yescharacteristic
81C22×C3×Q8—31no—
82C22×Dic6C22×C311yescharacteristic
83C3×#137—62no—
84C3×#137—62no—
85C2×C3×Q16—32no—
86C2×C3×SD16—32no—
87C3×#131—24no—
88C3×#131—24no—
89C3×#131—22no—
90C3×#131—22no—
91C3×#131—22no—
92C3×#131—22no—
93C2×C3×M16—24no—
94C2×C3×M16—22no—
95C2×C3×M16—22no—
96C2×C3×M16—21no—
97C2×C3×M16—21no—
98C2×C3×M16—31no—
99C2×C3×M16—31no—
100C2×C3×M16C2×D311yescharacteristic
101C2×C3×M16C2×D311yescharacteristic
102C2×C3×M16C22×C311yescharacteristic
103C2×C3×M16C22×C311yescharacteristic
104C3×#117—31no—
105#951—22no—
106#951—22no—
107C2×#216C3×C412yescomplemented
108C2×#214C3×C412yescomplemented
109#926C22×C311yescharacteristic
110C2×#208C22×C311yescharacteristic
111C2×#208C22×C311yescharacteristic
112C3×#104—62no—
113C3×#103—62no—
114C3×#101—62no—
115#837—22no—
116#824—22no—
117#816—22no—
118C23×C32—21no—
119C23×C32#4111yescharacteristic
120C32×Q8—21no—
121C32×Q8C22⋊C412yes—
122C32×D4—22no—
123C32×D4—22no—
124C32×D4—21no—
125C32×D4C22⋊C412yes—
126C2×C32×C4—21no—
127C2×C32×C4—22no—
128C2×C32×C4—22no—
129C2×C32×C4C22×C411yescharacteristic
130C2×C32×C4C2×D411yescharacteristic
131C2×C32×C4C2×D411yescharacteristic
132C2×C32×C4C2×D411yescharacteristic
133C2×C32×C4C2×D411yescharacteristic
134C2×C32×C4#4111yescharacteristic
135C2×C3×Dic3—22no—
136C2×C3×Dic3—41no—
137C2×C3×Dic3—41no—
138C2×C3×Dic3—41no—
139C3×Dic6—22no—
140C3×Dic6—22no—
141C3×Dic6—41no—
142C3×Dic6—41no—
143C32×C8—22no—
144C32×C8—22no—
145C3×C3⋊C8—22no—
146C3×C3⋊C8—41no—
147C2×#137—31no—
148#568C3×D311yescharacteristic
149#421—31no—
150#419—31no—
151C2×#101—31no—
152#344—31no—
153#338—31no—
154C3×#41—22no—
155C3×#41—22no—
156C3×#41—21no—
157C3×#41—21no—
158C3×#41—21no—
159C3×#41—21no—
160C3×#41—22no—
161C3×#41—22no—
162C3×#41D1212yes—
163C3×#41#6712yes—
164C3×#41C3×D412yes—
165C3×#41C3×D412yes—
166C2×C3×Q8—21no—
167C2×C3×Q8—32no—
168C2×C3×Q8—61no—
169C2×C3×Q8—61no—
170C2×C3×Q8—61no—
171C2×C3×Q8—61no—
172C2×C3×Q8—61no—
173C2×C3×Q8C4×D311yescharacteristic
174C2×C3×Q8C2×C3×C411yescharacteristic
175C2×C3×D4—22no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4C4×D311yescharacteristic
180C2×C3×D4C2×C3×C411yescharacteristic
181C22×C3×C4—22no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—61no—
186C22×C3×C4C22×D311yescharacteristic
187C22×C3×C4C23×C311yescharacteristic
188C22×Dic3—21no—
189C2×Dic6—21no—
190C2×Dic6—21no—
191C2×Dic6—21no—
192C2×Dic6—21no—
193C2×Dic6—21no—
194C2×Dic6C2×C3×C412yes—
195C3×Q16—64no—
196C3×SD16—64no—
197C3×M16—22no—
198C3×M16—28no—
199C3×M16—24no—
200C3×M16—24no—
201C3×M16—22no—
202C3×M16—22no—
203C3×M16—21no—
204C3×M16—21no—
205C3×M16—21no—
206C3×M16—21no—
207C3×M16—21no—
208C3×M16—21no—
209C3×M16—22no—
210C3×M16—61no—
211C3×M16—61no—
212C3×M16—61no—
213C3×M16—61no—
214C3×M16#6712yes—
215C3×M16C3×D412yes—
216C2×C3×C8—28no—
217C2×C3×C8—24no—
218C2×C3×C8—24no—
219C2×C3×C8—22no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—22no—
223C2×C3×C8—32no—
224C2×C3×C8—61no—
225C2×C3×C8#6712yes—
226C2×C3×C8C3×D412yes—
227C3×C4⋊C4—61no—
228C3×C22⋊C4—61no—
229C3×C42—61no—
230#217—21no—
231#216—24no—
232#214—24no—
233C4⋊Dic3—21no—
234C4×Dic3—21no—
235#208—21no—
236#208—21no—
237#208—21no—
238#208—21no—
239C2×C3⋊C8—21no—
240C2×C3⋊C8C2×C3×C412yes—
241C22×C32—22no—
242C22×C32—41no—
243C22×C32—81no—
244C22×C32C4×D411yescharacteristic
245C22×C32#12111yescharacteristic
246C22×C32#12311yescharacteristic
247C32×C4—22no—
248C32×C4C2×C22⋊C411yescharacteristic
249C32×C4C4×D411yescharacteristic
250C32×C4#12011yescharacteristic
251C32×C4#12111yescharacteristic
252C3×Dic3—42no—
253C3×Dic3—81no—
254C2×#41C2×C3×D311yescharacteristic
255C22×Q8—31no—
256#137—62no—
257#137—62no—
258C2×Q16—32no—
259C2×SD16—32no—
260#131—22no—
261#131—22no—
262C2×M16—22no—
263C2×M16—31no—
264C2×M16—31no—
265C2×M16C2×C3×D311yescharacteristic
266C2×M16C2×C3×D311yescharacteristic
267#117—31no—
268#104—62no—
269#103—62no—
270#101—62no—
271C23×C3—22no—
272C23×C3—21no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3#23511yescharacteristic
276C23×C3C3×#4111yescharacteristic
277C3×Q8—22no—
278C3×Q8—21no—
279C3×Q8—21no—
280C3×Q8—22no—
281C3×Q8—62no—
282C3×Q8—62no—
283C3×Q8—121no—
284C3×Q8—121no—
285C3×Q8(C2×D3)⋊C412yes—
286C3×Q8C3×C22⋊C412yes—
287C3×D4—24no—
288C3×D4—24no—
289C3×D4—22no—
290C3×D4—22no—
291C3×D4—22no—
292C3×D4—22no—
293C3×D4—22no—
294C3×D4—21no—
295C3×D4—21no—
296C3×D4—22no—
297C3×D4(C2×D3)⋊C412yes—
298C3×D4C3×C22⋊C412yes—
299C2×C3×C4—24no—
300C2×C3×C4—22no—
301C2×C3×C4—24no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—62no—
315C2×C3×C4—121no—
316C2×C3×C4—121no—
317C2×C3×C4—121no—
318C2×C3×C4C2×C4×D311yescharacteristic
319C2×C3×C4C2×D1211yescharacteristic
320C2×C3×C4#23511yescharacteristic
321C2×C3×C4C2×#6711yescharacteristic
322C2×C3×C4C2×#6711yescharacteristic
323C2×C3×C4C2×#6711yescharacteristic
324C2×C3×C4C22×C3×C411yescharacteristic
325C2×C3×C4C2×C3×D411yescharacteristic
326C2×C3×C4C2×C3×D411yescharacteristic
327C2×C3×C4C2×C3×D411yescharacteristic
328C2×C3×C4C2×C3×D411yescharacteristic
329C2×C3×C4C3×#4111yescharacteristic
330C2×Dic3—22no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334Dic6—22no—
335Dic6—22no—
336Dic6—41no—
337Dic6—41no—
338C3×C8—24no—
339C3×C8—24no—
340C3×C8—22no—
341C3×C8—22no—
342C3×C8—22no—
343C3×C8—22no—
344C3×C8—62no—
345C3×C8—121no—
346C3⋊C8—22no—
347C3⋊C8—41no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—21no—
351C2×C32—42no—
352C2×C32#38611yescharacteristic
353#41—21no—
354#41—21no—
355#41C3×D1212yes—
356#41C3×#6712yes—
357C2×Q8—32no—
358C2×Q8—61no—
359C2×Q8—61no—
360C2×Q8—61no—
361C2×Q8—61no—
362C2×Q8—61no—
363C2×Q8C3×C4×D311yescharacteristic
364C2×D4—21no—
365C2×D4C3×C4×D311yescharacteristic
366C22×C4—21no—
367C22×C4—61no—
368C22×C4C22×C3×D311yescharacteristic
369Q16—64no—
370SD16—64no—
371M16—24no—
372M16—21no—
373M16—21no—
374M16—21no—
375M16—61no—
376M16—61no—
377M16—61no—
378M16—61no—
379M16C3×#6712yes—
380C2×C8—24no—
381C2×C8—21no—
382C2×C8—32no—
383C2×C8—61no—
384C2×C8C3×#6712yes—
385C4⋊C4—61no—
386C22⋊C4—61no—
387C42—61no—
388C22×C3—24no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—41no—
395C22×C3—41no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—82no—
400C22×C3C4×#6711yescharacteristic
401C22×C3#92911yescharacteristic
402C22×C3#93011yescharacteristic
403C22×C3C3×C4×D411yescharacteristic
404C22×C3C3×#12111yescharacteristic
405C22×C3C3×#12311yescharacteristic
406C3×C4—24no—
407C3×C4—22no—
408C3×C4—22no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—122no—
414C3×C4—241no—
415C3×C4C2×(C2×D3)⋊C411yescharacteristic
416C3×C4C4×#6711yescharacteristic
417C3×C4#93011yescharacteristic
418C3×C4#95311yescharacteristic
419C3×C4C2×C3×C22⋊C411yescharacteristic
420C3×C4C3×C4×D411yescharacteristic
421C3×C4C3×#12011yescharacteristic
422C3×C4C3×#12111yescharacteristic
423Dic3—42no—
424Dic3—81no—
425C32#190211yescharacteristic, complemented, Sylow
426C23—21no—
427C23C3×#23511yescharacteristic
428Q8—21no—
429Q8—62no—
430Q8—62no—
431Q8—121no—
432Q8—121no—
433Q8C3×(C2×D3)⋊C412yes—
434D4—22no—
435D4—22no—
436D4—21no—
437D4C3×(C2×D3)⋊C412yes—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—62no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4C2×C3×C4×D311yescharacteristic
446C2×C4C2×C3×D1211yescharacteristic
447C2×C4C3×#23511yescharacteristic
448C2×C4C2×C3×#6711yescharacteristic
449C2×C4C2×C3×#6711yescharacteristic
450C2×C4C2×C3×#6711yescharacteristic
451C8—22no—
452C8—22no—
453C8—62no—
454C8—121no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—41no—
465C2×C3—44no—
466C2×C3—42no—
467C2×C3—42no—
468C2×C3#521311yescharacteristic
469C2×C3C3×#38611yescharacteristic
470C22—22no—
471C22—41no—
472C22—81no—
473C22C3×C4×#6711yescharacteristic
474C22C3×#92911yescharacteristic
475C22C3×#93011yescharacteristic
476C4—22no—
477C4—122no—
478C4—241no—
479C4C2×C3×(C2×D3)⋊C411yescharacteristic
480C4C3×C4×#6711yescharacteristic
481C4C3×#93011yescharacteristic
482C4C3×#95311yescharacteristic
483C3—21no—
484C3#7202211yescharacteristic, direct product factor
485C3C3×#190211yescharacteristic, complemented
486C2—21no—
487C2—21no—
488C2—21no—
489C2—42no—
490C2C3×#521311yescharacteristic
subgroupG/H
centerC2×C3#5213
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#568C2
socleC2×C32#386
prime radicalC2×C32×#41C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#568D3
hypocenterC3C3×#1902
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#568

Complements

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

NH ≅ G/Ntype
4. C2×C3×#824C2semidirect
5. C3×#5120C2semidirect
6. C3×C3⋊#338C2semidirect
7. C3×#4559C2semidirect
9. #72022C3direct (N × H)
19. C2×C3×#216C4semidirect
20. C2×C3×#214C4semidirect
36. C2×#824C2×C3semidirect
37. #5120C2×C3semidirect
40. C3⋊#338C2×C3semidirect
41. #4559C2×C3semidirect
107. C2×#216C3×C4semidirect
108. C2×#214C3×C4semidirect
425. C32#1902semidirect
484. C3#72022direct (N × H)
485. C3C3×#1902semidirect

Maximal subgroups

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

subgroupindex
C32×#5682
C2×C3×#9512
C3×#52362
C2×C3×#8242
C3×#51202
C3×C3⋊#3382
C3×#45592
C3×#19023
#720223