Finite Group Databasebeta

← C3×#71662 #11788515

Subgroups — C3×#71662 #11788515

Order 1152.

753 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#519C211yesmaximal, characteristic, complemented
2C2×C3×#942C211yesmaximal, characteristic
3C3×#5225C211yesmaximal, characteristic, complemented
4C3×#5006C211yesmaximal, characteristic
5C3×#5005C211yesmaximal, characteristic
6C3×#4533C211yesmaximal, characteristic, complemented
7C3×#4531C211yesmaximal, characteristic, complemented
8C3×#1905—31nomaximal
9#71662C311yesmaximal, characteristic, direct product factor
10C22×C32×Q8C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C32×C4×Q8—22no—
13C32×C4×Q8—22no—
14C32×#117—22no—
15C32×#117C2211yescharacteristic
16C32×#117C2211yescharacteristic
17C2×C3×#216C412yescomplemented
18C3×#942—22no—
19C3×#942—22no—
20C2×C3×#215C412yescomplemented
21C3×#926C2211yescharacteristic
22C2×C3×#208C2211yescharacteristic
23C2×C3×#208C2211yescharacteristic
24C3×#835—22no—
25C3×#806—22no—
26C3×#805—22no—
27C2×C3×#137—31no—
28C3×#519—21no—
29C3×#519D311yescharacteristic, complemented
30C3×#519C2×C311yescharacteristic, complemented
31C2×#942C2×C311yescharacteristic
32C3×#421—31no—
33C3×#419—31no—
34C3×#410—31no—
35#5225C2×C311yescharacteristic, complemented
36#5006C2×C311yescharacteristic
37#5005C2×C311yescharacteristic
38C3×#344—31no—
39C3×H(3,ℤ/4)—31no—
40#4533C2×C311yescharacteristic, complemented
41#4531C2×C311yescharacteristic, complemented
42C2×C32×Q8—21no—
43C2×C32×Q8—21no—
44C2×C32×Q8—21no—
45C2×C32×Q8C2×C412yes—
46C2×C32×Q8D412yes—
47C2×C32×Q8D412yes—
48C22×C32×C4—21no—
49C22×C32×C4C2311yescharacteristic
50C2×C3×#67—21no—
51C3×#235—21no—
52C3×#235—21no—
53C3×#235—21no—
54C3×#235—21no—
55C2×C3×D12C2×C411yescharacteristic
56C2×C3×C4×D3—21no—
57C2×C3×Dic6C2×C411yescharacteristic
58C32×C4⋊C4—24no—
59C32×C4⋊C4—21no—
60C32×C4⋊C4—21no—
61C32×C22⋊C4—21no—
62C32×C22⋊C4D412yes—
63C32×C42—24no—
64C32×C42—21no—
65C32×C42D412yes—
66C3×#217—21no—
67C3×#216—24no—
68C3×#215—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#1905—31noSylow
78C2×C3×#41—31no—
79C22×C3×Q8—21no—
80C22×C3×Q8C2×D311yescharacteristic
81C22×C3×Q8C22×C311yescharacteristic
82C2×#235C22×C311yescharacteristic
83C3×#137—62no—
84C3×#137—62no—
85C2×C3×Q16—32no—
86C2×C3×SD16—32no—
87C2×C3×M16—31no—
88C2×C3×M16—31no—
89C3×C4×Q8—24no—
90C3×C4×Q8—24no—
91C3×C4×Q8—22no—
92C3×C4×Q8—22no—
93C3×C4×Q8—22no—
94C3×C4×Q8—22no—
95C3×#117—24no—
96C3×#117—22no—
97C3×#117—22no—
98C3×#117—21no—
99C3×#117—21no—
100C3×#117—31no—
101C3×#117C2×D311yescharacteristic
102C3×#117C2×D311yescharacteristic
103C3×#117C22×C311yescharacteristic
104C3×#117C22×C311yescharacteristic
105C2×#216C3×C412yescomplemented
106#942—22no—
107#942—22no—
108C2×#215C3×C412yescomplemented
109#926C22×C311yescharacteristic
110C2×#208C22×C311yescharacteristic
111C2×#208C22×C311yescharacteristic
112C3×#104—62no—
113C3×#103—62no—
114C3×#99—62no—
115#835—22no—
116#806—22no—
117#805—22no—
118C23×C32C2×D411yescharacteristic
119C22×C3×D3—21no—
120C32×Q8—22no—
121C32×Q8—22no—
122C32×Q8—22no—
123C32×Q8C22⋊C414yes—
124C2×C32×C4—22no—
125C2×C32×C4—22no—
126C2×C32×C4—22no—
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×C4#4111yescharacteristic
134C2×C32×C4#4111yescharacteristic
135C3×#67—41no—
136C3×#67—41no—
137C2×C3×Dic3—21no—
138C2×C3×Dic3—41no—
139C3×D12—21no—
140C3×D12—21no—
141C3×C4×D3—41no—
142C3×C4×D3—41no—
143C3×Dic6—21no—
144C3×Dic6—21no—
145C3×C3⋊C8—22no—
146C3×C3⋊C8—41no—
147C2×#137—31no—
148#519C3×D311yescharacteristic, complemented
149#421—31no—
150#419—31no—
151#410—31no—
152#344—31no—
153H(3,ℤ/4)—31no—
154C3×#41—61no—
155C3×#41—61no—
156C3×#41—61no—
157C3×#41—61no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—31no—
171C2×C3×Q8C4×D312yes—
172C2×C3×Q8D1212yes—
173C2×C3×Q8#6712yes—
174C2×C3×Q8C2×C3×C412yes—
175C2×C3×Q8C3×D412yes—
176C2×C3×Q8C3×D412yes—
177C2×C3×D4—31no—
178C2×C3×D4—61no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—61no—
184C22×C3×C4C22×D311yescharacteristic
185C22×C3×C4C23×C311yescharacteristic
186C2×#67—21no—
187#235—21no—
188#235—21no—
189#235—21no—
190#235—21no—
191C2×D12C2×C3×C411yescharacteristic
192C2×C4×D3—21no—
193C2×Dic6C2×C3×C411yescharacteristic
194C3×Q16—64no—
195C3×SD16—64no—
196C3×M16—61no—
197C3×M16—61no—
198C3×M16—61no—
199C3×M16—61no—
200C2×C3×C8—32no—
201C2×C3×C8—61no—
202C3×C4⋊C4—28no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—24no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—61no—
212C3×C22⋊C4—22no—
213C3×C22⋊C4—21no—
214C3×C22⋊C4—21no—
215C3×C22⋊C4—22no—
216C3×C22⋊C4—61no—
217C3×C22⋊C4#6712yes—
218C3×C22⋊C4C3×D412yes—
219C3×C42—28no—
220C3×C42—24no—
221C3×C42—24no—
222C3×C42—22no—
223C3×C42—21no—
224C3×C42—21no—
225C3×C42—22no—
226C3×C42—61no—
227C3×C42#6712yes—
228C3×C42C3×D412yes—
229#217—21no—
230#216—24no—
231#215—24no—
232C4⋊Dic3—21no—
233C4×Dic3—21no—
234#208—21no—
235#208—21no—
236#208—21no—
237#208—21no—
238C2×C3⋊C8—21no—
239C2×C3⋊C8C2×C3×C412yes—
240C22×C32—41no—
241C22×C32C4×D411yescharacteristic
242C22×C32#12011yescharacteristic
243C22×C32#12111yescharacteristic
244C2×C3×D3—41no—
245C2×C3×D3—81no—
246C32×C4—24no—
247C32×C4—42no—
248C32×C4—42no—
249C32×C4C2×C22⋊C411yescharacteristic
250C32×C4C4×D411yescharacteristic
251C32×C4#12111yescharacteristic
252C32×C4#12311yescharacteristic
253C3×Dic3—41no—
254C3×Dic3—81no—
255C2×#41—31no—
256C22×Q8C2×C3×D311yescharacteristic
257#137—62no—
258#137—62no—
259C2×Q16—32no—
260C2×SD16—32no—
261C2×M16—31no—
262C2×M16—31no—
263C4×Q8—22no—
264C4×Q8—22no—
265#117—22no—
266#117—31no—
267#117C2×C3×D311yescharacteristic
268#117C2×C3×D311yescharacteristic
269#104—62no—
270#103—62no—
271#99—62no—
272C23×C3—21no—
273C23×C3—61no—
274C23×C3C2×#6711yescharacteristic
275C23×C3C2×C3×D411yescharacteristic
276C22×D3—21no—
277C3×Q8—24no—
278C3×Q8—24no—
279C3×Q8—24no—
280C3×Q8—22no—
281C3×Q8—22no—
282C3×Q8—22no—
283C3×Q8—22no—
284C3×Q8—22no—
285C3×Q8—22no—
286C3×Q8—24no—
287C3×Q8—61no—
288C3×Q8—61no—
289C3×Q8(C2×D3)⋊C414yes—
290C3×Q8C3×C22⋊C414yes—
291C3×D4—61no—
292C3×D4—61no—
293C3×D4—121no—
294C3×D4—121no—
295C2×C3×C4—24no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—61no—
317C2×C3×C4—121no—
318C2×C3×C4—121no—
319C2×C3×C4—121no—
320C2×C3×C4C2×C4×D311yescharacteristic
321C2×C3×C4C2×D1211yescharacteristic
322C2×C3×C4#23511yescharacteristic
323C2×C3×C4#23511yescharacteristic
324C2×C3×C4C2×#6711yescharacteristic
325C2×C3×C4C2×#6711yescharacteristic
326C2×C3×C4C22×C3×C411yescharacteristic
327C2×C3×C4C2×C3×D411yescharacteristic
328C2×C3×C4C2×C3×D411yescharacteristic
329C2×C3×C4C2×C3×D411yescharacteristic
330C2×C3×C4C3×#4111yescharacteristic
331C2×C3×C4C3×#4111yescharacteristic
332#67—41no—
333#67—41no—
334C2×Dic3—21no—
335C2×Dic3—41no—
336D12—21no—
337D12—21no—
338C4×D3—41no—
339C4×D3—41no—
340Dic6—21no—
341Dic6—21no—
342C3×C8—62no—
343C3×C8—121no—
344C3⋊C8—22no—
345C3⋊C8—41no—
346C2×C32—21no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32#38611yescharacteristic
350C3×D3—81no—
351#41—61no—
352#41—61no—
353#41—61no—
354#41—61no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—31no—
359C2×Q8C3×C4×D312yes—
360C2×Q8C3×D1212yes—
361C2×Q8C3×#6712yes—
362C2×D4—31no—
363C2×D4—61no—
364C22×C4—21no—
365C22×C4—61no—
366C22×C4C22×C3×D311yescharacteristic
367Q16—64no—
368SD16—64no—
369M16—61no—
370M16—61no—
371M16—61no—
372M16—61no—
373C2×C8—32no—
374C2×C8—61no—
375C4⋊C4—24no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—61no—
379C22⋊C4—21no—
380C22⋊C4—61no—
381C22⋊C4C3×#6712yes—
382C42—24no—
383C42—21no—
384C42—61no—
385C42C3×#6712yes—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—41no—
390C22×C3—41no—
391C22×C3—42no—
392C22×C3—121no—
393C22×C3—241no—
394C22×C3C4×#6711yescharacteristic
395C22×C3#93011yescharacteristic
396C22×C3#95311yescharacteristic
397C22×C3C3×C4×D411yescharacteristic
398C22×C3C3×#12011yescharacteristic
399C22×C3C3×#12111yescharacteristic
400C2×D3—41no—
401C2×D3—81no—
402C3×C4—28no—
403C3×C4—24no—
404C3×C4—24no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—44no—
410C3×C4—44no—
411C3×C4—42no—
412C3×C4—42no—
413C3×C4—42no—
414C3×C4—42no—
415C3×C4—121no—
416C3×C4—241no—
417C3×C4C2×(C2×D3)⋊C411yescharacteristic
418C3×C4C4×#6711yescharacteristic
419C3×C4#92911yescharacteristic
420C3×C4#93011yescharacteristic
421C3×C4C2×C3×C22⋊C411yescharacteristic
422C3×C4C3×C4×D411yescharacteristic
423C3×C4C3×#12111yescharacteristic
424C3×C4C3×#12311yescharacteristic
425Dic3—41no—
426Dic3—81no—
427C32#190511yescharacteristic, complemented, Sylow
428C23—61no—
429C23C2×C3×#6711yescharacteristic
430Q8—22no—
431Q8—22no—
432Q8—22no—
433Q8—61no—
434Q8—61no—
435Q8C3×(C2×D3)⋊C414yes—
436D4—61no—
437D4—61no—
438D4—121no—
439D4—121no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—61no—
446C2×C4—121no—
447C2×C4—121no—
448C2×C4—121no—
449C2×C4C2×C3×C4×D311yescharacteristic
450C2×C4C2×C3×D1211yescharacteristic
451C2×C4C3×#23511yescharacteristic
452C2×C4C3×#23511yescharacteristic
453C2×C4C2×C3×#6711yescharacteristic
454C2×C4C2×C3×#6711yescharacteristic
455C8—62no—
456C8—121no—
457C2×C3—22no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—41no—
466C2×C3—41no—
467C2×C3—241no—
468C2×C3#521311yescharacteristic
469C2×C3C3×#38611yescharacteristic
470D3—81no—
471C22—41no—
472C22—121no—
473C22—241no—
474C22C3×C4×#6711yescharacteristic
475C22C3×#93011yescharacteristic
476C22C3×#95311yescharacteristic
477C4—24no—
478C4—42no—
479C4—42no—
480C4—121no—
481C4—241no—
482C4C2×C3×(C2×D3)⋊C411yescharacteristic
483C4C3×C4×#6711yescharacteristic
484C4C3×#92911yescharacteristic
485C4C3×#93011yescharacteristic
486C3—21no—
487C3#7166211yescharacteristic, direct product factor
488C3C3×#190511yescharacteristic, complemented
489C2—21no—
490C2—21no—
491C2—21no—
492C2—241no—
493C2C3×#521311yescharacteristic
subgroupG/H
centerC2×C3#5213
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#519C2
socleC2×C32#386
prime radicalC2×C3×#235C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#519D3
hypocenterC3C3×#1905
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#519

Complements

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

NH ≅ G/Ntype
1. C32×#519C2semidirect
3. C3×#5225C2semidirect
6. C3×#4533C2semidirect
7. C3×#4531C2semidirect
9. #71662C3direct (N × H)
17. C2×C3×#216C4semidirect
20. C2×C3×#215C4semidirect
29. C3×#519D3semidirect
30. C3×#519C2×C3semidirect
35. #5225C2×C3semidirect
40. #4533C2×C3semidirect
41. #4531C2×C3semidirect
105. C2×#216C3×C4semidirect
108. C2×#215C3×C4semidirect
148. #519C3×D3semidirect
427. C32#1905semidirect
487. C3#71662direct (N × H)
488. C3C3×#1905semidirect

Maximal subgroups

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

subgroupindex
C32×#5192
C2×C3×#9422
C3×#52252
C3×#50062
C3×#50052
C3×#45332
C3×#45312
C3×#19053
#716623