Finite Group Databasebeta

← C3×#83704 #11896026

Subgroups — C3×#83704 #11896026

Order 1152.

782 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×Q82C211yesmaximal, characteristic, complemented
2C3×#5584C211yesmaximal, characteristic
3C3×#5094C212yesmaximal, complemented
4C3×#5093C211yesmaximal, characteristic, complemented
5C3×#5089C212yesmaximal
6C3×#5084C211yesmaximal, characteristic
7C3×#5031C211yesmaximal, characteristic, complemented
8C3×#5030C212yesmaximal, complemented
9C3×#5028C212yesmaximal
10C3×C4×#215C211yesmaximal, characteristic
11C3×#5024C211yesmaximal, characteristic, complemented
12C3×#3369—31nomaximal
13#83704C311yesmaximal, characteristic, direct product factor
14C32×C4⋊Q8—21no—
15C32×C4⋊Q8—22no—
16C32×C4⋊Q8C2211yescharacteristic
17C32×C4⋊Q8C2212yes—
18C32×C4×Q8—21no—
19C32×C4×Q8C2211yescharacteristic
20C32×C4×Q8C2212yes—
21C32×C4×Q8C2211yescharacteristic
22C3×#946—21no—
23C2×C3×#216C2212yes—
24C2×C3×#215C2211yescharacteristic
25C3×#898—22no—
26C3×#892—21no—
27C3×#875C2212yes—
28C3×C4×D12C2211yescharacteristic
29C3×#869C2211yescharacteristic
30C3×C4×Dic6C2212yes—
31C3×#835C2211yescharacteristic
32C3×#810C2212yes—
33C3×#810C2212yes—
34C3×#810C2212yes—
35C3×#809C2211yescharacteristic
36C3×#809C2212yes—
37C3×#808C2211yescharacteristic
38C3×#808C2212yes—
39C3×C4⋊(C3⋊C8)C2212yes—
40C3×C4⋊(C3⋊C8)C2211yescharacteristic
41C3×#803C2212yes—
42C3×C4×C3⋊C8C2211yescharacteristic
43C3×Q82—21no—
44C3×Q82D311yescharacteristic, complemented
45C3×Q82C2×C311yescharacteristic, complemented
46C3×#556—31no—
47#5584C2×C311yescharacteristic
48C3×#489—32no—
49C3×#487—31no—
50C3×#476—31no—
51C3×#475—32no—
52C3×#464—32no—
53C3×#462—31no—
54C3×#446—31no—
55C3×#441—32no—
56C3×C4×SD16—31no—
57#5094C2×C312yescomplemented
58#5093C2×C311yescharacteristic, complemented
59#5089C2×C312yes—
60#5084C2×C311yescharacteristic
61#5031C2×C311yescharacteristic, complemented
62#5030C2×C312yescomplemented
63#5028C2×C312yes—
64C4×#215C2×C311yescharacteristic
65#5024C2×C311yescharacteristic, complemented
66C2×C32×Q8—22no—
67C2×C32×Q8—21no—
68C2×C32×Q8D411yescharacteristic
69C2×C32×Q8C2311yescharacteristic
70C2×C3×D12C2311yescharacteristic
71C2×C3×C4×D3—21no—
72C2×C3×Dic6C2312yes—
73C32×C4⋊C4—22no—
74C32×C4⋊C4—21no—
75C32×C4⋊C4—22no—
76C32×C4⋊C4—21no—
77C32×C4⋊C4D411yescharacteristic
78C32×C4⋊C4D412yes—
79C32×C4⋊C4C2311yescharacteristic
80C32×C4⋊C4C2312yes—
81C32×C42—22no—
82C32×C42—21no—
83C32×C42C2311yescharacteristic
84C32×C42C2312yes—
85C3×#216—24no—
86C3×#215—22no—
87C3×(C2×D3)⋊C4—22no—
88C3×(C2×D3)⋊C4—21no—
89C3×(C2×D3)⋊C4—22no—
90C3×C4⋊Dic3—21no—
91C3×C4⋊Dic3—21no—
92C3×C4⋊Dic3C2311yescharacteristic
93C3×Dic3⋊C4—22no—
94C3×C4×Dic3—22no—
95C2×C3×C3⋊C8C2311yescharacteristic
96C2×C3×C3⋊C8C2312yes—
97C2×C3×C3⋊C8C2311yescharacteristic
98#3369—31noSylow
99C2×C3×Q16—32no—
100C2×C3×SD16—31no—
101C3×C4⋊Q8—22no—
102C3×C4⋊Q8—24no—
103C3×C4⋊Q8—21no—
104C3×C4⋊Q8—22no—
105C3×C4⋊Q8—21no—
106C3×C4⋊Q8—22no—
107C3×C4⋊Q8—22no—
108C3×C4⋊Q8—21no—
109C3×C4⋊Q8—31no—
110C3×C4⋊Q8C2×D312yes—
111C3×C4⋊Q8C2×D311yescharacteristic
112C3×C4⋊Q8C22×C311yescharacteristic
113C3×C4⋊Q8C22×C312yes—
114C3×#126—62no—
115C3×#124—32no—
116C3×C22⋊Q8—61no—
117C3×C4×Q8—22no—
118C3×C4×Q8—21no—
119C3×C4×Q8—21no—
120C3×C4×Q8—21no—
121C3×C4×Q8—22no—
122C3×C4×Q8—21no—
123C3×C4×Q8—32no—
124C3×C4×Q8C2×D311yescharacteristic
125C3×C4×Q8C2×D312yes—
126C3×C4×Q8C2×D311yescharacteristic
127C3×C4×Q8C22×C311yescharacteristic
128C3×C4×Q8C22×C312yes—
129C3×C4×Q8C22×C311yescharacteristic
130C3×C4×D4—31no—
131C3×#117—61no—
132#946—21no—
133C2×#216C22×C312yes—
134C2×#215C22×C311yescharacteristic
135#898—22no—
136#892—21no—
137#875C22×C312yes—
138C4×D12C22×C311yescharacteristic
139#869C22×C311yescharacteristic
140C4×Dic6C22×C312yes—
141C3×#106—31no—
142C3×#106—32no—
143C3×C4⋊C8—31no—
144C3×C4⋊C8—32no—
145C3×#103—31no—
146C3×#103—32no—
147C3×#103—32no—
148C3×#103—32no—
149C3×#102—31no—
150C3×#102—32no—
151C3×#97—32no—
152C3×C4×C8—31no—
153#835C22×C311yescharacteristic
154#810C22×C312yes—
155#810C22×C312yes—
156#810C22×C312yes—
157#809C22×C311yescharacteristic
158#809C22×C312yes—
159#808C22×C311yescharacteristic
160#808C22×C312yes—
161C4⋊(C3⋊C8)C22×C312yes—
162C4⋊(C3⋊C8)C22×C311yescharacteristic
163#803C22×C312yes—
164C4×C3⋊C8C22×C311yescharacteristic
165C22×C3×D3—21no—
166C32×Q8—22no—
167C32×Q8—21no—
168C32×Q8—82no—
169C32×Q8—81no—
170C32×Q8#4112yes—
171C2×C32×C4—21no—
172C2×C32×C4—22no—
173C2×C32×C4—21no—
174C2×C32×C4C2×D412yes—
175C2×C32×C4C2×D412yes—
176C2×C32×C4C2×D411yescharacteristic
177C2×C32×C4C2×D411yescharacteristic
178C2×C32×C4C2411yescharacteristic
179C2×C3×Dic3—21no—
180C2×C3×Dic3—22no—
181C3×D12—21no—
182C3×D12—21no—
183C3×C4×D3—42no—
184C3×Dic6—22no—
185C3×Dic6—22no—
186C3×C3⋊C8—21no—
187C3×C3⋊C8—22no—
188C3×C3⋊C8—21no—
189Q82C3×D311yescharacteristic, complemented
190#556—31no—
191#489—32no—
192#487—31no—
193#476—31no—
194#475—32no—
195#464—32no—
196#462—31no—
197#446—31no—
198#441—32no—
199C4×SD16—31no—
200C2×C3×Q8—24no—
201C2×C3×Q8—22no—
202C2×C3×Q8—22no—
203C2×C3×Q8—21no—
204C2×C3×Q8—22no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—32no—
209C2×C3×Q8#6711yescharacteristic
210C2×C3×Q8C3×D411yescharacteristic
211C2×C3×Q8C22×D311yescharacteristic
212C2×C3×Q8C23×C311yescharacteristic
213C2×C3×D4—31no—
214C22×C3×C4—61no—
215C2×D12C23×C311yescharacteristic
216C2×C4×D3—21no—
217C2×Dic6C23×C312yes—
218C3×Q16—64no—
219C3×SD16—62no—
220C2×C3×C8—31no—
221C2×C3×C8—31no—
222C2×C3×C8—32no—
223C3×C4⋊C4—24no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—24no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—31no—
240C3×C4⋊C4—62no—
241C3×C4⋊C4—61no—
242C3×C4⋊C4—61no—
243C3×C4⋊C4#6711yescharacteristic
244C3×C4⋊C4#6712yes—
245C3×C4⋊C4C3×D411yescharacteristic
246C3×C4⋊C4C3×D412yes—
247C3×C4⋊C4C22×D311yescharacteristic
248C3×C4⋊C4C22×D312yes—
249C3×C4⋊C4C23×C311yescharacteristic
250C3×C4⋊C4C23×C312yes—
251C3×C22⋊C4—62no—
252C3×C22⋊C4—61no—
253C3×C22⋊C4—62no—
254C3×C42—24no—
255C3×C42—22no—
256C3×C42—22no—
257C3×C42—21no—
258C3×C42—22no—
259C3×C42—21no—
260C3×C42—21no—
261C3×C42—22no—
262C3×C42—62no—
263C3×C42C22×D311yescharacteristic
264C3×C42C22×D312yes—
265C3×C42C23×C311yescharacteristic
266C3×C42C23×C312yes—
267#216—24no—
268#215—22no—
269(C2×D3)⋊C4—21no—
270(C2×D3)⋊C4—22no—
271(C2×D3)⋊C4—22no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3C23×C311yescharacteristic
275Dic3⋊C4—22no—
276C4×Dic3—22no—
277C2×C3⋊C8C23×C311yescharacteristic
278C2×C3⋊C8C23×C311yescharacteristic
279C2×C3⋊C8C23×C312yes—
280C22×C32C22×D411yescharacteristic
281C2×C3×D3—41no—
282C2×C3×D3—41no—
283C2×C3×D3—41no—
284C32×C4—22no—
285C32×C4—22no—
286C32×C4—22no—
287C32×C4—21no—
288C32×C4—42no—
289C32×C4—41no—
290C32×C4#13712yes—
291C32×C4C2×#4111yescharacteristic
292C32×C42−1+411yescharacteristic
293C3×Dic3—42no—
294C3×Dic3—41no—
295C2×Q16—32no—
296C2×SD16—31no—
297C4⋊Q8—21no—
298C4⋊Q8—22no—
299C4⋊Q8—31no—
300C4⋊Q8C2×C3×D312yes—
301C4⋊Q8C2×C3×D311yescharacteristic
302#126—62no—
303#124—32no—
304C22⋊Q8—61no—
305C4×Q8—21no—
306C4×Q8—32no—
307C4×Q8C2×C3×D311yescharacteristic
308C4×Q8C2×C3×D312yes—
309C4×Q8C2×C3×D311yescharacteristic
310C4×D4—31no—
311#117—61no—
312#106—31no—
313#106—32no—
314C4⋊C8—31no—
315C4⋊C8—32no—
316#103—31no—
317#103—32no—
318#103—32no—
319#103—32no—
320#102—31no—
321#102—32no—
322#97—32no—
323C4×C8—31no—
324C23×C3—61no—
325C22×D3—21no—
326C3×Q8—24no—
327C3×Q8—22no—
328C3×Q8—22no—
329C3×Q8—22no—
330C3×Q8—21no—
331C3×Q8—21no—
332C3×Q8—22no—
333C3×Q8—62no—
334C3×Q8—62no—
335C3×Q8—82no—
336C3×Q8—81no—
337C3×Q8—82no—
338C3×Q8—81no—
339C3×Q8—84no—
340C3×Q8—82no—
341C3×Q8#23912yes—
342C3×Q8C3×#4112yes—
343C3×D4—61no—
344C3×D4—61no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—24no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—21no—
359C2×C3×C4—61no—
360C2×C3×C4—62no—
361C2×C3×C4—122no—
362C2×C3×C4C2×#6712yes—
363C2×C3×C4C2×#6712yes—
364C2×C3×C4C2×#6711yescharacteristic
365C2×C3×C4C2×#6711yescharacteristic
366C2×C3×C4C2×C3×D412yes—
367C2×C3×C4C2×C3×D412yes—
368C2×C3×C4C2×C3×D411yescharacteristic
369C2×C3×C4C2×C3×D411yescharacteristic
370C2×C3×C4C23×D311yescharacteristic
371C2×C3×C4C24×C311yescharacteristic
372C2×Dic3—21no—
373C2×Dic3—22no—
374D12—21no—
375D12—21no—
376C4×D3—42no—
377Dic6—22no—
378Dic6—22no—
379C3×C8—61no—
380C3×C8—62no—
381C3×C8—61no—
382C3⋊C8—21no—
383C3⋊C8—22no—
384C3⋊C8—21no—
385C2×C32#54811yescharacteristic
386C2×C32C2×#13711yescharacteristic
387C2×C32#57711yescharacteristic
388C3×D3—81no—
389C2×Q8—22no—
390C2×Q8—21no—
391C2×Q8—32no—
392C2×Q8C3×#6711yescharacteristic
393C2×Q8C22×C3×D311yescharacteristic
394C2×D4—31no—
395C22×C4—61no—
396Q16—64no—
397SD16—62no—
398C2×C8—31no—
399C2×C8—31no—
400C2×C8—32no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—31no—
406C4⋊C4—61no—
407C4⋊C4—62no—
408C4⋊C4—61no—
409C4⋊C4C3×#6711yescharacteristic
410C4⋊C4C3×#6712yes—
411C4⋊C4C22×C3×D311yescharacteristic
412C4⋊C4C22×C3×D312yes—
413C22⋊C4—62no—
414C22⋊C4—61no—
415C22⋊C4—62no—
416C42—22no—
417C42—21no—
418C42—62no—
419C42C22×C3×D311yescharacteristic
420C42C22×C3×D312yes—
421C22×C3—21no—
422C22×C3—121no—
423C22×C3—121no—
424C22×C3—121no—
425C22×C3C22×#6711yescharacteristic
426C22×C3C22×C3×D411yescharacteristic
427C2×D3—41no—
428C2×D3—41no—
429C2×D3—41no—
430C3×C4—24no—
431C3×C4—24no—
432C3×C4—22no—
433C3×C4—22no—
434C3×C4—21no—
435C3×C4—22no—
436C3×C4—22no—
437C3×C4—22no—
438C3×C4—21no—
439C3×C4—22no—
440C3×C4—22no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—42no—
444C3×C4—41no—
445C3×C4—44no—
446C3×C4—42no—
447C3×C4—42no—
448C3×C4—41no—
449C3×C4—42no—
450C3×C4—41no—
451C3×C4—121no—
452C3×C4—122no—
453C3×C4#94212yes—
454C3×C4C3×#13712yes—
455C3×C4C2×#23911yescharacteristic
456C3×C4#100711yescharacteristic
457C3×C4C2×C3×#4111yescharacteristic
458C3×C4C3×2−1+411yescharacteristic
459Dic3—41no—
460Dic3—42no—
461C32#336911yescharacteristic, complemented, Sylow
462C23—61no—
463Q8—22no—
464Q8—21no—
465Q8—62no—
466Q8—62no—
467Q8—82no—
468Q8—81no—
469Q8C3×#23912yes—
470D4—61no—
471D4—61no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—61no—
476C2×C4—62no—
477C2×C4—122no—
478C2×C4C2×C3×#6712yes—
479C2×C4C2×C3×#6712yes—
480C2×C4C2×C3×#6711yescharacteristic
481C2×C4C2×C3×#6711yescharacteristic
482C2×C4C23×C3×D311yescharacteristic
483C8—61no—
484C8—62no—
485C8—61no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—241no—
490C2×C3C2×#94211yescharacteristic
491C2×C3#581711yescharacteristic
492C2×C3#583611yescharacteristic
493C2×C3C3×#54811yescharacteristic
494C2×C3C2×C3×#13711yescharacteristic
495C2×C3C3×#57711yescharacteristic
496D3—81no—
497C22—121no—
498C22—121no—
499C22—121no—
500C22C22×C3×#6711yescharacteristic
501C4—22no—
502C4—22no—
503C4—22no—
504C4—21no—
505C4—42no—
506C4—41no—
507C4—121no—
508C4—122no—
509C4C3×#94212yes—
510C4C2×C3×#23911yescharacteristic
511C4C3×#100711yescharacteristic
512C3—21no—
513C3#8370411yescharacteristic, direct product factor
514C3C3×#336911yescharacteristic, complemented
515C2—241no—
516C2C2×C3×#94211yescharacteristic
517C2C3×#581711yescharacteristic
518C2C3×#583611yescharacteristic
subgroupG/H
centerC22×C3C22×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×Q82C2
socleC22×C32C22×D4
prime radicalC2×C3×D12C23
hypercenterC3×Q82D3
hypocenterC3C3×#3369
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C4×Q8 ◁ C3×Q82

Complements

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

NH ≅ G/Ntype
1. C32×Q82C2semidirect
3. C3×#5094C2semidirect
4. C3×#5093C2semidirect
7. C3×#5031C2semidirect
8. C3×#5030C2semidirect
11. C3×#5024C2semidirect
13. #83704C3direct (N × H)
44. C3×Q82D3semidirect
45. C3×Q82C2×C3semidirect
57. #5094C2×C3semidirect
58. #5093C2×C3semidirect
61. #5031C2×C3semidirect
62. #5030C2×C3semidirect
65. #5024C2×C3semidirect
189. Q82C3×D3semidirect
461. C32#3369semidirect
513. C3#83704direct (N × H)
514. C3C3×#3369semidirect

Maximal subgroups

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

subgroupindex
C32×Q822
C3×#55842
C3×#50942
C3×#50932
C3×#50892
C3×#50842
C3×#50312
C3×#50302
C3×#50282
C3×C4×#2152
C3×#50242
C3×#33693
#837043