Finite Group Databasebeta

← C3×#79720 #11860790

Subgroups — C3×#79720 #11860790

Order 1152.

967 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D3×Q16C211yesmaximal, characteristic
2C3×#5572C212yesmaximal
3C32×C4×Q16C211yesmaximal, characteristic, complemented
4C3×Dic3×Q16C211yesmaximal, characteristic, complemented
5C3×#5030C212yesmaximal, complemented
6C3×#4880C211yesmaximal, characteristic
7C3×#4873C211yesmaximal, characteristic, complemented
8C3×#4802C212yesmaximal
9C3×#4789C212yesmaximal, complemented
10C3×C4×Dic12C211yesmaximal, characteristic, complemented
11C3×#4689C211yesmaximal, characteristic
12C3×#2950—31nomaximal
13#79720C311yesmaximal, characteristic, direct product factor
14C2×C3×D3×Q8C2212yes—
15C2×C32×Q16C2211yescharacteristic
16C32×C4×Q8C2212yes—
17C3×Q8×Dic3C2212yes—
18C2×C3×#216C2212yes—
19C3×D3×Q16C418yescomplemented
20C2×C3×Dic12C2211yescharacteristic
21C2×C3×ΓL(1,𝔽25)C2211yescharacteristic
22C2×C3×D3×C8C2211yescharacteristic
23C3×#892—22no—
24C3×#892C2212yes—
25C3×#887—22no—
26C3×#887C2212yes—
27C3×#872—22no—
28C3×#872C2211yescharacteristic
29C3×C4×Dic6—22no—
30C3×C4×Dic6C2212yes—
31C32×#107C2211yescharacteristic
32C32×#103C2212yes—
33C32×C4×C8C2211yescharacteristic
34C3×#835C2212yes—
35C3×#818C2211yescharacteristic
36C3×#816C2212yes—
37C3×#815C2211yescharacteristic
38C3×C8×Dic3C2211yescharacteristic
39C3×#810C2212yes—
40C3×#808C2212yes—
41C3×#803C2211yescharacteristic
42C22×C3×Q16—31no—
43C3×#519—32no—
44C2×D3×Q16C2×C311yescharacteristic
45#5572C2×C312yes—
46C3×#441—32no—
47C3×#441—32no—
48C3×C4×Q16—21no—
49C3×C4×Q16—31no—
50C3×C4×Q16—31no—
51C3×C4×Q16—31no—
52C3×C4×Q16D311yescharacteristic, complemented
53C3×C4×Q16C2×C311yescharacteristic, complemented
54C3×#427—31no—
55C3×#419—32no—
56C3×#405—31no—
57Dic3×Q16C2×C311yescharacteristic, complemented
58#5030C2×C312yescomplemented
59#4880C2×C311yescharacteristic
60#4873C2×C311yescharacteristic, complemented
61#4802C2×C312yes—
62#4789C2×C312yescomplemented
63C4×Dic12C2×C311yescharacteristic, complemented
64#4689C2×C311yescharacteristic
65C2×C32×Q8C2312yes—
66C3×D3×Q8—24no—
67C3×D3×Q8C2×C418yes—
68C2×C3×C4×D3—22no—
69C2×C3×C4×D3C2311yescharacteristic
70C2×C3×Dic6—22no—
71C2×C3×Dic6C2312yes—
72C32×Q16C2×C414yes—
73C2×C32×C8C2311yescharacteristic
74C2×C32×C8C2311yescharacteristic
75C32×C4⋊C4—22no—
76C32×C4⋊C4C2312yes—
77C32×C42—22no—
78C32×C42C2311yescharacteristic
79C3×#216C2×C418yes—
80C3×(C2×D3)⋊C4—22no—
81C3×(C2×D3)⋊C4D411yescharacteristic
82C3×(C2×D3)⋊C4D411yescharacteristic
83C3×C4⋊Dic3—22no—
84C3×C4⋊Dic3C2312yes—
85C3×Dic3⋊C4—22no—
86C3×Dic3⋊C4—22no—
87C3×Dic3⋊C4D411yescharacteristic
88C3×Dic3⋊C4D411yescharacteristic
89C3×C4×Dic3—22no—
90C3×C4×Dic3—22no—
91C3×C4×Dic3—22no—
92C3×C4×Dic3—22no—
93C3×C4×Dic3C2311yescharacteristic
94C2×C3×C3⋊C8C2311yescharacteristic
95C2×C3×C3⋊C8C2311yescharacteristic
96C3×Dic12C2×C414yes—
97C3×ΓL(1,𝔽25)—22no—
98C3×D3×C8C2×C414yes—
99#2950—31noSylow
100C22×C3×Q8—32no—
101C2×D3×Q8C22×C312yes—
102C2×C3×Q16—21no—
103C2×C3×Q16—32no—
104C2×C3×Q16—38no—
105C2×C3×Q16—31no—
106C2×C3×Q16C2×D311yescharacteristic
107C2×C3×Q16C22×C311yescharacteristic
108C2×C3×M16—31no—
109C22×C3×C8—31no—
110C3×C4×Q8—22no—
111C3×C4×Q8—32no—
112C3×C4×Q8—32no—
113C3×C4×Q8—32no—
114C3×C4×Q8—62no—
115C3×C4×Q8—62no—
116C3×C4×Q8C2×D312yes—
117C3×C4×Q8C22×C312yes—
118C3×#117—31no—
119C3×#117—32no—
120C3×#117—62no—
121C3×#117—62no—
122Q8×Dic3C22×C312yes—
123C2×#216C22×C312yes—
124D3×Q16C3×C418yescomplemented
125C2×Dic12C22×C311yescharacteristic
126C2×ΓL(1,𝔽25)C22×C311yescharacteristic
127C2×D3×C8C22×C311yescharacteristic
128#892—22no—
129#892C22×C312yes—
130#887—22no—
131#887C22×C312yes—
132#872—22no—
133#872C22×C311yescharacteristic
134C4×Dic6—22no—
135C4×Dic6C22×C312yes—
136C3×#107—21no—
137C3×#107—31no—
138C3×#107C2×D311yescharacteristic
139C3×#107C22×C311yescharacteristic
140C3×#106—32no—
141C3×#103—22no—
142C3×#103—32no—
143C3×#103—32no—
144C3×#103—32no—
145C3×#103C2×D312yes—
146C3×#103C22×C312yes—
147C3×#97—31no—
148C3×#97—31no—
149C3×C4×C8—21no—
150C3×C4×C8—31no—
151C3×C4×C8C2×D311yescharacteristic
152C3×C4×C8C22×C311yescharacteristic
153#835C22×C312yes—
154#818C22×C311yescharacteristic
155#816C22×C312yes—
156#815C22×C311yescharacteristic
157C8×Dic3C22×C311yescharacteristic
158#810C22×C312yes—
159#808C22×C312yes—
160#803C22×C311yescharacteristic
161C22×C3×D3C2×D411yescharacteristic
162C32×Q8—22no—
163C32×Q8C22×C414yes—
164C2×C32×C4—22no—
165C2×C32×C4—22no—
166C2×C32×C4C2×D411yescharacteristic
167C2×C32×C4C2×D411yescharacteristic
168C2×C32×C4C2411yescharacteristic
169C2×C3×Dic3—22no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3C2×D411yescharacteristic
172C2×C3×Dic3C2×D411yescharacteristic
173C2×C3×Dic3C2×D411yescharacteristic
174C3×C4×D3—28no—
175C3×C4×D3C22×C412yes—
176C3×C4×D3#4112yes—
177C3×Dic6—22no—
178C3×Dic6—28no—
179C3×Dic6C22×C414yes—
180C32×C8—21no—
181C32×C8C22×C412yes—
182C3×C3⋊C8—21no—
183C3×C3⋊C8C22×C412yes—
184C22×Q16—31no—
185#519—32no—
186#441—32no—
187#441—32no—
188C4×Q16—31no—
189C4×Q16—31no—
190C4×Q16—31no—
191C4×Q16C3×D311yescharacteristic, complemented
192#427—31no—
193#419—32no—
194#405—31no—
195C2×C3×Q8—22no—
196C2×C3×Q8—38no—
197C2×C3×Q8—32no—
198C2×C3×Q8—62no—
199C2×C3×Q8—64no—
200C2×C3×Q8C22×D312yes—
201C2×C3×Q8C23×C312yes—
202C22×C3×C4—31no—
203C22×C3×C4—62no—
204D3×Q8—24no—
205D3×Q8C2×C3×C418yes—
206C2×C4×D3—22no—
207C2×C4×D3C23×C311yescharacteristic
208C2×Dic6—22no—
209C2×Dic6C23×C312yes—
210C3×Q16—24no—
211C3×Q16—38no—
212C3×Q16—34no—
213C3×Q16C4×D314yes—
214C3×Q16C2×C3×C414yes—
215C3×M16—62no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—31no—
219C2×C3×C8—34no—
220C2×C3×C8—31no—
221C2×C3×C8C22×D311yescharacteristic
222C2×C3×C8C22×D311yescharacteristic
223C2×C3×C8C23×C311yescharacteristic
224C2×C3×C8C23×C311yescharacteristic
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—31no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—32no—
231C3×C4⋊C4—42no—
232C3×C4⋊C4—62no—
233C3×C4⋊C4—62no—
234C3×C4⋊C4—62no—
235C3×C4⋊C4C22×D312yes—
236C3×C4⋊C4C23×C312yes—
237C3×C22⋊C4—31no—
238C3×C22⋊C4—31no—
239C3×C22⋊C4—62no—
240C3×C42—22no—
241C3×C42—22no—
242C3×C42—21no—
243C3×C42—31no—
244C3×C42—42no—
245C3×C42—62no—
246C3×C42—62no—
247C3×C42—62no—
248C3×C42—62no—
249C3×C42C22×D311yescharacteristic
250C3×C42C23×C311yescharacteristic
251#216C2×C3×C418yes—
252(C2×D3)⋊C4—22no—
253(C2×D3)⋊C4C3×D411yescharacteristic
254(C2×D3)⋊C4C3×D411yescharacteristic
255C4⋊Dic3—22no—
256C4⋊Dic3C23×C312yes—
257Dic3⋊C4—22no—
258Dic3⋊C4—22no—
259Dic3⋊C4C3×D411yescharacteristic
260Dic3⋊C4C3×D411yescharacteristic
261C4×Dic3—22no—
262C4×Dic3—22no—
263C4×Dic3—22no—
264C4×Dic3—22no—
265C4×Dic3C23×C311yescharacteristic
266C2×C3⋊C8C23×C311yescharacteristic
267C2×C3⋊C8C23×C311yescharacteristic
268Dic12C2×C3×C414yes—
269ΓL(1,𝔽25)—22no—
270D3×C8C2×C3×C414yes—
271C22×C32C22×D411yescharacteristic
272C2×C3×D3—21no—
273C2×C3×D3—21no—
274C2×C3×D3C4×D412yes—
275C32×C4—21no—
276C32×C4—24no—
277C32×C4—21no—
278C32×C4—42no—
279C32×C4C23×C411yescharacteristic
280C32×C4C2×#4111yescharacteristic
281C3×Dic3—24no—
282C3×Dic3—21no—
283C3×Dic3—21no—
284C3×Dic3—42no—
285C3×Dic3C4×D412yes—
286C22×Q8—32no—
287C2×Q16—32no—
288C2×Q16—38no—
289C2×Q16—31no—
290C2×Q16C2×C3×D311yescharacteristic
291C2×M16—31no—
292C22×C8—31no—
293C4×Q8—32no—
294C4×Q8—32no—
295C4×Q8—32no—
296C4×Q8—62no—
297C4×Q8—62no—
298C4×Q8C2×C3×D312yes—
299#117—31no—
300#117—32no—
301#117—62no—
302#117—62no—
303#107—31no—
304#107C2×C3×D311yescharacteristic
305#106—32no—
306#103—32no—
307#103—32no—
308#103—32no—
309#103C2×C3×D312yes—
310#97—31no—
311#97—31no—
312C4×C8—31no—
313C4×C8C2×C3×D311yescharacteristic
314C23×C3—31no—
315C22×D3C2×C3×D411yescharacteristic
316C3×Q8—22no—
317C3×Q8—22no—
318C3×Q8—24no—
319C3×Q8—34no—
320C3×Q8—42no—
321C3×Q8—68no—
322C3×Q8—62no—
323C3×Q8C2×C4×D314yes—
324C3×Q8C22×C3×C414yes—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—31no—
333C2×C3×C4—32no—
334C2×C3×C4—32no—
335C2×C3×C4—31no—
336C2×C3×C4—31no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—68no—
340C2×C3×C4—62no—
341C2×C3×C4—62no—
342C2×C3×C4D3×D411yescharacteristic
343C2×C3×C4D3×D411yescharacteristic
344C2×C3×C4C2×C3×D411yescharacteristic
345C2×C3×C4C2×C3×D411yescharacteristic
346C2×C3×C4C23×D311yescharacteristic
347C2×C3×C4C24×C311yescharacteristic
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3C2×C3×D411yescharacteristic
351C2×Dic3C2×C3×D411yescharacteristic
352C2×Dic3C2×C3×D411yescharacteristic
353C4×D3—28no—
354C4×D3C22×C3×C412yes—
355C4×D3C3×#4112yes—
356Dic6—22no—
357Dic6—28no—
358Dic6C22×C3×C414yes—
359C3×C8—21no—
360C3×C8—21no—
361C3×C8—22no—
362C3×C8—32no—
363C3×C8—41no—
364C3×C8—61no—
365C3×C8C2×C4×D312yes—
366C3×C8C22×C3×C412yes—
367C3⋊C8—21no—
368C3⋊C8C22×C3×C412yes—
369C2×C32C2×C4×D411yescharacteristic
370C2×C32#57811yescharacteristic
371C2×C32#57811yescharacteristic
372C3×D3—22no—
373C2×Q8—38no—
374C2×Q8—32no—
375C2×Q8—62no—
376C2×Q8—64no—
377C2×Q8C22×C3×D312yes—
378C22×C4—31no—
379C22×C4—62no—
380Q16—38no—
381Q16—34no—
382Q16C3×C4×D314yes—
383M16—62no—
384C2×C8—31no—
385C2×C8—34no—
386C2×C8—31no—
387C2×C8C22×C3×D311yescharacteristic
388C2×C8C22×C3×D311yescharacteristic
389C4⋊C4—22no—
390C4⋊C4—31no—
391C4⋊C4—31no—
392C4⋊C4—32no—
393C4⋊C4—62no—
394C4⋊C4—62no—
395C4⋊C4—62no—
396C4⋊C4C22×C3×D312yes—
397C22⋊C4—31no—
398C22⋊C4—31no—
399C22⋊C4—62no—
400C42—22no—
401C42—31no—
402C42—62no—
403C42—62no—
404C42—62no—
405C42—62no—
406C42C22×C3×D311yescharacteristic
407C22×C3—21no—
408C22×C3—32no—
409C22×C3—61no—
410C22×C3—61no—
411C22×C3C2×D3×D411yescharacteristic
412C22×C3C22×C3×D411yescharacteristic
413C2×D3—21no—
414C2×D3—21no—
415C2×D3C3×C4×D412yes—
416C3×C4—22no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—24no—
422C3×C4—24no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—32no—
426C3×C4—41no—
427C3×C4—44no—
428C3×C4—42no—
429C3×C4—42no—
430C3×C4—61no—
431C3×C4—61no—
432C3×C4—64no—
433C3×C4—82no—
434C3×C4—122no—
435C3×C4C22×C4×D311yescharacteristic
436C3×C4D3×#4111yescharacteristic
437C3×C4C23×C3×C411yescharacteristic
438C3×C4C2×C3×#4111yescharacteristic
439Dic3—21no—
440Dic3—21no—
441Dic3—24no—
442Dic3—42no—
443Dic3C3×C4×D412yes—
444C32#295011yescharacteristic, complemented, Sylow
445C23—31no—
446Q8—22no—
447Q8—34no—
448Q8—68no—
449Q8—62no—
450Q8C2×C3×C4×D314yes—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—31no—
454C2×C4—32no—
455C2×C4—32no—
456C2×C4—31no—
457C2×C4—31no—
458C2×C4—68no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4C3×D3×D411yescharacteristic
462C2×C4C3×D3×D411yescharacteristic
463C2×C4C23×C3×D311yescharacteristic
464C8—21no—
465C8—32no—
466C8—61no—
467C8C2×C3×C4×D312yes—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—62no—
472C2×C3C4×D3×D411yescharacteristic
473C2×C3#576611yescharacteristic
474C2×C3#577111yescharacteristic
475C2×C3C2×C3×C4×D411yescharacteristic
476C2×C3C3×#57811yescharacteristic
477C2×C3C3×#57811yescharacteristic
478D3—22no—
479C22—32no—
480C22—61no—
481C22—61no—
482C22C2×C3×D3×D411yescharacteristic
483C4—21no—
484C4—21no—
485C4—24no—
486C4—32no—
487C4—42no—
488C4—61no—
489C4—61no—
490C4—64no—
491C4—122no—
492C4C22×C3×C4×D311yescharacteristic
493C4C3×D3×#4111yescharacteristic
494C3—21no—
495C3#7972011yescharacteristic, direct product factor
496C3C3×#295011yescharacteristic, complemented
497C2—62no—
498C2C3×C4×D3×D411yescharacteristic
499C2C3×#576611yescharacteristic
500C2C3×#577111yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC32×C4×Q16C2
socleC22×C32C22×D4
prime radicalC22×C3×D3C2×D4
cosocleC2×C3×C4C24×C3
hypercenterC3×C4×Q16D3
hypocenterC3C3×#2950
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×Q16

Complements

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

NH ≅ G/Ntype
3. C32×C4×Q16C2semidirect
4. C3×Dic3×Q16C2semidirect
5. C3×#5030C2semidirect
7. C3×#4873C2semidirect
9. C3×#4789C2semidirect
10. C3×C4×Dic12C2semidirect
13. #79720C3direct (N × H)
19. C3×D3×Q16C4semidirect
52. C3×C4×Q16D3semidirect
53. C3×C4×Q16C2×C3semidirect
57. Dic3×Q16C2×C3semidirect
58. #5030C2×C3semidirect
60. #4873C2×C3semidirect
62. #4789C2×C3semidirect
63. C4×Dic12C2×C3semidirect
124. D3×Q16C3×C4semidirect
191. C4×Q16C3×D3semidirect
444. C32#2950semidirect
495. C3#79720direct (N × H)
496. C3C3×#2950semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×Q162
C3×#55722
C32×C4×Q162
C3×Dic3×Q162
C3×#50302
C3×#48802
C3×#48732
C3×#48022
C3×#47892
C3×C4×Dic122
C3×#46892
C3×#29503
#797203