Finite Group Databasebeta

← C3×#78656 #11859402

Subgroups — C3×#78656 #11859402

Order 1152.

1063 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5828C211yesmaximal, characteristic
2C2×C3×#948C211yesmaximal, characteristic, complemented
3C22×C3×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
4C32×#408C211yesmaximal, characteristic, complemented
5C3×#5126C211yesmaximal, characteristic, complemented
6C3×#5122C211yesmaximal, characteristic, complemented
7C3×#5102C211yesmaximal, characteristic, complemented
8C3×#4730C214yesmaximal, complemented
9C3×#4729C214yesmaximal, complemented
10C3×#2930—31nomaximal
11#78656C311yesmaximal, characteristic, direct product factor
12C2×C32×#41C2211yescharacteristic
13C22×C3×C4×D3C2211yescharacteristic
14C2×C32×M16C2211yescharacteristic
15C22×C32×C8C2211yescharacteristic
16C3×#948—24no—
17C3×#946C412yes—
18C3×#938C412yes—
19C3×#937C412yes—
20C3×#935C412yes—
21C3×C4×#67C2214yes—
22C3×#926C2211yescharacteristic
23C2×C3×#208C2211yescharacteristic
24C2×C3×#208C2212yescomplemented
25C22×C3×C3⋊C8C2211yescharacteristic
26C22×C3×C3⋊C8C2212yescomplemented
27C2×C3×ΓL(1,𝔽25)—24no—
28C2×C3×ΓL(1,𝔽25)C2212yes—
29C2×C3×ΓL(1,𝔽25)C2212yes—
30C32×C22⋊C8C2214yescomplemented
31C3×(C2×D3)⋊C8C2214yes—
32C3×#815C2212yescomplemented
33C3×Dic3⋊C8C2214yescomplemented
34C3×C8×Dic3C2212yescomplemented
35C2×C3×#131—31no—
36C22×C3×M16—31no—
37C3×#525—31no—
38#5828C2×C311yescharacteristic
39C2×#948C2×C311yescharacteristic, complemented
40C22×ΓL(1,𝔽25)C2×C311yescharacteristic, complemented
41C3×#436—34no—
42C3×#435—34no—
43C3×#424—31no—
44C3×#408—21no—
45C3×#408D311yescharacteristic, complemented
46C3×#408C2×C311yescharacteristic, complemented
47C3×#407—31no—
48C3×#405—31no—
49#5126C2×C311yescharacteristic, complemented
50#5122C2×C311yescharacteristic, complemented
51#5102C2×C311yescharacteristic, complemented
52#4730C2×C314yescomplemented
53#4729C2×C314yescomplemented
54C23×C3×D3C2×C411yescharacteristic
55C32×#41—22no—
56C32×#41—22no—
57C2×C32×Q8C2×C411yescharacteristic
58C2×C32×D4C2×C412yes—
59C2×C32×D4C2×C411yescharacteristic
60C22×C32×C4C2312yes—
61C22×C32×C4C2311yescharacteristic
62C2×C3×#67C2×C414yes—
63C22×C3×Dic3C2×C411yescharacteristic
64C2×C3×C4×D3—22no—
65C2×C3×C4×D3—22no—
66C2×C3×C4×D3C2×C412yes—
67C2×C3×C4×D3C2312yes—
68C32×M16—22no—
69C2×C32×C8—22no—
70C2×C32×C8C2312yes—
71C2×C32×C8C2312yes—
72C3×#217C2×C412yes—
73C3×#217C2×C414yes—
74C3×(C2×D3)⋊C4C2×C414yes—
75C3×C4⋊Dic3C2×C412yes—
76C3×Dic3⋊C4C2×C414yes—
77C3×C4×Dic3C2312yes—
78C3×#208—22no—
79C3×#208—24no—
80C2×C3×C3⋊C8—24no—
81C2×C3×C3⋊C8D414yescomplemented
82C2×C3×C3⋊C8C2312yes—
83C2×C3×C3⋊C8C2311yescharacteristic
84C2×C3×C3⋊C8C2311yescharacteristic
85C3×ΓL(1,𝔽25)—24no—
86C3×ΓL(1,𝔽25)—24no—
87#2930—31noSylow
88C2×C3×#41—21no—
89C2×C3×#41C2×D311yescharacteristic
90C2×C3×#41C22×C311yescharacteristic
91C23×C3×C4—31no—
92C22×C4×D3C22×C311yescharacteristic
93C3×#131—64no—
94C2×C3×M16—21no—
95C2×C3×M16—31no—
96C2×C3×M16—32no—
97C2×C3×M16—32no—
98C2×C3×M16—32no—
99C2×C3×M16—64no—
100C2×C3×M16C2×D311yescharacteristic
101C2×C3×M16C22×C311yescharacteristic
102C22×C3×C8—21no—
103C22×C3×C8—31no—
104C22×C3×C8—32no—
105C22×C3×C8C2×D311yescharacteristic
106C22×C3×C8C22×C311yescharacteristic
107C3×#123—32no—
108C3×C22⋊Q8—32no—
109C3×#121—32no—
110C3×#120—32no—
111C3×C4×D4—34no—
112C3×#117—31no—
113#948—24no—
114#946C3×C412yes—
115#938C3×C412yes—
116#937C3×C412yes—
117#935C3×C412yes—
118C4×#67C22×C314yes—
119#926C22×C311yescharacteristic
120C2×#208C22×C311yescharacteristic
121C2×#208C22×C312yescomplemented
122C22×C3⋊C8C22×C311yescharacteristic
123C22×C3⋊C8C22×C312yescomplemented
124C2×ΓL(1,𝔽25)—24no—
125C2×ΓL(1,𝔽25)C22×C312yes—
126C2×ΓL(1,𝔽25)C22×C312yes—
127C3×C4⋊C8—34no—
128C3×C22⋊C8—24no—
129C3×C22⋊C8—34no—
130C3×C22⋊C8C2×D314yescomplemented
131C3×C22⋊C8C22×C314yescomplemented
132C3×#97—32no—
133C3×C4×C8—32no—
134(C2×D3)⋊C8C22×C314yes—
135#815C22×C312yescomplemented
136Dic3⋊C8C22×C314yescomplemented
137C8×Dic3C22×C312yescomplemented
138C23×C32C22×C412yes—
139C23×C32C22×C411yescharacteristic
140C22×C3×D3—22no—
141C22×C3×D3—42no—
142C22×C3×D3C22×C412yes—
143C32×Q8—22no—
144C32×D4—24no—
145C32×D4—22no—
146C2×C32×C4—22no—
147C2×C32×C4—22no—
148C2×C32×C4C22×C412yes—
149C2×C32×C4C22×C411yescharacteristic
150C2×C32×C4C2×D412yes—
151C2×C32×C4#4112yes—
152C2×C32×C4C2411yescharacteristic
153C3×#67—44no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3C22×C414yes—
156C2×C3×Dic3C22×C412yes—
157C3×C4×D3—24no—
158C3×C4×D3—22no—
159C3×C4×D3—22no—
160C32×C8—22no—
161C32×C8—22no—
162C3×C3⋊C8—22no—
163C3×C3⋊C8C2×D412yes—
164C3×C3⋊C8C2×D412yes—
165C2×#131—31no—
166C22×M16—31no—
167#525—31no—
168#436—34no—
169#435—34no—
170#424—31no—
171#408C3×D311yescharacteristic, complemented
172#407—31no—
173#405—31no—
174C24×C3—31no—
175C23×D3C2×C3×C411yescharacteristic
176C3×#41—22no—
177C3×#41—22no—
178C3×#41—22no—
179C3×#41—22no—
180C3×#41—42no—
181C3×#41—42no—
182C2×C3×Q8—21no—
183C2×C3×Q8C4×D311yescharacteristic
184C2×C3×Q8C2×C3×C411yescharacteristic
185C2×C3×D4—22no—
186C2×C3×D4—21no—
187C2×C3×D4—34no—
188C2×C3×D4C4×D312yes—
189C2×C3×D4C4×D311yescharacteristic
190C2×C3×D4C2×C3×C412yes—
191C2×C3×D4C2×C3×C411yescharacteristic
192C22×C3×C4—21no—
193C22×C3×C4—22no—
194C22×C3×C4—31no—
195C22×C3×C4—32no—
196C22×C3×C4—32no—
197C22×C3×C4—62no—
198C22×C3×C4—62no—
199C22×C3×C4C22×D312yes—
200C22×C3×C4C22×D311yescharacteristic
201C22×C3×C4C23×C312yes—
202C22×C3×C4C23×C311yescharacteristic
203C2×#67C2×C3×C414yes—
204C22×Dic3C2×C3×C411yescharacteristic
205C2×C4×D3—22no—
206C2×C4×D3—22no—
207C2×C4×D3C2×C3×C412yes—
208C2×C4×D3C23×C312yes—
209C3×M16—22no—
210C3×M16—22no—
211C3×M16—42no—
212C3×M16—64no—
213C3×M16—64no—
214C3×M16—64no—
215C3×M16—62no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—34no—
221C2×C3×C8—31no—
222C2×C3×C8—31no—
223C2×C3×C8—32no—
224C2×C3×C8—42no—
225C2×C3×C8—64no—
226C2×C3×C8C22×D312yes—
227C2×C3×C8C22×D312yes—
228C2×C3×C8C23×C312yes—
229C2×C3×C8C23×C312yes—
230C3×C4⋊C4—34no—
231C3×C4⋊C4—32no—
232C3×C22⋊C4—32no—
233C3×C22⋊C4—34no—
234C3×C22⋊C4—34no—
235C3×C42—32no—
236#217C2×C3×C412yes—
237#217C2×C3×C414yes—
238(C2×D3)⋊C4C2×C3×C414yes—
239C4⋊Dic3C2×C3×C412yes—
240Dic3⋊C4C2×C3×C414yes—
241C4×Dic3C23×C312yes—
242#208—22no—
243#208—24no—
244C2×C3⋊C8—24no—
245C2×C3⋊C8C3×D414yescomplemented
246C2×C3⋊C8C23×C312yes—
247C2×C3⋊C8C23×C311yescharacteristic
248C2×C3⋊C8C23×C311yescharacteristic
249ΓL(1,𝔽25)—24no—
250ΓL(1,𝔽25)—24no—
251C22×C32—22no—
252C22×C32—22no—
253C22×C32—24no—
254C22×C32C4×D412yes—
255C22×C32C23×C411yescharacteristic
256C2×C3×D3—22no—
257C2×C3×D3—22no—
258C2×C3×D3—22no—
259C2×C3×D3—44no—
260C32×C4—22no—
261C32×C4C4×D412yes—
262C32×C4C22×D411yescharacteristic
263C32×C4C2×#4111yescharacteristic
264C3×Dic3—22no—
265C3×Dic3—24no—
266C2×#41C2×C3×D311yescharacteristic
267C23×C4—31no—
268#131—64no—
269C2×M16—31no—
270C2×M16—32no—
271C2×M16—32no—
272C2×M16—32no—
273C2×M16—64no—
274C2×M16C2×C3×D311yescharacteristic
275C22×C8—31no—
276C22×C8—32no—
277C22×C8C2×C3×D311yescharacteristic
278#123—32no—
279C22⋊Q8—32no—
280#121—32no—
281#120—32no—
282C4×D4—34no—
283#117—31no—
284C4⋊C8—34no—
285C22⋊C8—34no—
286C22⋊C8C2×C3×D314yescomplemented
287#97—32no—
288C4×C8—32no—
289C23×C3—21no—
290C23×C3—22no—
291C23×C3—32no—
292C23×C3—62no—
293C23×C3—122no—
294C23×C3C2×C4×D312yes—
295C23×C3C2×C4×D311yescharacteristic
296C23×C3C22×C3×C412yes—
297C23×C3C22×C3×C411yescharacteristic
298C22×D3—22no—
299C22×D3—42no—
300C22×D3C22×C3×C412yes—
301C3×Q8—22no—
302C3×Q8—22no—
303C3×Q8—42no—
304C3×D4—24no—
305C3×D4—24no—
306C3×D4—22no—
307C3×D4—22no—
308C3×D4—42no—
309C3×D4—44no—
310C3×D4—124no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—34no—
321C2×C3×C4—32no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—62no—
325C2×C3×C4—64no—
326C2×C3×C4—62no—
327C2×C3×C4—62no—
328C2×C3×C4C2×C4×D312yes—
329C2×C3×C4C2×C4×D311yescharacteristic
330C2×C3×C4D3×D412yes—
331C2×C3×C4#23712yes—
332C2×C3×C4C22×C3×C412yes—
333C2×C3×C4C22×C3×C411yescharacteristic
334C2×C3×C4C2×C3×D412yes—
335C2×C3×C4C3×#4112yes—
336C2×C3×C4C23×D311yescharacteristic
337C2×C3×C4C24×C311yescharacteristic
338#67—44no—
339C2×Dic3—22no—
340C2×Dic3C22×C3×C414yes—
341C2×Dic3C22×C3×C412yes—
342C4×D3—24no—
343C4×D3—22no—
344C4×D3—22no—
345C3×C8—22no—
346C3×C8—22no—
347C3×C8—22no—
348C3×C8—22no—
349C3×C8—32no—
350C3×C8—32no—
351C3×C8—42no—
352C3×C8—42no—
353C3×C8—62no—
354C3⋊C8—22no—
355C3⋊C8C2×C3×D412yes—
356C3⋊C8C2×C3×D412yes—
357C2×C32—22no—
358C2×C32—42no—
359C2×C32C2×C4×D411yescharacteristic
360C2×C32#56812yes—
361C3×D3—42no—
362C24—31no—
363#41—22no—
364#41—22no—
365C2×Q8C3×C4×D311yescharacteristic
366C2×D4—34no—
367C2×D4C3×C4×D312yes—
368C2×D4C3×C4×D311yescharacteristic
369C22×C4—31no—
370C22×C4—32no—
371C22×C4—32no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4C22×C3×D312yes—
375C22×C4C22×C3×D311yescharacteristic
376M16—22no—
377M16—64no—
378M16—64no—
379M16—64no—
380M16—62no—
381C2×C8—22no—
382C2×C8—31no—
383C2×C8—31no—
384C2×C8—34no—
385C2×C8—32no—
386C2×C8—64no—
387C2×C8C22×C3×D312yes—
388C2×C8C22×C3×D312yes—
389C4⋊C4—32no—
390C4⋊C4—34no—
391C22⋊C4—32no—
392C22⋊C4—34no—
393C22⋊C4—34no—
394C42—32no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—24no—
398C22×C3—22no—
399C22×C3—24no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—21no—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3—44no—
406C22×C3—62no—
407C22×C3—62no—
408C22×C3—62no—
409C22×C3—124no—
410C22×C3#67⋊C412yes—
411C22×C3C3×C4×D412yes—
412C22×C3C22×C4×D311yescharacteristic
413C22×C3C23×C3×C411yescharacteristic
414C2×D3—22no—
415C2×D3—22no—
416C2×D3—22no—
417C2×D3—44no—
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—22no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—42no—
424C3×C4—62no—
425C3×C4—64no—
426C3×C4#67⋊C412yes—
427C3×C4C3×C4×D412yes—
428C3×C4C2×D3×D411yescharacteristic
429C3×C4C2×#23711yescharacteristic
430C3×C4C22×C3×D411yescharacteristic
431C3×C4C2×C3×#4111yescharacteristic
432Dic3—24no—
433Dic3—22no—
434C32#293011yescharacteristic, complemented, Sylow
435C23—32no—
436C23—62no—
437C23—122no—
438C23C2×C3×C4×D312yes—
439C23C2×C3×C4×D311yescharacteristic
440Q8—22no—
441D4—24no—
442D4—22no—
443D4—124no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—34no—
447C2×C4—32no—
448C2×C4—62no—
449C2×C4—64no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4C2×C3×C4×D312yes—
453C2×C4C2×C3×C4×D311yescharacteristic
454C2×C4C3×D3×D412yes—
455C2×C4C3×#23712yes—
456C2×C4C23×C3×D311yescharacteristic
457C8—22no—
458C8—22no—
459C8—32no—
460C8—32no—
461C8—62no—
462C2×C3—22no—
463C2×C3—22no—
464C2×C3—22no—
465C2×C3—21no—
466C2×C3—42no—
467C2×C3—42no—
468C2×C3—42no—
469C2×C3—82no—
470C2×C3—122no—
471C2×C3C2×#67⋊C411yescharacteristic
472C2×C3#575012yes—
473C2×C3C2×C3×C4×D411yescharacteristic
474C2×C3C3×#56812yes—
475D3—42no—
476C22—22no—
477C22—24no—
478C22—22no—
479C22—62no—
480C22—62no—
481C22—62no—
482C22—124no—
483C22C3×#67⋊C412yes—
484C22C22×C3×C4×D311yescharacteristic
485C4—22no—
486C4—64no—
487C4—62no—
488C4C3×#67⋊C412yes—
489C4C2×C3×D3×D411yescharacteristic
490C4C2×C3×#23711yescharacteristic
491C3—21no—
492C3#7865611yescharacteristic, direct product factor
493C3C3×#293011yescharacteristic, complemented
494C2—22no—
495C2—42no—
496C2—122no—
497C2C2×C3×#67⋊C411yescharacteristic
498C2C3×#575012yes—
subgroupG/H
centerC2×C3×C4C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC32×#408C2
socleC22×C32C23×C4
prime radicalC3×#5828C2
cosocleC2×C3×C4C24×C3
hypercenterC3×#408D3
hypocenterC3C3×#2930
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C3×#408

Complements

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

NH ≅ G/Ntype
2. C2×C3×#948C2semidirect
3. C22×C3×ΓL(1,𝔽25)C2semidirect
4. C32×#408C2semidirect
5. C3×#5126C2semidirect
6. C3×#5122C2semidirect
7. C3×#5102C2semidirect
8. C3×#4730C2semidirect
9. C3×#4729C2semidirect
11. #78656C3direct (N × H)
24. C2×C3×#208C22semidirect
26. C22×C3×C3⋊C8C22semidirect
30. C32×C22⋊C8C22semidirect
32. C3×#815C22semidirect
33. C3×Dic3⋊C8C22semidirect
34. C3×C8×Dic3C22semidirect
39. C2×#948C2×C3semidirect
40. C22×ΓL(1,𝔽25)C2×C3semidirect
45. C3×#408D3semidirect
46. C3×#408C2×C3semidirect
49. #5126C2×C3semidirect
50. #5122C2×C3semidirect
51. #5102C2×C3semidirect
52. #4730C2×C3semidirect
53. #4729C2×C3semidirect
81. C2×C3×C3⋊C8D4semidirect
121. C2×#208C22×C3semidirect
123. C22×C3⋊C8C22×C3semidirect
130. C3×C22⋊C8C2×D3semidirect
131. C3×C22⋊C8C22×C3semidirect
135. #815C22×C3semidirect
136. Dic3⋊C8C22×C3semidirect
137. C8×Dic3C22×C3semidirect
171. #408C3×D3semidirect
245. C2×C3⋊C8C3×D4semidirect
286. C22⋊C8C2×C3×D3semidirect
434. C32#2930semidirect
492. C3#78656direct (N × H)
493. C3C3×#2930semidirect

Maximal subgroups

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

subgroupindex
C3×#58282
C2×C3×#9482
C22×C3×ΓL(1,𝔽25)2
C32×#4082
C3×#51262
C3×#51222
C3×#51022
C3×#47302
C3×#47292
C3×#29303
#786563