Finite Group Databasebeta

← C3×#78627 #11859381

Subgroups — C3×#78627 #11859381

Order 1152.

1079 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5791C211yesmaximal, characteristic
2C22×C3×#208C211yesmaximal, characteristic, complemented
3C2×C3×#907C211yesmaximal, characteristic, complemented
4C32×#407C211yesmaximal, characteristic, complemented
5C3×#5129C211yesmaximal, characteristic, complemented
6C3×#5122C211yesmaximal, characteristic, complemented
7C3×#5119C211yesmaximal, characteristic, complemented
8C3×#4729C214yesmaximal, complemented
9C3×#4728C214yesmaximal, complemented
10C3×#2930—31nomaximal
11#78627C311yesmaximal, characteristic, direct product factor
12C23×C32×C4C2211yescharacteristic
13C2×C3×#235C2211yescharacteristic
14C2×C32×M16C2211yescharacteristic
15C2×C32×M16C2211yescharacteristic
16C3×#953C412yes—
17C3×#930C412yes—
18C3×#929C412yes—
19C3×C4×#67C2214yes—
20C3×#926C2211yescharacteristic
21C3×#924C412yes—
22C2×C3×#208—24no—
23C2×C3×#208C2212yes—
24C2×C3×#208C2212yes—
25C22×C3×C3⋊C8C2211yescharacteristic
26C22×C3×C3⋊C8C2211yescharacteristic
27C3×#907—24no—
28C2×C3×ΓL(1,𝔽25)C2212yescomplemented
29C2×C3×D3×C8C2212yescomplemented
30C32×C22⋊C8C2214yes—
31C3×(C2×D3)⋊C8C2214yescomplemented
32C3×#815C2212yescomplemented
33C3×Dic3⋊C8C2214yescomplemented
34C3×C8×Dic3C2212yescomplemented
35C2×C3×#131—31no—
36C22×C3×M16—31no—
37C3×#525—31no—
38#5791C2×C311yescharacteristic
39C22×#208C2×C311yescharacteristic, complemented
40C2×#907C2×C311yescharacteristic, complemented
41C3×#436—34no—
42C3×#435—34no—
43C3×#424—31no—
44C3×#408—31no—
45C3×#407—21no—
46C3×#407D311yescharacteristic, complemented
47C3×#407C2×C311yescharacteristic, complemented
48C3×#405—31no—
49#5129C2×C311yescharacteristic, complemented
50#5122C2×C311yescharacteristic, complemented
51#5119C2×C311yescharacteristic, complemented
52#4729C2×C314yescomplemented
53#4728C2×C314yescomplemented
54C24×C32C2×C411yescharacteristic
55C22×C32×C4—22no—
56C22×C32×C4—22no—
57C22×C32×C4C2×C412yes—
58C22×C32×C4C2×C411yescharacteristic
59C22×C32×C4C2312yes—
60C22×C32×C4C2311yescharacteristic
61C2×C3×#67C2×C412yes—
62C2×C3×#67C2×C414yes—
63C3×#235—22no—
64C3×#235—22no—
65C2×C3×D12C2×C411yescharacteristic
66C2×C3×C4×D3C2312yes—
67C2×C3×Dic6C2×C411yescharacteristic
68C32×M16—22no—
69C32×M16—22no—
70C2×C32×C8C2312yes—
71C2×C32×C8C2312yes—
72C3×#217C2×C414yes—
73C3×#217C2×C412yes—
74C3×(C2×D3)⋊C4C2×C414yes—
75C3×C4⋊Dic3C2×C412yes—
76C3×Dic3⋊C4C2×C414yes—
77C3×C4×Dic3C2312yes—
78C3×#208—24no—
79C3×#208—24no—
80C2×C3×C3⋊C8—22no—
81C2×C3×C3⋊C8D414yescomplemented
82C2×C3×C3⋊C8C2312yes—
83C2×C3×C3⋊C8C2311yescharacteristic
84C2×C3×C3⋊C8C2311yescharacteristic
85C3×ΓL(1,𝔽25)—24no—
86C3×D3×C8—24no—
87#2930—31noSylow
88C2×C3×#41—31no—
89C23×C3×C4—21no—
90C23×C3×C4C2×D311yescharacteristic
91C23×C3×C4C22×C311yescharacteristic
92C2×#235C22×C311yescharacteristic
93C3×#131—64no—
94C2×C3×M16—21no—
95C2×C3×M16—21no—
96C2×C3×M16—32no—
97C2×C3×M16—32no—
98C2×C3×M16—32no—
99C2×C3×M16—64no—
100C2×C3×M16C2×D311yescharacteristic
101C2×C3×M16C2×D311yescharacteristic
102C2×C3×M16C22×C311yescharacteristic
103C2×C3×M16C22×C311yescharacteristic
104C22×C3×C8—31no—
105C22×C3×C8—31no—
106C22×C3×C8—32no—
107C3×#123—32no—
108C3×C22⋊Q8—32no—
109C3×#121—32no—
110C3×#120—32no—
111C3×C4×D4—34no—
112C3×#117—31no—
113#953C3×C412yes—
114#930C3×C412yes—
115#929C3×C412yes—
116C4×#67C22×C314yes—
117#926C22×C311yescharacteristic
118#924C3×C412yes—
119C2×#208—24no—
120C2×#208C22×C312yes—
121C2×#208C22×C312yes—
122C22×C3⋊C8C22×C311yescharacteristic
123C22×C3⋊C8C22×C311yescharacteristic
124#907—24no—
125C2×ΓL(1,𝔽25)C22×C312yescomplemented
126C2×D3×C8C22×C312yescomplemented
127C3×C4⋊C8—34no—
128C3×C22⋊C8—24no—
129C3×C22⋊C8—34no—
130C3×C22⋊C8C2×D314yes—
131C3×C22⋊C8C22×C314yes—
132C3×#97—32no—
133C3×C4×C8—32no—
134(C2×D3)⋊C8C22×C314yescomplemented
135#815C22×C312yescomplemented
136Dic3⋊C8C22×C314yescomplemented
137C8×Dic3C22×C312yescomplemented
138C23×C32—22no—
139C23×C32—42no—
140C23×C32C22×C412yes—
141C23×C32C22×C411yescharacteristic
142C22×C3×D3C22×C412yes—
143C2×C32×C4—22no—
144C2×C32×C4—24no—
145C2×C32×C4—22no—
146C2×C32×C4—22no—
147C2×C32×C4C22×C412yes—
148C2×C32×C4C22×C411yescharacteristic
149C2×C32×C4C2×D412yes—
150C2×C32×C4#4112yes—
151C2×C32×C4C2411yescharacteristic
152C3×#67—24no—
153C3×#67—44no—
154C2×C3×Dic3C22×C414yes—
155C2×C3×Dic3C22×C412yes—
156C3×D12—22no—
157C3×C4×D3—22no—
158C3×C4×D3—22no—
159C3×Dic6—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#408—31no—
172#407C3×D311yescharacteristic, complemented
173#405—31no—
174C24×C3—21no—
175C24×C3C4×D311yescharacteristic
176C24×C3C2×C3×C411yescharacteristic
177C3×#41—62no—
178C3×#41—62no—
179C2×C3×Q8—31no—
180C2×C3×D4—32no—
181C2×C3×D4—34no—
182C2×C3×D4—31no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—32no—
192C22×C3×C4—42no—
193C22×C3×C4—42no—
194C22×C3×C4C4×D312yes—
195C22×C3×C4C4×D311yescharacteristic
196C22×C3×C4C2×C3×C412yes—
197C22×C3×C4C2×C3×C411yescharacteristic
198C22×C3×C4C22×D312yes—
199C22×C3×C4C22×D311yescharacteristic
200C22×C3×C4C23×C312yes—
201C22×C3×C4C23×C311yescharacteristic
202C2×#67C2×C3×C412yes—
203C2×#67C2×C3×C414yes—
204#235—22no—
205#235—22no—
206C2×D12C2×C3×C411yescharacteristic
207C2×C4×D3C23×C312yes—
208C2×Dic6C2×C3×C411yescharacteristic
209C3×M16—22no—
210C3×M16—22no—
211C3×M16—22no—
212C3×M16—22no—
213C3×M16—42no—
214C3×M16—42no—
215C3×M16—64no—
216C3×M16—64no—
217C3×M16—64no—
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—64no—
225C2×C3×C8—62no—
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×C414yes—
237#217C2×C3×C412yes—
238(C2×D3)⋊C4C2×C3×C414yes—
239C4⋊Dic3C2×C3×C412yes—
240Dic3⋊C4C2×C3×C414yes—
241C4×Dic3C23×C312yes—
242#208—24no—
243#208—24no—
244C2×C3⋊C8—22no—
245C2×C3⋊C8C3×D414yescomplemented
246C2×C3⋊C8C23×C312yes—
247C2×C3⋊C8C23×C311yescharacteristic
248C2×C3⋊C8C23×C311yescharacteristic
249ΓL(1,𝔽25)—24no—
250D3×C8—24no—
251C22×C32—22no—
252C22×C32—22no—
253C22×C32—22no—
254C22×C32—22no—
255C22×C32—44no—
256C22×C32C4×D412yes—
257C22×C32C23×C411yescharacteristic
258C2×C3×D3—22no—
259C2×C3×D3—24no—
260C32×C4—22no—
261C32×C4C4×D412yes—
262C32×C4C22×D411yescharacteristic
263C32×C4C2×#4111yescharacteristic
264C3×Dic3—22no—
265C3×Dic3—24no—
266C2×#41—31no—
267C23×C4C2×C3×D311yescharacteristic
268#131—64no—
269C2×M16—32no—
270C2×M16—32no—
271C2×M16—32no—
272C2×M16—64no—
273C2×M16C2×C3×D311yescharacteristic
274C2×M16C2×C3×D311yescharacteristic
275C22×C8—31no—
276C22×C8—31no—
277C22×C8—32no—
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×D314yes—
287#97—32no—
288C4×C8—32no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—21no—
293C23×C3—32no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—44no—
298C23×C3C2×C4×D312yes—
299C23×C3C2×C4×D311yescharacteristic
300C23×C3C22×C3×C412yes—
301C23×C3C22×C3×C411yescharacteristic
302C22×D3C22×C3×C412yes—
303C3×Q8—62no—
304C3×D4—64no—
305C3×D4—62no—
306C3×D4—124no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
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—44no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
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—24no—
339#67—44no—
340C2×Dic3C22×C3×C414yes—
341C2×Dic3C22×C3×C412yes—
342D12—22no—
343C4×D3—22no—
344C4×D3—22no—
345Dic6—22no—
346C3×C8—22no—
347C3×C8—22no—
348C3×C8—22no—
349C3×C8—22no—
350C3×C8—32no—
351C3×C8—32no—
352C3×C8—42no—
353C3×C8—42no—
354C3×C8—62no—
355C3⋊C8—22no—
356C3⋊C8C2×C3×D412yes—
357C3⋊C8C2×C3×D412yes—
358C2×C32—22no—
359C2×C32—42no—
360C2×C32C2×C4×D411yescharacteristic
361C2×C32#56812yes—
362C3×D3—42no—
363C24C3×C4×D311yescharacteristic
364#41—62no—
365#41—62no—
366C2×Q8—31no—
367C2×D4—32no—
368C2×D4—34no—
369C2×D4—31no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—32no—
373C22×C4C3×C4×D312yes—
374C22×C4C3×C4×D311yescharacteristic
375C22×C4C22×C3×D312yes—
376C22×C4C22×C3×D311yescharacteristic
377M16—22no—
378M16—22no—
379M16—64no—
380M16—64no—
381M16—64no—
382C2×C8—31no—
383C2×C8—31no—
384C2×C8—34no—
385C2×C8—32no—
386C2×C8—64no—
387C2×C8—62no—
388C2×C8C22×C3×D312yes—
389C2×C8C22×C3×D312yes—
390C4⋊C4—32no—
391C4⋊C4—34no—
392C22⋊C4—32no—
393C22⋊C4—34no—
394C22⋊C4—34no—
395C42—32no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—24no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—21no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—44no—
411C22×C3—44no—
412C22×C3—48no—
413C22×C3—62no—
414C22×C3—64no—
415C22×C3#67⋊C412yes—
416C22×C3C3×C4×D412yes—
417C22×C3C22×C4×D311yescharacteristic
418C22×C3C23×C3×C411yescharacteristic
419C2×D3—24no—
420C2×D3—22no—
421C3×C4—22no—
422C3×C4—22no—
423C3×C4—22no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—42no—
427C3×C4—62no—
428C3×C4—64no—
429C3×C4#67⋊C412yes—
430C3×C4C3×C4×D412yes—
431C3×C4C2×D3×D411yescharacteristic
432C3×C4C2×#23711yescharacteristic
433C3×C4C22×C3×D411yescharacteristic
434C3×C4C2×C3×#4111yescharacteristic
435Dic3—24no—
436Dic3—22no—
437C32#293011yescharacteristic, complemented, Sylow
438C23—22no—
439C23—32no—
440C23—42no—
441C23C2×C3×C4×D312yes—
442C23C2×C3×C4×D311yescharacteristic
443Q8—62no—
444D4—64no—
445D4—62no—
446D4—124no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—34no—
452C2×C4—32no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4C2×C3×C4×D312yes—
456C2×C4C2×C3×C4×D311yescharacteristic
457C2×C4C3×D3×D412yes—
458C2×C4C3×#23712yes—
459C2×C4C23×C3×D311yescharacteristic
460C8—22no—
461C8—22no—
462C8—32no—
463C8—32no—
464C8—62no—
465C2×C3—22no—
466C2×C3—22no—
467C2×C3—22no—
468C2×C3—21no—
469C2×C3—42no—
470C2×C3—44no—
471C2×C3—42no—
472C2×C3—42no—
473C2×C3—122no—
474C2×C3C2×#67⋊C411yescharacteristic
475C2×C3#574512yes—
476C2×C3C2×C3×C4×D411yescharacteristic
477C2×C3C3×#56812yes—
478D3—42no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—44no—
484C22—64no—
485C22—62no—
486C22C3×#67⋊C412yes—
487C22C22×C3×C4×D311yescharacteristic
488C4—22no—
489C4—64no—
490C4—62no—
491C4C3×#67⋊C412yes—
492C4C2×C3×D3×D411yescharacteristic
493C4C2×C3×#23711yescharacteristic
494C3—21no—
495C3#7862711yescharacteristic, direct product factor
496C3C3×#293011yescharacteristic, complemented
497C2—22no—
498C2—42no—
499C2—122no—
500C2C2×C3×#67⋊C411yescharacteristic
501C2C3×#574512yes—
subgroupG/H
centerC2×C3×C4C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC32×#407C2
socleC22×C32C23×C4
prime radicalC3×#5791C2
cosocleC2×C3×C4C24×C3
hypercenterC3×#407D3
hypocenterC3C3×#2930
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C3×#407

Complements

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

NH ≅ G/Ntype
2. C22×C3×#208C2semidirect
3. C2×C3×#907C2semidirect
4. C32×#407C2semidirect
5. C3×#5129C2semidirect
6. C3×#5122C2semidirect
7. C3×#5119C2semidirect
8. C3×#4729C2semidirect
9. C3×#4728C2semidirect
11. #78627C3direct (N × H)
28. C2×C3×ΓL(1,𝔽25)C22semidirect
29. C2×C3×D3×C8C22semidirect
31. C3×(C2×D3)⋊C8C22semidirect
32. C3×#815C22semidirect
33. C3×Dic3⋊C8C22semidirect
34. C3×C8×Dic3C22semidirect
39. C22×#208C2×C3semidirect
40. C2×#907C2×C3semidirect
46. C3×#407D3semidirect
47. C3×#407C2×C3semidirect
49. #5129C2×C3semidirect
50. #5122C2×C3semidirect
51. #5119C2×C3semidirect
52. #4729C2×C3semidirect
53. #4728C2×C3semidirect
81. C2×C3×C3⋊C8D4semidirect
125. C2×ΓL(1,𝔽25)C22×C3semidirect
126. C2×D3×C8C22×C3semidirect
134. (C2×D3)⋊C8C22×C3semidirect
135. #815C22×C3semidirect
136. Dic3⋊C8C22×C3semidirect
137. C8×Dic3C22×C3semidirect
172. #407C3×D3semidirect
245. C2×C3⋊C8C3×D4semidirect
437. C32#2930semidirect
495. C3#78627direct (N × H)
496. C3C3×#2930semidirect

Maximal subgroups

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

subgroupindex
C3×#57912
C22×C3×#2082
C2×C3×#9072
C32×#4072
C3×#51292
C3×#51222
C3×#51192
C3×#47292
C3×#47282
C3×#29303
#786273