Finite Group Databasebeta

← C3×#68758 #11786576

Subgroups — C3×#68758 #11786576

Order 1152.

703 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5521C211yesmaximal, characteristic
2C2×C3×#823C211yesmaximal, characteristic
3C3×#5121C211yesmaximal, characteristic, complemented
4C3×#4995C211yesmaximal, characteristic, complemented
5C3×#4939C211yesmaximal, characteristic
6C32×#339C211yesmaximal, characteristic, complemented
7C3×#4523C211yesmaximal, characteristic, complemented
8C3×#1856—31nomaximal
9#68758C311yesmaximal, characteristic, direct product factor
10C22×C3×D12C2211yescharacteristic
11C2×C32×M16C2211yescharacteristic
12C32×#117C2211yescharacteristic
13C2×C32×C42C2211yescharacteristic
14C2×C3×(C2×D3)⋊C4—22no—
15C3×#926C2211yescharacteristic
16C2×C3×C4⋊Dic3C2211yescharacteristic
17C2×C3×#208C2211yescharacteristic
18C3×#893—21no—
19C3×#893—21no—
20C3×D3×C22⋊C4—22no—
21C3×C4×D12—21no—
22C3×C4×D12—21no—
23C3×#823C414yescomplemented
24C3×#818—21no—
25C3×#817—21no—
26C3×#808—21no—
27C3×#807—21no—
28C3×#518—31no—
29#5521C2×C311yescharacteristic
30C3×#428—31no—
31C3×#428—31no—
32C2×C3×#100—31no—
33C3×#390—31no—
34C2×#823C2×C311yescharacteristic
35#5121C2×C311yescharacteristic, complemented
36#4995C2×C311yescharacteristic, complemented
37#4939C2×C311yescharacteristic
38C3×#339—21no—
39C3×#339—31no—
40C3×#339D311yescharacteristic, complemented
41C3×#339C2×C311yescharacteristic, complemented
42#4523C2×C311yescharacteristic, complemented
43C23×C3×D3D412yes—
44C22×C32×C4—21no—
45C22×C32×C4C2311yescharacteristic
46C22×C3×Dic3—21no—
47C2×C3×D12—21no—
48C2×C3×D12—21no—
49C2×C3×D12—21no—
50C2×C3×D12—21no—
51C2×C3×D12C2×C412yes—
52C2×C3×D12Q812yes—
53C2×C3×C4×D3—22no—
54C2×C3×C4×D3—22no—
55C32×M16—21no—
56C32×M16C2×C412yescomplemented
57C2×C32×C8—21no—
58C32×C4⋊C4D411yescharacteristic
59C32×C4⋊C4D411yescharacteristic
60C32×C22⋊C4—21no—
61C32×C42—21no—
62C32×C42—21no—
63C32×C42—21no—
64C3×#217—21no—
65C3×(C2×D3)⋊C4—22no—
66C3×(C2×D3)⋊C4—22no—
67C3×(C2×D3)⋊C4—44no—
68C3×C4⋊Dic3—21no—
69C3×C4⋊Dic3—21no—
70C3×C4⋊Dic3D411yescharacteristic
71C3×C4⋊Dic3D411yescharacteristic
72C3×C4×Dic3—21no—
73C3×#208—21no—
74C3×#208C2×C412yescomplemented
75C2×C3×C3⋊C8—21no—
76#1856—31noSylow
77C22×C3×D4—31no—
78C22×D12C22×C311yescharacteristic
79C2×C3×M16—21no—
80C2×C3×M16—31no—
81C2×C3×M16C2×D311yescharacteristic
82C2×C3×M16C22×C311yescharacteristic
83C3×C4×D4—61no—
84C3×C4×D4—61no—
85C3×C4×D4—61no—
86C3×C4×D4—61no—
87C3×#117—21no—
88C3×#117—31no—
89C3×#117C2×D311yescharacteristic
90C3×#117C22×C311yescharacteristic
91C2×C3×C4⋊C4—31no—
92C2×C3×C22⋊C4—62no—
93C2×C3×C22⋊C4—62no—
94C2×C3×C42—21no—
95C2×C3×C42C2×D311yescharacteristic
96C2×C3×C42C22×C311yescharacteristic
97C2×(C2×D3)⋊C4—22no—
98#926C22×C311yescharacteristic
99C2×C4⋊Dic3C22×C311yescharacteristic
100C2×#208C22×C311yescharacteristic
101#893—21no—
102#893—21no—
103D3×C22⋊C4—22no—
104C4×D12—21no—
105C4×D12—21no—
106C3×#107—61no—
107C3×#107—61no—
108C3×#106—61no—
109C3×#106—61no—
110C3×#100—34no—
111#823C3×C414yescomplemented
112#818—21no—
113#817—21no—
114#808—21no—
115#807—21no—
116C23×C32C2×D411yescharacteristic
117C22×C3×D3—22no—
118C22×C3×D3—22no—
119C22×C3×D3—42no—
120C22×C3×D3—42no—
121C22×C3×D3C4⋊C414yes—
122C2×C32×C4—22no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—41no—
126C2×C32×C4—41no—
127C2×C32×C4C22×C411yescharacteristic
128C2×C32×C4C2×D411yescharacteristic
129C2×C32×C4C2×D411yescharacteristic
130C2×C32×C4C2×Q811yescharacteristic
131C2×C32×C4#4111yescharacteristic
132C2×C32×C4#4111yescharacteristic
133C2×C3×Dic3—21no—
134C2×C3×Dic3—21no—
135C2×C3×Dic3—42no—
136C2×C3×Dic3—41no—
137C3×D12—22no—
138C3×D12—22no—
139C3×D12—22no—
140C3×D12—22no—
141C3×C4×D3—82no—
142C3×C4×D3—82no—
143C32×C8—22no—
144C3×C3⋊C8—22no—
145#518—31no—
146#428—31no—
147#428—31no—
148C2×#100—31no—
149#390—31no—
150#339—31no—
151#339C3×D311yescharacteristic, complemented
152C24×C3—32no—
153C23×D3C3×D412yes—
154C2×C3×D4—32no—
155C2×C3×D4—32no—
156C2×C3×D4—61no—
157C2×C3×D4—61no—
158C2×C3×D4—61no—
159C2×C3×D4—61no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—41no—
164C22×C3×C4—62no—
165C22×C3×C4—62no—
166C22×C3×C4—61no—
167C22×C3×C4C22×D311yescharacteristic
168C22×C3×C4C23×C311yescharacteristic
169C22×Dic3—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12C2×C3×C412yes—
175C2×D12C3×Q812yes—
176C2×C4×D3—22no—
177C2×C4×D3—22no—
178C3×M16—21no—
179C3×M16—21no—
180C3×M16—22no—
181C3×M16—32no—
182C3×M16—41no—
183C3×M16—61no—
184C3×M16C4×D312yescomplemented
185C3×M16C2×C3×C412yescomplemented
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C2×C3×C8—41no—
189C2×C3×C8—61no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—31no—
193C3×C4⋊C4—31no—
194C3×C4⋊C4—61no—
195C3×C4⋊C4—61no—
196C3×C4⋊C4D1211yescharacteristic
197C3×C4⋊C4D1211yescharacteristic
198C3×C4⋊C4C3×D411yescharacteristic
199C3×C4⋊C4C3×D411yescharacteristic
200C3×C22⋊C4—21no—
201C3×C22⋊C4—21no—
202C3×C22⋊C4—41no—
203C3×C22⋊C4—62no—
204C3×C22⋊C4—62no—
205C3×C22⋊C4—61no—
206C3×C22⋊C4—124no—
207C3×C42—21no—
208C3×C42—21no—
209C3×C42—22no—
210C3×C42—21no—
211C3×C42—21no—
212C3×C42—21no—
213C3×C42—21no—
214C3×C42—41no—
215C3×C42—41no—
216C3×C42—61no—
217#217—21no—
218(C2×D3)⋊C4—22no—
219(C2×D3)⋊C4—22no—
220(C2×D3)⋊C4—44no—
221C4⋊Dic3—21no—
222C4⋊Dic3—21no—
223C4⋊Dic3C3×D411yescharacteristic
224C4⋊Dic3C3×D411yescharacteristic
225C4×Dic3—21no—
226#208—21no—
227#208C2×C3×C412yescomplemented
228C2×C3⋊C8—21no—
229C22×C32—21no—
230C22×C32—21no—
231C22×C32C2×C4⋊C411yescharacteristic
232C22×C32#12011yescharacteristic
233C22×C32#12311yescharacteristic
234C2×C3×D3—24no—
235C2×C3×D3—42no—
236C2×C3×D3—42no—
237C2×C3×D3—44no—
238C2×C3×D3—82no—
239C32×C4—42no—
240C32×C4—41no—
241C32×C4—41no—
242C32×C4C4×D411yescharacteristic
243C32×C4C4×D411yescharacteristic
244C32×C4C22⋊Q811yescharacteristic
245C32×C4C22⋊Q811yescharacteristic
246C3×Dic3—41no—
247C3×Dic3—41no—
248C3×Dic3—81no—
249C22×D4—31no—
250C2×M16—31no—
251C2×M16C2×C3×D311yescharacteristic
252C4×D4—61no—
253C4×D4—61no—
254C4×D4—61no—
255C4×D4—61no—
256#117—31no—
257#117C2×C3×D311yescharacteristic
258C2×C4⋊C4—31no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C42C2×C3×D311yescharacteristic
262#107—61no—
263#107—61no—
264#106—61no—
265#106—61no—
266#100—34no—
267C23×C3—21no—
268C23×C3—34no—
269C23×C3—62no—
270C23×C3—62no—
271C23×C3—122no—
272C23×C3—122no—
273C23×C3D3×D411yescharacteristic
274C23×C3C2×C3×D411yescharacteristic
275C22×D3—22no—
276C22×D3—22no—
277C22×D3—42no—
278C22×D3—42no—
279C22×D3C3×C4⋊C414yes—
280C3×D4—62no—
281C3×D4—62no—
282C3×D4—62no—
283C3×D4—62no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—42no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—122no—
308C2×C3×C4—121no—
309C2×C3×C4—242no—
310C2×C3×C4—242no—
311C2×C3×C4C2×C4×D311yescharacteristic
312C2×C3×C4C2×D1211yescharacteristic
313C2×C3×C4#23511yescharacteristic
314C2×C3×C4D3×D411yescharacteristic
315C2×C3×C4#23711yescharacteristic
316C2×C3×C4D3×Q811yescharacteristic
317C2×C3×C4C22×C3×C411yescharacteristic
318C2×C3×C4C2×C3×D411yescharacteristic
319C2×C3×C4C2×C3×D411yescharacteristic
320C2×C3×C4C2×C3×Q811yescharacteristic
321C2×C3×C4C3×#4111yescharacteristic
322C2×C3×C4C3×#4111yescharacteristic
323C2×Dic3—21no—
324C2×Dic3—21no—
325C2×Dic3—42no—
326C2×Dic3—41no—
327D12—22no—
328D12—22no—
329D12—22no—
330D12—22no—
331C4×D3—82no—
332C4×D3—82no—
333C3×C8—22no—
334C3×C8—22no—
335C3×C8—42no—
336C3×C8—62no—
337C3⋊C8—22no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32#38511yescharacteristic
341C2×C32#45311yescharacteristic
342C2×C32#45511yescharacteristic
343C3×D3—44no—
344C24—32no—
345C2×D4—32no—
346C2×D4—32no—
347C2×D4—61no—
348C2×D4—61no—
349C2×D4—61no—
350C2×D4—61no—
351C22×C4—21no—
352C22×C4—62no—
353C22×C4—62no—
354C22×C4—61no—
355C22×C4C22×C3×D311yescharacteristic
356M16—21no—
357M16—32no—
358M16—61no—
359M16C3×C4×D312yescomplemented
360C2×C8—21no—
361C2×C8—61no—
362C4⋊C4—31no—
363C4⋊C4—31no—
364C4⋊C4—61no—
365C4⋊C4—61no—
366C4⋊C4C3×D1211yescharacteristic
367C4⋊C4C3×D1211yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—62no—
370C22⋊C4—62no—
371C22⋊C4—61no—
372C22⋊C4—124no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—61no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—41no—
385C22×C3—41no—
386C22×C3—64no—
387C22×C3—124no—
388C22×C3—122no—
389C22×C3—122no—
390C22×C3—242no—
391C22×C3#88211yescharacteristic
392C22×C3#88311yescharacteristic
393C22×C3D3×C4⋊C411yescharacteristic
394C22×C3C2×C3×C4⋊C411yescharacteristic
395C22×C3C3×#12011yescharacteristic
396C22×C3C3×#12311yescharacteristic
397C2×D3—24no—
398C2×D3—42no—
399C2×D3—42no—
400C2×D3—44no—
401C2×D3—82no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4—44no—
409C3×C4—41no—
410C3×C4—41no—
411C3×C4—41no—
412C3×C4—41no—
413C3×C4—81no—
414C3×C4—81no—
415C3×C4—121no—
416C3×C4—121no—
417C3×C4—241no—
418C3×C4C4×D1211yescharacteristic
419C3×C4#67⋊C411yescharacteristic
420C3×C4#89611yescharacteristic
421C3×C4#89711yescharacteristic
422C3×C4C3×C4×D411yescharacteristic
423C3×C4C3×C4×D411yescharacteristic
424C3×C4C3×C22⋊Q811yescharacteristic
425C3×C4C3×C22⋊Q811yescharacteristic
426Dic3—41no—
427Dic3—41no—
428Dic3—81no—
429C32#185611yescharacteristic, complemented, Sylow
430C23—34no—
431C23—62no—
432C23—62no—
433C23—122no—
434C23—122no—
435C23C3×D3×D411yescharacteristic
436D4—62no—
437D4—62no—
438D4—62no—
439D4—62no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—122no—
448C2×C4—121no—
449C2×C4—242no—
450C2×C4—242no—
451C2×C4C2×C3×C4×D311yescharacteristic
452C2×C4C2×C3×D1211yescharacteristic
453C2×C4C3×#23511yescharacteristic
454C2×C4C3×D3×D411yescharacteristic
455C2×C4C3×#23711yescharacteristic
456C2×C4C3×D3×Q811yescharacteristic
457C8—22no—
458C8—62no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—41no—
467C2×C3—41no—
468C2×C3—124no—
469C2×C3#466811yescharacteristic
470C2×C3#482211yescharacteristic
471C2×C3#482511yescharacteristic
472C2×C3C3×#38511yescharacteristic
473C2×C3C3×#45311yescharacteristic
474C2×C3C3×#45511yescharacteristic
475D3—44no—
476C22—21no—
477C22—21no—
478C22—64no—
479C22—124no—
480C22—122no—
481C22—122no—
482C22—242no—
483C22C3×#88211yescharacteristic
484C22C3×#88311yescharacteristic
485C22C3×D3×C4⋊C411yescharacteristic
486C4—42no—
487C4—41no—
488C4—41no—
489C4—121no—
490C4—121no—
491C4—241no—
492C4C3×C4×D1211yescharacteristic
493C4C3×#67⋊C411yescharacteristic
494C4C3×#89611yescharacteristic
495C4C3×#89711yescharacteristic
496C3—21no—
497C3#6875811yescharacteristic, direct product factor
498C3C3×#185611yescharacteristic, complemented
499C2—21no—
500C2—21no—
501C2—124no—
502C2C3×#466811yescharacteristic
503C2C3×#482211yescharacteristic
504C2C3×#482511yescharacteristic
subgroupG/H
centerC22×C3#882
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#339C2
socleC22×C32#120
prime radicalC22×C3×D12C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#339D3
hypocenterC3C3×#1856
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#339

Complements

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

NH ≅ G/Ntype
3. C3×#5121C2semidirect
4. C3×#4995C2semidirect
6. C32×#339C2semidirect
7. C3×#4523C2semidirect
9. #68758C3direct (N × H)
23. C3×#823C4semidirect
35. #5121C2×C3semidirect
36. #4995C2×C3semidirect
40. C3×#339D3semidirect
41. C3×#339C2×C3semidirect
42. #4523C2×C3semidirect
56. C32×M16C2×C4semidirect
74. C3×#208C2×C4semidirect
111. #823C3×C4semidirect
151. #339C3×D3semidirect
184. C3×M16C4×D3semidirect
185. C3×M16C2×C3×C4semidirect
227. #208C2×C3×C4semidirect
359. M16C3×C4×D3semidirect
429. C32#1856semidirect
497. C3#68758direct (N × H)
498. C3C3×#1856semidirect

Maximal subgroups

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

subgroupindex
C3×#55212
C2×C3×#8232
C3×#51212
C3×#49952
C3×#49392
C32×#3392
C3×#45232
C3×#18563
#687583