Finite Group Databasebeta

← C3×#69007 #11788665

Subgroups — C3×#69007 #11788665

Order 1152.

673 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#909C211yesmaximal, characteristic
2C2×C3×#907C211yesmaximal, characteristic
3C2×C3×#824C211yesmaximal, characteristic, complemented
4C2×C3×#810C211yesmaximal, characteristic, complemented
5C3×#4966C211yesmaximal, characteristic
6C32×#343C211yesmaximal, characteristic, complemented
7C3×#4558C211yesmaximal, characteristic, complemented
8C3×#1908—31nomaximal
9#69007C311yesmaximal, characteristic, direct product factor
10C2×C3×#235C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C2×C32×M16C2211yescharacteristic
14C2×C32×C4⋊C4C2211yescharacteristic
15C2×C3×#208C2211yescharacteristic
16C22×C3×C3⋊C8C2211yescharacteristic
17C3×#909—22no—
18C3×#909C414yescomplemented
19C3×#907—22no—
20C3×#907—22no—
21C2×C3×Dic12—21no—
22C2×C3×#204—21no—
23C2×C3×ΓL(1,𝔽25)—21no—
24C2×C3×D3×C8—21no—
25C3×#824—22no—
26C3×#810—21no—
27C3×#810—21no—
28C3×#810—21no—
29C3×#809—21no—
30C2×C3×#137—31no—
31C2×C3×#131—31no—
32C2×#909C2×C311yescharacteristic
33C2×#907C2×C311yescharacteristic
34C3×#417—31no—
35C2×C3×#103—31no—
36C2×C3×#101—31no—
37C2×#824C2×C311yescharacteristic, complemented
38C2×#810C2×C311yescharacteristic, complemented
39#4966C2×C311yescharacteristic
40C3×#343—21no—
41C3×#343D311yescharacteristic, complemented
42C3×#343C2×C311yescharacteristic, complemented
43C3×#341—31no—
44#4558C2×C311yescharacteristic, complemented
45C22×C32×C4—21no—
46C22×C32×C4C2311yescharacteristic
47C2×C3×#67—21no—
48C22×C3×Dic3—21no—
49C3×#235—21no—
50C3×#235—21no—
51C3×#235C2×C412yes—
52C3×#235D412yes—
53C2×C3×D12D411yescharacteristic
54C2×C3×C4×D3—21no—
55C2×C3×Dic6—21no—
56C2×C3×Dic6—21no—
57C2×C3×Dic6—21no—
58C2×C3×Dic6—21no—
59C2×C3×Dic6—21no—
60C2×C3×Dic6C2×C412yes—
61C2×C3×Dic6D411yescharacteristic
62C32×M16—21no—
63C32×M16—21no—
64C32×M16C2×C412yes—
65C32×M16D412yes—
66C2×C32×C8—21no—
67C2×C32×C8—21no—
68C32×C4⋊C4—21no—
69C32×C4⋊C4—21no—
70C3×#208—21no—
71C3×#208—21no—
72C2×C3×C3⋊C8—21no—
73C2×C3×C3⋊C8—21no—
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8D411yescharacteristic
76C2×C3×C3⋊C8D411yescharacteristic
77C3×Dic12—24no—
78C3×#204—24no—
79C3×ΓL(1,𝔽25)—24no—
80C3×D3×C8—24no—
81#1908—31noSylow
82C2×C3×#41—31no—
83C22×C3×Q8—31no—
84C2×#235C22×C311yescharacteristic
85C22×Dic6C22×C311yescharacteristic
86C3×#137—34no—
87C3×#137—62no—
88C2×C3×Q16—61no—
89C2×C3×SD16—61no—
90C3×#131—62no—
91C3×#131—62no—
92C2×C3×M16—21no—
93C2×C3×M16—21no—
94C2×C3×M16—31no—
95C2×C3×M16—61no—
96C2×C3×M16C2×D311yescharacteristic
97C2×C3×M16C2×D311yescharacteristic
98C2×C3×M16C22×C311yescharacteristic
99C2×C3×M16C22×C311yescharacteristic
100C22×C3×C8—31no—
101C22×C3×C8—61no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4C2×D311yescharacteristic
104C2×C3×C4⋊C4C22×C311yescharacteristic
105C2×#208C22×C311yescharacteristic
106C22×C3⋊C8C22×C311yescharacteristic
107#909—22no—
108#909C3×C414yescomplemented
109#907—22no—
110#907—22no—
111C2×Dic12—21no—
112C2×#204—21no—
113C2×ΓL(1,𝔽25)—21no—
114C2×D3×C8—21no—
115C3×#103—61no—
116C3×#103—61no—
117C3×#103—61no—
118C3×#102—61no—
119C3×#101—62no—
120#824—22no—
121#810—21no—
122#810—21no—
123#810—21no—
124#809—21no—
125C23×C32#4111yescharacteristic
126C22×C3×D3—21no—
127C2×C32×C4—41no—
128C2×C32×C4—81no—
129C2×C32×C4C22×C411yescharacteristic
130C2×C32×C4C2×D411yescharacteristic
131C2×C32×C4C2×D411yescharacteristic
132C2×C32×C4C2×D411yescharacteristic
133C2×C32×C4C2×D411yescharacteristic
134C2×C32×C4#4111yescharacteristic
135C3×#67—22no—
136C3×#67—22no—
137C2×C3×Dic3—21no—
138C2×C3×Dic3—22no—
139C2×C3×Dic3—41no—
140C2×C3×Dic3—41no—
141C3×D12—21no—
142C3×D12C22⋊C412yes—
143C3×C4×D3—22no—
144C3×C4×D3—22no—
145C3×Dic6—21no—
146C3×Dic6—22no—
147C3×Dic6—22no—
148C3×Dic6—41no—
149C3×Dic6—41no—
150C3×Dic6C22⋊C412yes—
151C32×C8—22no—
152C32×C8—22no—
153C3×C3⋊C8—21no—
154C3×C3⋊C8—21no—
155C3×C3⋊C8—41no—
156C2×#137—31no—
157C2×#131—31no—
158#417—31no—
159C2×#103—31no—
160C2×#101—31no—
161#343C3×D311yescharacteristic, complemented
162#341—31no—
163C3×#41—32no—
164C3×#41—32no—
165C3×#41—61no—
166C3×#41—61no—
167C2×C3×Q8—32no—
168C2×C3×Q8—31no—
169C2×C3×Q8—61no—
170C2×C3×Q8—61no—
171C2×C3×Q8—61no—
172C2×C3×Q8—61no—
173C2×C3×Q8—61no—
174C2×C3×D4—31no—
175C2×C3×D4—61no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—61no—
181C22×C3×C4—61no—
182C22×C3×C4C22×D311yescharacteristic
183C22×C3×C4C23×C311yescharacteristic
184C2×#67—21no—
185C22×Dic3—21no—
186#235—21no—
187#235—21no—
188#235C2×C3×C412yes—
189#235C3×D412yes—
190C2×D12C3×D411yescharacteristic
191C2×C4×D3—21no—
192C2×Dic6—21no—
193C2×Dic6—21no—
194C2×Dic6—21no—
195C2×Dic6—21no—
196C2×Dic6—21no—
197C2×Dic6C2×C3×C412yes—
198C2×Dic6C3×D411yescharacteristic
199C3×Q16—64no—
200C3×SD16—64no—
201C3×M16—21no—
202C3×M16—21no—
203C3×M16—21no—
204C3×M16—21no—
205C3×M16—22no—
206C3×M16—22no—
207C3×M16—41no—
208C3×M16—41no—
209C3×M16—64no—
210C3×M16—61no—
211C3×M16—61no—
212C3×M16C4×D312yes—
213C3×M16D1212yes—
214C3×M16C2×C3×C412yes—
215C3×M16C3×D412yes—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—31no—
221C2×C3×C8—31no—
222C2×C3×C8—41no—
223C2×C3×C8—41no—
224C2×C3×C8—64no—
225C2×C3×C8—61no—
226C2×C3×C8—61no—
227C2×C3×C8—61no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—41no—
233C3×C4⋊C4—41no—
234#208—21no—
235#208—21no—
236C2×C3⋊C8—21no—
237C2×C3⋊C8—21no—
238C2×C3⋊C8—21no—
239C2×C3⋊C8C3×D411yescharacteristic
240C2×C3⋊C8C3×D411yescharacteristic
241Dic12—24no—
242#204—24no—
243ΓL(1,𝔽25)—24no—
244D3×C8—24no—
245C22×C32—21no—
246C22×C32—21no—
247C22×C32C4×D411yescharacteristic
248C22×C32#12111yescharacteristic
249C22×C32#12311yescharacteristic
250C2×C3×D3—22no—
251C2×C3×D3—41no—
252C2×C3×D3—41no—
253C32×C4—81no—
254C32×C4C2×C22⋊C411yescharacteristic
255C32×C4C4×D411yescharacteristic
256C32×C4#12011yescharacteristic
257C32×C4#12111yescharacteristic
258C3×Dic3—22no—
259C3×Dic3—42no—
260C2×#41—31no—
261C22×Q8—31no—
262#137—34no—
263#137—62no—
264C2×Q16—61no—
265C2×SD16—61no—
266#131—62no—
267#131—62no—
268C2×M16—31no—
269C2×M16—61no—
270C2×M16C2×C3×D311yescharacteristic
271C2×M16C2×C3×D311yescharacteristic
272C22×C8—31no—
273C22×C8—61no—
274C2×C4⋊C4C2×C3×D311yescharacteristic
275#103—61no—
276#103—61no—
277#103—61no—
278#102—61no—
279#101—62no—
280C23×C3—21no—
281C23×C3—61no—
282C23×C3#23511yescharacteristic
283C23×C3C3×#4111yescharacteristic
284C22×D3—21no—
285C3×Q8—32no—
286C3×Q8—61no—
287C3×Q8—62no—
288C3×Q8—62no—
289C3×Q8—121no—
290C3×Q8—121no—
291C3×D4—32no—
292C3×D4—62no—
293C3×D4—62no—
294C3×D4—61no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—42no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—61no—
305C2×C3×C4—62no—
306C2×C3×C4—62no—
307C2×C3×C4—62no—
308C2×C3×C4—81no—
309C2×C3×C4—81no—
310C2×C3×C4—82no—
311C2×C3×C4—121no—
312C2×C3×C4—121no—
313C2×C3×C4C2×C4×D311yescharacteristic
314C2×C3×C4C2×D1211yescharacteristic
315C2×C3×C4D3×D411yescharacteristic
316C2×C3×C4D3×D411yescharacteristic
317C2×C3×C4D3×D411yescharacteristic
318C2×C3×C4#23911yescharacteristic
319C2×C3×C4C22×C3×C411yescharacteristic
320C2×C3×C4C2×C3×D411yescharacteristic
321C2×C3×C4C2×C3×D411yescharacteristic
322C2×C3×C4C2×C3×D411yescharacteristic
323C2×C3×C4C2×C3×D411yescharacteristic
324C2×C3×C4C3×#4111yescharacteristic
325#67—22no—
326#67—22no—
327C2×Dic3—21no—
328C2×Dic3—22no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331D12—21no—
332D12C3×C22⋊C412yes—
333C4×D3—22no—
334C4×D3—22no—
335Dic6—21no—
336Dic6—22no—
337Dic6—22no—
338Dic6—41no—
339Dic6—41no—
340Dic6C3×C22⋊C412yes—
341C3×C8—22no—
342C3×C8—22no—
343C3×C8—22no—
344C3×C8—22no—
345C3×C8—42no—
346C3×C8—42no—
347C3×C8—61no—
348C3×C8—61no—
349C3×C8—121no—
350C3⋊C8—21no—
351C3⋊C8—21no—
352C3⋊C8—41no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32#38611yescharacteristic
356C2×C32#47111yescharacteristic
357C2×C32#47311yescharacteristic
358C3×D3—42no—
359#41—32no—
360#41—32no—
361#41—61no—
362#41—61no—
363C2×Q8—32no—
364C2×Q8—31no—
365C2×Q8—61no—
366C2×Q8—61no—
367C2×Q8—61no—
368C2×Q8—61no—
369C2×Q8—61no—
370C2×D4—31no—
371C2×D4—61no—
372C22×C4—21no—
373C22×C4—61no—
374C22×C4—61no—
375C22×C4C22×C3×D311yescharacteristic
376Q16—64no—
377SD16—64no—
378M16—21no—
379M16—21no—
380M16—64no—
381M16—61no—
382M16—61no—
383M16C3×C4×D312yes—
384M16C3×D1212yes—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—31no—
388C2×C8—31no—
389C2×C8—64no—
390C2×C8—61no—
391C2×C8—61no—
392C2×C8—61no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3—62no—
405C22×C3—121no—
406C22×C3—121no—
407C22×C3C4×D1211yescharacteristic
408C22×C3#88411yescharacteristic
409C22×C3#89411yescharacteristic
410C22×C3C3×C4×D411yescharacteristic
411C22×C3C3×#12111yescharacteristic
412C22×C3C3×#12311yescharacteristic
413C2×D3—22no—
414C2×D3—41no—
415C2×D3—41no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—62no—
421C3×C4—82no—
422C3×C4—81no—
423C3×C4—81no—
424C3×C4—122no—
425C3×C4D3×C22⋊C411yescharacteristic
426C3×C4#88211yescharacteristic
427C3×C4#89311yescharacteristic
428C3×C4#89511yescharacteristic
429C3×C4C2×C3×C22⋊C411yescharacteristic
430C3×C4C3×C4×D411yescharacteristic
431C3×C4C3×#12011yescharacteristic
432C3×C4C3×#12111yescharacteristic
433Dic3—22no—
434Dic3—42no—
435C32#190811yescharacteristic, complemented, Sylow
436C23—61no—
437C23C3×#23511yescharacteristic
438Q8—32no—
439Q8—61no—
440Q8—62no—
441Q8—62no—
442Q8—121no—
443Q8—121no—
444D4—32no—
445D4—62no—
446D4—62no—
447D4—61no—
448C2×C4—41no—
449C2×C4—61no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—81no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4C2×C3×C4×D311yescharacteristic
457C2×C4C2×C3×D1211yescharacteristic
458C2×C4C3×D3×D411yescharacteristic
459C2×C4C3×D3×D411yescharacteristic
460C2×C4C3×D3×D411yescharacteristic
461C2×C4C3×#23911yescharacteristic
462C8—22no—
463C8—22no—
464C8—61no—
465C8—61no—
466C8—121no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—41no—
475C2×C3—41no—
476C2×C3—122no—
477C2×C3#466511yescharacteristic
478C2×C3#474811yescharacteristic
479C2×C3#475511yescharacteristic
480C2×C3C3×#38611yescharacteristic
481C2×C3C3×#47111yescharacteristic
482C2×C3C3×#47311yescharacteristic
483D3—42no—
484C22—21no—
485C22—21no—
486C22—62no—
487C22—121no—
488C22—121no—
489C22C3×C4×D1211yescharacteristic
490C22C3×#88411yescharacteristic
491C22C3×#89411yescharacteristic
492C4—62no—
493C4—81no—
494C4—122no—
495C4C3×D3×C22⋊C411yescharacteristic
496C4C3×#88211yescharacteristic
497C4C3×#89311yescharacteristic
498C4C3×#89511yescharacteristic
499C3—21no—
500C3#6900711yescharacteristic, direct product factor
501C3C3×#190811yescharacteristic, complemented
502C2—21no—
503C2—21no—
504C2—122no—
505C2C3×#466511yescharacteristic
506C2C3×#474811yescharacteristic
507C2C3×#475511yescharacteristic
subgroupG/H
centerC22×C3#884
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#343C2
socleC22×C32#121
prime radicalC2×C3×#235C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#343D3
hypocenterC3C3×#1908
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#343

Complements

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

NH ≅ G/Ntype
3. C2×C3×#824C2semidirect
4. C2×C3×#810C2semidirect
6. C32×#343C2semidirect
7. C3×#4558C2semidirect
9. #69007C3direct (N × H)
18. C3×#909C4semidirect
37. C2×#824C2×C3semidirect
38. C2×#810C2×C3semidirect
41. C3×#343D3semidirect
42. C3×#343C2×C3semidirect
44. #4558C2×C3semidirect
108. #909C3×C4semidirect
161. #343C3×D3semidirect
435. C32#1908semidirect
500. C3#69007direct (N × H)
501. C3C3×#1908semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9092
C2×C3×#9072
C2×C3×#8242
C2×C3×#8102
C3×#49662
C32×#3432
C3×#45582
C3×#19083
#690073