Finite Group Databasebeta

← C3×#87018 #11906330

Subgroups — C3×#87018 #11906330

Order 1152.

2220 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#557C211yesmaximal, characteristic, complemented
2C3×#5727C213yesmaximal, complemented
3C3×D3×C4⋊Q8C213yesmaximal, complemented
4C3×#5690C216yesmaximal, complemented
5C3×#5689C216yesmaximal, complemented
6C3×#5578C213yesmaximal, complemented
7C3×Q8×D12C213yesmaximal, complemented
8C3×#5574C212yesmaximal
9C3×C4×#239C211yesmaximal, characteristic
10C3×#5570C211yesmaximal, characteristic, complemented
11C3×Q8×Dic6C212yesmaximal, complemented
12C3×#3503—31nomaximal
13#87018C311yesmaximal, characteristic, direct product factor
14C2×C3×#239C2211yescharacteristic
15C2×C3×D3×Q8C2216yes—
16C32×C4⋊Q8C2213yes—
17C32×#125C2216yes—
18C32×C4×Q8C2212yes—
19C32×C4×Q8C2211yescharacteristic
20C32×C4×Q8C2213yes—
21C3×#946C2213yes—
22C3×Q8×Dic3C2211yescharacteristic
23C3×Q8×Dic3C2212yes—
24C3×#944C2213yes—
25C3×#897C2213yes—
26C3×#897C22112yes—
27C3×#896C2216yes—
28C3×#896C22112yes—
29C3×#893C2213yes—
30C3×#893C2216yes—
31C3×#892C2213yes—
32C3×D3×C4⋊C4C2216yes—
33C3×D3×C4⋊C4C2216yes—
34C3×#890C2216yes—
35C3×#889C2216yes—
36C3×#888C2216yes—
37C3×#888C2216yes—
38C3×#888C2211yescharacteristic
39C3×#888C2216yes—
40C3×C4×D12C2213yes—
41C3×C4×D12C2216yes—
42C3×#872C2216yes—
43C3×C42×D3C2213yes—
44C3×#869C2216yes—
45C3×#869C2213yes—
46C3×C4×Dic6C2216yes—
47C3×C4×Dic6C2213yes—
48C3×Q82—32no—
49C3×#557—21no—
50C3×#557—31no—
51C3×#557D311yescharacteristic, complemented
52C3×#557C2×C311yescharacteristic, complemented
53C3×#554—36no—
54C3×D4×Q8—33no—
55C3×D4×Q8—33no—
56C3×#544—36no—
57C3×#533—33no—
58C2×C3×C4⋊Q8—33no—
59C3×#520—32no—
60C3×C4×#41—31no—
61#5727C2×C313yescomplemented
62D3×C4⋊Q8C2×C313yescomplemented
63#5690C2×C316yescomplemented
64#5689C2×C316yescomplemented
65#5578C2×C313yescomplemented
66Q8×D12C2×C313yescomplemented
67#5574C2×C312yes—
68C4×#239C2×C311yescharacteristic
69#5570C2×C311yescharacteristic, complemented
70Q8×Dic6C2×C312yescomplemented
71C2×C32×Q8C2313yes—
72C2×C32×Q8C2311yescharacteristic
73C3×#239Q818yescomplemented
74C3×D3×Q8—412no—
75C2×C3×D12C2313yes—
76C2×C3×C4×D3C2313yes—
77C2×C3×C4×D3C2313yes—
78C2×C3×C4×D3C23112yes—
79C2×C3×C4×D3C2313yes—
80C2×C3×Dic6C2312yes—
81C2×C3×Dic6C2313yes—
82C2×C3×Dic6C23112yes—
83C32×C4⋊C4C2316yes—
84C32×C4⋊C4C2316yes—
85C32×C4⋊C4C2311yescharacteristic
86C32×C4⋊C4C2316yes—
87C32×C4⋊C4C2313yes—
88C32×C42C2316yes—
89C32×C42C2313yes—
90C3×(C2×D3)⋊C4C2316yes—
91C3×(C2×D3)⋊C4C2316yes—
92C3×(C2×D3)⋊C4C2316yes—
93C3×C4⋊Dic3C2313yes—
94C3×C4⋊Dic3C2316yes—
95C3×C4⋊Dic3C2316yes—
96C3×C4⋊Dic3C2311yescharacteristic
97C3×C4⋊Dic3C2316yes—
98C3×Dic3⋊C4C2316yes—
99C3×Dic3⋊C4C2316yes—
100C3×Dic3⋊C4C2316yes—
101C3×Dic3⋊C4C2312yes—
102C3×Dic3⋊C4C2316yes—
103C3×C4×Dic3C2313yes—
104C3×C4×Dic3C2313yes—
105C3×C4×Dic3C2316yes—
106C3×C4×Dic3C2311yescharacteristic
107#3503—31noSylow
108C2×C3×#41—31no—
109C22×C3×Q8—36no—
110C2×#239C22×C311yescharacteristic
111C2×D3×Q8C22×C316yes—
112C3×C4⋊Q8—23no—
113C3×C4⋊Q8—33no—
114C3×C4⋊Q8—36no—
115C3×C4⋊Q8—36no—
116C3×C4⋊Q8—36no—
117C3×C4⋊Q8—31no—
118C3×C4⋊Q8—36no—
119C3×C4⋊Q8—33no—
120C3×C4⋊Q8C2×D313yes—
121C3×C4⋊Q8C22×C313yes—
122C3×#125—26no—
123C3×#125—36no—
124C3×#125—36no—
125C3×#125C2×D316yes—
126C3×#125C22×C316yes—
127C3×C22⋊Q8—33no—
128C3×C22⋊Q8—36no—
129C3×C22⋊Q8—312no—
130C3×C22⋊Q8—33no—
131C3×C22⋊Q8—312no—
132C3×C4×Q8—21no—
133C3×C4×Q8—22no—
134C3×C4×Q8—23no—
135C3×C4×Q8—31no—
136C3×C4×Q8—32no—
137C3×C4×Q8—36no—
138C3×C4×Q8—33no—
139C3×C4×Q8C2×D312yes—
140C3×C4×Q8C2×D311yescharacteristic
141C3×C4×Q8C2×D313yes—
142C3×C4×Q8C22×C312yes—
143C3×C4×Q8C22×C311yescharacteristic
144C3×C4×Q8C22×C313yes—
145C3×C4×D4—33no—
146C3×C4×D4—36no—
147C3×C4×D4—33no—
148C3×C4×D4—36no—
149C3×#117—33no—
150C3×#117—36no—
151C2×C3×C4⋊C4—36no—
152C2×C3×C4⋊C4—36no—
153C2×C3×C42—33no—
154#946C22×C313yes—
155Q8×Dic3C22×C311yescharacteristic
156Q8×Dic3C22×C312yes—
157#944C22×C313yes—
158#897C22×C3112yes—
159#897C22×C313yes—
160#896C22×C3112yes—
161#896C22×C316yes—
162#893C22×C313yes—
163#893C22×C316yes—
164#892C22×C313yes—
165D3×C4⋊C4C22×C316yes—
166D3×C4⋊C4C22×C316yes—
167#890C22×C316yes—
168#889C22×C316yes—
169#888C22×C316yes—
170#888C22×C316yes—
171#888C22×C316yes—
172#888C22×C311yescharacteristic
173C4×D12C22×C316yes—
174C4×D12C22×C313yes—
175#872C22×C316yes—
176C42×D3C22×C313yes—
177#869C22×C316yes—
178#869C22×C313yes—
179C4×Dic6C22×C316yes—
180C4×Dic6C22×C313yes—
181C22×C3×D3C2413yes—
182C32×Q8—26no—
183C32×Q8C2×Q814yescomplemented
184C2×C32×C4C2413yes—
185C2×C32×C4C2416yes—
186C2×C32×C4C2412yes—
187C2×C32×C4C2413yes—
188C2×C32×C4C2411yescharacteristic
189C2×C3×Dic3C2412yes—
190C2×C3×Dic3C2413yes—
191C2×C3×Dic3C2416yes—
192C2×C3×Dic3C2411yescharacteristic
193C2×C3×Dic3C2411yescharacteristic
194C3×D12C2×Q8112yescomplemented
195C3×C4×D3—26no—
196C3×C4×D3—26no—
197C3×C4×D3—412no—
198C3×C4×D3C2×Q8112yescomplemented
199C3×Dic6—26no—
200C3×Dic6—24no—
201C3×Dic6—412no—
202Q82—32no—
203#557—31no—
204#557C3×D311yescharacteristic, complemented
205#554—36no—
206D4×Q8—33no—
207D4×Q8—33no—
208#544—36no—
209#533—33no—
210C2×C4⋊Q8—33no—
211#520—32no—
212C4×#41—31no—
213C3×#41—38no—
214C2×C3×Q8—21no—
215C2×C3×Q8—23no—
216C2×C3×Q8—32no—
217C2×C3×Q8—33no—
218C2×C3×Q8—312no—
219C2×C3×Q8—1212no—
220C2×C3×Q8C22×D313yes—
221C2×C3×Q8C22×D311yescharacteristic
222C2×C3×Q8C23×C313yes—
223C2×C3×Q8C23×C311yescharacteristic
224C2×C3×D4—33no—
225C22×C3×C4—33no—
226C22×C3×C4—33no—
227C22×C3×C4—33no—
228C22×C3×C4—312no—
229#239C3×Q818yescomplemented
230D3×Q8—412no—
231C2×D12C23×C313yes—
232C2×C4×D3C23×C313yes—
233C2×C4×D3C23×C313yes—
234C2×C4×D3C23×C3112yes—
235C2×C4×D3C23×C313yes—
236C2×Dic6C23×C312yes—
237C2×Dic6C23×C313yes—
238C2×Dic6C23×C3112yes—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—23no—
241C3×C4⋊C4—26no—
242C3×C4⋊C4—26no—
243C3×C4⋊C4—26no—
244C3×C4⋊C4—36no—
245C3×C4⋊C4—36no—
246C3×C4⋊C4—32no—
247C3×C4⋊C4—36no—
248C3×C4⋊C4—36no—
249C3×C4⋊C4—36no—
250C3×C4⋊C4—36no—
251C3×C4⋊C4—33no—
252C3×C4⋊C4—31no—
253C3×C4⋊C4—36no—
254C3×C4⋊C4C22×D316yes—
255C3×C4⋊C4C22×D316yes—
256C3×C4⋊C4C22×D313yes—
257C3×C4⋊C4C22×D311yescharacteristic
258C3×C4⋊C4C22×D316yes—
259C3×C4⋊C4C23×C316yes—
260C3×C4⋊C4C23×C316yes—
261C3×C4⋊C4C23×C311yescharacteristic
262C3×C4⋊C4C23×C316yes—
263C3×C4⋊C4C23×C313yes—
264C3×C22⋊C4—36no—
265C3×C22⋊C4—36no—
266C3×C22⋊C4—36no—
267C3×C42—26no—
268C3×C42—23no—
269C3×C42—31no—
270C3×C42—33no—
271C3×C42—33no—
272C3×C42—36no—
273C3×C42C22×D313yes—
274C3×C42C22×D316yes—
275C3×C42C23×C313yes—
276C3×C42C23×C316yes—
277(C2×D3)⋊C4C23×C316yes—
278(C2×D3)⋊C4C23×C316yes—
279(C2×D3)⋊C4C23×C316yes—
280C4⋊Dic3C23×C311yescharacteristic
281C4⋊Dic3C23×C316yes—
282C4⋊Dic3C23×C313yes—
283C4⋊Dic3C23×C316yes—
284C4⋊Dic3C23×C316yes—
285Dic3⋊C4C23×C316yes—
286Dic3⋊C4C23×C316yes—
287Dic3⋊C4C23×C316yes—
288Dic3⋊C4C23×C312yes—
289Dic3⋊C4C23×C316yes—
290C4×Dic3C23×C313yes—
291C4×Dic3C23×C316yes—
292C4×Dic3C23×C311yescharacteristic
293C4×Dic3C23×C313yes—
294C22×C32C2511yescharacteristic
295C2×C3×D3—23no—
296C2×C3×D3—23no—
297C2×C3×D3C22×Q816yes—
298C32×C4—26no—
299C32×C4—22no—
300C32×C4—23no—
301C32×C4C22×Q816yes—
302C32×C4H(5,𝔽2)12yes—
303C3×Dic3—26no—
304C3×Dic3—23no—
305C3×Dic3—22no—
306C3×Dic3C22×Q812yes—
307C3×Dic32−1+412yes—
308C2×#41—31no—
309C22×Q8—36no—
310C4⋊Q8—33no—
311C4⋊Q8—36no—
312C4⋊Q8—36no—
313C4⋊Q8—31no—
314C4⋊Q8—36no—
315C4⋊Q8—33no—
316C4⋊Q8—36no—
317C4⋊Q8C2×C3×D313yes—
318#125—36no—
319#125—36no—
320#125C2×C3×D316yes—
321C22⋊Q8—33no—
322C22⋊Q8—33no—
323C22⋊Q8—36no—
324C22⋊Q8—312no—
325C22⋊Q8—312no—
326C4×Q8—31no—
327C4×Q8—32no—
328C4×Q8—33no—
329C4×Q8—36no—
330C4×Q8C2×C3×D311yescharacteristic
331C4×Q8C2×C3×D312yes—
332C4×Q8C2×C3×D313yes—
333C4×D4—33no—
334C4×D4—36no—
335C4×D4—33no—
336C4×D4—36no—
337#117—33no—
338#117—36no—
339C2×C4⋊C4—36no—
340C2×C4⋊C4—36no—
341C2×C42—33no—
342C23×C3—33no—
343C22×D3C24×C313yes—
344C3×Q8—26no—
345C3×Q8—26no—
346C3×Q8—24no—
347C3×Q8—46no—
348C3×Q8—64no—
349C3×Q8—66no—
350C3×Q8—1212no—
351C3×Q8D3×Q814yescomplemented
352C3×Q8C2×C3×Q814yescomplemented
353C3×D4—312no—
354C2×C3×C4—23no—
355C2×C3×C4—21no—
356C2×C3×C4—22no—
357C2×C3×C4—23no—
358C2×C3×C4—26no—
359C2×C3×C4—31no—
360C2×C3×C4—31no—
361C2×C3×C4—312no—
362C2×C3×C4—32no—
363C2×C3×C4—33no—
364C2×C3×C4—36no—
365C2×C3×C4—66no—
366C2×C3×C4—66no—
367C2×C3×C4—1212no—
368C2×C3×C4C23×D313yes—
369C2×C3×C4C23×D312yes—
370C2×C3×C4C23×D316yes—
371C2×C3×C4C23×D313yes—
372C2×C3×C4C23×D311yescharacteristic
373C2×C3×C4C24×C313yes—
374C2×C3×C4C24×C316yes—
375C2×C3×C4C24×C312yes—
376C2×C3×C4C24×C313yes—
377C2×C3×C4C24×C311yescharacteristic
378C2×Dic3C24×C312yes—
379C2×Dic3C24×C313yes—
380C2×Dic3C24×C316yes—
381C2×Dic3C24×C311yescharacteristic
382C2×Dic3C24×C311yescharacteristic
383D12C2×C3×Q8112yescomplemented
384C4×D3—26no—
385C4×D3—26no—
386C4×D3—412no—
387C4×D3C2×C3×Q8112yescomplemented
388Dic6—26no—
389Dic6—24no—
390Dic6—412no—
391C2×C32C23×Q811yescharacteristic
392C2×C32C2×H(5,𝔽2)11yescharacteristic
393C2×C32C2×2−1+411yescharacteristic
394C3×D3—26no—
395#41—38no—
396C2×Q8—32no—
397C2×Q8—33no—
398C2×Q8—312no—
399C2×Q8—1212no—
400C2×Q8C22×C3×D313yes—
401C2×Q8C22×C3×D311yescharacteristic
402C2×D4—33no—
403C22×C4—33no—
404C22×C4—33no—
405C22×C4—33no—
406C22×C4—312no—
407C4⋊C4—36no—
408C4⋊C4—36no—
409C4⋊C4—36no—
410C4⋊C4—36no—
411C4⋊C4—32no—
412C4⋊C4—33no—
413C4⋊C4—31no—
414C4⋊C4—36no—
415C4⋊C4—36no—
416C4⋊C4—36no—
417C4⋊C4C22×C3×D316yes—
418C4⋊C4C22×C3×D316yes—
419C4⋊C4C22×C3×D313yes—
420C4⋊C4C22×C3×D311yescharacteristic
421C4⋊C4C22×C3×D316yes—
422C22⋊C4—36no—
423C22⋊C4—36no—
424C22⋊C4—36no—
425C42—31no—
426C42—33no—
427C42—33no—
428C42—36no—
429C42C22×C3×D313yes—
430C42C22×C3×D316yes—
431C22×C3—21no—
432C22×C3—36no—
433C22×C3—63no—
434C22×C3—63no—
435C22×C3C24×D311yescharacteristic
436C22×C3C25×C311yescharacteristic
437C2×D3—23no—
438C2×D3—23no—
439C2×D3C22×C3×Q816yes—
440C3×C4—26no—
441C3×C4—23no—
442C3×C4—22no—
443C3×C4—26no—
444C3×C4—22no—
445C3×C4—23no—
446C3×C4—26no—
447C3×C4—22no—
448C3×C4—32no—
449C3×C4—32no—
450C3×C4—46no—
451C3×C4—43no—
452C3×C4—42no—
453C3×C4—63no—
454C3×C4—66no—
455C3×C4—62no—
456C3×C4C2×D3×Q816yes—
457C3×C4#100912yes—
458C3×C4C22×C3×Q816yes—
459C3×C4C3×H(5,𝔽2)12yes—
460Dic3—26no—
461Dic3—23no—
462Dic3—22no—
463Dic3C22×C3×Q812yes—
464Dic3C3×2−1+412yes—
465C32#350311yescharacteristic, complemented, Sylow
466C23—33no—
467Q8—26no—
468Q8—64no—
469Q8—66no—
470Q8—1212no—
471Q8C3×D3×Q814yescomplemented
472D4—312no—
473C2×C4—31no—
474C2×C4—31no—
475C2×C4—312no—
476C2×C4—32no—
477C2×C4—33no—
478C2×C4—36no—
479C2×C4—66no—
480C2×C4—66no—
481C2×C4—1212no—
482C2×C4C23×C3×D313yes—
483C2×C4C23×C3×D312yes—
484C2×C4C23×C3×D316yes—
485C2×C4C23×C3×D313yes—
486C2×C4C23×C3×D311yescharacteristic
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—66no—
491C2×C3C22×D3×Q811yescharacteristic
492C2×C3C2×#100911yescharacteristic
493C2×C3D3×2−1+411yescharacteristic
494C2×C3C23×C3×Q811yescharacteristic
495C2×C3C2×C3×H(5,𝔽2)11yescharacteristic
496C2×C3C2×C3×2−1+411yescharacteristic
497D3—26no—
498C22—36no—
499C22—63no—
500C22—63no—
501C22C24×C3×D311yescharacteristic
502C4—26no—
503C4—23no—
504C4—22no—
505C4—32no—
506C4—32no—
507C4—63no—
508C4—66no—
509C4—62no—
510C4C2×C3×D3×Q816yes—
511C4C3×#100912yes—
512C3—21no—
513C3#8701811yescharacteristic, direct product factor
514C3C3×#350311yescharacteristic, complemented
515C2—66no—
516C2C22×C3×D3×Q811yescharacteristic
517C2C2×C3×#100911yescharacteristic
518C2C3×D3×2−1+411yescharacteristic
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×#557C2
socleC22×C32C25
prime radicalC2×C3×#239C22
hypercenterC3×#557D3
hypocenterC3C3×#3503
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×#557

Complements

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

NH ≅ G/Ntype
1. C32×#557C2semidirect
2. C3×#5727C2semidirect
3. C3×D3×C4⋊Q8C2semidirect
4. C3×#5690C2semidirect
5. C3×#5689C2semidirect
6. C3×#5578C2semidirect
7. C3×Q8×D12C2semidirect
10. C3×#5570C2semidirect
11. C3×Q8×Dic6C2semidirect
13. #87018C3direct (N × H)
51. C3×#557D3semidirect
52. C3×#557C2×C3semidirect
61. #5727C2×C3semidirect
62. D3×C4⋊Q8C2×C3semidirect
63. #5690C2×C3semidirect
64. #5689C2×C3semidirect
65. #5578C2×C3semidirect
66. Q8×D12C2×C3semidirect
69. #5570C2×C3semidirect
70. Q8×Dic6C2×C3semidirect
73. C3×#239Q8semidirect
183. C32×Q8C2×Q8semidirect
194. C3×D12C2×Q8semidirect
198. C3×C4×D3C2×Q8semidirect
204. #557C3×D3semidirect
229. #239C3×Q8semidirect
351. C3×Q8D3×Q8semidirect
352. C3×Q8C2×C3×Q8semidirect
383. D12C2×C3×Q8semidirect
387. C4×D3C2×C3×Q8semidirect
465. C32#3503semidirect
471. Q8C3×D3×Q8semidirect
513. C3#87018direct (N × H)
514. C3C3×#3503semidirect

Maximal subgroups

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

subgroupindex
C32×#5572
C3×#57272
C3×D3×C4⋊Q82
C3×#56902
C3×#56892
C3×#55782
C3×Q8×D122
C3×#55742
C3×C4×#2392
C3×#55702
C3×Q8×Dic62
C3×#35033
#870183