Finite Group Databasebeta

← C3×#87292 #11906004

Subgroups — C3×#87292 #11906004

Order 1152.

3349 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
1C2×C32×2−1+4C211yesmaximal, characteristic, complemented
2C2×C3×#1010C211yesmaximal, characteristic, complemented
3C3×#5832C213yesmaximal, complemented
4C3×Dic3×#41C211yesmaximal, characteristic, complemented
5C3×#5825C213yesmaximal, complemented
6C3×#5824C2112yesmaximal, complemented
7C3×#5816C213yesmaximal, complemented
8C3×Q8×#67C214yesmaximal, complemented
9C2×C3×#944C213yesmaximal, complemented
10C3×#3499—31nomaximal
11#87292C311yesmaximal, characteristic, direct product factor
12C32×2−1+4—28no—
13C2×C32×#41C2216yes—
14C2×C32×#41C2213yes—
15C2×C32×#41C2211yescharacteristic
16C22×C32×Q8C2213yes—
17C22×C32×Q8C2212yes—
18C3×#1010—28no—
19C2×C3×D3×Q8C2212yes—
20C2×C3×#237C2216yes—
21C2×C3×#235C2213yes—
22C22×C3×Dic6C2213yes—
23C3×#947C2213yes—
24C3×#946C22112yes—
25C3×Q8×Dic3C2212yes—
26C3×#944C2213yes—
27C3×#944C22112yes—
28C3×#944C2216yes—
29C3×#940C2211yescharacteristic
30C3×#939C22112yes—
31C3×#936C2213yes—
32C3×#935C22112yes—
33C3×D4×Dic3C2216yes—
34C3×#929C22112yes—
35C3×C4×#67C22112yes—
36C3×#926C2213yes—
37C3×#924C22112yes—
38C2×C3×Dic3⋊C4C22112yes—
39C2×C3×C4×Dic3C2213yes—
40C2×C3×2−1+4—21no—
41C2×C3×2−1+4—31no—
42C2×C3×2−1+4D311yescharacteristic, complemented
43C2×C3×2−1+4C2×C311yescharacteristic, complemented
44C2×#1010C2×C311yescharacteristic, complemented
45C3×D4×Q8—34no—
46C3×#547—312no—
47C3×#536—33no—
48C3×#536—33no—
49C3×#532—33no—
50C2×C3×C4⋊Q8—33no—
51C3×C4×#41—31no—
52#5832C2×C313yescomplemented
53Dic3×#41C2×C311yescharacteristic, complemented
54#5825C2×C313yescomplemented
55#5824C2×C3112yescomplemented
56#5816C2×C313yescomplemented
57Q8×#67C2×C314yescomplemented
58C2×#944C2×C313yescomplemented
59C32×#41—224no—
60C32×#41—212no—
61C32×#41D418yescomplemented
62C2×C32×Q8—28no—
63C2×C32×Q8—212no—
64C2×C32×Q8C2313yes—
65C2×C32×Q8C2316yes—
66C2×C32×Q8C2311yescharacteristic
67C2×C32×D4C2316yes—
68C2×C32×D4C2311yescharacteristic
69C2×C32×D4C2313yes—
70C22×C32×C4C2316yes—
71C22×C32×C4C2316yes—
72C22×C32×C4C2313yes—
73C2×C3×#67C2314yes—
74C2×C3×#67C2316yes—
75C22×C3×Dic3C2316yes—
76C22×C3×Dic3C2316yes—
77C3×D3×Q8—28no—
78C3×#237—224no—
79C3×#235—212no—
80C2×C3×D12C2311yescharacteristic
81C2×C3×C4×D3C2316yes—
82C2×C3×Dic6—212no—
83C2×C3×Dic6C2316yes—
84C2×C3×Dic6C2313yes—
85C3×#217C23112yes—
86C3×#217C2316yes—
87C3×(C2×D3)⋊C4C23112yes—
88C3×C4⋊Dic3C2316yes—
89C3×Dic3⋊C4C23112yes—
90C3×Dic3⋊C4C23124yes—
91C3×C4×Dic3C2311yescharacteristic
92C3×C4×Dic3C2313yes—
93C3×C4×Dic3C2316yes—
94#3499—31noSylow
95C3×2−1+4—216no—
96C3×2−1+4—28no—
97C3×2−1+4—28no—
98C3×2−1+4—68no—
99C2×C3×#41—23no—
100C2×C3×#41—26no—
101C2×C3×#41—21no—
102C2×C3×#41—36no—
103C2×C3×#41—33no—
104C2×C3×#41C2×D313yes—
105C2×C3×#41C2×D316yes—
106C2×C3×#41C2×D311yescharacteristic
107C2×C3×#41C22×C316yes—
108C2×C3×#41C22×C313yes—
109C2×C3×#41C22×C311yescharacteristic
110C22×C3×Q8—23no—
111C22×C3×Q8—22no—
112C22×C3×Q8—32no—
113C22×C3×Q8—33no—
114C22×C3×Q8C2×D313yes—
115C22×C3×Q8C2×D312yes—
116C22×C3×Q8C22×C313yes—
117C22×C3×Q8C22×C312yes—
118#1010—28no—
119C2×D3×Q8C22×C312yes—
120C2×#237C22×C316yes—
121C2×#235C22×C313yes—
122C22×Dic6C22×C313yes—
123C3×C4⋊Q8—33no—
124C3×C4⋊Q8—36no—
125C3×C4⋊Q8—312no—
126C3×#127—31no—
127C3×#124—33no—
128C3×#124—33no—
129C3×#123—312no—
130C3×#123—312no—
131C3×C22⋊Q8—312no—
132C3×C22⋊Q8—312no—
133C3×#121—312no—
134C3×C4×Q8—32no—
135C3×C4×D4—36no—
136C3×C4×D4—312no—
137C3×#117—33no—
138C2×C3×C4⋊C4—312no—
139C2×C3×C42—33no—
140#947C22×C313yes—
141#946C22×C3112yes—
142Q8×Dic3C22×C312yes—
143#944C22×C313yes—
144#944C22×C3112yes—
145#944C22×C316yes—
146#940C22×C311yescharacteristic
147#939C22×C3112yes—
148#936C22×C313yes—
149#935C22×C3112yes—
150D4×Dic3C22×C316yes—
151#929C22×C3112yes—
152C4×#67C22×C3112yes—
153#926C22×C313yes—
154#924C22×C3112yes—
155C2×Dic3⋊C4C22×C3112yes—
156C2×C4×Dic3C22×C313yes—
157C23×C32C2412yes—
158C23×C32C2413yes—
159C22×C3×D3C2412yes—
160C32×Q8—26no—
161C32×Q8—212no—
162C32×Q8C2×D414yes—
163C32×D4—212no—
164C32×D4—22no—
165C32×D4C2×D4112yes—
166C2×C32×C4—212no—
167C2×C32×C4—212no—
168C2×C32×C4C2×D4112yes—
169C2×C32×C4C2416yes—
170C2×C32×C4C2411yescharacteristic
171C2×C32×C4C2413yes—
172C3×#67—212no—
173C3×#67—28no—
174C2×C3×Dic3—212no—
175C2×C3×Dic3—212no—
176C2×C3×Dic3C2416yes—
177C2×C3×Dic3C2416yes—
178C2×C3×Dic3C2412yes—
179C3×D12—22no—
180C3×C4×D3—212no—
181C3×Dic6—26no—
182C3×Dic6—212no—
183C2×2−1+4—31no—
184C2×2−1+4C3×D311yescharacteristic, complemented
185D4×Q8—34no—
186#547—312no—
187#536—33no—
188#536—33no—
189#532—33no—
190C2×C4⋊Q8—33no—
191C4×#41—31no—
192C3×#41—248no—
193C3×#41—224no—
194C3×#41—224no—
195C3×#41—224no—
196C3×#41—212no—
197C3×#41—212no—
198C3×#41—28no—
199C3×#41—624no—
200C3×#41—612no—
201C3×#41#6718yescomplemented
202C3×#41C3×D418yescomplemented
203C2×C3×Q8—216no—
204C2×C3×Q8—224no—
205C2×C3×Q8—28no—
206C2×C3×Q8—28no—
207C2×C3×Q8—212no—
208C2×C3×Q8—212no—
209C2×C3×Q8—26no—
210C2×C3×Q8—23no—
211C2×C3×Q8—21no—
212C2×C3×Q8—36no—
213C2×C3×Q8—33no—
214C2×C3×Q8—68no—
215C2×C3×Q8—612no—
216C2×C3×Q8C22×D316yes—
217C2×C3×Q8C22×D313yes—
218C2×C3×Q8C22×D311yescharacteristic
219C2×C3×Q8C23×C313yes—
220C2×C3×Q8C23×C316yes—
221C2×C3×Q8C23×C311yescharacteristic
222C2×C3×D4—26no—
223C2×C3×D4—21no—
224C2×C3×D4—23no—
225C2×C3×D4—34no—
226C2×C3×D4—36no—
227C2×C3×D4—31no—
228C2×C3×D4C22×D316yes—
229C2×C3×D4C22×D311yescharacteristic
230C2×C3×D4C22×D313yes—
231C2×C3×D4C23×C316yes—
232C2×C3×D4C23×C311yescharacteristic
233C2×C3×D4C23×C313yes—
234C22×C3×C4—26no—
235C22×C3×C4—26no—
236C22×C3×C4—23no—
237C22×C3×C4—36no—
238C22×C3×C4—36no—
239C22×C3×C4—36no—
240C22×C3×C4C22×D316yes—
241C22×C3×C4C22×D316yes—
242C22×C3×C4C22×D313yes—
243C22×C3×C4C23×C316yes—
244C22×C3×C4C23×C316yes—
245C22×C3×C4C23×C313yes—
246C2×#67C23×C314yes—
247C2×#67C23×C316yes—
248C22×Dic3C23×C316yes—
249C22×Dic3C23×C316yes—
250D3×Q8—28no—
251#237—224no—
252#235—212no—
253C2×D12C23×C311yescharacteristic
254C2×C4×D3C23×C316yes—
255C2×Dic6—212no—
256C2×Dic6C23×C316yes—
257C2×Dic6C23×C313yes—
258C3×C4⋊C4—312no—
259C3×C4⋊C4—324no—
260C3×C4⋊C4—36no—
261C3×C22⋊C4—36no—
262C3×C22⋊C4—312no—
263C3×C22⋊C4—312no—
264C3×C42—31no—
265C3×C42—33no—
266C3×C42—36no—
267#217C23×C3112yes—
268#217C23×C316yes—
269(C2×D3)⋊C4C23×C3112yes—
270C4⋊Dic3C23×C316yes—
271Dic3⋊C4C23×C3112yes—
272Dic3⋊C4C23×C3124yes—
273C4×Dic3C23×C311yescharacteristic
274C4×Dic3C23×C313yes—
275C4×Dic3C23×C316yes—
276C22×C32—22no—
277C22×C32—26no—
278C22×C32—24no—
279C22×C32C22×D416yes—
280C22×C32C2511yescharacteristic
281C2×C3×D3—22no—
282C2×C3×D3—24no—
283C32×C4—26no—
284C32×C4C22×D412yes—
285C32×C4C22×D416yes—
286C3×Dic3—26no—
287C3×Dic3—26no—
288C3×Dic32−1+414yescomplemented
2892−1+4—28no—
2902−1+4—68no—
291C2×#41—36no—
292C2×#41—33no—
293C2×#41C2×C3×D313yes—
294C2×#41C2×C3×D316yes—
295C2×#41C2×C3×D311yescharacteristic
296C22×Q8—32no—
297C22×Q8—33no—
298C22×Q8C2×C3×D313yes—
299C22×Q8C2×C3×D312yes—
300C4⋊Q8—33no—
301C4⋊Q8—36no—
302C4⋊Q8—312no—
303#127—31no—
304#124—33no—
305#124—33no—
306#123—312no—
307#123—312no—
308C22⋊Q8—312no—
309C22⋊Q8—312no—
310#121—312no—
311C4×Q8—32no—
312C4×D4—36no—
313C4×D4—312no—
314#117—33no—
315C2×C4⋊C4—312no—
316C2×C42—33no—
317C23×C3—22no—
318C23×C3—23no—
319C23×C3—32no—
320C23×C3C23×D312yes—
321C23×C3C23×D313yes—
322C23×C3C24×C312yes—
323C23×C3C24×C313yes—
324C22×D3C24×C312yes—
325C3×Q8—224no—
326C3×Q8—212no—
327C3×Q8—212no—
328C3×Q8—26no—
329C3×Q8—26no—
330C3×Q8—212no—
331C3×Q8—24no—
332C3×Q8—612no—
333C3×Q8—66no—
334C3×Q8C2×#6714yes—
335C3×Q8C2×C3×D414yes—
336C3×D4—224no—
337C3×D4—24no—
338C3×D4—212no—
339C3×D4—212no—
340C3×D4—22no—
341C3×D4—22no—
342C3×D4—212no—
343C3×D4—68no—
344C3×D4—612no—
345C3×D4—62no—
346C3×D4C2×#67112yes—
347C3×D4C2×C3×D4112yes—
348C2×C3×C4—224no—
349C2×C3×C4—224no—
350C2×C3×C4—212no—
351C2×C3×C4—212no—
352C2×C3×C4—212no—
353C2×C3×C4—212no—
354C2×C3×C4—26no—
355C2×C3×C4—21no—
356C2×C3×C4—23no—
357C2×C3×C4—212no—
358C2×C3×C4—32no—
359C2×C3×C4—36no—
360C2×C3×C4—36no—
361C2×C3×C4—612no—
362C2×C3×C4—612no—
363C2×C3×C4—612no—
364C2×C3×C4C2×#67112yes—
365C2×C3×C4C2×C3×D4112yes—
366C2×C3×C4C23×D316yes—
367C2×C3×C4C23×D311yescharacteristic
368C2×C3×C4C23×D313yes—
369C2×C3×C4C24×C316yes—
370C2×C3×C4C24×C311yescharacteristic
371C2×C3×C4C24×C313yes—
372#67—212no—
373#67—28no—
374C2×Dic3—212no—
375C2×Dic3—212no—
376C2×Dic3C24×C316yes—
377C2×Dic3C24×C316yes—
378C2×Dic3C24×C312yes—
379D12—22no—
380C4×D3—212no—
381Dic6—26no—
382Dic6—212no—
383C2×C32—26no—
384C2×C32—42no—
385C2×C32C23×D411yescharacteristic
386C2×C32C2×2−1+412yes—
387C3×D3—42no—
388#41—224no—
389#41—212no—
390#41—624no—
391#41—612no—
392#41C3×#6718yescomplemented
393C2×Q8—28no—
394C2×Q8—212no—
395C2×Q8—36no—
396C2×Q8—33no—
397C2×Q8—68no—
398C2×Q8—612no—
399C2×Q8C22×C3×D316yes—
400C2×Q8C22×C3×D313yes—
401C2×Q8C22×C3×D311yescharacteristic
402C2×D4—34no—
403C2×D4—36no—
404C2×D4—31no—
405C2×D4C22×C3×D316yes—
406C2×D4C22×C3×D311yescharacteristic
407C2×D4C22×C3×D313yes—
408C22×C4—36no—
409C22×C4—36no—
410C22×C4—36no—
411C22×C4C22×C3×D316yes—
412C22×C4C22×C3×D316yes—
413C22×C4C22×C3×D313yes—
414C4⋊C4—312no—
415C4⋊C4—324no—
416C4⋊C4—36no—
417C22⋊C4—36no—
418C22⋊C4—312no—
419C22⋊C4—312no—
420C42—31no—
421C42—33no—
422C42—36no—
423C22×C3—26no—
424C22×C3—26no—
425C22×C3—24no—
426C22×C3—24no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—24no—
430C22×C3—26no—
431C22×C3—21no—
432C22×C3—46no—
433C22×C3—44no—
434C22×C3—64no—
435C22×C3—62no—
436C22×C3C22×#6716yes—
437C22×C3C22×C3×D416yes—
438C22×C3C24×D311yescharacteristic
439C22×C3C25×C311yescharacteristic
440C2×D3—22no—
441C2×D3—24no—
442C3×C4—212no—
443C3×C4—26no—
444C3×C4—26no—
445C3×C4—26no—
446C3×C4—22no—
447C3×C4—34no—
448C3×C4—66no—
449C3×C4—66no—
450C3×C4C22×#6716yes—
451C3×C4C22×#6712yes—
452C3×C4C22×C3×D412yes—
453C3×C4C22×C3×D416yes—
454Dic3—26no—
455Dic3—26no—
456Dic3C3×2−1+414yescomplemented
457C32#349911yescharacteristic, complemented, Sylow
458C23—32no—
459C23C23×C3×D312yes—
460C23C23×C3×D313yes—
461Q8—212no—
462Q8—26no—
463Q8—612no—
464Q8—66no—
465Q8C2×C3×#6714yes—
466D4—212no—
467D4—22no—
468D4—68no—
469D4—612no—
470D4—62no—
471D4C2×C3×#67112yes—
472C2×C4—212no—
473C2×C4—212no—
474C2×C4—32no—
475C2×C4—36no—
476C2×C4—36no—
477C2×C4—612no—
478C2×C4—612no—
479C2×C4—612no—
480C2×C4C2×C3×#67112yes—
481C2×C4C23×C3×D316yes—
482C2×C4C23×C3×D311yescharacteristic
483C2×C4C23×C3×D313yes—
484C2×C3—26no—
485C2×C3—26no—
486C2×C3—22no—
487C2×C3—21no—
488C2×C3—46no—
489C2×C3—44no—
490C2×C3—42no—
491C2×C3—42no—
492C2×C3—122no—
493C2×C3D3×2−1+412yes—
494C2×C3C23×#6711yescharacteristic
495C2×C3C23×C3×D411yescharacteristic
496C2×C3C2×C3×2−1+412yes—
497D3—42no—
498C22—26no—
499C22—24no—
500C22—22no—
501C22—64no—
502C22—62no—
503C22C22×C3×#6716yes—
504C22C24×C3×D311yescharacteristic
505C4—26no—
506C4—34no—
507C4—66no—
508C4—66no—
509C4C22×C3×#6716yes—
510C4C22×C3×#6712yes—
511C3—21no—
512C3#8729211yescharacteristic, direct product factor
513C3C3×#349911yescharacteristic, complemented
514C2—26no—
515C2—42no—
516C2—122no—
517C2C3×D3×2−1+412yes—
518C2C23×C3×#6711yescharacteristic
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×2−1+4C2
socleC22×C32C25
hypercenterC2×C3×2−1+4D3
hypocenterC3C3×#3499
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×C32×2−1+4C2semidirect
2. C2×C3×#1010C2semidirect
3. C3×#5832C2semidirect
4. C3×Dic3×#41C2semidirect
5. C3×#5825C2semidirect
6. C3×#5824C2semidirect
7. C3×#5816C2semidirect
8. C3×Q8×#67C2semidirect
9. C2×C3×#944C2semidirect
11. #87292C3direct (N × H)
42. C2×C3×2−1+4D3semidirect
43. C2×C3×2−1+4C2×C3semidirect
44. C2×#1010C2×C3semidirect
52. #5832C2×C3semidirect
53. Dic3×#41C2×C3semidirect
54. #5825C2×C3semidirect
55. #5824C2×C3semidirect
56. #5816C2×C3semidirect
57. Q8×#67C2×C3semidirect
58. C2×#944C2×C3semidirect
61. C32×#41D4semidirect
184. C2×2−1+4C3×D3semidirect
201. C3×#41#67semidirect
202. C3×#41C3×D4semidirect
288. C3×Dic32−1+4semidirect
392. #41C3×#67semidirect
456. Dic3C3×2−1+4semidirect
457. C32#3499semidirect
512. C3#87292direct (N × H)
513. C3C3×#3499semidirect

Maximal subgroups

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

subgroupindex
C2×C32×2−1+42
C2×C3×#10102
C3×#58322
C3×Dic3×#412
C3×#58252
C3×#58242
C3×#58162
C3×Q8×#672
C2×C3×#9442
C3×#34993
#872923