Finite Group Databasebeta

← C3×#83320 #11882234

Subgroups — C3×#83320 #11882234

Order 1152.

939 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#532C211yesmaximal, characteristic
2C3×#5470C211yesmaximal, characteristic, complemented
3C3×#5094C212yesmaximal, complemented
4C3×#5075C212yesmaximal, complemented
5C3×#5041C214yesmaximal, complemented
6C3×#5019C212yesmaximal, complemented
7C3×#5008C212yesmaximal, complemented
8C2×C3×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
9C3×#3246—31nomaximal
10#83320C311yesmaximal, characteristic, direct product factor
11C2×C32×#41—21no—
12C32×C4⋊Q8C2211yescharacteristic
13C32×#127C2211yescharacteristic
14C32×#124—21no—
15C32×#121—21no—
16C32×#121C2212yes—
17C32×C4×D4—21no—
18C32×C4×D4C2212yes—
19C2×C32×C42C2211yescharacteristic
20C2×C3×#214C2214yes—
21C3×#926—21no—
22C3×#924—21no—
23C3×#924C2212yes—
24C22×C3×C3⋊C8C2212yes—
25C3×#870—21no—
26C3×#869C2211yescharacteristic
27C3×#869C2211yescharacteristic
28C3×C4×Dic6—21no—
29C3×C4×Dic6C2212yes—
30C3×#832C2214yes—
31C3×#810C2214yes—
32C3×#808C2214yes—
33C3×C4⋊(C3⋊C8)C2212yes—
34C3×C4⋊(C3⋊C8)C2212yes—
35C3×#533—31no—
36C3×#532—21no—
37C3×#532D311yescharacteristic
38C3×#532C2×C311yescharacteristic
39#5470C2×C311yescharacteristic, complemented
40C3×#476—32no—
41C3×#475—32no—
42C3×#465—34no—
43C3×#461—32no—
44C3×#460—32no—
45C2×C3×C4⋊C8—31no—
46#5094C2×C312yescomplemented
47#5075C2×C312yescomplemented
48#5041C2×C314yescomplemented
49#5019C2×C312yescomplemented
50#5008C2×C312yescomplemented
51C2×C4⋊(C3⋊C8)C2×C311yescharacteristic, complemented
52C32×#41—44no—
53C2×C32×Q8—21no—
54C2×C32×D4—22no—
55C2×C32×D4—21no—
56C2×C32×D4—21no—
57C2×C32×D4C2312yes—
58C22×C32×C4—22no—
59C22×C32×C4D411yescharacteristic
60C22×C32×C4D411yescharacteristic
61C22×C32×C4C2311yescharacteristic
62C2×C3×Dic6—21no—
63C2×C3×Dic6C2312yes—
64C32×C4⋊C4—21no—
65C32×C4⋊C4C2312yes—
66C32×C22⋊C4—22no—
67C32×C22⋊C4—22no—
68C32×C42D411yescharacteristic
69C32×C42D411yescharacteristic
70C32×C42C2311yescharacteristic
71C32×C42C2311yescharacteristic
72C3×#217—22no—
73C3×#214—44no—
74C3×C4⋊Dic3—21no—
75C3×C4⋊Dic3—21no—
76C3×C4⋊Dic3—21no—
77C3×C4⋊Dic3C2312yes—
78C3×Dic3⋊C4—22no—
79C3×Dic3⋊C4—22no—
80C3×C4×Dic3—22no—
81C2×C3×C3⋊C8—42no—
82C2×C3×C3⋊C8C2314yes—
83#3246—31noSylow
84C2×C3×#41—22no—
85C2×C3×#41—21no—
86C2×C3×#41—21no—
87C2×C3×SD16—34no—
88C22×C3×C8—32no—
89C3×C4⋊Q8—21no—
90C3×C4⋊Q8—31no—
91C3×C4⋊Q8—31no—
92C3×C4⋊Q8C2×D311yescharacteristic
93C3×C4⋊Q8C22×C311yescharacteristic
94C3×#127—21no—
95C3×#127C2×D311yescharacteristic
96C3×#127C22×C311yescharacteristic
97C3×#125—61no—
98C3×#124—22no—
99C3×#124—21no—
100C3×#124—21no—
101C3×C22⋊Q8—32no—
102C3×C22⋊Q8—61no—
103C3×#121—22no—
104C3×#121—21no—
105C3×#121—21no—
106C3×#121—22no—
107C3×#121C2×D312yes—
108C3×#121C22×C312yes—
109C3×C4×Q8—32no—
110C3×C4×Q8—61no—
111C3×C4×D4—22no—
112C3×C4×D4—21no—
113C3×C4×D4—21no—
114C3×C4×D4—22no—
115C3×C4×D4C2×D312yes—
116C3×C4×D4C22×C312yes—
117C3×#117—61no—
118C2×C3×C42—21no—
119C2×C3×C42C2×D311yescharacteristic
120C2×C3×C42C22×C311yescharacteristic
121C2×#214C22×C314yes—
122#926—21no—
123#924—21no—
124#924C22×C312yes—
125C22×C3⋊C8C22×C312yes—
126#870—21no—
127#869C22×C311yescharacteristic
128#869C22×C311yescharacteristic
129C4×Dic6—21no—
130C4×Dic6C22×C312yes—
131C3×#106—34no—
132C3×C4⋊C8—32no—
133C3×C4⋊C8—32no—
134C3×#103—34no—
135C3×#102—34no—
136#832C22×C314yes—
137#810C22×C314yes—
138#808C22×C314yes—
139C4⋊(C3⋊C8)C22×C312yes—
140C4⋊(C3⋊C8)C22×C312yes—
141C23×C32—22no—
142C23×C32C2×D411yescharacteristic
143C32×Q8—42no—
144C32×D4—22no—
145C32×D4—22no—
146C32×D4—42no—
147C32×D4—44no—
148C32×D4—42no—
149C2×C32×C4—22no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—41no—
153C2×C32×C4—44no—
154C2×C32×C4SD1614yescomplemented
155C2×C32×C4C2×D411yescharacteristic
156C2×C32×C4C2×D411yescharacteristic
157C2×C32×C4C2×D411yescharacteristic
158C2×C32×C4C2×D411yescharacteristic
159C2×C32×C4C2×D411yescharacteristic
160C2×C32×C4C2411yescharacteristic
161C2×C3×Dic3—22no—
162C2×C3×Dic3—22no—
163C3×Dic6—22no—
164C3×Dic6—22no—
165C3×Dic6—42no—
166C3×C3⋊C8—24no—
167#533—31no—
168#532C3×D311yescharacteristic
169#476—32no—
170#475—32no—
171#465—34no—
172#461—32no—
173#460—32no—
174C2×C4⋊C8—31no—
175C3×#41—48no—
176C3×#41—44no—
177C3×#41—44no—
178C2×C3×Q8—22no—
179C2×C3×Q8—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—32no—
182C2×C3×Q8—61no—
183C2×C3×D4—22no—
184C2×C3×D4—24no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—22no—
193C2×C3×D4C22×D312yes—
194C2×C3×D4C23×C312yes—
195C22×C3×C4—24no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4#6711yescharacteristic
202C22×C3×C4#6711yescharacteristic
203C22×C3×C4C3×D411yescharacteristic
204C22×C3×C4C3×D411yescharacteristic
205C22×C3×C4C22×D311yescharacteristic
206C22×C3×C4C23×C311yescharacteristic
207C2×Dic6—21no—
208C2×Dic6C23×C312yes—
209C3×SD16—124no—
210C2×C3×C8—34no—
211C2×C3×C8—122no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—32no—
217C3×C4⋊C4—62no—
218C3×C4⋊C4—62no—
219C3×C4⋊C4—61no—
220C3×C4⋊C4—61no—
221C3×C4⋊C4—61no—
222C3×C4⋊C4C22×D312yes—
223C3×C4⋊C4C23×C312yes—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—22no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—62no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—62no—
236C3×C42#6711yescharacteristic
237C3×C42#6711yescharacteristic
238C3×C42C3×D411yescharacteristic
239C3×C42C3×D411yescharacteristic
240C3×C42C22×D311yescharacteristic
241C3×C42C22×D311yescharacteristic
242C3×C42C23×C311yescharacteristic
243C3×C42C23×C311yescharacteristic
244#217—22no—
245#214—44no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3C23×C312yes—
250Dic3⋊C4—22no—
251Dic3⋊C4—22no—
252C4×Dic3—22no—
253C2×C3⋊C8—42no—
254C2×C3⋊C8C23×C314yes—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—42no—
258C22×C32—42no—
259C22×C32—42no—
260C22×C32C2×SD1612yes—
261C22×C32C22×D411yescharacteristic
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—42no—
266C32×C4C2×SD1612yes—
267C32×C4C2×SD1612yes—
268C32×C4H(5,𝔽2)11yescharacteristic
269C32×C42−1+411yescharacteristic
270C3×Dic3—42no—
271C3×Dic3—42no—
272C2×#41—21no—
273C2×SD16—34no—
274C22×C8—32no—
275C4⋊Q8—31no—
276C4⋊Q8—31no—
277C4⋊Q8C2×C3×D311yescharacteristic
278#127C2×C3×D311yescharacteristic
279#125—61no—
280#124—21no—
281C22⋊Q8—32no—
282C22⋊Q8—61no—
283#121—21no—
284#121C2×C3×D312yes—
285C4×Q8—32no—
286C4×Q8—61no—
287C4×D4—21no—
288C4×D4C2×C3×D312yes—
289#117—61no—
290C2×C42C2×C3×D311yescharacteristic
291#106—34no—
292C4⋊C8—32no—
293C4⋊C8—32no—
294#103—34no—
295#102—34no—
296C23×C3—24no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—21no—
300C23×C3C2×#6711yescharacteristic
301C23×C3C2×C3×D411yescharacteristic
302C3×Q8—44no—
303C3×Q8—42no—
304C3×Q8—42no—
305C3×Q8—62no—
306C3×Q8—62no—
307C3×Q8—122no—
308C3×D4—22no—
309C3×D4—22no—
310C3×D4—22no—
311C3×D4—22no—
312C3×D4—42no—
313C3×D4—42no—
314C3×D4—44no—
315C3×D4—42no—
316C3×D4—42no—
317C3×D4—44no—
318C3×D4—48no—
319C3×D4—44no—
320C3×D4—44no—
321C3×D4—42no—
322C3×D4—42no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—24no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—21no—
331C2×C3×C4—24no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—42no—
342C2×C3×C4—48no—
343C2×C3×C4—44no—
344C2×C3×C4—44no—
345C2×C3×C4—62no—
346C2×C3×C4—62no—
347C2×C3×C4#21414yescomplemented
348C2×C3×C4C3×SD1614yescomplemented
349C2×C3×C4C2×#6711yescharacteristic
350C2×C3×C4C2×#6711yescharacteristic
351C2×C3×C4C2×#6711yescharacteristic
352C2×C3×C4C2×#6711yescharacteristic
353C2×C3×C4C2×#6711yescharacteristic
354C2×C3×C4C2×C3×D411yescharacteristic
355C2×C3×C4C2×C3×D411yescharacteristic
356C2×C3×C4C2×C3×D411yescharacteristic
357C2×C3×C4C2×C3×D411yescharacteristic
358C2×C3×C4C2×C3×D411yescharacteristic
359C2×C3×C4C23×D311yescharacteristic
360C2×C3×C4C24×C311yescharacteristic
361C2×Dic3—22no—
362C2×Dic3—22no—
363Dic6—22no—
364Dic6—22no—
365Dic6—42no—
366C3×C8—64no—
367C3⋊C8—24no—
368C2×C32—22no—
369C2×C32—82no—
370C2×C32#53711yescharacteristic
371C2×C32C22×SD1611yescharacteristic
372C2×C32#57511yescharacteristic
373#41—44no—
374C2×Q8—21no—
375C2×Q8—32no—
376C2×Q8—61no—
377C2×D4—22no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4C22×C3×D312yes—
381C22×C4—22no—
382C22×C4C3×#6711yescharacteristic
383C22×C4C3×#6711yescharacteristic
384C22×C4C22×C3×D311yescharacteristic
385SD16—124no—
386C2×C8—34no—
387C2×C8—122no—
388C4⋊C4—21no—
389C4⋊C4—32no—
390C4⋊C4—61no—
391C4⋊C4—61no—
392C4⋊C4—62no—
393C4⋊C4—62no—
394C4⋊C4—61no—
395C4⋊C4C22×C3×D312yes—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—62no—
399C42—62no—
400C42C3×#6711yescharacteristic
401C42C3×#6711yescharacteristic
402C42C22×C3×D311yescharacteristic
403C42C22×C3×D311yescharacteristic
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—22no—
409C22×C3—21no—
410C22×C3—41no—
411C22×C3—41no—
412C22×C3—44no—
413C22×C3—44no—
414C22×C3—44no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3C2×#21412yes—
422C22×C3C2×C3×SD1612yes—
423C22×C3C22×#6711yescharacteristic
424C22×C3C22×C3×D411yescharacteristic
425C3×C4—22no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—22no—
433C3×C4—22no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—41no—
437C3×C4—41no—
438C3×C4—44no—
439C3×C4—42no—
440C3×C4—42no—
441C3×C4—122no—
442C3×C4—122no—
443C3×C4C2×#21412yes—
444C3×C4C2×#21412yes—
445C3×C4C2×C3×SD1612yes—
446C3×C4C2×C3×SD1612yes—
447C3×C4#100911yescharacteristic
448C3×C4#101011yescharacteristic
449C3×C4C3×H(5,𝔽2)11yescharacteristic
450C3×C4C3×2−1+411yescharacteristic
451Dic3—42no—
452Dic3—42no—
453C32#324611yescharacteristic, complemented, Sylow
454C23—22no—
455C23C2×C3×#6711yescharacteristic
456Q8—42no—
457Q8—62no—
458Q8—62no—
459Q8—122no—
460D4—22no—
461D4—22no—
462D4—42no—
463D4—44no—
464D4—42no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—41no—
469C2×C4—44no—
470C2×C4—62no—
471C2×C4—62no—
472C2×C4C3×#21414yescomplemented
473C2×C4C2×C3×#6711yescharacteristic
474C2×C4C2×C3×#6711yescharacteristic
475C2×C4C2×C3×#6711yescharacteristic
476C2×C4C2×C3×#6711yescharacteristic
477C2×C4C2×C3×#6711yescharacteristic
478C2×C4C23×C3×D311yescharacteristic
479C8—64no—
480C2×C3—22no—
481C2×C3—22no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—42no—
486C2×C3—84no—
487C2×C3—82no—
488C2×C3—82no—
489C2×C3C22×#21411yescharacteristic
490C2×C3#582211yescharacteristic
491C2×C3#583311yescharacteristic
492C2×C3C3×#53711yescharacteristic
493C2×C3C22×C3×SD1611yescharacteristic
494C2×C3C3×#57511yescharacteristic
495C22—21no—
496C22—21no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22C2×C3×#21412yes—
501C22C22×C3×#6711yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—42no—
506C4—122no—
507C4—122no—
508C4C2×C3×#21412yes—
509C4C2×C3×#21412yes—
510C4C3×#100911yescharacteristic
511C4C3×#101011yescharacteristic
512C3—21no—
513C3#8332011yescharacteristic, direct product factor
514C3C3×#324611yescharacteristic, complemented
515C2—22no—
516C2—82no—
517C2C22×C3×#21411yescharacteristic
518C2C3×#582211yescharacteristic
519C2C3×#583311yescharacteristic
subgroupG/H
centerC22×C3C22×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
Fitting, prime radicalC32×#532C2
socleC22×C32C22×D4
hypercenterC3×#532D3
hypocenterC3C3×#3246
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C3×#532

Complements

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

NH ≅ G/Ntype
2. C3×#5470C2semidirect
3. C3×#5094C2semidirect
4. C3×#5075C2semidirect
5. C3×#5041C2semidirect
6. C3×#5019C2semidirect
7. C3×#5008C2semidirect
8. C2×C3×C4⋊(C3⋊C8)C2semidirect
10. #83320C3direct (N × H)
39. #5470C2×C3semidirect
46. #5094C2×C3semidirect
47. #5075C2×C3semidirect
48. #5041C2×C3semidirect
49. #5019C2×C3semidirect
50. #5008C2×C3semidirect
51. C2×C4⋊(C3⋊C8)C2×C3semidirect
154. C2×C32×C4SD16semidirect
347. C2×C3×C4#214semidirect
348. C2×C3×C4C3×SD16semidirect
453. C32#3246semidirect
472. C2×C4C3×#214semidirect
513. C3#83320direct (N × H)
514. C3C3×#3246semidirect

Maximal subgroups

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

subgroupindex
C32×#5322
C3×#54702
C3×#50942
C3×#50752
C3×#50412
C3×#50192
C3×#50082
C2×C3×C4⋊(C3⋊C8)2
C3×#32463
#833203