Finite Group Databasebeta

← C3×#83706 #11897609

Subgroups — C3×#83706 #11897609

Order 1152.

786 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×Q82C211yesmaximal, characteristic, complemented
2C3×#5579C211yesmaximal, characteristic
3C3×#5087C212yesmaximal
4C3×#5084C211yesmaximal, characteristic
5C3×#5081C212yesmaximal, complemented
6C3×#5078C211yesmaximal, characteristic, complemented
7C3×#5026C212yesmaximal
8C3×C4×#215C211yesmaximal, characteristic
9C3×#5024C211yesmaximal, characteristic, complemented
10C3×#5022C212yesmaximal, complemented
11C3×#5021C211yesmaximal, characteristic, complemented
12C3×#3385—31nomaximal
13#83706C311yesmaximal, characteristic, direct product factor
14C32×C4⋊Q8—21no—
15C32×C4⋊Q8—22no—
16C32×C4⋊Q8C2211yescharacteristic
17C32×C4⋊Q8C2212yes—
18C32×C4×Q8—21no—
19C32×C4×Q8C2211yescharacteristic
20C32×C4×Q8C2212yes—
21C32×C4×Q8C2211yescharacteristic
22C3×#946—21no—
23C2×C3×#215C2211yescharacteristic
24C3×#896—22no—
25C3×D3×C4⋊C4—21no—
26C3×C4×D12C2211yescharacteristic
27C3×C4×D12C2212yes—
28C3×#870C2212yes—
29C3×#869C2211yescharacteristic
30C3×#835C2212yes—
31C3×#835C2211yescharacteristic
32C3×#809C2211yescharacteristic
33C3×#809C2212yes—
34C3×#808C2212yes—
35C3×#808C2211yescharacteristic
36C3×#807C2212yes—
37C3×#807C2212yes—
38C3×#807C2212yes—
39C3×C4⋊(C3⋊C8)C2211yescharacteristic
40C3×C4⋊(C3⋊C8)C2212yes—
41C3×#803C2212yes—
42C3×C4×C3⋊C8C2211yescharacteristic
43C3×Q82—21no—
44C3×Q82D311yescharacteristic, complemented
45C3×Q82C2×C311yescharacteristic, complemented
46C3×#554—31no—
47#5579C2×C311yescharacteristic
48C3×#501—32no—
49C3×#500—31no—
50C3×#479—32no—
51C3×#477—31no—
52C3×#476—31no—
53C3×#474—32no—
54C3×#446—31no—
55C3×#440—32no—
56C3×C4×SD16—31no—
57#5087C2×C312yes—
58#5084C2×C311yescharacteristic
59#5081C2×C312yescomplemented
60#5078C2×C311yescharacteristic, complemented
61#5026C2×C312yes—
62C4×#215C2×C311yescharacteristic
63#5024C2×C311yescharacteristic, complemented
64#5022C2×C312yescomplemented
65#5021C2×C311yescharacteristic, complemented
66C2×C32×Q8—22no—
67C2×C32×Q8—21no—
68C2×C32×Q8D411yescharacteristic
69C2×C32×Q8C2311yescharacteristic
70C2×C3×D12C2311yescharacteristic
71C2×C3×C4×D3—21no—
72C2×C3×C4×D3—22no—
73C2×C3×Dic6—21no—
74C32×C4⋊C4—22no—
75C32×C4⋊C4—21no—
76C32×C4⋊C4—22no—
77C32×C4⋊C4—21no—
78C32×C4⋊C4D411yescharacteristic
79C32×C4⋊C4D412yes—
80C32×C4⋊C4C2311yescharacteristic
81C32×C4⋊C4C2312yes—
82C32×C42—22no—
83C32×C42—21no—
84C32×C42C2311yescharacteristic
85C32×C42C2312yes—
86C3×#215Q814yescomplemented
87C3×(C2×D3)⋊C4—21no—
88C3×(C2×D3)⋊C4—22no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3C2312yes—
91C3×C4⋊Dic3C2311yescharacteristic
92C3×Dic3⋊C4—22no—
93C3×Dic3⋊C4—22no—
94C2×C3×C3⋊C8C2311yescharacteristic
95C2×C3×C3⋊C8C2312yes—
96C2×C3×C3⋊C8C2311yescharacteristic
97#3385—31noSylow
98C2×C3×SD16—31no—
99C3×C4⋊Q8—22no—
100C3×C4⋊Q8—24no—
101C3×C4⋊Q8—21no—
102C3×C4⋊Q8—22no—
103C3×C4⋊Q8—21no—
104C3×C4⋊Q8—22no—
105C3×C4⋊Q8—22no—
106C3×C4⋊Q8—21no—
107C3×C4⋊Q8—31no—
108C3×C4⋊Q8C2×D312yes—
109C3×C4⋊Q8C2×D311yescharacteristic
110C3×C4⋊Q8C22×C311yescharacteristic
111C3×C4⋊Q8C22×C312yes—
112C3×#125—32no—
113C3×C22⋊Q8—61no—
114C3×C22⋊Q8—62no—
115C3×C4×Q8—22no—
116C3×C4×Q8—21no—
117C3×C4×Q8—21no—
118C3×C4×Q8—21no—
119C3×C4×Q8—22no—
120C3×C4×Q8—21no—
121C3×C4×Q8C2×D311yescharacteristic
122C3×C4×Q8C2×D312yes—
123C3×C4×Q8C2×D311yescharacteristic
124C3×C4×Q8C22×C311yescharacteristic
125C3×C4×Q8C22×C312yes—
126C3×C4×Q8C22×C311yescharacteristic
127C3×C4×D4—31no—
128C3×C4×D4—32no—
129C2×C3×C4⋊C4—61no—
130#946—21no—
131C2×#215C22×C311yescharacteristic
132#896—22no—
133D3×C4⋊C4—21no—
134C4×D12C22×C311yescharacteristic
135C4×D12C22×C312yes—
136#870C22×C312yes—
137#869C22×C311yescharacteristic
138C3×#107—32no—
139C3×#107—32no—
140C3×#107—32no—
141C3×#106—31no—
142C3×#106—32no—
143C3×C4⋊C8—32no—
144C3×C4⋊C8—31no—
145C3×#103—31no—
146C3×#103—32no—
147C3×#102—31no—
148C3×#102—32no—
149C3×#97—32no—
150C3×C4×C8—31no—
151#835C22×C312yes—
152#835C22×C311yescharacteristic
153#809C22×C311yescharacteristic
154#809C22×C312yes—
155#808C22×C312yes—
156#808C22×C311yescharacteristic
157#807C22×C312yes—
158#807C22×C312yes—
159#807C22×C312yes—
160C4⋊(C3⋊C8)C22×C311yescharacteristic
161C4⋊(C3⋊C8)C22×C312yes—
162#803C22×C312yes—
163C4×C3⋊C8C22×C311yescharacteristic
164C22×C3×D3—21no—
165C32×Q8—22no—
166C32×Q8—21no—
167C32×Q8—81no—
168C32×Q8—82no—
169C32×Q8C2×Q812yes—
170C2×C32×C4—21no—
171C2×C32×C4—22no—
172C2×C32×C4—21no—
173C2×C32×C4C2×D412yes—
174C2×C32×C4C2×D412yes—
175C2×C32×C4C2×D411yescharacteristic
176C2×C32×C4C2×D411yescharacteristic
177C2×C32×C4C2411yescharacteristic
178C2×C3×Dic3—21no—
179C2×C3×Dic3—22no—
180C3×D12—21no—
181C3×D12C2×Q812yes—
182C3×C4×D3—42no—
183C3×C4×D3—82no—
184C3×Dic6—81no—
185C3×C3⋊C8—21no—
186C3×C3⋊C8—22no—
187C3×C3⋊C8C2×Q812yes—
188Q82C3×D311yescharacteristic, complemented
189#554—31no—
190#501—32no—
191#500—31no—
192#479—32no—
193#477—31no—
194#476—31no—
195#474—32no—
196#446—31no—
197#440—32no—
198C4×SD16—31no—
199C2×C3×Q8—22no—
200C2×C3×Q8—24no—
201C2×C3×Q8—21no—
202C2×C3×Q8—22no—
203C2×C3×Q8—22no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—61no—
208C2×C3×Q8#6711yescharacteristic
209C2×C3×Q8C3×D411yescharacteristic
210C2×C3×Q8C22×D311yescharacteristic
211C2×C3×Q8C23×C311yescharacteristic
212C2×C3×D4—31no—
213C22×C3×C4—61no—
214C22×C3×C4—62no—
215C2×D12C23×C311yescharacteristic
216C2×C4×D3—21no—
217C2×C4×D3—22no—
218C2×Dic6—21no—
219C3×SD16—34no—
220C2×C3×C8—31no—
221C2×C3×C8—31no—
222C2×C3×C8—32no—
223C3×C4⋊C4—24no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—24no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—31no—
240C3×C4⋊C4—32no—
241C3×C4⋊C4—62no—
242C3×C4⋊C4—62no—
243C3×C4⋊C4—61no—
244C3×C4⋊C4#6711yescharacteristic
245C3×C4⋊C4#6712yes—
246C3×C4⋊C4C3×D411yescharacteristic
247C3×C4⋊C4C3×D412yes—
248C3×C4⋊C4C22×D311yescharacteristic
249C3×C4⋊C4C22×D312yes—
250C3×C4⋊C4C23×C311yescharacteristic
251C3×C4⋊C4C23×C312yes—
252C3×C22⋊C4—61no—
253C3×C22⋊C4—62no—
254C3×C42—24no—
255C3×C42—22no—
256C3×C42—22no—
257C3×C42—21no—
258C3×C42—22no—
259C3×C42—21no—
260C3×C42—21no—
261C3×C42—22no—
262C3×C42C22×D311yescharacteristic
263C3×C42C22×D312yes—
264C3×C42C23×C311yescharacteristic
265C3×C42C23×C312yes—
266#215C3×Q814yescomplemented
267(C2×D3)⋊C4—21no—
268(C2×D3)⋊C4—22no—
269C4⋊Dic3—21no—
270C4⋊Dic3C23×C312yes—
271C4⋊Dic3C23×C311yescharacteristic
272Dic3⋊C4—22no—
273Dic3⋊C4—22no—
274C2×C3⋊C8C23×C311yescharacteristic
275C2×C3⋊C8C23×C312yes—
276C2×C3⋊C8C23×C311yescharacteristic
277C22×C32C22×D411yescharacteristic
278C2×C3×D3—22no—
279C2×C3×D3—41no—
280C2×C3×D3—41no—
281C32×C4—22no—
282C32×C4—22no—
283C32×C4—21no—
284C32×C4—22no—
285C32×C4—42no—
286C32×C4—41no—
287C32×C4#13712yes—
288C32×C4C22×Q811yescharacteristic
289C32×C42−1+411yescharacteristic
290C3×Dic3—41no—
291C3×Dic3—42no—
292C2×SD16—31no—
293C4⋊Q8—21no—
294C4⋊Q8—22no—
295C4⋊Q8—31no—
296C4⋊Q8C2×C3×D312yes—
297C4⋊Q8C2×C3×D311yescharacteristic
298#125—32no—
299C22⋊Q8—61no—
300C22⋊Q8—62no—
301C4×Q8—21no—
302C4×Q8C2×C3×D311yescharacteristic
303C4×Q8C2×C3×D312yes—
304C4×Q8C2×C3×D311yescharacteristic
305C4×D4—31no—
306C4×D4—32no—
307C2×C4⋊C4—61no—
308#107—32no—
309#107—32no—
310#107—32no—
311#106—31no—
312#106—32no—
313C4⋊C8—32no—
314C4⋊C8—31no—
315#103—31no—
316#103—32no—
317#102—31no—
318#102—32no—
319#97—32no—
320C4×C8—31no—
321C23×C3—61no—
322C22×D3—21no—
323C3×Q8—24no—
324C3×Q8—22no—
325C3×Q8—22no—
326C3×Q8—22no—
327C3×Q8—21no—
328C3×Q8—21no—
329C3×Q8—22no—
330C3×Q8—81no—
331C3×Q8—82no—
332C3×Q8—82no—
333C3×Q8—81no—
334C3×Q8—84no—
335C3×Q8—82no—
336C3×Q8—241no—
337C3×Q8D3×Q812yes—
338C3×Q8C2×C3×Q812yes—
339C3×D4—32no—
340C3×D4—61no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—24no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—22no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—22no—
355C2×C3×C4—61no—
356C2×C3×C4—62no—
357C2×C3×C4—122no—
358C2×C3×C4—242no—
359C2×C3×C4C2×#6712yes—
360C2×C3×C4C2×#6711yescharacteristic
361C2×C3×C4C2×#6712yes—
362C2×C3×C4C2×#6711yescharacteristic
363C2×C3×C4C2×C3×D412yes—
364C2×C3×C4C2×C3×D412yes—
365C2×C3×C4C2×C3×D411yescharacteristic
366C2×C3×C4C2×C3×D411yescharacteristic
367C2×C3×C4C23×D311yescharacteristic
368C2×C3×C4C24×C311yescharacteristic
369C2×Dic3—21no—
370C2×Dic3—22no—
371D12—21no—
372D12C2×C3×Q812yes—
373C4×D3—42no—
374C4×D3—82no—
375Dic6—81no—
376C3×C8—32no—
377C3×C8—61no—
378C3×C8—62no—
379C3⋊C8—21no—
380C3⋊C8—22no—
381C3⋊C8C2×C3×Q812yes—
382C2×C32D4×Q811yescharacteristic
383C2×C32C2×#13711yescharacteristic
384C2×C32#57611yescharacteristic
385C3×D3—42no—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—61no—
389C2×Q8C3×#6711yescharacteristic
390C2×Q8C22×C3×D311yescharacteristic
391C2×D4—31no—
392C22×C4—61no—
393C22×C4—62no—
394SD16—34no—
395C2×C8—31no—
396C2×C8—31no—
397C2×C8—32no—
398C4⋊C4—22no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—31no—
403C4⋊C4—32no—
404C4⋊C4—62no—
405C4⋊C4—61no—
406C4⋊C4—62no—
407C4⋊C4C3×#6711yescharacteristic
408C4⋊C4C3×#6712yes—
409C4⋊C4C22×C3×D311yescharacteristic
410C4⋊C4C22×C3×D312yes—
411C22⋊C4—61no—
412C22⋊C4—62no—
413C42—22no—
414C42—21no—
415C42C22×C3×D311yescharacteristic
416C42C22×C3×D312yes—
417C22×C3—21no—
418C22×C3—62no—
419C22×C3—121no—
420C22×C3—121no—
421C22×C3C22×#6711yescharacteristic
422C22×C3C22×C3×D411yescharacteristic
423C2×D3—22no—
424C2×D3—41no—
425C2×D3—41no—
426C3×C4—24no—
427C3×C4—24no—
428C3×C4—22no—
429C3×C4—21no—
430C3×C4—22no—
431C3×C4—22no—
432C3×C4—22no—
433C3×C4—22no—
434C3×C4—21no—
435C3×C4—22no—
436C3×C4—22no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—42no—
440C3×C4—41no—
441C3×C4—42no—
442C3×C4—44no—
443C3×C4—42no—
444C3×C4—41no—
445C3×C4—41no—
446C3×C4—42no—
447C3×C4—121no—
448C3×C4—122no—
449C3×C4#94212yes—
450C3×C4C3×#13712yes—
451C3×C4C2×D3×Q811yescharacteristic
452C3×C4#100711yescharacteristic
453C3×C4C22×C3×Q811yescharacteristic
454C3×C4C3×2−1+411yescharacteristic
455Dic3—41no—
456Dic3—42no—
457C32#338511yescharacteristic, complemented, Sylow
458C23—61no—
459Q8—22no—
460Q8—21no—
461Q8—82no—
462Q8—81no—
463Q8—241no—
464Q8C3×D3×Q812yes—
465D4—32no—
466D4—61no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—61no—
471C2×C4—62no—
472C2×C4—122no—
473C2×C4—242no—
474C2×C4C2×C3×#6712yes—
475C2×C4C2×C3×#6711yescharacteristic
476C2×C4C2×C3×#6712yes—
477C2×C4C2×C3×#6711yescharacteristic
478C2×C4C23×C3×D311yescharacteristic
479C8—32no—
480C8—61no—
481C8—62no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—122no—
486C2×C3C2×#94211yescharacteristic
487C2×C3Q8×#6711yescharacteristic
488C2×C3#583511yescharacteristic
489C2×C3C3×D4×Q811yescharacteristic
490C2×C3C2×C3×#13711yescharacteristic
491C2×C3C3×#57611yescharacteristic
492D3—42no—
493C22—62no—
494C22—121no—
495C22—121no—
496C22C22×C3×#6711yescharacteristic
497C4—22no—
498C4—22no—
499C4—22no—
500C4—21no—
501C4—41no—
502C4—42no—
503C4—121no—
504C4—122no—
505C4C3×#94212yes—
506C4C2×C3×D3×Q811yescharacteristic
507C4C3×#100711yescharacteristic
508C3—21no—
509C3#8370611yescharacteristic, direct product factor
510C3C3×#338511yescharacteristic, complemented
511C2—122no—
512C2C2×C3×#94211yescharacteristic
513C2C3×Q8×#6711yescharacteristic
514C2C3×#583511yescharacteristic
subgroupG/H
centerC22×C3C22×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×Q82C2
socleC22×C32C22×D4
prime radicalC2×C3×D12C23
hypercenterC3×Q82D3
hypocenterC3C3×#3385
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C4×Q8 ◁ C3×Q82

Complements

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

NH ≅ G/Ntype
1. C32×Q82C2semidirect
5. C3×#5081C2semidirect
6. C3×#5078C2semidirect
9. C3×#5024C2semidirect
10. C3×#5022C2semidirect
11. C3×#5021C2semidirect
13. #83706C3direct (N × H)
44. C3×Q82D3semidirect
45. C3×Q82C2×C3semidirect
59. #5081C2×C3semidirect
60. #5078C2×C3semidirect
63. #5024C2×C3semidirect
64. #5022C2×C3semidirect
65. #5021C2×C3semidirect
86. C3×#215Q8semidirect
188. Q82C3×D3semidirect
266. #215C3×Q8semidirect
457. C32#3385semidirect
509. C3#83706direct (N × H)
510. C3C3×#3385semidirect

Maximal subgroups

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

subgroupindex
C32×Q822
C3×#55792
C3×#50872
C3×#50842
C3×#50812
C3×#50782
C3×#50262
C3×C4×#2152
C3×#50242
C3×#50222
C3×#50212
C3×#33853
#837063