Finite Group Databasebeta

← C3×#10591623 #13158625

Subgroups — C3×#10591623 #13158625

Order 1728.

582 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#944C211yesmaximal, characteristic
2C3×#63918C211yesmaximal, characteristic
3C3×#63662C211yesmaximal, characteristic
4C32×#835C211yesmaximal, characteristic
5C3×#63566C211yesmaximal, characteristic
6C3×#63565C211yesmaximal, characteristic
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic
8C3×#5079—31nomaximal
9C3×#4792—31nomaximal
10#10591623C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C3×#627—21no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×Dic3⋊C4—21no—
18C32×Dic3⋊C4—21no—
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22C32×C4⋊Q8—31no—
23C3×#944—21no—
24C3×#944D311yescharacteristic
25C3×#944C2×C311yescharacteristic
26C3×#888—31no—
27C3×#869—31no—
28#63918C2×C311yescharacteristic
29C32×#103—31no—
30#63662C2×C311yescharacteristic
31C3×#835—21no—
32C3×#835—31no—
33C3×#835—31no—
34C3×#835—31no—
35C3×#835—31no—
36C3×#835—61no—
37C3×#835D311yescharacteristic
38C3×#835C2×C311yescharacteristic
39C3×#816—31no—
40C3×C8×Dic3—31no—
41C3×#810—31no—
42C3×C4×C3⋊C8—31no—
43#63566C2×C311yescharacteristic
44#63565C2×C311yescharacteristic
45Dic3×C3⋊C8C2×C311yescharacteristic
46C33×Q8—21no—
47C33×Q8—21no—
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C32×Dic6—21no—
56C32×Dic6—21no—
57C3×#627—42no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60C3×#487—91no—
61#5079—31no—
62#4792—31no—
63C2×C32×Q8—21no—
64C2×C32×Q8—21no—
65C2×C32×Q8—21no—
66C2×C32×Q8—31no—
67C2×C32×Q8—41no—
68C2×C32×Q8C2×D311yescharacteristic
69C2×C32×Q8C2×D311yescharacteristic
70C2×C32×Q8C22×C311yescharacteristic
71C2×C3×Dic6—21no—
72C2×C3×Dic6—61no—
73C2×C3×Dic6—61no—
74C2×C3×Dic6C2×D311yescharacteristic
75C2×C3×Dic6C22×C311yescharacteristic
76C2×#627—21no—
77C2×C32×C8—31no—
78C32×C4⋊C4—31no—
79C32×C4⋊C4—61no—
80C32×C4⋊C4—61no—
81C32×C42—31no—
82C4⋊(C3⋊Dic3)C22×C311yescharacteristic
83C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
84C3×C4⋊Dic3—21no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—61no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C22×C311yescharacteristic
91C3×Dic3⋊C4—22no—
92C3×Dic3⋊C4—22no—
93C3×Dic3⋊C4—21no—
94C3×Dic3⋊C4—21no—
95C3×Dic3⋊C4—21no—
96C3×Dic3⋊C4—21no—
97C3×Dic3⋊C4—61no—
98C3×C4×Dic3—21no—
99C3×C4×Dic3C2×D311yescharacteristic
100C3×C4×Dic3C22×C311yescharacteristic
101C2×C3×C3⋊C8—21no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—61no—
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C2×C3×C3⋊C8C22×C311yescharacteristic
107Dic3⋊Dic3—21no—
108C22×C33C2×D411yescharacteristic
109C33×C4—41no—
110C33×C4#4111yescharacteristic
111C33×C4#4111yescharacteristic
112C3×C3⋊Dic3—41no—
113C32×Dic3—41no—
114C32×Dic3—41no—
115C32×Dic3SD1612yes—
116C32×Dic3Q1612yes—
117C3×C4⋊Q8—31no—
118C3×C4⋊Q8—31no—
119C3×C4⋊Q8—61no—
120C3×C4⋊Q8—91no—
121#944C3×D311yescharacteristic
122#888—31no—
123#869—31no—
124C3×#103—31no—
125C3×#103—31no—
126C3×#103—61no—
127C3×#103—91no—
128C3×#103—91no—
129C3×#103—91no—
130C3×C4×C8—91no—
131#835—31no—
132#835—31no—
133#835—31no—
134#835—31no—
135#835—61no—
136#835C3×D311yescharacteristic
137#816—31no—
138C8×Dic3—31no—
139#810—31no—
140C4×C3⋊C8—31no—
141C32×Q8—21no—
142C32×Q8—21no—
143C32×Q8—22no—
144C32×Q8—22no—
145C32×Q8—21no—
146C32×Q8—21no—
147C32×Q8—21no—
148C32×Q8—21no—
149C32×Q8—41no—
150C32×Q8—41no—
151C32×Q8—41no—
152C32×Q8—41no—
153C32×Q8—42no—
154C32×Q8—42no—
155C32×Q8—61no—
156C32×Q8—61no—
157C2×C32×C4—21no—
158C2×C32×C4—22no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—31no—
165C2×C32×C4—31no—
166C2×C32×C4—41no—
167C2×C32×C4—41no—
168C2×C32×C4—41no—
169C2×C32×C4—42no—
170C2×C32×C4—61no—
171C2×C32×C4—61no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C23×C311yescharacteristic
175C2×C3⋊Dic3—21no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—41no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—121no—
187C2×C3×Dic3#6711yescharacteristic
188C2×C3×Dic3#6711yescharacteristic
189C2×C3×Dic3C3×D411yescharacteristic
190C2×C3×Dic3C3×D411yescharacteristic
191C3×Dic6—21no—
192C3×Dic6—21no—
193C3×Dic6—21no—
194C3×Dic6—21no—
195C3×Dic6—41no—
196C3×Dic6—41no—
197C3×Dic6—122no—
198C3×Dic6—122no—
199#627—42no—
200C32×C8—61no—
201C3⋊(C3⋊C8)—21no—
202C3×C3⋊C8—21no—
203C3×C3⋊C8—21no—
204C3×C3⋊C8—41no—
205C3×C3⋊C8—61no—
206C3×C3⋊C8—61no—
207C3×C3⋊C8—121no—
208#487—91noSylow
209C2×C33#12411yescharacteristic
210C2×C33C2×SD1611yescharacteristic
211C2×C33C2×Q1611yescharacteristic
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8—31no—
216C2×C3×Q8—31no—
217C2×C3×Q8—41no—
218C2×C3×Q8—61no—
219C2×C3×Q8—181no—
220C2×C3×Q8D3211yescharacteristic
221C2×C3×Q8C2×C3×D311yescharacteristic
222C2×C3×Q8C2×C3×D311yescharacteristic
223C2×Dic6—61no—
224C2×Dic6—61no—
225C2×Dic6C2×C3×D311yescharacteristic
226C2×C3×C8—31no—
227C2×C3×C8—31no—
228C2×C3×C8—61no—
229C2×C3×C8—91no—
230C3×C4⋊C4—31no—
231C3×C4⋊C4—31no—
232C3×C4⋊C4—61no—
233C3×C4⋊C4—61no—
234C3×C4⋊C4—61no—
235C3×C4⋊C4—61no—
236C3×C4⋊C4—61no—
237C3×C4⋊C4—62no—
238C3×C4⋊C4—62no—
239C3×C4⋊C4—91no—
240C3×C4⋊C4—181no—
241C3×C42—31no—
242C3×C42—31no—
243C3×C42—61no—
244C4⋊Dic3—31no—
245C4⋊Dic3—31no—
246C4⋊Dic3—61no—
247C4⋊Dic3—61no—
248C4⋊Dic3C2×C3×D311yescharacteristic
249Dic3⋊C4—21no—
250Dic3⋊C4—21no—
251Dic3⋊C4—61no—
252C4×Dic3C2×C3×D311yescharacteristic
253C2×C3⋊C8—31no—
254C2×C3⋊C8—31no—
255C2×C3⋊C8—61no—
256C2×C3⋊C8C2×C3×D311yescharacteristic
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32—41no—
261C22×C32D3×D411yescharacteristic
262C22×C32C2×#6711yescharacteristic
263C22×C32C2×C3×D411yescharacteristic
264C32×C4—21no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—32no—
271C32×C4—32no—
272C32×C4—41no—
273C32×C4—41no—
274C32×C4—42no—
275C32×C4—42no—
276C32×C4—42no—
277C32×C4—44no—
278C32×C4—41no—
279C32×C4—41no—
280C32×C4—41no—
281C32×C4—121no—
282C32×C4—121no—
283C32×C4#23511yescharacteristic
284C32×C4#23711yescharacteristic
285C32×C4#23711yescharacteristic
286C32×C4#23711yescharacteristic
287C32×C4C3×#4111yescharacteristic
288C32×C4C3×#4111yescharacteristic
289C3⋊Dic3—41no—
290C3×Dic3—22no—
291C3×Dic3—22no—
292C3×Dic3—42no—
293C3×Dic3—41no—
294C3×Dic3—41no—
295C3×Dic3—41no—
296C3×Dic3—41no—
297C3×Dic3—81no—
298C3×Dic3—121no—
299C3×Dic3—121no—
300C3×Dic3—241no—
301C3×Dic3#21512yes—
302C3×Dic3#21612yes—
303C3×Dic3C3×SD1612yes—
304C3×Dic3C3×Q1612yes—
305C4⋊Q8—31no—
306C4⋊Q8—91no—
307#103—31no—
308#103—91no—
309#103—91no—
310#103—91no—
311C4×C8—91no—
312C33#48711yescharacteristic, complemented, Sylow
313C3×Q8—22no—
314C3×Q8—22no—
315C3×Q8—21no—
316C3×Q8—21no—
317C3×Q8—21no—
318C3×Q8—21no—
319C3×Q8—21no—
320C3×Q8—21no—
321C3×Q8—41no—
322C3×Q8—41no—
323C3×Q8—41no—
324C3×Q8—41no—
325C3×Q8—42no—
326C3×Q8—42no—
327C3×Q8—61no—
328C3×Q8—61no—
329C3×Q8—61no—
330C3×Q8—61no—
331C3×Q8—121no—
332C3×Q8—121no—
333C3×Q8—362no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—31no—
342C2×C3×C4—31no—
343C2×C3×C4—31no—
344C2×C3×C4—31no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—41no—
349C2×C3×C4—61no—
350C2×C3×C4—61no—
351C2×C3×C4—62no—
352C2×C3×C4—61no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—121no—
357C2×C3×C4—181no—
358C2×C3×C4C2×D3211yescharacteristic
359C2×C3×C4C22×C3×D311yescharacteristic
360C2×C3×C4C22×C3×D311yescharacteristic
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—61no—
364C2×Dic3—61no—
365C2×Dic3—121no—
366C2×Dic3C3×#6711yescharacteristic
367C2×Dic3C3×#6711yescharacteristic
368Dic6—21no—
369Dic6—21no—
370Dic6—122no—
371Dic6—122no—
372C3×C8—61no—
373C3×C8—61no—
374C3×C8—121no—
375C3×C8—181no—
376C3⋊C8—21no—
377C3⋊C8—61no—
378C3⋊C8—61no—
379C3⋊C8—121no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—41no—
390C2×C32—41no—
391C2×C32—41no—
392C2×C32#88511yescharacteristic
393C2×C32D3×SD1611yescharacteristic
394C2×C32D3×Q1611yescharacteristic
395C2×C32#93611yescharacteristic
396C2×C32C2×#21511yescharacteristic
397C2×C32C2×#21611yescharacteristic
398C2×C32C3×#12411yescharacteristic
399C2×C32C2×C3×SD1611yescharacteristic
400C2×C32C2×C3×Q1611yescharacteristic
401C2×Q8—31no—
402C2×Q8—181no—
403C2×Q8C3×D3211yescharacteristic
404C2×C8—31no—
405C2×C8—91no—
406C4⋊C4—31no—
407C4⋊C4—61no—
408C4⋊C4—61no—
409C4⋊C4—91no—
410C4⋊C4—181no—
411C42—31no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—41no—
416C22×C3D3×#6711yescharacteristic
417C22×C3C3×D3×D411yescharacteristic
418C22×C3C2×C3×#6711yescharacteristic
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—32no—
426C3×C4—32no—
427C3×C4—32no—
428C3×C4—32no—
429C3×C4—42no—
430C3×C4—42no—
431C3×C4—42no—
432C3×C4—44no—
433C3×C4—41no—
434C3×C4—41no—
435C3×C4—41no—
436C3×C4—41no—
437C3×C4—41no—
438C3×C4—62no—
439C3×C4—62no—
440C3×C4—122no—
441C3×C4—121no—
442C3×C4—121no—
443C3×C4—121no—
444C3×C4—121no—
445C3×C4—241no—
446C3×C4—361no—
447C3×C4#380411yescharacteristic
448C3×C4#380511yescharacteristic
449C3×C4C3×#23511yescharacteristic
450C3×C4C3×#23711yescharacteristic
451C3×C4C3×#23711yescharacteristic
452C3×C4C3×#23711yescharacteristic
453Dic3—41no—
454Dic3—41no—
455Dic3—121no—
456Dic3—121no—
457Dic3—241no—
458Dic3C3×#21512yes—
459Dic3C3×#21612yes—
460C32—21no—
461C32—21no—
462C32—21no—
463C32—41no—
464C32#479211yescharacteristic, complemented
465C32#507911yescharacteristic, complemented
466C32C3×#48711yescharacteristic, complemented
467Q8—21no—
468Q8—21no—
469Q8—61no—
470Q8—61no—
471Q8—362no—
472C2×C4—21no—
473C2×C4—31no—
474C2×C4—31no—
475C2×C4—61no—
476C2×C4—61no—
477C2×C4—181no—
478C2×C4C2×C3×D3211yescharacteristic
479C8—61no—
480C8—181no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—41no—
491C2×C3—41no—
492C2×C3—41no—
493C2×C3D3×#21511yescharacteristic
494C2×C3D3×#21611yescharacteristic
495C2×C3#6395911yescharacteristic
496C2×C3C3×#88511yescharacteristic
497C2×C3C3×D3×SD1611yescharacteristic
498C2×C3C3×D3×Q1611yescharacteristic
499C2×C3C3×#93611yescharacteristic
500C2×C3C2×C3×#21511yescharacteristic
501C2×C3C2×C3×#21611yescharacteristic
502C22C3×D3×#6711yescharacteristic
503C4—32no—
504C4—32no—
505C4—41no—
506C4—121no—
507C4—121no—
508C4—361no—
509C4C3×#380411yescharacteristic
510C4C3×#380511yescharacteristic
511C3—21no—
512C3—21no—
513C3—21no—
514C3—41no—
515C3#1059162311yescharacteristic, direct product factor
516C3C3×#479211yescharacteristic, complemented
517C3C3×#507911yescharacteristic, complemented
518C2C3×D3×#21511yescharacteristic
519C2C3×D3×#21611yescharacteristic
520C2C3×#6395911yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×C3×Q8D32
hypocenterC32C3×#487
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
10. #10591623C3direct (N × H)
312. C33#487semidirect
464. C32#4792semidirect
465. C32#5079semidirect
466. C32C3×#487semidirect
515. C3#10591623direct (N × H)
516. C3C3×#4792semidirect
517. C3C3×#5079semidirect

Maximal subgroups

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

subgroupindex
C32×#9442
C3×#639182
C3×#636622
C32×#8352
C3×#635662
C3×#635652
C3×Dic3×C3⋊C82
C3×#50793
C3×#47923
#105916233