Finite Group Databasebeta

← C3×#10590689 #13157690

Subgroups — C3×#10590689 #13157690

Order 1728.

566 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63900C211yesmaximal, characteristic
2C3×#63871C211yesmaximal, characteristic
3C3×#63647C211yesmaximal, characteristic
4C32×#818C211yesmaximal, characteristic, complemented
5C32×#816C211yesmaximal, characteristic, complemented
6C3×#63576C211yesmaximal, characteristic, complemented
7C3×#63565C211yesmaximal, characteristic, complemented
8C3×#4884—31nomaximal
9C3×#4811—31nomaximal
10#10590689C311yesmaximal, characteristic, direct product factor
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C3×#624—21no—
14C2×C33×C8C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C3×#3724—21no—
20C3×Dic3⋊Dic3—21no—
21C3×#3722—21no—
22C3×#946—31no—
23C3×#897—31no—
24C3×D3×C4⋊C4—31no—
25C3×D3×C4⋊C4—31no—
26#63900C2×C311yescharacteristic
27#63871C2×C311yescharacteristic
28C32×#107—31no—
29C32×#103—31no—
30#63647C2×C311yescharacteristic
31C3×#835—31no—
32C3×(C2×D3)⋊C8—31no—
33C3×(C2×D3)⋊C8—31no—
34C3×(C2×D3)⋊C8—61no—
35C3×#818—21no—
36C3×#818D311yescharacteristic, complemented
37C3×#818C2×C311yescharacteristic, complemented
38C3×#816—21no—
39C3×#816D311yescharacteristic, complemented
40C3×#816C2×C311yescharacteristic, complemented
41C3×#810—31no—
42C3×#807—31no—
43C3×#807—31no—
44#63576C2×C311yescharacteristic, complemented
45#63565C2×C311yescharacteristic, complemented
46C22×C3×C3⋊D3D411yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3D411yescharacteristic
49C3×C4×C3⋊D3—21no—
50C3×C4×C3⋊D3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55C32×Dic6—21no—
56C32×Dic6—21no—
57C3×#624—42no—
58C33×C8—21no—
59C3×C3⋊(C3⋊C8)—21no—
60C3×#484—91no—
61#4884—31no—
62#4811—31no—
63C2×C32×Q8—31no—
64C2×C4×C3⋊D3C22×C311yescharacteristic
65C2×C3×C4×D3—31no—
66C2×C3×C4×D3—31no—
67C2×C3×C4×D3—61no—
68C2×C3×C4×D3—61no—
69C2×C3×C4×D3—61no—
70C2×C3×Dic6—21no—
71C2×C3×Dic6C2×D311yescharacteristic
72C2×C3×Dic6C22×C311yescharacteristic
73C2×#624—21no—
74C2×C32×C8—21no—
75C2×C32×C8—21no—
76C2×C32×C8—21no—
77C2×C32×C8—41no—
78C2×C32×C8C2×D311yescharacteristic
79C2×C32×C8C2×D311yescharacteristic
80C2×C32×C8C22×C311yescharacteristic
81C32×C4⋊C4—31no—
82C32×C4⋊C4—31no—
83C32×C4⋊C4—31no—
84C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
85C3×(C2×D3)⋊C4—61no—
86C3×(C2×D3)⋊C4—61no—
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3—61no—
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4⋊Dic3C22×C311yescharacteristic
95C3×C4⋊Dic3C22×C311yescharacteristic
96C3×C4⋊Dic3C22×C311yescharacteristic
97C3×Dic3⋊C4—61no—
98C3×Dic3⋊C4—61no—
99C3×Dic3⋊C4—61no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—61no—
103#3724—21no—
104Dic3⋊Dic3—21no—
105#3722—21no—
106C22×C33C2×D411yescharacteristic
107C2×C3×C3⋊D3—21no—
108C2×C3×C3⋊D3—21no—
109C2×C3×C3⋊D3Q1612yes—
110C33×C4#4111yescharacteristic
111C33×C4#4111yescharacteristic
112C3×C3⋊Dic3—21no—
113C32×Dic3—41no—
114C32×Dic3—41no—
115C32×Dic3—41no—
116C32×Dic3—41no—
117C3×C22⋊Q8—91no—
118C2×C3×C4⋊C4—91no—
119#946—31no—
120#897—31no—
121D3×C4⋊C4—31no—
122D3×C4⋊C4—31no—
123C3×#107—31no—
124C3×#107—31no—
125C3×#107—61no—
126C3×#107—91no—
127C3×#103—31no—
128C3×#103—31no—
129C3×#103—61no—
130C3×#103—91no—
131C3×C22⋊C8—91no—
132#835—31no—
133(C2×D3)⋊C8—31no—
134(C2×D3)⋊C8—31no—
135(C2×D3)⋊C8—61no—
136#818C3×D311yescharacteristic, complemented
137#816C3×D311yescharacteristic, complemented
138#810—31no—
139#807—31no—
140#807—31no—
141C22×C3⋊D3C3×D411yescharacteristic
142C22×C3×D3—31no—
143C22×C3×D3—31no—
144C22×C3×D3—61no—
145C32×Q8—61no—
146C32×Q8—61no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—41no—
151C2×C32×C4—61no—
152C2×C32×C4—61no—
153C2×C32×C4—61no—
154C2×C32×C4—61no—
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C23×C311yescharacteristic
158C2×C3⋊Dic3C3×D411yescharacteristic
159C4×C3⋊D3—21no—
160C4×C3⋊D3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—21no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—31no—
170C2×C3×Dic3—31no—
171C2×C3×Dic3—41no—
172C2×C3×Dic3—41no—
173C2×C3×Dic3—41no—
174C2×C3×Dic3—41no—
175C2×C3×Dic3—61no—
176C3×C4×D3—61no—
177C3×C4×D3—61no—
178C3×C4×D3—61no—
179C3×C4×D3—61no—
180C3×C4×D3—121no—
181C3×C4×D3—121no—
182C3×C4×D3—122no—
183C3×C4×D3—122no—
184C3×Dic6—22no—
185C3×Dic6—21no—
186C3×Dic6—21no—
187C3×Dic6—21no—
188C3×Dic6—21no—
189C3×Dic6—41no—
190#624—42no—
191C32×C8—21no—
192C32×C8—22no—
193C32×C8—21no—
194C32×C8—21no—
195C32×C8—41no—
196C32×C8—41no—
197C32×C8—42no—
198C3⋊(C3⋊C8)—21no—
199C3×C3⋊C8—61no—
200C3×C3⋊C8—61no—
201C3×C3⋊C8—121no—
202#484—91noSylow
203C2×C33#12311yescharacteristic
204C2×C33C2×Q1611yescharacteristic
205C2×C33AGL(1,ℤ/8)11yescharacteristic
206C3×C3⋊D3—22no—
207C2×C3×Q8—31no—
208C2×C3×Q8—31no—
209C2×C3×Q8—61no—
210C22×C3×C4—91no—
211C22×C3×C4—181no—
212C2×C4×D3—31no—
213C2×C4×D3—31no—
214C2×C4×D3—61no—
215C2×C4×D3—61no—
216C2×C4×D3—61no—
217C2×Dic6C2×C3×D311yescharacteristic
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—41no—
222C2×C3×C8—91no—
223C2×C3×C8D3211yescharacteristic
224C2×C3×C8C2×C3×D311yescharacteristic
225C2×C3×C8C2×C3×D311yescharacteristic
226C3×C4⋊C4—31no—
227C3×C4⋊C4—31no—
228C3×C4⋊C4—31no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—31no—
231C3×C4⋊C4—31no—
232C3×C4⋊C4—61no—
233C3×C4⋊C4—61no—
234C3×C4⋊C4—61no—
235C3×C4⋊C4—181no—
236C3×C4⋊C4—181no—
237C3×C22⋊C4—181no—
238(C2×D3)⋊C4—61no—
239(C2×D3)⋊C4—61no—
240C4⋊Dic3—61no—
241C4⋊Dic3C2×C3×D311yescharacteristic
242C4⋊Dic3C2×C3×D311yescharacteristic
243C4⋊Dic3C2×C3×D311yescharacteristic
244Dic3⋊C4—61no—
245Dic3⋊C4—61no—
246Dic3⋊C4—61no—
247C2×C3⋊C8—31no—
248C2×C3⋊C8—31no—
249C2×C3⋊C8—61no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—21no—
253C22×C32—41no—
254C22×C32D3×D411yescharacteristic
255C22×C32D3×D411yescharacteristic
256C22×C32C2×C3×D411yescharacteristic
257C2×C3⋊D3—21no—
258C2×C3⋊D3—21no—
259C2×C3⋊D3C3×Q1612yes—
260C2×C3×D3—32no—
261C2×C3×D3—32no—
262C2×C3×D3—62no—
263C2×C3×D3—61no—
264C2×C3×D3—61no—
265C2×C3×D3—61no—
266C2×C3×D3—61no—
267C2×C3×D3—121no—
268C2×C3×D3—121no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—41no—
276C32×C4—41no—
277C32×C4—121no—
278C32×C4—121no—
279C32×C4—121no—
280C32×C4—121no—
281C32×C4#23511yescharacteristic
282C32×C4#23511yescharacteristic
283C32×C4#23711yescharacteristic
284C32×C4#23911yescharacteristic
285C32×C4C3×#4111yescharacteristic
286C32×C4C3×#4111yescharacteristic
287C3⋊Dic3—21no—
288C3×Dic3—42no—
289C3×Dic3—41no—
290C3×Dic3—41no—
291C3×Dic3—41no—
292C3×Dic3—41no—
293C3×Dic3—41no—
294C3×Dic3—41no—
295C3×Dic3—41no—
296C3×Dic3—41no—
297C3×Dic3—61no—
298C3×Dic3—61no—
299C3×Dic3—81no—
300C3×Dic3—81no—
301C3×Dic3—81no—
302C3×Dic3—121no—
303C22⋊Q8—91no—
304C2×C4⋊C4—91no—
305#107—31no—
306#107—91no—
307#103—31no—
308#103—91no—
309C22⋊C8—91no—
310C33#48411yescharacteristic, complemented, Sylow
311C23×C3—91no—
312C22×D3—31no—
313C22×D3—31no—
314C22×D3—61no—
315C3×Q8—62no—
316C3×Q8—61no—
317C3×Q8—61no—
318C3×Q8—61no—
319C3×Q8—61no—
320C3×Q8—121no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—41no—
325C2×C3×C4—61no—
326C2×C3×C4—61no—
327C2×C3×C4—61no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4—61no—
333C2×C3×C4—91no—
334C2×C3×C4—121no—
335C2×C3×C4—121no—
336C2×C3×C4—121no—
337C2×C3×C4—121no—
338C2×C3×C4—181no—
339C2×C3×C4—181no—
340C2×C3×C4—362no—
341C2×C3×C4C2×D3211yescharacteristic
342C2×C3×C4C22×C3×D311yescharacteristic
343C2×C3×C4C22×C3×D311yescharacteristic
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—31no—
349C2×Dic3—31no—
350C2×Dic3—61no—
351C4×D3—61no—
352C4×D3—61no—
353C4×D3—61no—
354C4×D3—61no—
355C4×D3—121no—
356C4×D3—121no—
357C4×D3—122no—
358C4×D3—122no—
359Dic6—21no—
360Dic6—21no—
361C3×C8—22no—
362C3×C8—21no—
363C3×C8—21no—
364C3×C8—21no—
365C3×C8—41no—
366C3×C8—41no—
367C3×C8—42no—
368C3×C8—181no—
369C3⋊C8—61no—
370C3⋊C8—61no—
371C3⋊C8—121no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—41no—
382C2×C32—41no—
383C2×C32—41no—
384C2×C32#88311yescharacteristic
385C2×C32#89411yescharacteristic
386C2×C32#91111yescharacteristic
387C2×C32#91411yescharacteristic
388C2×C32D3×Q1611yescharacteristic
389C2×C32D3×Q1611yescharacteristic
390C2×C32C3×#12311yescharacteristic
391C2×C32C2×C3×Q1611yescharacteristic
392C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
393C3⋊D3—22no—
394C3×D3—62no—
395C3×D3—62no—
396C3×D3—122no—
397C2×Q8—31no—
398C22×C4—91no—
399C22×C4—181no—
400C2×C8—91no—
401C2×C8C3×D3211yescharacteristic
402C4⋊C4—31no—
403C4⋊C4—31no—
404C4⋊C4—31no—
405C4⋊C4—181no—
406C4⋊C4—181no—
407C22⋊C4—181no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—41no—
412C22×C3—92no—
413C22×C3—181no—
414C22×C3—181no—
415C22×C3#380211yescharacteristic
416C22×C3C3×D3×D411yescharacteristic
417C22×C3C3×D3×D411yescharacteristic
418C2×D3—32no—
419C2×D3—32no—
420C2×D3—62no—
421C2×D3—61no—
422C2×D3—61no—
423C2×D3—61no—
424C2×D3—61no—
425C2×D3—121no—
426C2×D3—121no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—41no—
434C3×C4—41no—
435C3×C4—122no—
436C3×C4—121no—
437C3×C4—121no—
438C3×C4—121no—
439C3×C4—121no—
440C3×C4—121no—
441C3×C4—121no—
442C3×C4—121no—
443C3×C4—121no—
444C3×C4—181no—
445C3×C4—241no—
446C3×C4—241no—
447C3×C4—241no—
448C3×C4#379911yescharacteristic
449C3×C4#380411yescharacteristic
450C3×C4C3×#23511yescharacteristic
451C3×C4C3×#23511yescharacteristic
452C3×C4C3×#23711yescharacteristic
453C3×C4C3×#23911yescharacteristic
454Dic3—41no—
455Dic3—41no—
456Dic3—41no—
457Dic3—41no—
458Dic3—61no—
459Dic3—61no—
460Dic3—121no—
461C32—21no—
462C32—21no—
463C32—21no—
464C32—41no—
465C32#481111yescharacteristic, complemented
466C32#488411yescharacteristic, complemented
467C32C3×#48411yescharacteristic, complemented
468C23—91no—
469Q8—61no—
470Q8—61no—
471C2×C4—61no—
472C2×C4—61no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—91no—
476C2×C4—181no—
477C2×C4—181no—
478C2×C4—362no—
479C2×C4C2×C3×D3211yescharacteristic
480C8—21no—
481C8—181no—
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—21no—
491C2×C3—41no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—182no—
495C2×C3#6379811yescharacteristic
496C2×C3#6380211yescharacteristic
497C2×C3#6385411yescharacteristic
498C2×C3C3×#88311yescharacteristic
499C2×C3C3×#89411yescharacteristic
500C2×C3C3×#91111yescharacteristic
501C2×C3C3×#91411yescharacteristic
502C2×C3C3×D3×Q1611yescharacteristic
503C2×C3C3×D3×Q1611yescharacteristic
504D3—62no—
505D3—62no—
506D3—122no—
507C22—92no—
508C22—181no—
509C22—181no—
510C22C3×#380211yescharacteristic
511C4—121no—
512C4—121no—
513C4—121no—
514C4—121no—
515C4—181no—
516C4C3×#379911yescharacteristic
517C4C3×#380411yescharacteristic
518C3—21no—
519C3—21no—
520C3—21no—
521C3—41no—
522C3#1059068911yescharacteristic, direct product factor
523C3C3×#481111yescharacteristic, complemented
524C3C3×#488411yescharacteristic, complemented
525C2—182no—
526C2C3×#6379811yescharacteristic
527C2C3×#6380211yescharacteristic
528C2C3×#6385411yescharacteristic
subgroupG/H
centerC22×C3#3802
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×C3×C8D32
hypocenterC32C3×#484
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
4. C32×#818C2semidirect
5. C32×#816C2semidirect
6. C3×#63576C2semidirect
7. C3×#63565C2semidirect
10. #10590689C3direct (N × H)
36. C3×#818D3semidirect
37. C3×#818C2×C3semidirect
39. C3×#816D3semidirect
40. C3×#816C2×C3semidirect
44. #63576C2×C3semidirect
45. #63565C2×C3semidirect
136. #818C3×D3semidirect
137. #816C3×D3semidirect
310. C33#484semidirect
465. C32#4811semidirect
466. C32#4884semidirect
467. C32C3×#484semidirect
522. C3#10590689direct (N × H)
523. C3C3×#4811semidirect
524. C3C3×#4884semidirect

Maximal subgroups

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

subgroupindex
C3×#639002
C3×#638712
C3×#636472
C32×#8182
C32×#8162
C3×#635762
C3×#635652
C3×#48843
C3×#48113
#105906893