Finite Group Databasebeta

← C3×#10590539 #13157515

Subgroups — C3×#10590539 #13157515

Order 1728.

558 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64099C211yesmaximal, characteristic, complemented
2C3×#63863C211yesmaximal, characteristic
3C32×#810C211yesmaximal, characteristic, complemented
4C32×#809C211yesmaximal, characteristic
5C3×#63566C211yesmaximal, characteristic, complemented
6C3×#63563C211yesmaximal, characteristic
7C3×#63553C211yesmaximal, characteristic, complemented
8C3×#4800—31nomaximal
9C3×#4768—31nomaximal
10#10590539C311yesmaximal, characteristic, direct product factor
11C2×C32×D12C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3750—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×(C2×D3)⋊Dic3—21no—
21C3×(C2×D3)⋊Dic3—21no—
22#64099C2×C311yescharacteristic, complemented
23C3×#947—31no—
24C3×#936—31no—
25C3×#890—31no—
26C3×#890—31no—
27C3×#890—61no—
28#63863C2×C311yescharacteristic
29C32×#103—31no—
30C32×#102—31no—
31C3×#835—31no—
32C3×#832—31no—
33C3×#821—31no—
34C3×#816—31no—
35C3×C8×Dic3—31no—
36C3×C8×Dic3—31no—
37C3×#810—21no—
38C3×#810D311yescharacteristic, complemented
39C3×#810C2×C311yescharacteristic, complemented
40C3×#809—21no—
41C3×#809D311yescharacteristic
42C3×#809C2×C311yescharacteristic
43#63566C2×C311yescharacteristic, complemented
44#63563C2×C311yescharacteristic
45#63553C2×C311yescharacteristic, complemented
46C22×C32×D3—21no—
47C2×C33×C4—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C2×C3×C3⋊Dic3D411yescharacteristic
52C2×C32×Dic3—21no—
53C32×D12—21no—
54C32×D12—21no—
55C32×Dic6—21no—
56C32×Dic6—21no—
57C32×C3⋊C8—21no—
58C32×C3⋊C8—21no—
59C3×#488—91no—
60#4800—31no—
61#4768—31no—
62C2×C32×Q8—31no—
63C2×C32×D4—31no—
64C2×C3×D12—21no—
65C2×C3×D12C2×D311yescharacteristic
66C2×C3×D12C22×C311yescharacteristic
67C2×C3×Dic6—21no—
68C2×C3×Dic6C2×D311yescharacteristic
69C2×C3×Dic6C22×C311yescharacteristic
70C2×C32×C8—31no—
71C2×C32×C8—31no—
72C32×C4⋊C4—21no—
73C32×C4⋊C4—21no—
74C32×C4⋊C4—21no—
75C32×C4⋊C4—41no—
76C32×C4⋊C4C2×D311yescharacteristic
77C32×C4⋊C4C2×D311yescharacteristic
78C32×C4⋊C4C22×C311yescharacteristic
79C4⋊(C3⋊Dic3)—21no—
80C4⋊(C3⋊Dic3)C22×C311yescharacteristic
81#3750—21no—
82C4×C3⋊Dic3C22×C311yescharacteristic
83C3×#217—61no—
84C3×#217—61no—
85C3×(C2×D3)⋊C4—61no—
86C3×(C2×D3)⋊C4—61no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3—61no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3—62no—
93C3×Dic3⋊C4—61no—
94C3×Dic3⋊C4—61no—
95C3×Dic3⋊C4—62no—
96C3×C4×Dic3—31no—
97C3×C4×Dic3—31no—
98C3×C4×Dic3—61no—
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C22×C311yescharacteristic
104C2×C3×C3⋊C8C22×C311yescharacteristic
105(C2×D3)⋊Dic3—21no—
106(C2×D3)⋊Dic3—21no—
107C22×C33C2×D411yescharacteristic
108C2×C32×D3—41no—
109C2×C32×D3—41no—
110C2×C32×D3—41no—
111C33×C4—41no—
112C33×C4#4111yescharacteristic
113C33×C4#4111yescharacteristic
114C3×C3⋊Dic3—21no—
115C3×C3⋊Dic3—21no—
116C3×C3⋊Dic3—41no—
117C32×Dic3—41no—
118C3×#125—91no—
119C3×#124—91no—
120#947—31no—
121#936—31no—
122#890—31no—
123#890—31no—
124#890—61no—
125C3×#103—31no—
126C3×#103—31no—
127C3×#103—61no—
128C3×#103—91no—
129C3×#102—31no—
130C3×#102—31no—
131C3×#102—61no—
132C3×#102—91no—
133C3×C4×C8—91no—
134#835—31no—
135#832—31no—
136#821—31no—
137#816—31no—
138C8×Dic3—31no—
139C8×Dic3—31no—
140#810C3×D311yescharacteristic, complemented
141#809C3×D311yescharacteristic
142C23×C32—61no—
143C22×C3×D3—21no—
144C22×C3×D3—21no—
145C22×C3×D3—41no—
146C32×Q8—61no—
147C32×Q8—61no—
148C32×D4—61no—
149C32×D4—61no—
150C2×C32×C4—21no—
151C2×C32×C4—22no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—41no—
158C2×C32×C4—41no—
159C2×C32×C4—41no—
160C2×C32×C4—42no—
161C2×C32×C4—61no—
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C23×C311yescharacteristic
165C2×C3⋊Dic3—21no—
166C2×C3⋊Dic3C3×D411yescharacteristic
167C2×C3⋊Dic3C3×D411yescharacteristic
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—31no—
171C2×C3×Dic3—31no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—41no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—62no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C3×D12—22no—
181C3×D12—21no—
182C3×D12—21no—
183C3×D12—21no—
184C3×D12—21no—
185C3×D12—41no—
186C3×Dic6—22no—
187C3×Dic6—21no—
188C3×Dic6—21no—
189C3×Dic6—21no—
190C3×Dic6—21no—
191C3×Dic6—41no—
192C32×C8—61no—
193C32×C8—61no—
194C3×C3⋊C8—21no—
195C3×C3⋊C8—21no—
196C3×C3⋊C8—21no—
197C3×C3⋊C8—21no—
198C3×C3⋊C8—41no—
199C3×C3⋊C8—41no—
200#488—91noSylow
201C2×C33#12411yescharacteristic
202C2×C33#13511yescharacteristic
203C2×C33#13511yescharacteristic
204C32×D3—81no—
205C2×C3×Q8—31no—
206C2×C3×Q8—31no—
207C2×C3×Q8—61no—
208C2×C3×D4—31no—
209C2×C3×D4—31no—
210C2×C3×D4—61no—
211C2×D12C2×C3×D311yescharacteristic
212C2×Dic6C2×C3×D311yescharacteristic
213C2×C3×C8—31no—
214C2×C3×C8—31no—
215C2×C3×C8—31no—
216C2×C3×C8—31no—
217C2×C3×C8—61no—
218C2×C3×C8—61no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—41no—
223C3×C4⋊C4—91no—
224C3×C4⋊C4—181no—
225C3×C4⋊C4—181no—
226C3×C4⋊C4D3211yescharacteristic
227C3×C4⋊C4C2×C3×D311yescharacteristic
228C3×C4⋊C4C2×C3×D311yescharacteristic
229C3×C22⋊C4—181no—
230C3×C22⋊C4—181no—
231C3×C42—91no—
232#217—61no—
233#217—61no—
234(C2×D3)⋊C4—61no—
235(C2×D3)⋊C4—61no—
236C4⋊Dic3—31no—
237C4⋊Dic3—31no—
238C4⋊Dic3—61no—
239C4⋊Dic3—61no—
240C4⋊Dic3—61no—
241C4⋊Dic3—62no—
242Dic3⋊C4—61no—
243Dic3⋊C4—61no—
244Dic3⋊C4—62no—
245C4×Dic3—31no—
246C4×Dic3—31no—
247C4×Dic3—61no—
248C2×C3⋊C8C2×C3×D311yescharacteristic
249C2×C3⋊C8C2×C3×D311yescharacteristic
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—21no—
253C22×C32—41no—
254C22×C32—121no—
255C22×C32—121no—
256C22×C32—121no—
257C22×C32D3×D411yescharacteristic
258C22×C32D3×D411yescharacteristic
259C22×C32C2×C3×D411yescharacteristic
260C2×C3×D3—42no—
261C2×C3×D3—41no—
262C2×C3×D3—41no—
263C2×C3×D3—41no—
264C2×C3×D3—41no—
265C2×C3×D3—41no—
266C2×C3×D3—41no—
267C2×C3×D3—81no—
268C2×C3×D3—81no—
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—41no—
278C32×C4—42no—
279C32×C4—41no—
280C32×C4—41no—
281C32×C4—81no—
282C32×C4—81no—
283C32×C4—82no—
284C32×C4—121no—
285C32×C4#23511yescharacteristic
286C32×C4#23511yescharacteristic
287C32×C4#23711yescharacteristic
288C32×C4#23711yescharacteristic
289C32×C4C3×#4111yescharacteristic
290C32×C4C3×#4111yescharacteristic
291C3⋊Dic3—21no—
292C3⋊Dic3—21no—
293C3⋊Dic3—41no—
294C3×Dic3—42no—
295C3×Dic3—41no—
296C3×Dic3—41no—
297C3×Dic3—61no—
298C3×Dic3—61no—
299C3×Dic3—61no—
300C3×Dic3—61no—
301C3×Dic3—121no—
302C3×Dic3—121no—
303C3×Dic3—122no—
304C3×Dic3—121no—
305C3×Dic3—121no—
306#125—91no—
307#124—91no—
308#103—31no—
309#103—91no—
310#102—31no—
311#102—91no—
312C4×C8—91no—
313C33#48811yescharacteristic, complemented, Sylow
314C23×C3—61no—
315C23×C3—61no—
316C23×C3—121no—
317C22×D3—21no—
318C3×Q8—62no—
319C3×Q8—61no—
320C3×Q8—61no—
321C3×Q8—61no—
322C3×Q8—61no—
323C3×Q8—121no—
324C3×D4—62no—
325C3×D4—61no—
326C3×D4—61no—
327C3×D4—61no—
328C3×D4—61no—
329C3×D4—121no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
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—42no—
340C2×C3×C4—41no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—91no—
344C2×C3×C4—91no—
345C2×C3×C4—121no—
346C2×C3×C4—181no—
347C2×C3×C4C2×D3211yescharacteristic
348C2×C3×C4C22×C3×D311yescharacteristic
349C2×C3×C4C22×C3×D311yescharacteristic
350C2×Dic3—21no—
351C2×Dic3—31no—
352C2×Dic3—31no—
353C2×Dic3—31no—
354C2×Dic3—31no—
355C2×Dic3—61no—
356C2×Dic3—61no—
357C2×Dic3—62no—
358C2×Dic3—61no—
359C2×Dic3—61no—
360D12—21no—
361D12—21no—
362Dic6—21no—
363Dic6—21no—
364C3×C8—61no—
365C3×C8—61no—
366C3×C8—61no—
367C3×C8—61no—
368C3×C8—121no—
369C3×C8—121no—
370C3⋊C8—21no—
371C3⋊C8—21no—
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—241no—
385C2×C32#88511yescharacteristic
386C2×C32#88511yescharacteristic
387C2×C32#91211yescharacteristic
388C2×C32#91611yescharacteristic
389C2×C32#91611yescharacteristic
390C2×C32#91911yescharacteristic
391C2×C32C3×#12411yescharacteristic
392C2×C32C3×#13511yescharacteristic
393C2×C32C3×#13511yescharacteristic
394C3×D3—82no—
395C3×D3—81no—
396C3×D3—81no—
397C2×Q8—31no—
398C2×D4—31no—
399C2×C8—31no—
400C2×C8—31no—
401C4⋊C4—91no—
402C4⋊C4—181no—
403C4⋊C4—181no—
404C4⋊C4C3×D3211yescharacteristic
405C22⋊C4—181no—
406C22⋊C4—181no—
407C42—91no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—41no—
412C22×C3—122no—
413C22×C3—121no—
414C22×C3—121no—
415C22×C3—121no—
416C22×C3—121no—
417C22×C3—121no—
418C22×C3—121no—
419C22×C3—241no—
420C22×C3—241no—
421C22×C3#381111yescharacteristic
422C22×C3C3×D3×D411yescharacteristic
423C22×C3C3×D3×D411yescharacteristic
424C2×D3—41no—
425C2×D3—41no—
426C2×D3—41no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—42no—
434C3×C4—41no—
435C3×C4—41no—
436C3×C4—41no—
437C3×C4—41no—
438C3×C4—41no—
439C3×C4—81no—
440C3×C4—81no—
441C3×C4—82no—
442C3×C4—122no—
443C3×C4—121no—
444C3×C4—121no—
445C3×C4—181no—
446C3×C4—181no—
447C3×C4—361no—
448C3×C4#379511yescharacteristic
449C3×C4#379511yescharacteristic
450C3×C4C3×#23511yescharacteristic
451C3×C4C3×#23511yescharacteristic
452C3×C4C3×#23711yescharacteristic
453C3×C4C3×#23711yescharacteristic
454Dic3—41no—
455Dic3—61no—
456Dic3—61no—
457Dic3—61no—
458Dic3—61no—
459Dic3—121no—
460Dic3—121no—
461Dic3—122no—
462Dic3—121no—
463Dic3—121no—
464C32—21no—
465C32—21no—
466C32—21no—
467C32—41no—
468C32#476811yescharacteristic, complemented
469C32#480011yescharacteristic, complemented
470C32C3×#48811yescharacteristic, complemented
471C23—61no—
472Q8—61no—
473Q8—61no—
474D4—61no—
475D4—61no—
476C2×C4—21no—
477C2×C4—61no—
478C2×C4—91no—
479C2×C4—91no—
480C2×C4—181no—
481C2×C4C2×C3×D3211yescharacteristic
482C8—61no—
483C8—61no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—41no—
494C2×C3—41no—
495C2×C3—41no—
496C2×C3—242no—
497C2×C3—241no—
498C2×C3—241no—
499C2×C3#6391311yescharacteristic
500C2×C3#6393611yescharacteristic
501C2×C3#6395011yescharacteristic
502C2×C3C3×#88511yescharacteristic
503C2×C3C3×#88511yescharacteristic
504C2×C3C3×#91211yescharacteristic
505C2×C3C3×#91611yescharacteristic
506C2×C3C3×#91611yescharacteristic
507C2×C3C3×#91911yescharacteristic
508D3—81no—
509C22—121no—
510C22—121no—
511C22—121no—
512C22C3×#381111yescharacteristic
513C4—41no—
514C4—121no—
515C4—181no—
516C4—181no—
517C4—361no—
518C4C3×#379511yescharacteristic
519C4C3×#379511yescharacteristic
520C3—21no—
521C3—21no—
522C3—21no—
523C3—41no—
524C3#1059053911yescharacteristic, direct product factor
525C3C3×#476811yescharacteristic, complemented
526C3C3×#480011yescharacteristic, complemented
527C2—241no—
528C2C3×#6391311yescharacteristic
529C2C3×#6393611yescharacteristic
530C2C3×#6395011yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#488
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#64099C2semidirect
3. C32×#810C2semidirect
5. C3×#63566C2semidirect
7. C3×#63553C2semidirect
10. #10590539C3direct (N × H)
22. #64099C2×C3semidirect
38. C3×#810D3semidirect
39. C3×#810C2×C3semidirect
43. #63566C2×C3semidirect
45. #63553C2×C3semidirect
140. #810C3×D3semidirect
313. C33#488semidirect
468. C32#4768semidirect
469. C32#4800semidirect
470. C32C3×#488semidirect
524. C3#10590539direct (N × H)
525. C3C3×#4768semidirect
526. C3C3×#4800semidirect

Maximal subgroups

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

subgroupindex
C3×#640992
C3×#638632
C32×#8102
C32×#8092
C3×#635662
C3×#635632
C3×#635532
C3×#48003
C3×#47683
#105905393