Finite Group Databasebeta

← C3×#10591625 #13158628

Subgroups — C3×#10591625 #13158628

Order 1728.

660 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3756C211yesmaximal, characteristic
2C32×Q8×Dic3C211yesmaximal, characteristic
3C2×C32×#216C211yesmaximal, characteristic
4C3×#63866C211yesmaximal, characteristic
5C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic
6C3×#63567C211yesmaximal, characteristic
7C3×#63565C211yesmaximal, characteristic
8C3×#5182—31nomaximal
9C3×#5031—31nomaximal
10#10591625C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C3×#3756—22no—
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×#216—22no—
17C32×C4⋊Dic3—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4×Dic3—21no—
20C32×C4×Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×Dic3⋊Dic3—21no—
23C3×Dic3⋊Dic3—21no—
24C2×C32×Q16—31no—
25C32×C4×Q8—31no—
26C2×#3756C2×C311yescharacteristic
27C3×Q8×Dic3—21no—
28C3×Q8×Dic3D311yescharacteristic
29C3×Q8×Dic3C2×C311yescharacteristic
30C3×#944—31no—
31C2×C3×#216—21no—
32C2×C3×#216—31no—
33C2×C3×#216—31no—
34C2×C3×#216—61no—
35C2×C3×#216D311yescharacteristic
36C2×C3×#216C2×C311yescharacteristic
37C3×#869—31no—
38#63866C2×C311yescharacteristic
39C3×#835—31no—
40C3×#816—31no—
41C3×Dic3⋊C8—31no—
42C3×#810—31no—
43C3×#810—31no—
44C3×C4⋊(C3⋊C8)—31no—
45Dic3⋊(C3⋊C8)C2×C311yescharacteristic
46#63567C2×C311yescharacteristic
47#63565C2×C311yescharacteristic
48C33×Q8—21no—
49C33×Q8D412yes—
50C2×C33×C4—21no—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C3×#634—21no—
54C3×#634—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D411yescharacteristic
58C2×C32×Dic3D411yescharacteristic
59C32×Dic6—21no—
60C32×Dic6—21no—
61C3×C3⋊(C3⋊C8)—21no—
62C32×C3⋊C8—21no—
63C3×#462—91no—
64#5182—31no—
65#5031—31no—
66C2×C32×Q8—21no—
67C2×C32×Q8—21no—
68C2×C32×Q8—21no—
69C2×C32×Q8—31no—
70C2×C32×Q8—41no—
71C2×C32×Q8C2×D311yescharacteristic
72C2×C32×Q8C2×D311yescharacteristic
73C2×C32×Q8C22×C311yescharacteristic
74C2×#634C22×C311yescharacteristic
75C2×C3×Dic6—21no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—61no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×C3×Dic6C22×C311yescharacteristic
81C32×Q16—62no—
82C2×C32×C8—31no—
83C32×C4⋊C4—31no—
84C32×C4⋊C4—61no—
85C32×C42—31no—
86C32×C42—61no—
87#3756—22no—
88C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
89C3×#216—24no—
90C3×#216—22no—
91C3×#216—22no—
92C3×#216—64no—
93C3×#216—62no—
94C3×#216—62no—
95C3×C4⋊Dic3—22no—
96C3×C4⋊Dic3—21no—
97C3×C4⋊Dic3—21no—
98C3×C4⋊Dic3—21no—
99C3×C4⋊Dic3—61no—
100C3×C4⋊Dic3—61no—
101C3×C4⋊Dic3C2×D311yescharacteristic
102C3×C4⋊Dic3C22×C311yescharacteristic
103C3×Dic3⋊C4—61no—
104C3×Dic3⋊C4—61no—
105C3×C4×Dic3—22no—
106C3×C4×Dic3—21no—
107C3×C4×Dic3—21no—
108C3×C4×Dic3—21no—
109C3×C4×Dic3C2×D311yescharacteristic
110C3×C4×Dic3C22×C311yescharacteristic
111C2×C3×C3⋊C8—21no—
112C2×C3×C3⋊C8—31no—
113C2×C3×C3⋊C8—31no—
114C2×C3×C3⋊C8—61no—
115C2×C3×C3⋊C8C2×D311yescharacteristic
116C2×C3×C3⋊C8C22×C311yescharacteristic
117Dic3⋊Dic3—21no—
118Dic3⋊Dic3—21no—
119C22×C33C2×D411yescharacteristic
120C33×C4—22no—
121C33×C4C2×D411yescharacteristic
122C33×C4#4111yescharacteristic
123C3×C3⋊Dic3—41no—
124C32×Dic3—21no—
125C32×Dic3—41no—
126C32×Dic3—41no—
127C32×Dic3Q1612yescomplemented
128C2×C3×Q16—31no—
129C2×C3×Q16—31no—
130C2×C3×Q16—61no—
131C2×C3×Q16—91no—
132C3×C4⋊Q8—91no—
133C3×C4×Q8—31no—
134C3×C4×Q8—31no—
135C3×C4×Q8—61no—
136Q8×Dic3C3×D311yescharacteristic
137#944—31no—
138C2×#216—31no—
139C2×#216—31no—
140C2×#216—61no—
141C2×#216C3×D311yescharacteristic
142#869—31no—
143C3×C4⋊C8—91no—
144C3×#103—91no—
145C3×#103—91no—
146#835—31no—
147#816—31no—
148Dic3⋊C8—31no—
149#810—31no—
150#810—31no—
151C4⋊(C3⋊C8)—31no—
152C32×Q8—21no—
153C32×Q8—22no—
154C32×Q8—21no—
155C32×Q8—21no—
156C32×Q8—22no—
157C32×Q8—22no—
158C32×Q8—22no—
159C32×Q8—42no—
160C32×Q8—41no—
161C32×Q8—41no—
162C32×Q8—42no—
163C32×Q8—61no—
164C32×Q8—61no—
165C32×Q8D1212yes—
166C32×Q8#6712yes—
167C32×Q8C3×D412yes—
168C2×C32×C4—21no—
169C2×C32×C4—22no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—31no—
176C2×C32×C4—31no—
177C2×C32×C4—41no—
178C2×C32×C4—41no—
179C2×C32×C4—41no—
180C2×C32×C4—42no—
181C2×C32×C4—61no—
182C2×C32×C4—61no—
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C23×C311yescharacteristic
186C2×C3⋊Dic3—21no—
187#634—21no—
188#634—21no—
189C2×C3×Dic3—22no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—41no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—121no—
200C2×C3×Dic3#6711yescharacteristic
201C2×C3×Dic3#6711yescharacteristic
202C2×C3×Dic3C3×D411yescharacteristic
203C2×C3×Dic3C3×D411yescharacteristic
204C3×Dic6—22no—
205C3×Dic6—21no—
206C3×Dic6—21no—
207C3×Dic6—21no—
208C3×Dic6—21no—
209C3×Dic6—41no—
210C3×Dic6—62no—
211C3×Dic6—61no—
212C3×Dic6—61no—
213C3×Dic6—61no—
214C3×Dic6—61no—
215C3×Dic6—121no—
216C32×C8—61no—
217C3⋊(C3⋊C8)—21no—
218C3×C3⋊C8—22no—
219C3×C3⋊C8—21no—
220C3×C3⋊C8—21no—
221C3×C3⋊C8—62no—
222C3×C3⋊C8—61no—
223C3×C3⋊C8—61no—
224#462—91noSylow
225C2×C33#12111yescharacteristic
226C2×C33C2×Q1611yescharacteristic
227C2×C33#13711yescharacteristic
228C2×C3×Q8—21no—
229C2×C3×Q8—21no—
230C2×C3×Q8—21no—
231C2×C3×Q8—31no—
232C2×C3×Q8—31no—
233C2×C3×Q8—41no—
234C2×C3×Q8—61no—
235C2×C3×Q8—91no—
236C2×C3×Q8D3211yescharacteristic
237C2×C3×Q8C2×C3×D311yescharacteristic
238C2×C3×Q8C2×C3×D311yescharacteristic
239C2×Dic6—31no—
240C2×Dic6—31no—
241C2×Dic6—61no—
242C2×Dic6C2×C3×D311yescharacteristic
243C3×Q16—64no—
244C3×Q16—62no—
245C3×Q16—62no—
246C3×Q16—182no—
247C2×C3×C8—31no—
248C2×C3×C8—31no—
249C2×C3×C8—61no—
250C2×C3×C8—91no—
251C3×C4⋊C4—31no—
252C3×C4⋊C4—31no—
253C3×C4⋊C4—61no—
254C3×C4⋊C4—61no—
255C3×C4⋊C4—61no—
256C3×C4⋊C4—62no—
257C3×C4⋊C4—181no—
258C3×C4⋊C4—181no—
259C3×C42—31no—
260C3×C42—31no—
261C3×C42—61no—
262C3×C42—61no—
263C3×C42—61no—
264C3×C42—62no—
265#216—22no—
266#216—64no—
267#216—62no—
268#216—62no—
269C4⋊Dic3—21no—
270C4⋊Dic3—61no—
271C4⋊Dic3—61no—
272C4⋊Dic3C2×C3×D311yescharacteristic
273Dic3⋊C4—61no—
274Dic3⋊C4—61no—
275C4×Dic3—21no—
276C4×Dic3C2×C3×D311yescharacteristic
277C2×C3⋊C8—31no—
278C2×C3⋊C8—31no—
279C2×C3⋊C8—61no—
280C2×C3⋊C8C2×C3×D311yescharacteristic
281C22×C32—21no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—41no—
285C22×C32D3×D411yescharacteristic
286C22×C32C2×#6711yescharacteristic
287C22×C32C2×C3×D411yescharacteristic
288C32×C4—22no—
289C32×C4—24no—
290C32×C4—22no—
291C32×C4—22no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—32no—
299C32×C4—41no—
300C32×C4—41no—
301C32×C4—42no—
302C32×C4—42no—
303C32×C4—44no—
304C32×C4—61no—
305C32×C4—121no—
306C32×C4—121no—
307C32×C4C2×D1211yescharacteristic
308C32×C4#23511yescharacteristic
309C32×C4#23711yescharacteristic
310C32×C4C2×#6711yescharacteristic
311C32×C4C2×C3×D411yescharacteristic
312C32×C4C3×#4111yescharacteristic
313C3⋊Dic3—41no—
314C3×Dic3—21no—
315C3×Dic3—21no—
316C3×Dic3—22no—
317C3×Dic3—41no—
318C3×Dic3—42no—
319C3×Dic3—42no—
320C3×Dic3—41no—
321C3×Dic3—41no—
322C3×Dic3—41no—
323C3×Dic3—41no—
324C3×Dic3—122no—
325C3×Dic3—121no—
326C3×Dic3—121no—
327C3×Dic3#21612yescomplemented
328C3×Dic3C3×Q1612yescomplemented
329C2×Q16—31no—
330C2×Q16—91no—
331C4⋊Q8—91no—
332C4×Q8—31no—
333C4⋊C8—91no—
334#103—91no—
335#103—91no—
336C33#46211yescharacteristic, complemented, Sylow
337C3×Q8—22no—
338C3×Q8—21no—
339C3×Q8—21no—
340C3×Q8—21no—
341C3×Q8—22no—
342C3×Q8—22no—
343C3×Q8—22no—
344C3×Q8—41no—
345C3×Q8—41no—
346C3×Q8—42no—
347C3×Q8—42no—
348C3×Q8—62no—
349C3×Q8—61no—
350C3×Q8—61no—
351C3×Q8—61no—
352C3×Q8—61no—
353C3×Q8—121no—
354C3×Q8—181no—
355C3×Q8—181no—
356C3×Q8#62612yes—
357C3×Q8C3×D1212yes—
358C3×Q8C3×#6712yes—
359C2×C3×C4—22no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—31no—
367C2×C3×C4—31no—
368C2×C3×C4—31no—
369C2×C3×C4—31no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—42no—
373C2×C3×C4—41no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—62no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—61no—
381C2×C3×C4—121no—
382C2×C3×C4—181no—
383C2×C3×C4C2×D3211yescharacteristic
384C2×C3×C4C22×C3×D311yescharacteristic
385C2×C3×C4C22×C3×D311yescharacteristic
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—61no—
389C2×Dic3—61no—
390C2×Dic3—121no—
391C2×Dic3C3×#6711yescharacteristic
392C2×Dic3C3×#6711yescharacteristic
393Dic6—21no—
394Dic6—21no—
395Dic6—62no—
396Dic6—61no—
397Dic6—61no—
398Dic6—61no—
399Dic6—61no—
400Dic6—121no—
401C3×C8—62no—
402C3×C8—61no—
403C3×C8—61no—
404C3×C8—181no—
405C3⋊C8—21no—
406C3⋊C8—62no—
407C3⋊C8—61no—
408C3⋊C8—61no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—21no—
416C2×C32—21no—
417C2×C32—21no—
418C2×C32—41no—
419C2×C32—41no—
420C2×C32—41no—
421C2×C32D3×Q1611yescharacteristic
422C2×C32#91811yescharacteristic
423C2×C32#93011yescharacteristic
424C2×C32#93911yescharacteristic
425C2×C32C2×#21611yescharacteristic
426C2×C32#95111yescharacteristic
427C2×C32C3×#12111yescharacteristic
428C2×C32C2×C3×Q1611yescharacteristic
429C2×C32C3×#13711yescharacteristic
430C2×Q8—31no—
431C2×Q8—91no—
432C2×Q8C3×D3211yescharacteristic
433Q16—62no—
434Q16—182no—
435C2×C8—31no—
436C2×C8—91no—
437C4⋊C4—31no—
438C4⋊C4—61no—
439C4⋊C4—181no—
440C4⋊C4—181no—
441C42—31no—
442C42—61no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—41no—
447C22×C3D3×#6711yescharacteristic
448C22×C3C3×D3×D411yescharacteristic
449C22×C3C2×C3×#6711yescharacteristic
450C3×C4—24no—
451C3×C4—22no—
452C3×C4—22no—
453C3×C4—22no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—32no—
461C3×C4—32no—
462C3×C4—42no—
463C3×C4—42no—
464C3×C4—44no—
465C3×C4—41no—
466C3×C4—41no—
467C3×C4—62no—
468C3×C4—61no—
469C3×C4—61no—
470C3×C4—121no—
471C3×C4—122no—
472C3×C4—122no—
473C3×C4—121no—
474C3×C4—121no—
475C3×C4—121no—
476C3×C4—121no—
477C3×C4—361no—
478C3×C4#380411yescharacteristic
479C3×C4C2×#62611yescharacteristic
480C3×C4C2×C3×D1211yescharacteristic
481C3×C4C3×#23511yescharacteristic
482C3×C4C3×#23711yescharacteristic
483C3×C4C2×C3×#6711yescharacteristic
484Dic3—21no—
485Dic3—41no—
486Dic3—41no—
487Dic3—122no—
488Dic3—121no—
489Dic3—121no—
490Dic3C3×#21612yescomplemented
491C32—21no—
492C32—21no—
493C32—21no—
494C32—41no—
495C32#503111yescharacteristic, complemented
496C32#518211yescharacteristic, complemented
497C32C3×#46211yescharacteristic, complemented
498Q8—21no—
499Q8—61no—
500Q8—61no—
501Q8—181no—
502Q8—181no—
503Q8C3×#62612yes—
504C2×C4—21no—
505C2×C4—31no—
506C2×C4—31no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—181no—
510C2×C4C2×C3×D3211yescharacteristic
511C8—61no—
512C8—181no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—41no—
523C2×C3—41no—
524C2×C3—41no—
525C2×C3D3×#21611yescharacteristic
526C2×C3#6394311yescharacteristic
527C2×C3#6397811yescharacteristic
528C2×C3C3×D3×Q1611yescharacteristic
529C2×C3C3×#91811yescharacteristic
530C2×C3C3×#93011yescharacteristic
531C2×C3C3×#93911yescharacteristic
532C2×C3C2×C3×#21611yescharacteristic
533C2×C3C3×#95111yescharacteristic
534C22C3×D3×#6711yescharacteristic
535C4—22no—
536C4—32no—
537C4—61no—
538C4—121no—
539C4—121no—
540C4—361no—
541C4C3×#380411yescharacteristic
542C4C2×C3×#62611yescharacteristic
543C3—21no—
544C3—21no—
545C3—21no—
546C3—41no—
547C3#1059162511yescharacteristic, direct product factor
548C3C3×#503111yescharacteristic, complemented
549C3C3×#518211yescharacteristic, complemented
550C2C3×D3×#21611yescharacteristic
551C2C3×#6394311yescharacteristic
552C2C3×#6397811yescharacteristic
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×#462
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. #10591625C3direct (N × H)
127. C32×Dic3Q16semidirect
327. C3×Dic3#216semidirect
328. C3×Dic3C3×Q16semidirect
336. C33#462semidirect
490. Dic3C3×#216semidirect
495. C32#5031semidirect
496. C32#5182semidirect
497. C32C3×#462semidirect
547. C3#10591625direct (N × H)
548. C3C3×#5031semidirect
549. C3C3×#5182semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37562
C32×Q8×Dic32
C2×C32×#2162
C3×#638662
C3×Dic3⋊(C3⋊C8)2
C3×#635672
C3×#635652
C3×#51823
C3×#50313
#105916253