Finite Group Databasebeta

← C3×#10591572 #13158575

Subgroups — C3×#10591572 #13158575

Order 1728.

698 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×C32×#215C211yesmaximal, characteristic
3C3×#63898C211yesmaximal, characteristic
4C32×#835C211yesmaximal, characteristic, complemented
5C3×#63575C211yesmaximal, characteristic, complemented
6C3×#63563C211yesmaximal, characteristic
7C3×#63555C211yesmaximal, characteristic, complemented
8C3×#5164—31nomaximal
9C3×#4817—31nomaximal
10#10591572C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C3×D3×Dic3—21no—
14C3×C4⋊(C3⋊Dic3)—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4×C3⋊Dic3—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×#215C414yescomplemented
19C32×C4⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×(C2×D3)⋊Dic3—21no—
23C2×C32×SD16—31no—
24Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
25C3×Q8×Dic3—31no—
26C3×Q8×Dic3—31no—
27C3×Q8×Dic3—61no—
28C2×C3×#215—21no—
29C2×C3×#215D311yescharacteristic
30C2×C3×#215C2×C311yescharacteristic
31C3×D4×Dic3—31no—
32C3×#893—31no—
33#63898C2×C311yescharacteristic
34C32×#103—31no—
35C3×#835—21no—
36C3×#835D311yescharacteristic, complemented
37C3×#835C2×C311yescharacteristic, complemented
38C3×#832—31no—
39C3×#821—31no—
40C3×#818—31no—
41C3×#815—31no—
42C3×#815—31no—
43C3×#807—31no—
44#63575C2×C311yescharacteristic, complemented
45#63563C2×C311yescharacteristic
46#63555C2×C311yescharacteristic, complemented
47C22×C32×D3—21no—
48C33×Q8—21no—
49C33×Q8C2×C412yescomplemented
50C2×C33×C4—21no—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3D411yescharacteristic
54C2×C3×C3⋊Dic3D411yescharacteristic
55C2×C32×Dic3—21no—
56C32×D12—21no—
57C32×D12C2×C412yes—
58C3×D3×Dic3—42no—
59C32×C3⋊C8—21no—
60C32×C3⋊C8C2×C412yescomplemented
61C3×#440—91no—
62#5164—31no—
63#4817—31no—
64C2×C32×Q8—21no—
65C2×C32×Q8—21no—
66C2×C32×Q8—21no—
67C2×C32×Q8—41no—
68C2×C32×Q8C2×D311yescharacteristic
69C2×C32×Q8C2×D311yescharacteristic
70C2×C32×Q8C22×C311yescharacteristic
71C2×C32×D4—31no—
72C22×C3×Dic3—61no—
73C2×C3×D12—21no—
74C2×C3×D12C2×D311yescharacteristic
75C2×C3×D12C22×C311yescharacteristic
76C2×C3×C4×D3—61no—
77C2×D3×Dic3—21no—
78C32×SD16—34no—
79C2×C32×C8—31no—
80C2×C32×C8—31no—
81C32×C4⋊C4—31no—
82C4⋊(C3⋊Dic3)—21no—
83C4⋊(C3⋊Dic3)C22×C311yescharacteristic
84C4×C3⋊Dic3—21no—
85C4×C3⋊Dic3C22×C311yescharacteristic
86C3×#217—61no—
87C3×#215—24no—
88C3×#215Dic314yescomplemented
89C3×#215C3×C414yescomplemented
90C3×(C2×D3)⋊C4—61no—
91C3×C4⋊Dic3—21no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—61no—
95C3×C4⋊Dic3—61no—
96C3×C4⋊Dic3—61no—
97C3×C4⋊Dic3—62no—
98C3×C4⋊Dic3C2×D311yescharacteristic
99C3×C4⋊Dic3C22×C311yescharacteristic
100C3×C4×Dic3—31no—
101C3×C4×Dic3—31no—
102C3×C4×Dic3—61no—
103C3×C4×Dic3—61no—
104C3×C4×Dic3—61no—
105C3×C4×Dic3—62no—
106C2×C3×C3⋊C8—21no—
107C2×C3×C3⋊C8—21no—
108C2×C3×C3⋊C8C2×D311yescharacteristic
109C2×C3×C3⋊C8C2×D311yescharacteristic
110C2×C3×C3⋊C8C22×C311yescharacteristic
111C2×C3×C3⋊C8C22×C311yescharacteristic
112(C2×D3)⋊Dic3—21no—
113C22×C33C2×D411yescharacteristic
114C2×C32×D3—22no—
115C2×C32×D3—41no—
116C2×C32×D3—41no—
117C33×C4—22no—
118C33×C4C22×C411yescharacteristic
119C33×C4#4111yescharacteristic
120C3×C3⋊Dic3—21no—
121C3×C3⋊Dic3—21no—
122C3×C3⋊Dic3—41no—
123C32×Dic3—41no—
124C2×C3×SD16—31no—
125C2×C3×SD16—31no—
126C2×C3×SD16—61no—
127C3×C4×Q8—91no—
128C3×C4×D4—91no—
129Q8×Dic3—31no—
130Q8×Dic3—31no—
131Q8×Dic3—61no—
132C2×#215C3×D311yescharacteristic
133D4×Dic3—31no—
134#893—31no—
135C3×#107—91no—
136C3×#103—31no—
137C3×#103—31no—
138C3×#103—61no—
139C3×#102—91no—
140C3×#97—91no—
141#835C3×D311yescharacteristic, complemented
142#832—31no—
143#821—31no—
144#818—31no—
145#815—31no—
146#815—31no—
147#807—31no—
148C23×C32—61no—
149C22×C3×D3—21no—
150C22×C3×D3—21no—
151C22×C3×D3—41no—
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×Q8C4×D312yescomplemented
164C32×Q8C2×Dic312yescomplemented
165C32×Q8C2×C3×C412yescomplemented
166C32×D4—32no—
167C32×D4—61no—
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—41no—
176C2×C32×C4—41no—
177C2×C32×C4—41no—
178C2×C32×C4—42no—
179C2×C32×C4—61no—
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C23×C311yescharacteristic
183C2×C3⋊Dic3—21no—
184C2×C3⋊Dic3C3×D411yescharacteristic
185C2×C3⋊Dic3C3×D411yescharacteristic
186C2×C3×Dic3—21no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—31no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3—31no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3—41no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—62no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—122no—
199C3×D12—21no—
200C3×D12—21no—
201C3×D12—22no—
202C3×D12—41no—
203C3×D12C2×Dic312yes—
204C3×D12C2×C3×C412yes—
205C3×C4×D3—122no—
206D3×Dic3—42no—
207C32×C8—32no—
208C32×C8—61no—
209C3×C3⋊C8—21no—
210C3×C3⋊C8—21no—
211C3×C3⋊C8—22no—
212C3×C3⋊C8—41no—
213C3×C3⋊C8C2×Dic312yescomplemented
214C3×C3⋊C8C2×C3×C412yescomplemented
215#440—91noSylow
216C2×C33C4×D411yescharacteristic
217C2×C33AGL(1,ℤ/8)11yescharacteristic
218C2×C33#13711yescharacteristic
219C32×D3—42no—
220C2×C3×Q8—21no—
221C2×C3×Q8—21no—
222C2×C3×Q8—21no—
223C2×C3×Q8—41no—
224C2×C3×Q8D3211yescharacteristic
225C2×C3×Q8C2×C3×D311yescharacteristic
226C2×C3×Q8C2×C3×D311yescharacteristic
227C2×C3×D4—31no—
228C2×C3×D4—31no—
229C2×C3×D4—61no—
230C22×C3×C4—181no—
231C22×Dic3—61no—
232C2×D12C2×C3×D311yescharacteristic
233C2×C4×D3—61no—
234C3×SD16—34no—
235C3×SD16—34no—
236C3×SD16—64no—
237C2×C3×C8—31no—
238C2×C3×C8—31no—
239C2×C3×C8—31no—
240C2×C3×C8—31no—
241C2×C3×C8—61no—
242C2×C3×C8—61no—
243C3×C4⋊C4—31no—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—61no—
246C3×C4⋊C4—91no—
247C3×C4⋊C4—181no—
248C3×C22⋊C4—181no—
249C3×C42—91no—
250C3×C42—181no—
251#217—61no—
252#215C3×Dic314yescomplemented
253(C2×D3)⋊C4—61no—
254C4⋊Dic3—31no—
255C4⋊Dic3—31no—
256C4⋊Dic3—61no—
257C4⋊Dic3—61no—
258C4⋊Dic3—61no—
259C4⋊Dic3—62no—
260C4⋊Dic3C2×C3×D311yescharacteristic
261C4×Dic3—31no—
262C4×Dic3—31no—
263C4×Dic3—61no—
264C4×Dic3—61no—
265C4×Dic3—61no—
266C4×Dic3—62no—
267C2×C3⋊C8C2×C3×D311yescharacteristic
268C2×C3⋊C8C2×C3×D311yescharacteristic
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—41no—
273C22×C32—62no—
274C22×C32—121no—
275C22×C32—121no—
276C22×C32D3×D411yescharacteristic
277C22×C32D3×D411yescharacteristic
278C22×C32C2×C3×D411yescharacteristic
279C2×C3×D3—22no—
280C2×C3×D3—22no—
281C2×C3×D3—42no—
282C2×C3×D3—41no—
283C2×C3×D3—41no—
284C2×C3×D3—41no—
285C2×C3×D3—41no—
286C2×C3×D3—81no—
287C2×C3×D3—81no—
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—41no—
299C32×C4—41no—
300C32×C4—42no—
301C32×C4—42no—
302C32×C4—44no—
303C32×C4—121no—
304C32×C4C2×C4×D311yescharacteristic
305C32×C4#23711yescharacteristic
306C32×C4#23711yescharacteristic
307C32×C4C22×Dic311yescharacteristic
308C32×C4C22×C3×C411yescharacteristic
309C32×C4C3×#4111yescharacteristic
310C3⋊Dic3—21no—
311C3⋊Dic3—21no—
312C3⋊Dic3—41no—
313C3×Dic3—41no—
314C3×Dic3—41no—
315C3×Dic3—62no—
316C3×Dic3—61no—
317C3×Dic3—61no—
318C3×Dic3—61no—
319C3×Dic3—61no—
320C3×Dic3—81no—
321C3×Dic3—121no—
322C3×Dic3—122no—
323C3×Dic3—121no—
324C3×Dic3—121no—
325C2×SD16—31no—
326C4×Q8—91no—
327C4×D4—91no—
328#107—91no—
329#103—31no—
330#102—91no—
331#97—91no—
332C33#44011yescharacteristic, complemented, Sylow
333C23×C3—61no—
334C23×C3—61no—
335C23×C3—121no—
336C22×D3—21no—
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×Q8D3×Dic312yescomplemented
349C3×Q8C3×C4×D312yescomplemented
350C3×Q8C2×C3×Dic312yescomplemented
351C3×D4—32no—
352C3×D4—32no—
353C3×D4—62no—
354C3×D4—61no—
355C3×D4—61no—
356C3×D4—121no—
357C2×C3×C4—22no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—42no—
367C2×C3×C4—41no—
368C2×C3×C4—61no—
369C2×C3×C4—61no—
370C2×C3×C4—91no—
371C2×C3×C4—91no—
372C2×C3×C4—121no—
373C2×C3×C4—181no—
374C2×C3×C4—362no—
375C2×C3×C4C2×D3211yescharacteristic
376C2×C3×C4C22×C3×D311yescharacteristic
377C2×C3×C4C22×C3×D311yescharacteristic
378C2×Dic3—21no—
379C2×Dic3—31no—
380C2×Dic3—31no—
381C2×Dic3—31no—
382C2×Dic3—31no—
383C2×Dic3—61no—
384C2×Dic3—61no—
385C2×Dic3—62no—
386C2×Dic3—61no—
387C2×Dic3—61no—
388C2×Dic3—122no—
389D12—21no—
390D12C2×C3×Dic312yes—
391C4×D3—122no—
392C3×C8—32no—
393C3×C8—32no—
394C3×C8—62no—
395C3×C8—61no—
396C3×C8—61no—
397C3×C8—121no—
398C3⋊C8—21no—
399C3⋊C8C2×C3×Dic312yescomplemented
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—41no—
410C2×C32—41no—
411C2×C32—41no—
412C2×C32—122no—
413C2×C32#67⋊C411yescharacteristic
414C2×C32#91411yescharacteristic
415C2×C32#91411yescharacteristic
416C2×C32#91511yescharacteristic
417C2×C32#91811yescharacteristic
418C2×C32D4×Dic311yescharacteristic
419C2×C32C3×C4×D411yescharacteristic
420C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
421C2×C32C3×#13711yescharacteristic
422C3×D3—42no—
423C3×D3—42no—
424C3×D3—82no—
425C2×Q8C3×D3211yescharacteristic
426C2×D4—31no—
427C22×C4—181no—
428SD16—34no—
429C2×C8—31no—
430C2×C8—31no—
431C4⋊C4—31no—
432C4⋊C4—91no—
433C4⋊C4—181no—
434C22⋊C4—181no—
435C42—91no—
436C42—181no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—41no—
441C22×C3—62no—
442C22×C3—62no—
443C22×C3—122no—
444C22×C3—121no—
445C22×C3—121no—
446C22×C3—121no—
447C22×C3—121no—
448C22×C3—241no—
449C22×C3—241no—
450C22×C3#381111yescharacteristic
451C22×C3C3×D3×D411yescharacteristic
452C22×C3C3×D3×D411yescharacteristic
453C2×D3—22no—
454C2×D3—41no—
455C2×D3—41no—
456C3×C4—24no—
457C3×C4—22no—
458C3×C4—22no—
459C3×C4—22no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—42no—
467C3×C4—42no—
468C3×C4—44no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—121no—
472C3×C4—121no—
473C3×C4—181no—
474C3×C4—181no—
475C3×C4—241no—
476C3×C4—361no—
477C3×C4C2×D3×Dic311yescharacteristic
478C3×C4#380511yescharacteristic
479C3×C4C2×C3×C4×D311yescharacteristic
480C3×C4C3×#23711yescharacteristic
481C3×C4C3×#23711yescharacteristic
482C3×C4C22×C3×Dic311yescharacteristic
483Dic3—41no—
484Dic3—62no—
485Dic3—61no—
486Dic3—61no—
487Dic3—61no—
488Dic3—61no—
489Dic3—121no—
490Dic3—122no—
491Dic3—121no—
492Dic3—121no—
493C32—21no—
494C32—21no—
495C32—21no—
496C32—41no—
497C32#481711yescharacteristic, complemented
498C32#516411yescharacteristic, complemented
499C32C3×#44011yescharacteristic, complemented
500C23—61no—
501Q8—21no—
502Q8C3×D3×Dic312yescomplemented
503D4—32no—
504D4—61no—
505C2×C4—21no—
506C2×C4—61no—
507C2×C4—91no—
508C2×C4—91no—
509C2×C4—181no—
510C2×C4—362no—
511C2×C4C2×C3×D3211yescharacteristic
512C8—32no—
513C8—61no—
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—21no—
523C2×C3—41no—
524C2×C3—41no—
525C2×C3—41no—
526C2×C3—122no—
527C2×C3—122no—
528C2×C3—242no—
529C2×C3#6394111yescharacteristic
530C2×C3#6395111yescharacteristic
531C2×C3#67⋊Dic311yescharacteristic
532C2×C3C3×#67⋊C411yescharacteristic
533C2×C3C3×#91411yescharacteristic
534C2×C3C3×#91411yescharacteristic
535C2×C3C3×#91511yescharacteristic
536C2×C3C3×#91811yescharacteristic
537C2×C3C3×D4×Dic311yescharacteristic
538D3—42no—
539C22—62no—
540C22—121no—
541C22—121no—
542C22C3×#381111yescharacteristic
543C4—22no—
544C4—121no—
545C4—181no—
546C4—181no—
547C4—361no—
548C4C2×C3×D3×Dic311yescharacteristic
549C4C3×#380511yescharacteristic
550C3—21no—
551C3—21no—
552C3—21no—
553C3—41no—
554C3#1059157211yescharacteristic, direct product factor
555C3C3×#481711yescharacteristic, complemented
556C3C3×#516411yescharacteristic, complemented
557C2—122no—
558C2C3×#6394111yescharacteristic
559C2C3×#6395111yescharacteristic
560C2C3×#67⋊Dic311yescharacteristic
subgroupG/H
centerC22×C3#3811
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×Q8D32
hypocenterC32C3×#440
LCSG ▷ 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
1. C3×Q8×C3⋊Dic3C2semidirect
4. C32×#835C2semidirect
5. C3×#63575C2semidirect
7. C3×#63555C2semidirect
10. #10591572C3direct (N × H)
18. C32×#215C4semidirect
24. Q8×C3⋊Dic3C2×C3semidirect
36. C3×#835D3semidirect
37. C3×#835C2×C3semidirect
44. #63575C2×C3semidirect
46. #63555C2×C3semidirect
49. C33×Q8C2×C4semidirect
60. C32×C3⋊C8C2×C4semidirect
88. C3×#215Dic3semidirect
89. C3×#215C3×C4semidirect
141. #835C3×D3semidirect
163. C32×Q8C4×D3semidirect
164. C32×Q8C2×Dic3semidirect
165. C32×Q8C2×C3×C4semidirect
213. C3×C3⋊C8C2×Dic3semidirect
214. C3×C3⋊C8C2×C3×C4semidirect
252. #215C3×Dic3semidirect
332. C33#440semidirect
348. C3×Q8D3×Dic3semidirect
349. C3×Q8C3×C4×D3semidirect
350. C3×Q8C2×C3×Dic3semidirect
399. C3⋊C8C2×C3×Dic3semidirect
497. C32#4817semidirect
498. C32#5164semidirect
499. C32C3×#440semidirect
502. Q8C3×D3×Dic3semidirect
554. C3#10591572direct (N × H)
555. C3C3×#4817semidirect
556. C3C3×#5164semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C3⋊Dic32
C2×C32×#2152
C3×#638982
C32×#8352
C3×#635752
C3×#635632
C3×#635552
C3×#51643
C3×#48173
#105915723