Finite Group Databasebeta

← C3×#10589733 #13152912

Subgroups — C3×#10589733 #13152912

Order 1728.

950 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3701C211yesmaximal, characteristic, complemented
2C2×C3×#3697C211yesmaximal, characteristic, complemented
3C3×C9×C22⋊Q8C211yesmaximal, characteristic, complemented
4C3×#63506C211yesmaximal, characteristic, complemented
5C3×C4×#611C211yesmaximal, characteristic, complemented
6C3×#63459C211yesmaximal, characteristic, complemented
7C3×#63456C212yesmaximal, complemented
8C3×#63455C211yesmaximal, characteristic, complemented
9C3×C4⋊C4×D9C212yesmaximal, complemented
10C3×#63446C212yesmaximal, complemented
11C3×#63445C212yesmaximal, complemented
12C3×#5631—31nomaximal
13#10589733C311yesmaximal, characteristic, direct product factor
14C2×C3×Q8×C9C2211yescharacteristic
15C22×C3×C4×C9C2211yescharacteristic
16C2×C3×#611C2212yes—
17C2×C3×#611C2211yescharacteristic
18C3×#3701—24no—
19C3×#3697—24no—
20C2×C3×D36C2212yes—
21C2×C3×D36C2211yescharacteristic
22C2×C3×C4×D9C2212yes—
23C2×C3×C4×D9C2211yescharacteristic
24C2×C3×C4×D9C2212yes—
25C2×C3×C4×D9C2212yes—
26C2×C3×Dic18C2211yescharacteristic
27C3×C9×C4⋊C4C2212yes—
28C3×C9×C4⋊C4C2211yescharacteristic
29C3×C9×C22⋊C4C2212yes—
30C3×#3676C2211yescharacteristic
31C3×(C2×D9)⋊C4C2211yescharacteristic
32C3×(C2×D9)⋊C4C2212yes—
33C3×(C2×D9)⋊C4C2212yes—
34C3×C4⋊Dic9C2212yes—
35C3×Dic9⋊C4C2211yescharacteristic
36C3×Dic9⋊C4C2212yes—
37C3×Dic9⋊C4C2212yes—
38C3×C4×Dic9C2211yescharacteristic
39C2×C3×#239—31no—
40C2×C3×#235—31no—
41C32×C22⋊Q8D311yescharacteristic
42C3×#944—31no—
43C3×C4×#67—31no—
44C3×#897—31no—
45C3×#894—32no—
46C3×#893—31no—
47C3×D3×C4⋊C4—32no—
48C3×#884—32no—
49C3×#883—32no—
50C2×#3701C2×C311yescharacteristic, complemented
51C2×#3697C2×C311yescharacteristic, complemented
52C9×C22⋊Q8—21no—
53C9×C22⋊Q8C2×C311yescharacteristic, complemented
54#63506C2×C311yescharacteristic, complemented
55C4×#611C2×C311yescharacteristic, complemented
56#63459C2×C311yescharacteristic, complemented
57#63456C2×C312yescomplemented
58#63455C2×C311yescharacteristic, complemented
59C4⋊C4×D9C2×C312yescomplemented
60#63446C2×C312yescomplemented
61#63445C2×C312yescomplemented
62C23×C3×C9C2311yescharacteristic
63C22×C3×D9C2311yescharacteristic
64C22×C3×D9C2312yes—
65C3×Q8×C9—22no—
66C2×C3×C4×C9—22no—
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3×C4×C9C2312yes—
69C2×C3×C4×C9C2312yes—
70C2×C3×C4×C9C2311yescharacteristic
71C3×#611—24no—
72C3×#611—22no—
73C2×C3×Dic9C2312yes—
74C2×C3×Dic9C2311yescharacteristic
75C2×C3×Dic9C2311yescharacteristic
76C2×C3×Dic9C2311yescharacteristic
77C3×D36—24no—
78C3×D36D414yescomplemented
79C3×C4×D9—24no—
80C3×C4×D9—24no—
81C3×C4×D9—22no—
82C3×C4×D9—24no—
83C3×Dic18—22no—
84C3×#547—91no—
85#5631—31no—
86C2×C32×Q8C2×D311yescharacteristic
87C22×C32×C4C2×D311yescharacteristic
88C2×C3×#67—32no—
89C2×C3×#67—31no—
90C3×#239—64no—
91C3×#235—64no—
92C2×C3×D12—31no—
93C2×C3×D12—32no—
94C2×C3×C4×D3—32no—
95C2×C3×C4×D3—32no—
96C2×C3×C4×D3—32no—
97C2×C3×C4×D3—31no—
98C2×C3×Dic6—31no—
99C32×C4⋊C4C2×D312yes—
100C32×C4⋊C4C2×D311yescharacteristic
101C32×C22⋊C4C2×D312yes—
102C3×#217—31no—
103C3×(C2×D3)⋊C4—32no—
104C3×(C2×D3)⋊C4—32no—
105C3×(C2×D3)⋊C4—31no—
106C3×C4⋊Dic3—32no—
107C3×Dic3⋊C4—32no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—31no—
110C3×C4×Dic3—31no—
111C2×Q8×C9—21no—
112C2×Q8×C9C22×C311yescharacteristic
113C22×C4×C9—21no—
114C22×C4×C9C22×C311yescharacteristic
115C2×#611C22×C311yescharacteristic
116C2×#611C22×C312yes—
117#3701—24no—
118#3697—24no—
119C2×D36C22×C312yes—
120C2×D36C22×C311yescharacteristic
121C2×C4×D9C22×C311yescharacteristic
122C2×C4×D9C22×C312yes—
123C2×C4×D9C22×C312yes—
124C2×C4×D9C22×C312yes—
125C2×Dic18C22×C311yescharacteristic
126C9×C4⋊C4—21no—
127C9×C4⋊C4—22no—
128C9×C4⋊C4C22×C312yes—
129C9×C4⋊C4C22×C311yescharacteristic
130C9×C22⋊C4—22no—
131C9×C22⋊C4C22×C312yes—
132#3676C22×C311yescharacteristic
133(C2×D9)⋊C4C22×C311yescharacteristic
134(C2×D9)⋊C4C22×C312yes—
135(C2×D9)⋊C4C22×C312yes—
136C4⋊Dic9C22×C312yes—
137Dic9⋊C4C22×C311yescharacteristic
138Dic9⋊C4C22×C312yes—
139Dic9⋊C4C22×C312yes—
140C4×Dic9C22×C311yescharacteristic
141C22×C3×C9—21no—
142C22×C3×C9—21no—
143C22×C3×C9—21no—
144C22×C3×C9C2411yescharacteristic
145C2×C3×D9—22no—
146C2×C3×D9—22no—
147C2×C3×D9—21no—
148C2×C3×D9—21no—
149C2×C3×D9—21no—
150C2×C3×D9C2×D414yes—
151C3×C4×C9—21no—
152C3×C4×C9—22no—
153C3×C4×C9—22no—
154C3×C4×C9C2×D412yes—
155C3×Dic9—22no—
156C3×Dic9—21no—
157C3×Dic9—21no—
158C3×Dic9#4112yescomplemented
159C2×C3×#41—91no—
160C2×C3×#41—91no—
161C2×#239—31no—
162C2×#235—31no—
163C3×C4⋊Q8—91no—
164C3×#123—92no—
165C3×#123—92no—
166C3×C22⋊Q8—21no—
167C3×C22⋊Q8—91no—
168C3×C22⋊Q8D911yescharacteristic, complemented
169C3×C22⋊Q8C3×D311yescharacteristic
170C3×#121—92no—
171C3×C4×D4—91no—
172C3×C4×D4—91no—
173C2×C3×C4⋊C4—92no—
174#944—31no—
175C4×#67—31no—
176#897—31no—
177#894—32no—
178#893—31no—
179D3×C4⋊C4—32no—
180#884—32no—
181#883—32no—
182C23×C32C22×D311yescharacteristic
183C22×C3×D3—32no—
184C22×C3×D3—31no—
185C32×Q8—22no—
186C2×C32×C4—22no—
187C2×C32×C4C22×D312yes—
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D312yes—
190C2×C32×C4C22×D311yescharacteristic
191C3×#67—62no—
192C3×#67—64no—
193C2×C3×Dic3—31no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—31no—
197C3×D12—34no—
198C3×D12—64no—
199C3×C4×D3—64no—
200C3×C4×D3—64no—
201C3×C4×D3—64no—
202C3×C4×D3—62no—
203C3×Dic6—62no—
204C23×C9—21no—
205C23×C9C23×C311yescharacteristic
206C22×D9C23×C311yescharacteristic
207C22×D9C23×C312yes—
208Q8×C9—22no—
209Q8×C9—42no—
210C2×C4×C9—22no—
211C2×C4×C9—21no—
212C2×C4×C9—22no—
213C2×C4×C9—22no—
214C2×C4×C9—21no—
215C2×C4×C9—42no—
216C2×C4×C9C23×C312yes—
217C2×C4×C9C23×C311yescharacteristic
218C2×C4×C9C23×C312yes—
219C2×C4×C9C23×C311yescharacteristic
220#611—24no—
221#611—22no—
222C2×Dic9C23×C311yescharacteristic
223C2×Dic9C23×C312yes—
224C2×Dic9C23×C311yescharacteristic
225C2×Dic9C23×C311yescharacteristic
226D36—24no—
227D36C3×D414yescomplemented
228C4×D9—24no—
229C4×D9—24no—
230C4×D9—22no—
231C4×D9—24no—
232Dic18—22no—
233#547—91noSylow
234C2×C3×C9—41no—
235C2×C3×C9C22×D411yescharacteristic
236C2×C3×C9C2×#4111yescharacteristic
237C2×C3×C92−1+411yescharacteristic
238C3×D9—24no—
239C3×D9—41no—
240C3×#41—184no—
241C3×#41—184no—
242C2×C3×Q8—21no—
243C2×C3×Q8—91no—
244C2×C3×Q8C2×D911yescharacteristic
245C2×C3×Q8C2×C3×D311yescharacteristic
246C2×C3×D4—91no—
247C2×C3×D4—92no—
248C2×C3×D4—91no—
249C2×C3×D4—92no—
250C22×C3×C4—21no—
251C22×C3×C4—92no—
252C22×C3×C4—92no—
253C22×C3×C4—92no—
254C22×C3×C4—91no—
255C22×C3×C4C2×D911yescharacteristic
256C22×C3×C4C2×C3×D311yescharacteristic
257C2×#67—32no—
258C2×#67—31no—
259#239—64no—
260#235—64no—
261C2×D12—31no—
262C2×D12—32no—
263C2×C4×D3—32no—
264C2×C4×D3—32no—
265C2×C4×D3—32no—
266C2×C4×D3—31no—
267C2×Dic6—31no—
268C3×C4⋊C4—22no—
269C3×C4⋊C4—21no—
270C3×C4⋊C4—91no—
271C3×C4⋊C4—92no—
272C3×C4⋊C4—92no—
273C3×C4⋊C4—92no—
274C3×C4⋊C4C2×D912yes—
275C3×C4⋊C4C2×D911yescharacteristic
276C3×C4⋊C4C2×C3×D312yes—
277C3×C4⋊C4C2×C3×D311yescharacteristic
278C3×C22⋊C4—22no—
279C3×C22⋊C4—91no—
280C3×C22⋊C4—92no—
281C3×C22⋊C4—91no—
282C3×C22⋊C4—92no—
283C3×C22⋊C4C2×D912yes—
284C3×C22⋊C4C2×C3×D312yes—
285C3×C42—91no—
286#217—31no—
287(C2×D3)⋊C4—32no—
288(C2×D3)⋊C4—32no—
289(C2×D3)⋊C4—31no—
290C4⋊Dic3—32no—
291Dic3⋊C4—32no—
292Dic3⋊C4—32no—
293Dic3⋊C4—31no—
294C4×Dic3—31no—
295C22×C32—21no—
296C22×C32—21no—
297C22×C32—21no—
298C22×C32C23×D311yescharacteristic
299C2×C3×D3—34no—
300C2×C3×D3—62no—
301C2×C3×D3—62no—
302C2×C3×D3—61no—
303C2×C3×D3—61no—
304C2×C3×D3—61no—
305C32×C4—21no—
306C32×C4—22no—
307C32×C4—22no—
308C32×C4D3×D412yes—
309C3×Dic3—32no—
310C3×Dic3—62no—
311C3×Dic3—61no—
312C3×Dic3—61no—
313C22×C9—22no—
314C22×C9—21no—
315C22×C9—21no—
316C22×C9—21no—
317C22×C9—21no—
318C22×C9—41no—
319C22×C9—41no—
320C22×C9C24×C311yescharacteristic
321C2×D9—22no—
322C2×D9—22no—
323C2×D9—21no—
324C2×D9—21no—
325C2×D9—21no—
326C2×D9C2×C3×D414yes—
327C4×C9—21no—
328C4×C9—22no—
329C4×C9—22no—
330C4×C9—22no—
331C4×C9—42no—
332C4×C9—41no—
333C4×C9—42no—
334C4×C9C2×C3×D412yes—
335Dic9—22no—
336Dic9—21no—
337Dic9—21no—
338Dic9C3×#4112yescomplemented
339C2×#41—91no—
340C2×#41—91no—
341C4⋊Q8—91no—
342#123—92no—
343#123—92no—
344C22⋊Q8—91no—
345C22⋊Q8C3×D911yescharacteristic, complemented
346#121—92no—
347C4×D4—91no—
348C4×D4—91no—
349C2×C4⋊C4—92no—
350C3×C9#54711yescharacteristic, complemented, Sylow
351C23×C3—21no—
352C23×C3—92no—
353C23×C3—91no—
354C23×C3C22×D911yescharacteristic
355C23×C3C22×C3×D311yescharacteristic
356C22×D3—32no—
357C22×D3—31no—
358C3×Q8—22no—
359C3×Q8—22no—
360C3×Q8—42no—
361C3×Q8—182no—
362C3×D4—94no—
363C3×D4—184no—
364C3×D4—182no—
365C3×D4—184no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—42no—
373C2×C3×C4—91no—
374C2×C3×C4—91no—
375C2×C3×C4—91no—
376C2×C3×C4—92no—
377C2×C3×C4—184no—
378C2×C3×C4—184no—
379C2×C3×C4—184no—
380C2×C3×C4—182no—
381C2×C3×C4C22×D912yes—
382C2×C3×C4C22×D911yescharacteristic
383C2×C3×C4C22×D912yes—
384C2×C3×C4C22×D911yescharacteristic
385C2×C3×C4C22×C3×D312yes—
386C2×C3×C4C22×C3×D311yescharacteristic
387C2×C3×C4C22×C3×D312yes—
388C2×C3×C4C22×C3×D311yescharacteristic
389#67—62no—
390#67—64no—
391C2×Dic3—31no—
392C2×Dic3—31no—
393C2×Dic3—32no—
394C2×Dic3—31no—
395D12—34no—
396D12—64no—
397C4×D3—64no—
398C4×D3—64no—
399C4×D3—64no—
400C4×D3—62no—
401Dic6—62no—
402C2×C32—41no—
403C2×C32C2×D3×D411yescharacteristic
404C2×C32#100711yescharacteristic
405C2×C32D3×#4111yescharacteristic
406C3×D3—64no—
407C3×D3—121no—
408C2×C9—21no—
409C2×C9—21no—
410C2×C9—21no—
411C2×C9—42no—
412C2×C9—41no—
413C2×C9C22×C3×D411yescharacteristic
414C2×C9C2×C3×#4111yescharacteristic
415C2×C9C3×2−1+411yescharacteristic
416D9—24no—
417D9—41no—
418#41—184no—
419#41—184no—
420C2×Q8—91no—
421C2×Q8C2×C3×D911yescharacteristic
422C2×D4—91no—
423C2×D4—92no—
424C2×D4—91no—
425C2×D4—92no—
426C22×C4—92no—
427C22×C4—92no—
428C22×C4—92no—
429C22×C4—91no—
430C22×C4C2×C3×D911yescharacteristic
431C4⋊C4—91no—
432C4⋊C4—92no—
433C4⋊C4—92no—
434C4⋊C4—92no—
435C4⋊C4C2×C3×D912yes—
436C4⋊C4C2×C3×D911yescharacteristic
437C22⋊C4—91no—
438C22⋊C4—92no—
439C22⋊C4—91no—
440C22⋊C4—92no—
441C22⋊C4C2×C3×D912yes—
442C42—91no—
443C22×C3—22no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—41no—
452C22×C3—41no—
453C22×C3—94no—
454C22×C3—182no—
455C22×C3—182no—
456C22×C3—181no—
457C22×C3—181no—
458C22×C3—181no—
459C22×C3C23×D911yescharacteristic
460C22×C3C23×C3×D311yescharacteristic
461C2×D3—34no—
462C2×D3—62no—
463C2×D3—62no—
464C2×D3—61no—
465C2×D3—61no—
466C2×D3—61no—
467C3×C4—21no—
468C3×C4—22no—
469C3×C4—22no—
470C3×C4—21no—
471C3×C4—22no—
472C3×C4—22no—
473C3×C4—22no—
474C3×C4—41no—
475C3×C4—42no—
476C3×C4—42no—
477C3×C4—92no—
478C3×C4—181no—
479C3×C4—182no—
480C3×C4—181no—
481C3×C4D4×D912yes—
482C3×C4C3×D3×D412yes—
483Dic3—32no—
484Dic3—62no—
485Dic3—61no—
486Dic3—61no—
487C32#563111yescharacteristic
488C9—21no—
489C9C3×#54711yescharacteristic, complemented
490C23—92no—
491C23—91no—
492C23C22×C3×D911yescharacteristic
493Q8—22no—
494Q8—182no—
495D4—94no—
496D4—184no—
497D4—182no—
498D4—184no—
499C2×C4—22no—
500C2×C4—91no—
501C2×C4—91no—
502C2×C4—91no—
503C2×C4—92no—
504C2×C4—184no—
505C2×C4—184no—
506C2×C4—184no—
507C2×C4—182no—
508C2×C4C22×C3×D912yes—
509C2×C4C22×C3×D911yescharacteristic
510C2×C4C22×C3×D912yes—
511C2×C4C22×C3×D911yescharacteristic
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—42no—
516C2×C3—41no—
517C2×C3—41no—
518C2×C3—184no—
519C2×C3—361no—
520C2×C3C2×D4×D911yescharacteristic
521C2×C3#6371311yescharacteristic
522C2×C3#41×D911yescharacteristic
523C2×C3C2×C3×D3×D411yescharacteristic
524C2×C3C3×#100711yescharacteristic
525C2×C3C3×D3×#4111yescharacteristic
526D3—64no—
527D3—121no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—94no—
532C22—182no—
533C22—182no—
534C22—181no—
535C22—181no—
536C22—181no—
537C22C23×C3×D911yescharacteristic
538C4—21no—
539C4—22no—
540C4—22no—
541C4—92no—
542C4—181no—
543C4—182no—
544C4—181no—
545C4C3×D4×D912yes—
546C3—21no—
547C3#1058973311yescharacteristic, direct product factor
548C3C3×#563111yescharacteristic
549C2—41no—
550C2—184no—
551C2—361no—
552C2C2×C3×D4×D911yescharacteristic
553C2C3×#6371311yescharacteristic
554C2C3×#41×D911yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×C22⋊Q8C2
socleC22×C32C23×D3
hypercenterC3×C22⋊Q8D9
hypocenterC9C3×#547
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3701C2semidirect
2. C2×C3×#3697C2semidirect
3. C3×C9×C22⋊Q8C2semidirect
4. C3×#63506C2semidirect
5. C3×C4×#611C2semidirect
6. C3×#63459C2semidirect
7. C3×#63456C2semidirect
8. C3×#63455C2semidirect
9. C3×C4⋊C4×D9C2semidirect
10. C3×#63446C2semidirect
11. C3×#63445C2semidirect
13. #10589733C3direct (N × H)
50. C2×#3701C2×C3semidirect
51. C2×#3697C2×C3semidirect
53. C9×C22⋊Q8C2×C3semidirect
54. #63506C2×C3semidirect
55. C4×#611C2×C3semidirect
56. #63459C2×C3semidirect
57. #63456C2×C3semidirect
58. #63455C2×C3semidirect
59. C4⋊C4×D9C2×C3semidirect
60. #63446C2×C3semidirect
61. #63445C2×C3semidirect
78. C3×D36D4semidirect
158. C3×Dic9#41semidirect
168. C3×C22⋊Q8D9semidirect
227. D36C3×D4semidirect
338. Dic9C3×#41semidirect
345. C22⋊Q8C3×D9semidirect
350. C3×C9#547semidirect
489. C9C3×#547semidirect
547. C3#10589733direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37012
C2×C3×#36972
C3×C9×C22⋊Q82
C3×#635062
C3×C4×#6112
C3×#634592
C3×#634562
C3×#634552
C3×C4⋊C4×D92
C3×#634462
C3×#634452
C3×#56313
#105897333