Finite Group Databasebeta

← C3×#10589178 #13145474

Subgroups — C3×#10589178 #13145474

Order 1728.

610 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#940C211yesmaximal, characteristic
2C3×#63661C211yesmaximal, characteristic, complemented
3C32×#834C211yesmaximal, characteristic, complemented
4C3×#4545—31nomaximal
5C3×#4474—31nomaximal
6#10589178C311yesmaximal, characteristic, direct product factor
7C2×C33×D4C2211yescharacteristic
8C2×C32×#67—21no—
9C2×C32×#67—21no—
10C2×C32×D12C411yescharacteristic, complemented
11C3×#3757—21no—
12C32×#217—21no—
13C32×C4×Dic3C411yescharacteristic, complemented
14C32×#127—31no—
15C3×#940—21no—
16C3×#940D311yescharacteristic
17C3×#940C2×C311yescharacteristic
18C32×#99—31no—
19#63661C2×C311yescharacteristic, complemented
20C3×#834—21no—
21C3×#834—31no—
22C3×#834—31no—
23C3×#834—61no—
24C3×#834D311yescharacteristic, complemented
25C3×#834C2×C311yescharacteristic, complemented
26C23×C33D411yescharacteristic
27C23×C33D411yescharacteristic
28C22×C32×D3—21no—
29C33×D4—41no—
30C2×C33×C4C2×C411yescharacteristic
31C2×C3×C3⋊Dic3—41no—
32C32×#67—42no—
33C32×#67—42no—
34C2×C32×Dic3—21no—
35C2×C32×Dic3—41no—
36C32×D12—41no—
37C3×#353—91no—
38#4545—31no—
39#4474—31no—
40C2×C32×D4—21no—
41C2×C32×D4—21no—
42C2×C32×D4—21no—
43C2×C32×D4—31no—
44C2×C32×D4—41no—
45C2×C32×D4—61no—
46C2×C32×D4—61no—
47C2×C32×D4C2×D311yescharacteristic
48C2×C32×D4C2×D311yescharacteristic
49C2×C32×D4C22×C311yescharacteristic
50C2×C3×#67—22no—
51C2×C3×#67—22no—
52C2×C3×#67—21no—
53C2×C3×#67—21no—
54C2×C3×#67—21no—
55C2×C3×#67—21no—
56C2×C3×D12—21no—
57C2×C3×D12Dic311yescharacteristic, complemented
58C2×C3×D12C3×C411yescharacteristic, complemented
59C32×C22⋊C4—61no—
60C32×C42—31no—
61#3757—21no—
62C3×#217—22no—
63C3×#217—21no—
64C3×#217—21no—
65C3×#217—62no—
66C3×#217—61no—
67C3×#217—61no—
68C3×C4×Dic3—21no—
69C3×C4×Dic3Dic311yescharacteristic, complemented
70C3×C4×Dic3C3×C411yescharacteristic, complemented
71C22×C33—21no—
72C22×C33—21no—
73C22×C33—41no—
74C22×C33—41no—
75C22×C33C22⋊C411yescharacteristic
76C2×C32×D3—41no—
77C2×C32×D3—41no—
78C2×C32×D3—41no—
79C33×C4—21no—
80C3×C3⋊Dic3—81no—
81C32×Dic3—22no—
82C32×Dic3—81no—
83C3×#127—31no—
84C3×#127—31no—
85C3×#127—61no—
86#940C3×D311yescharacteristic
87C3×#99—31no—
88C3×#99—31no—
89C3×#99—61no—
90C3×#99—91no—
91#834—31no—
92#834—31no—
93#834—61no—
94#834C3×D311yescharacteristic, complemented
95C23×C32—21no—
96C23×C32—21no—
97C23×C32—21no—
98C23×C32—21no—
99C23×C32—21no—
100C23×C32—21no—
101C23×C32—41no—
102C23×C32—41no—
103C23×C32—61no—
104C23×C32D1211yescharacteristic
105C23×C32#6711yescharacteristic
106C23×C32#6711yescharacteristic
107C23×C32#6711yescharacteristic
108C23×C32C3×D411yescharacteristic
109C23×C32C3×D411yescharacteristic
110C22×C3×D3—22no—
111C22×C3×D3—21no—
112C22×C3×D3—21no—
113C32×D4—41no—
114C32×D4—41no—
115C32×D4—41no—
116C32×D4—42no—
117C32×D4—42no—
118C32×D4—42no—
119C32×D4—44no—
120C32×D4—121no—
121C32×D4—122no—
122C32×D4—122no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—41no—
127C2×C32×C4—61no—
128C2×C32×C4—121no—
129C2×C32×C4C4×D311yescharacteristic
130C2×C32×C4C2×Dic311yescharacteristic
131C2×C32×C4C2×C3×C411yescharacteristic
132C2×C3⋊Dic3—41no—
133C3×#67—42no—
134C3×#67—42no—
135C3×#67—42no—
136C3×#67—42no—
137C3×#67—44no—
138C3×#67—44no—
139C2×C3×Dic3—22no—
140C2×C3×Dic3—21no—
141C2×C3×Dic3—21no—
142C2×C3×Dic3—42no—
143C2×C3×Dic3—41no—
144C2×C3×Dic3—41no—
145C2×C3×Dic3—122no—
146C2×C3×Dic3—121no—
147C2×C3×Dic3—121no—
148C3×D12—41no—
149C3×D12—41no—
150C3×D12—42no—
151#353—91noSylow
152C2×C33—21no—
153C2×C33—41no—
154C2×C33—41no—
155C2×C33#9911yescharacteristic
156C32×D3—81no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—31no—
161C2×C3×D4—31no—
162C2×C3×D4—41no—
163C2×C3×D4—61no—
164C2×C3×D4—62no—
165C2×C3×D4—62no—
166C2×C3×D4—61no—
167C2×C3×D4—61no—
168C2×C3×D4—61no—
169C2×C3×D4—61no—
170C2×C3×D4D3211yescharacteristic
171C2×C3×D4C2×C3×D311yescharacteristic
172C2×C3×D4C2×C3×D311yescharacteristic
173C2×#67—21no—
174C2×#67—21no—
175C2×D12C3×Dic311yescharacteristic, complemented
176C3×C22⋊C4—62no—
177C3×C22⋊C4—61no—
178C3×C22⋊C4—61no—
179C3×C22⋊C4—181no—
180C3×C42—31no—
181C3×C42—31no—
182C3×C42—61no—
183#217—21no—
184#217—62no—
185#217—61no—
186#217—61no—
187C4×Dic3C3×Dic311yescharacteristic, complemented
188C22×C32—21no—
189C22×C32—21no—
190C22×C32—22no—
191C22×C32—22no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—41no—
200C22×C32—41no—
201C22×C32—41no—
202C22×C32—41no—
203C22×C32—41no—
204C22×C32—42no—
205C22×C32—42no—
206C22×C32—41no—
207C22×C32—41no—
208C22×C32—41no—
209C22×C32—41no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32—42no—
213C22×C32—42no—
214C22×C32—81no—
215C22×C32—81no—
216C22×C32—81no—
217C22×C32—81no—
218C22×C32—82no—
219C22×C32—82no—
220C22×C32—121no—
221C22×C32—121no—
222C22×C32—121no—
223C22×C32(C2×D3)⋊C411yescharacteristic
224C22×C32#21711yescharacteristic
225C22×C32C3×C22⋊C411yescharacteristic
226C2×C3×D3—42no—
227C2×C3×D3—42no—
228C2×C3×D3—42no—
229C2×C3×D3—41no—
230C2×C3×D3—41no—
231C2×C3×D3—41no—
232C2×C3×D3—41no—
233C2×C3×D3—41no—
234C2×C3×D3—41no—
235C32×C4—21no—
236C32×C4—22no—
237C32×C4—21no—
238C32×C4—21no—
239C32×C4—41no—
240C32×C4—41no—
241C32×C4—42no—
242C32×C4—62no—
243C32×C4—241no—
244C3⋊Dic3—81no—
245C3×Dic3—24no—
246C3×Dic3—22no—
247C3×Dic3—22no—
248C3×Dic3—82no—
249C3×Dic3—81no—
250C3×Dic3—81no—
251C3×Dic3—242no—
252C3×Dic3—241no—
253C3×Dic3—241no—
254#127—31no—
255#99—31no—
256#99—91no—
257C33#35311yescharacteristic, complemented, Sylow
258C23×C3—21no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—62no—
267C23×C3—61no—
268C23×C3—61no—
269C23×C3#62511yescharacteristic
270C23×C3#62611yescharacteristic
271C23×C3C3×D1211yescharacteristic
272C23×C3C3×#6711yescharacteristic
273C23×C3C3×#6711yescharacteristic
274C23×C3C3×#6711yescharacteristic
275C22×D3—21no—
276C3×D4—41no—
277C3×D4—41no—
278C3×D4—41no—
279C3×D4—42no—
280C3×D4—42no—
281C3×D4—42no—
282C3×D4—44no—
283C3×D4—121no—
284C3×D4—121no—
285C3×D4—122no—
286C3×D4—122no—
287C3×D4—122no—
288C3×D4—122no—
289C3×D4—122no—
290C3×D4—124no—
291C3×D4—124no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—41no—
296C2×C3×C4—62no—
297C2×C3×C4—61no—
298C2×C3×C4—61no—
299C2×C3×C4—122no—
300C2×C3×C4—121no—
301C2×C3×C4—121no—
302C2×C3×C4—361no—
303C2×C3×C4D3×Dic311yescharacteristic
304C2×C3×C4C3×C4×D311yescharacteristic
305C2×C3×C4C2×C3×Dic311yescharacteristic
306#67—42no—
307#67—42no—
308C2×Dic3—21no—
309C2×Dic3—41no—
310C2×Dic3—122no—
311C2×Dic3—121no—
312C2×Dic3—121no—
313D12—41no—
314C2×C32—21no—
315C2×C32—22no—
316C2×C32—21no—
317C2×C32—21no—
318C2×C32—21no—
319C2×C32—21no—
320C2×C32—21no—
321C2×C32—41no—
322C2×C32—41no—
323C2×C32—41no—
324C2×C32—42no—
325C2×C32—41no—
326C2×C32—41no—
327C2×C32—42no—
328C2×C32—42no—
329C2×C32—41no—
330C2×C32—41no—
331C2×C32—41no—
332C2×C32—41no—
333C2×C32—81no—
334C2×C32—81no—
335C2×C32—81no—
336C2×C32—81no—
337C2×C32—82no—
338C2×C32—82no—
339C2×C32—241no—
340C2×C32#80611yescharacteristic
341C2×C32#83411yescharacteristic
342C2×C32C3×#9911yescharacteristic
343C3×D3—82no—
344C3×D3—81no—
345C3×D3—81no—
346C2×D4—31no—
347C2×D4—61no—
348C2×D4—61no—
349C2×D4C3×D3211yescharacteristic
350C22⋊C4—61no—
351C22⋊C4—181no—
352C42—31no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—41no—
365C22×C3—41no—
366C22×C3—41no—
367C22×C3—41no—
368C22×C3—42no—
369C22×C3—42no—
370C22×C3—41no—
371C22×C3—41no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—41no—
375C22×C3—41no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—41no—
379C22×C3—81no—
380C22×C3—81no—
381C22×C3—81no—
382C22×C3—81no—
383C22×C3—82no—
384C22×C3—82no—
385C22×C3—122no—
386C22×C3—122no—
387C22×C3—121no—
388C22×C3—121no—
389C22×C3—121no—
390C22×C3—121no—
391C22×C3—122no—
392C22×C3—121no—
393C22×C3—121no—
394C22×C3(C2×D3)⋊Dic311yescharacteristic
395C22×C3C3×(C2×D3)⋊C411yescharacteristic
396C22×C3C3×#21711yescharacteristic
397C2×D3—41no—
398C2×D3—41no—
399C2×D3—41no—
400C3×C4—22no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—41no—
405C3×C4—41no—
406C3×C4—42no—
407C3×C4—64no—
408C3×C4—62no—
409C3×C4—62no—
410C3×C4—242no—
411C3×C4—241no—
412C3×C4—241no—
413C3×C4—721no—
414Dic3—22no—
415Dic3—81no—
416Dic3—242no—
417Dic3—241no—
418Dic3—241no—
419C32—21no—
420C32—21no—
421C32—21no—
422C32—41no—
423C32#447411yescharacteristic, complemented
424C32#454511yescharacteristic, complemented
425C32C3×#35311yescharacteristic, complemented
426C23—61no—
427C23C3×#62511yescharacteristic
428C23C3×#62611yescharacteristic
429D4—41no—
430D4—121no—
431D4—122no—
432D4—122no—
433C2×C4—61no—
434C2×C4—121no—
435C2×C4—361no—
436C2×C4C3×D3×Dic311yescharacteristic
437C2×C3—22no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—41no—
445C2×C3—41no—
446C2×C3—42no—
447C2×C3—42no—
448C2×C3—42no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3—41no—
452C2×C3—41no—
453C2×C3—41no—
454C2×C3—41no—
455C2×C3—41no—
456C2×C3—81no—
457C2×C3—81no—
458C2×C3—81no—
459C2×C3—81no—
460C2×C3—82no—
461C2×C3—82no—
462C2×C3—242no—
463C2×C3—241no—
464C2×C3—241no—
465C2×C3#6358011yescharacteristic
466C2×C3C3×#80611yescharacteristic
467C2×C3C3×#83411yescharacteristic
468D3—81no—
469C22—21no—
470C22—21no—
471C22—41no—
472C22—41no—
473C22—121no—
474C22—121no—
475C22—121no—
476C22C3×(C2×D3)⋊Dic311yescharacteristic
477C4—21no—
478C4—62no—
479C4—241no—
480C4—721no—
481C3—21no—
482C3—21no—
483C3—21no—
484C3—41no—
485C3#1058917811yescharacteristic, direct product factor
486C3C3×#447411yescharacteristic, complemented
487C3C3×#454511yescharacteristic, complemented
488C2—21no—
489C2—41no—
490C2—41no—
491C2—241no—
492C2C3×#6358011yescharacteristic
subgroupG/H
centerC2×C3#63580
commutatorC2×C32×C4C2×C3×C4
FrattiniC2×D4C3×D32
FittingC2×C33×D4C22
socleC2×C33#99
prime radicalC32×#940C2
cosocleC2×C32×D4C22×C3
hypercenterC2×C3×D4D32
hypocenterC32C3×#353
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C3×#63661C2semidirect
3. C32×#834C2semidirect
6. #10589178C3direct (N × H)
10. C2×C32×D12C4semidirect
13. C32×C4×Dic3C4semidirect
19. #63661C2×C3semidirect
24. C3×#834D3semidirect
25. C3×#834C2×C3semidirect
57. C2×C3×D12Dic3semidirect
58. C2×C3×D12C3×C4semidirect
69. C3×C4×Dic3Dic3semidirect
70. C3×C4×Dic3C3×C4semidirect
94. #834C3×D3semidirect
175. C2×D12C3×Dic3semidirect
187. C4×Dic3C3×Dic3semidirect
257. C33#353semidirect
423. C32#4474semidirect
424. C32#4545semidirect
425. C32C3×#353semidirect
485. C3#10589178direct (N × H)
486. C3C3×#4474semidirect
487. C3C3×#4545semidirect

Maximal subgroups

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

subgroupindex
C32×#9402
C3×#636612
C32×#8342
C3×#45453
C3×#44743
#105891783