Finite Group Databasebeta

← C3×#10594810 #13174645

Subgroups — C3×#10594810 #13174645

Order 1728.

1198 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
2C32×#924C212yesmaximal
3C3×#63982C212yesmaximal
4C3×#63957C212yesmaximal
5C3×#63919C211yesmaximal, characteristic, complemented
6C3×#63873C211yesmaximal, characteristic, complemented
7C3×#63870C212yesmaximal, complemented
8C3×#63843C212yesmaximal, complemented
9C3×#63838C212yesmaximal, complemented
10C3×#5622—32nomaximal
11#10594810C311yesmaximal, characteristic, direct product factor
12C22×C33×C4C2211yescharacteristic
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C2×C32×Dic6C2212yes—
16C2×C3×#627C2212yes—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C3×C4×C3⋊Dic3C2211yescharacteristic
20C3×C4×C3⋊Dic3C2211yescharacteristic
21C32×#217C2214yes—
22C32×C4⋊Dic3C2212yes—
23C32×Dic3⋊C4C2214yes—
24C3×#3724C2212yes—
25C3×#3724C2212yes—
26C3×#3724C2212yes—
27C3×Dic3⋊Dic3C2214yes—
28C3×Dic32C2214yes—
29C32×C22⋊Q8—32no—
30C2×C4×C3⋊Dic3C2×C311yescharacteristic
31C3×Q8×Dic3—32no—
32C3×#944—32no—
33C3×#924—22no—
34C3×#924D312yes—
35C3×#924C2×C312yes—
36C2×C3×C4×Dic3—32no—
37C2×C3×C4×Dic3—61no—
38C3×#889—34no—
39C3×#888—32no—
40C3×#887—32no—
41C3×#887—34no—
42C3×#878—34no—
43C3×#877—34no—
44#63982C2×C312yes—
45#63957C2×C312yes—
46#63919C2×C311yescharacteristic, complemented
47#63873C2×C311yescharacteristic, complemented
48#63870C2×C312yescomplemented
49#63843C2×C312yescomplemented
50#63838C2×C312yescomplemented
51C23×C33C2311yescharacteristic
52C2×C33×C4—22no—
53C2×C33×C4C2311yescharacteristic
54C2×C33×C4C2311yescharacteristic
55C2×C3×C3⋊Dic3—22no—
56C2×C3×C3⋊Dic3Q814yescomplemented
57C2×C3×C3⋊Dic3C2311yescharacteristic
58C2×C3×C3⋊Dic3C2311yescharacteristic
59C2×C3×C3⋊Dic3C2311yescharacteristic
60C2×C3×C3⋊Dic3C2311yescharacteristic
61C2×C32×Dic3C2314yes—
62C2×C32×Dic3C2314yes—
63C32×Dic6—24no—
64C3×#627—24no—
65C3×#533—91no—
66#5622—32no—
67C2×C32×Q8—32no—
68C22×C32×C4—21no—
69C22×C32×C4—22no—
70C22×C32×C4—41no—
71C22×C32×C4C2×D312yes—
72C22×C32×C4C22×C311yescharacteristic
73C22×C3⋊Dic3C22×C311yescharacteristic
74C22×C3⋊Dic3C22×C311yescharacteristic
75C22×C3×Dic3—32no—
76C22×C3×Dic3—32no—
77C22×C3×Dic3—61no—
78C22×C3×Dic3—61no—
79C2×C3×Dic6—22no—
80C2×C3×Dic6—34no—
81C2×C3×Dic6C2×D312yes—
82C2×C3×Dic6C22×C312yes—
83C2×#627C22×C312yes—
84C32×C4⋊C4—32no—
85C32×C4⋊C4—34no—
86C32×C22⋊C4—34no—
87C4×C3⋊Dic3C22×C311yescharacteristic
88C4×C3⋊Dic3C22×C311yescharacteristic
89C4×C3⋊Dic3C22×C311yescharacteristic
90C4×C3⋊Dic3C22×C311yescharacteristic
91C3×#217—24no—
92C3×#217C2×D314yes—
93C3×#217C22×C314yes—
94C3×C4⋊Dic3—22no—
95C3×C4⋊Dic3—34no—
96C3×C4⋊Dic3C2×D312yes—
97C3×C4⋊Dic3C22×C312yes—
98C3×Dic3⋊C4—24no—
99C3×Dic3⋊C4—34no—
100C3×Dic3⋊C4—34no—
101C3×Dic3⋊C4—34no—
102C3×Dic3⋊C4—34no—
103C3×Dic3⋊C4C2×D314yes—
104C3×Dic3⋊C4C22×C314yes—
105C3×C4×Dic3—32no—
106C3×C4×Dic3—32no—
107C3×C4×Dic3—32no—
108C3×C4×Dic3—32no—
109C3×C4×Dic3—34no—
110C3×C4×Dic3—34no—
111C3×C4×Dic3—61no—
112C3×C4×Dic3—61no—
113C3×C4×Dic3—61no—
114C3×C4×Dic3—61no—
115#3724C22×C312yes—
116#3724C22×C312yes—
117#3724C22×C312yes—
118Dic3⋊Dic3C22×C314yes—
119Dic32C22×C314yes—
120C22×C33—21no—
121C22×C33—21no—
122C22×C33C2×Q812yes—
123C22×C33C2411yescharacteristic
124C33×C4—21no—
125C33×C4#4112yes—
126C3×C3⋊Dic3—21no—
127C3×C3⋊Dic3C2×Q812yes—
128C3×C3⋊Dic3C2×Q812yes—
129C3×C3⋊Dic3#4112yes—
130C32×Dic3—24no—
131C32×Dic3—24no—
132C3×C4⋊Q8—91no—
133C3×C4⋊Q8—91no—
134C3×#125—92no—
135C3×C22⋊Q8—32no—
136C3×C22⋊Q8—32no—
137C3×C22⋊Q8—62no—
138C3×C22⋊Q8—92no—
139C3×C4×Q8—92no—
140C3×C4×Q8—92no—
141C3×#117—92no—
142C2×C3×C42—91no—
143Q8×Dic3—32no—
144#944—32no—
145#924C3×D312yes—
146C2×C4×Dic3—32no—
147C2×C4×Dic3—61no—
148#889—34no—
149#888—32no—
150#887—32no—
151#887—34no—
152#878—34no—
153#877—34no—
154C23×C32—21no—
155C23×C32—22no—
156C23×C32—41no—
157C23×C32C22×D312yes—
158C23×C32C23×C311yescharacteristic
159C32×Q8—64no—
160C2×C32×C4—22no—
161C2×C32×C4—24no—
162C2×C32×C4—24no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—22no—
166C2×C32×C4—22no—
167C2×C32×C4—34no—
168C2×C32×C4—34no—
169C2×C32×C4—41no—
170C2×C32×C4—41no—
171C2×C32×C4—44no—
172C2×C32×C4—44no—
173C2×C32×C4C22×D312yes—
174C2×C32×C4C22×D312yes—
175C2×C32×C4C23×C311yescharacteristic
176C2×C32×C4C23×C311yescharacteristic
177C2×C3⋊Dic3—22no—
178C2×C3⋊Dic3C3×Q814yescomplemented
179C2×C3⋊Dic3C23×C311yescharacteristic
180C2×C3⋊Dic3C23×C311yescharacteristic
181C2×C3⋊Dic3C23×C311yescharacteristic
182C2×C3⋊Dic3C23×C311yescharacteristic
183C2×C3×Dic3—24no—
184C2×C3×Dic3—24no—
185C2×C3×Dic3—38no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3—32no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3—32no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—64no—
195C2×C3×Dic3—64no—
196C2×C3×Dic3—64no—
197C2×C3×Dic3C22×D314yes—
198C2×C3×Dic3C22×D314yes—
199C2×C3×Dic3C23×C314yes—
200C2×C3×Dic3C23×C314yes—
201C3×Dic6—24no—
202C3×Dic6—24no—
203C3×Dic6—44no—
204C3×Dic6—68no—
205#627—24no—
206#533—91noSylow
207C2×C33—22no—
208C2×C33C22×Q811yescharacteristic
209C2×C33C2×#4111yescharacteristic
210C2×C33C2×#4111yescharacteristic
211C2×C3×Q8—32no—
212C2×C3×Q8—32no—
213C2×C3×Q8—62no—
214C2×C3×Q8—92no—
215C22×C3×C4—21no—
216C22×C3×C4—22no—
217C22×C3×C4—41no—
218C22×C3×C4—91no—
219C22×C3×C4—91no—
220C22×C3×C4D3211yescharacteristic
221C22×C3×C4C2×C3×D312yes—
222C22×Dic3—32no—
223C22×Dic3—32no—
224C22×Dic3—61no—
225C22×Dic3—61no—
226C2×Dic6—34no—
227C2×Dic6C2×C3×D312yes—
228C3×C4⋊C4—34no—
229C3×C4⋊C4—32no—
230C3×C4⋊C4—32no—
231C3×C4⋊C4—34no—
232C3×C4⋊C4—64no—
233C3×C4⋊C4—62no—
234C3×C4⋊C4—92no—
235C3×C4⋊C4—92no—
236C3×C4⋊C4—92no—
237C3×C4⋊C4—94no—
238C3×C22⋊C4—34no—
239C3×C22⋊C4—34no—
240C3×C22⋊C4—64no—
241C3×C42—94no—
242C3×C42—91no—
243C3×C42—91no—
244C3×C42—91no—
245C3×C42—91no—
246#217C2×C3×D314yes—
247C4⋊Dic3—34no—
248C4⋊Dic3C2×C3×D312yes—
249Dic3⋊C4—34no—
250Dic3⋊C4—34no—
251Dic3⋊C4—34no—
252Dic3⋊C4—34no—
253Dic3⋊C4C2×C3×D314yes—
254C4×Dic3—32no—
255C4×Dic3—32no—
256C4×Dic3—32no—
257C4×Dic3—32no—
258C4×Dic3—34no—
259C4×Dic3—34no—
260C4×Dic3—61no—
261C4×Dic3—61no—
262C4×Dic3—61no—
263C4×Dic3—61no—
264C22×C32—21no—
265C22×C32—21no—
266C22×C32—22no—
267C22×C32—22no—
268C22×C32—22no—
269C22×C32—22no—
270C22×C32—21no—
271C22×C32—22no—
272C22×C32—22no—
273C22×C32—24no—
274C22×C32—42no—
275C22×C32—41no—
276C22×C32—42no—
277C22×C32—42no—
278C22×C32—42no—
279C22×C32—42no—
280C22×C32D3×Q814yes—
281C22×C32C2×C3×Q812yes—
282C22×C32C23×D312yes—
283C22×C32C24×C311yescharacteristic
284C32×C4—21no—
285C32×C4—22no—
286C32×C4—22no—
287C32×C4—22no—
288C32×C4—24no—
289C32×C4—42no—
290C32×C4—42no—
291C32×C4—42no—
292C32×C4—64no—
293C32×C4—64no—
294C32×C4#23714yes—
295C32×C4C3×#4112yes—
296C3⋊Dic3—21no—
297C3⋊Dic3C2×C3×Q812yes—
298C3⋊Dic3C2×C3×Q812yes—
299C3⋊Dic3C3×#4112yes—
300C3×Dic3—24no—
301C3×Dic3—24no—
302C3×Dic3—24no—
303C3×Dic3—24no—
304C3×Dic3—34no—
305C3×Dic3—34no—
306C3×Dic3—34no—
307C3×Dic3—44no—
308C3×Dic3—44no—
309C3×Dic3—62no—
310C3×Dic3—62no—
311C3×Dic3—62no—
312C3×Dic3—62no—
313C3×Dic3—62no—
314C4⋊Q8—91no—
315C4⋊Q8—91no—
316#125—92no—
317C22⋊Q8—32no—
318C22⋊Q8—92no—
319C4×Q8—92no—
320C4×Q8—92no—
321#117—92no—
322C2×C42—91no—
323C33#53311yescharacteristic, complemented, Sylow
324C23×C3—21no—
325C23×C3—22no—
326C23×C3—41no—
327C23×C3C2×D3211yescharacteristic
328C23×C3C22×C3×D312yes—
329C3×Q8—64no—
330C3×Q8—64no—
331C3×Q8—124no—
332C3×Q8—184no—
333C2×C3×C4—24no—
334C2×C3×C4—22no—
335C2×C3×C4—24no—
336C2×C3×C4—22no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—22no—
340C2×C3×C4—34no—
341C2×C3×C4—34no—
342C2×C3×C4—34no—
343C2×C3×C4—34no—
344C2×C3×C4—44no—
345C2×C3×C4—44no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—64no—
349C2×C3×C4—64no—
350C2×C3×C4—91no—
351C2×C3×C4—91no—
352C2×C3×C4—91no—
353C2×C3×C4—94no—
354C2×C3×C4—91no—
355C2×C3×C4—182no—
356C2×C3×C4C2×D3211yescharacteristic
357C2×C3×C4C2×D3211yescharacteristic
358C2×C3×C4C22×C3×D312yes—
359C2×C3×C4C22×C3×D312yes—
360C2×Dic3—32no—
361C2×Dic3—32no—
362C2×Dic3—32no—
363C2×Dic3—38no—
364C2×Dic3—32no—
365C2×Dic3—64no—
366C2×Dic3—61no—
367C2×Dic3—61no—
368C2×Dic3—61no—
369C2×Dic3—61no—
370C2×Dic3—64no—
371C2×Dic3—64no—
372C2×Dic3C22×C3×D314yes—
373C2×Dic3C22×C3×D314yes—
374Dic6—24no—
375Dic6—68no—
376C2×C32—22no—
377C2×C32—24no—
378C2×C32—24no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—22no—
383C2×C32—22no—
384C2×C32—22no—
385C2×C32—41no—
386C2×C32—41no—
387C2×C32—41no—
388C2×C32—44no—
389C2×C32—44no—
390C2×C32C2×#23712yes—
391C2×C32C2×D3×Q812yes—
392C2×C32D3×#4112yes—
393C2×C32C22×C3×Q811yescharacteristic
394C2×C32C2×C3×#4111yescharacteristic
395C2×C32C2×C3×#4111yescharacteristic
396C2×Q8—32no—
397C2×Q8—92no—
398C22×C4—91no—
399C22×C4—91no—
400C22×C4C3×D3211yescharacteristic
401C4⋊C4—34no—
402C4⋊C4—32no—
403C4⋊C4—92no—
404C4⋊C4—92no—
405C4⋊C4—94no—
406C4⋊C4—92no—
407C22⋊C4—34no—
408C42—94no—
409C42—91no—
410C42—91no—
411C42—91no—
412C42—91no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—21no—
420C22×C3—22no—
421C22×C3—22no—
422C22×C3—24no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—41no—
428C22×C3—42no—
429C22×C3#379712yes—
430C22×C3C3×D3×Q814yes—
431C22×C3C22×D3211yescharacteristic
432C22×C3C23×C3×D312yes—
433C3×C4—22no—
434C3×C4—21no—
435C3×C4—22no—
436C3×C4—22no—
437C3×C4—24no—
438C3×C4—42no—
439C3×C4—42no—
440C3×C4—42no—
441C3×C4—64no—
442C3×C4—64no—
443C3×C4—64no—
444C3×C4—64no—
445C3×C4—92no—
446C3×C4—92no—
447C3×C4—92no—
448C3×C4—124no—
449C3×C4—124no—
450C3×C4—181no—
451C3×C4#380512yes—
452C3×C4C3×#23714yes—
453Dic3—24no—
454Dic3—24no—
455Dic3—34no—
456Dic3—34no—
457Dic3—34no—
458Dic3—62no—
459Dic3—62no—
460Dic3—62no—
461Dic3—62no—
462Dic3—62no—
463C32—22no—
464C32—21no—
465C32—41no—
466C32#562212yescomplemented
467C32C3×#53311yescharacteristic, complemented
468C23C2×C3×D3211yescharacteristic
469Q8—64no—
470Q8—184no—
471C2×C4—22no—
472C2×C4—34no—
473C2×C4—34no—
474C2×C4—91no—
475C2×C4—91no—
476C2×C4—91no—
477C2×C4—94no—
478C2×C4—91no—
479C2×C4—182no—
480C2×C4C2×C3×D3211yescharacteristic
481C2×C4C2×C3×D3211yescharacteristic
482C2×C3—24no—
483C2×C3—24no—
484C2×C3—22no—
485C2×C3—21no—
486C2×C3—22no—
487C2×C3—21no—
488C2×C3—22no—
489C2×C3—21no—
490C2×C3—22no—
491C2×C3—44no—
492C2×C3—44no—
493C2×C3—41no—
494C2×C3—41no—
495C2×C3—41no—
496C2×C3C2×#379711yescharacteristic
497C2×C3#6430611yescharacteristic
498C2×C3C2×#380511yescharacteristic
499C2×C3C2×C3×#23712yes—
500C2×C3C2×C3×D3×Q812yes—
501C2×C3C3×D3×#4112yes—
502C22—21no—
503C22—21no—
504C22C3×#379712yes—
505C22C22×C3×D3211yescharacteristic
506C4—21no—
507C4—64no—
508C4—64no—
509C4—92no—
510C4—92no—
511C4—92no—
512C4—181no—
513C4C3×#380512yes—
514C3—21no—
515C3—22no—
516C3—41no—
517C3#1059481011yescharacteristic, direct product factor
518C3C3×#562212yescomplemented
519C2—22no—
520C2C2×C3×#379711yescharacteristic
521C2C3×#6430611yescharacteristic
522C2C2×C3×#380511yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socleC22×C33C24
prime radicalC23×C33C23
hypercenterC22×C3×C4D32
hypocenterC32C3×#533
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
5. C3×#63919C2semidirect
6. C3×#63873C2semidirect
7. C3×#63870C2semidirect
8. C3×#63843C2semidirect
9. C3×#63838C2semidirect
11. #10594810C3direct (N × H)
46. #63919C2×C3semidirect
47. #63873C2×C3semidirect
48. #63870C2×C3semidirect
49. #63843C2×C3semidirect
50. #63838C2×C3semidirect
56. C2×C3×C3⋊Dic3Q8semidirect
178. C2×C3⋊Dic3C3×Q8semidirect
323. C33#533semidirect
466. C32#5622semidirect
467. C32C3×#533semidirect
517. C3#10594810direct (N × H)
518. C3C3×#5622semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C3⋊Dic32
C32×#9242
C3×#639822
C3×#639572
C3×#639192
C3×#638732
C3×#638702
C3×#638432
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