Finite Group Databasebeta

← C3×#10593485 #13166880

Subgroups — C3×#10593485 #13166880

Order 1728.

924 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C22×C3×#624C211yesmaximal, characteristic
2C2×C3×C22⋊#157C211yesmaximal, characteristic
3C2×C3×#3773C212yesmaximal, complemented
4C3×#63974C211yesmaximal, characteristic
5C3×#63737C212yesmaximal, complemented
6#10593485C311yesmaximal, characteristic, direct product factor
7C23×C3×C3⋊D3C2211yescharacteristic
8C22×C3×#157C2212yes—
9C22×C32×Dic3—21no—
10C2×C3×#624—21no—
11C2×C3×#624—24no—
12C2×C3×#624C2211yescharacteristic
13C2×C3×#624C2211yescharacteristic
14C2×C3×#624C2212yes—
15C3×C22⋊#157C414yescomplemented
16C3×#3773—24no—
17C32×#217—21no—
18C3×#3722—21no—
19C22×C3×C4×D3—61no—
20C22×#624C2×C311yescharacteristic
21C2×C22⋊#157C2×C311yescharacteristic
22C2×#3773C2×C312yescomplemented
23C3×D3×C22⋊C4—61no—
24#63974C2×C311yescharacteristic
25#63737C2×C312yescomplemented
26C23×C33D411yescharacteristic
27C22×C3×C3⋊D3—22no—
28C22×C3×C3⋊D3—22no—
29C22×C3×C3⋊D3C2×C412yes—
30C22×C3×C3⋊D3D411yescharacteristic
31C22×C3×C3⋊D3Q812yes—
32C22×C3×C3⋊D3C2311yescharacteristic
33C2×C3×#157—22no—
34C2×C3×#157—22no—
35C2×C3×#157C2×C414yescomplemented
36C2×C32×Dic3—24no—
37C2×C32×Dic3—22no—
38C2×C32×Dic3—21no—
39C2×C32×Dic3—21no—
40C3×#624—24no—
41C3×#624—22no—
42C3×#624—22no—
43C3×#624—22no—
44C3×#624D414yes—
45C3×#385—91nomaximal
46C23×C3⋊D3C22×C311yescharacteristic
47C23×C3×D3—61no—
48C23×C3×D3—61no—
49C22×#157C22×C312yes—
50C22×C32×C4—61no—
51C22×C3×Dic3—21no—
52C22×C3×Dic3—21no—
53C22×C3×Dic3—41no—
54C2×C3×C4×D3—62no—
55C2×C3×C4×D3—68no—
56C2×C3×C4×D3—62no—
57C2×C3×C4×D3—61no—
58C2×C3×C4×D3—61no—
59C2×#624—21no—
60C2×#624—24no—
61C2×#624C22×C311yescharacteristic
62C2×#624C22×C311yescharacteristic
63C2×#624C22×C312yes—
64C22⋊#157C3×C414yescomplemented
65#3773—24no—
66C32×C22⋊C4—61no—
67C3×#217—21no—
68C3×#217—21no—
69C3×#217—41no—
70C3×(C2×D3)⋊C4—61no—
71C3×(C2×D3)⋊C4—61no—
72#3722—21no—
73C22×C33—21no—
74C22×C33—21no—
75C22×C33C4⋊C412yescomplemented
76C22×C33C2×D411yescharacteristic
77C2×C3×C3⋊D3—21no—
78C2×C3×C3⋊D3—21no—
79C2×C3×C3⋊D3—24no—
80C2×C3×C3⋊D3—24no—
81C2×C3×C3⋊D3C4⋊C412yescomplemented
82C2×C3×C3⋊D3C22×C411yescharacteristic
83C2×C3×C3⋊D3C2×D412yes—
84C2×C3×C3⋊D3C2×Q811yescharacteristic
85C2×C3×C3⋊D3#4112yes—
86C3×#157—24no—
87C32×Dic3—24no—
88C32×Dic3—41no—
89C32×Dic3—41no—
90C23×C3×C4—91no—
91C22×C4×D3—61no—
92C2×C3×C4⋊C4—92no—
93C2×C3×C22⋊C4—91no—
94C2×C3×C22⋊C4—91no—
95D3×C22⋊C4—61no—
96C3×#95—92no—
97C23×C32—21no—
98C23×C32—21no—
99C23×C32—41no—
100C23×C32—41no—
101C23×C32C3×D411yescharacteristic
102C22×C3⋊D3—22no—
103C22×C3⋊D3—22no—
104C22×C3⋊D3C2×C3×C412yes—
105C22×C3⋊D3C3×D411yescharacteristic
106C22×C3⋊D3C3×Q812yes—
107C22×C3⋊D3C23×C311yescharacteristic
108C22×C3×D3—62no—
109C22×C3×D3—61no—
110C22×C3×D3—62no—
111C22×C3×D3—61no—
112C22×C3×D3—61no—
113C22×C3×D3—61no—
114C22×C3×D3—62no—
115C22×C3×D3—62no—
116C22×C3×D3—64no—
117C22×C3×D3—64no—
118C22×C3×D3—122no—
119C22×C3×D3—122no—
120C2×#157—22no—
121C2×#157—22no—
122C2×#157C2×C3×C414yescomplemented
123C2×C32×C4—64no—
124C2×C32×C4—62no—
125C2×C32×C4—61no—
126C2×C32×C4—61no—
127C2×C3×Dic3—24no—
128C2×C3×Dic3—24no—
129C2×C3×Dic3—22no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—21no—
134C2×C3×Dic3—21no—
135C2×C3×Dic3—42no—
136C2×C3×Dic3—44no—
137C2×C3×Dic3—41no—
138C2×C3×Dic3—41no—
139C3×C4×D3—64no—
140C3×C4×D3—64no—
141C3×C4×D3—68no—
142C3×C4×D3—122no—
143C3×C4×D3—122no—
144#624—24no—
145#624—22no—
146#624—22no—
147#624—22no—
148#624C3×D414yes—
149#385—91noSylow
150C2×C33—22no—
151C2×C33C2×C4⋊C411yescharacteristic
152C2×C33#12011yescharacteristic
153C2×C33#12311yescharacteristic
154C3×C3⋊D3—22no—
155C3×C3⋊D3C4×D412yes—
156C3×C3⋊D3C22⋊Q812yes—
157C24×C3—91no—
158C23×D3—61no—
159C23×D3—61no—
160C22×C3×C4—61no—
161C22×C3×C4—61no—
162C22×C3×C4—92no—
163C22×C3×C4—91no—
164C22×C3×C4—91no—
165C22×C3×C4—92no—
166C22×C3×C4—121no—
167C22×C3×C4—184no—
168C22×C3×C4—181no—
169C22×Dic3—21no—
170C2×C4×D3—62no—
171C2×C4×D3—61no—
172C2×C4×D3—61no—
173C2×C4×D3—68no—
174C2×C4×D3—62no—
175C3×C4⋊C4—184no—
176C3×C22⋊C4—61no—
177C3×C22⋊C4—61no—
178C3×C22⋊C4—94no—
179C3×C22⋊C4—121no—
180C3×C22⋊C4—181no—
181#217—21no—
182(C2×D3)⋊C4—61no—
183(C2×D3)⋊C4—61no—
184C22×C32—22no—
185C22×C32—22no—
186C22×C32—21no—
187C22×C32—21no—
188C22×C32—21no—
189C22×C32—21no—
190C22×C32—22no—
191C22×C32—22no—
192C22×C32—42no—
193C22×C32—41no—
194C22×C32—41no—
195C22×C32—42no—
196C22×C32—41no—
197C22×C32—41no—
198C22×C32—42no—
199C22×C32—42no—
200C22×C32—81no—
201C22×C32—81no—
202C22×C32C3×C4⋊C412yescomplemented
203C22×C32C2×C3×D411yescharacteristic
204C2×C3⋊D3—21no—
205C2×C3⋊D3—21no—
206C2×C3⋊D3—24no—
207C2×C3⋊D3—24no—
208C2×C3⋊D3C3×C4⋊C412yescomplemented
209C2×C3⋊D3C22×C3×C411yescharacteristic
210C2×C3⋊D3C2×C3×D412yes—
211C2×C3⋊D3C2×C3×Q811yescharacteristic
212C2×C3⋊D3C3×#4112yes—
213C2×C3×D3—62no—
214C2×C3×D3—61no—
215C2×C3×D3—61no—
216C2×C3×D3—62no—
217C2×C3×D3—62no—
218C2×C3×D3—61no—
219C2×C3×D3—61no—
220C2×C3×D3—62no—
221C2×C3×D3—62no—
222C2×C3×D3—62no—
223C2×C3×D3—62no—
224C2×C3×D3—68no—
225C2×C3×D3—68no—
226C2×C3×D3—62no—
227C2×C3×D3—124no—
228C2×C3×D3—124no—
229C2×C3×D3—121no—
230C2×C3×D3—121no—
231#157—24no—
232C32×C4—64no—
233C32×C4—121no—
234C32×C4—121no—
235C3×Dic3—24no—
236C3×Dic3—24no—
237C3×Dic3—44no—
238C3×Dic3—41no—
239C3×Dic3—41no—
240C3×Dic3—41no—
241C3×Dic3—41no—
242C3×Dic3—81no—
243C3×Dic3—81no—
244C23×C4—91no—
245C2×C4⋊C4—92no—
246C2×C22⋊C4—91no—
247C2×C22⋊C4—91no—
248#95—92no—
249C33#38511yescharacteristic, complemented, Sylow
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—41no—
253C23×C3—41no—
254C23×C3—91no—
255C23×C3—91no—
256C23×C3—92no—
257C23×C3—92no—
258C23×C3—182no—
259C23×C3—182no—
260C23×C3GO+(4,𝔽2)11yescharacteristic
261C22×D3—62no—
262C22×D3—61no—
263C22×D3—62no—
264C22×D3—61no—
265C22×D3—61no—
266C22×D3—62no—
267C22×D3—61no—
268C22×D3—62no—
269C22×D3—64no—
270C22×D3—64no—
271C22×D3—122no—
272C22×D3—122no—
273C2×C3×C4—64no—
274C2×C3×C4—62no—
275C2×C3×C4—64no—
276C2×C3×C4—62no—
277C2×C3×C4—61no—
278C2×C3×C4—61no—
279C2×C3×C4—61no—
280C2×C3×C4—61no—
281C2×C3×C4—94no—
282C2×C3×C4—94no—
283C2×C3×C4—124no—
284C2×C3×C4—122no—
285C2×C3×C4—121no—
286C2×C3×C4—121no—
287C2×C3×C4—182no—
288C2×C3×C4—184no—
289C2×C3×C4—182no—
290C2×C3×C4—182no—
291C2×C3×C4—182no—
292C2×C3×C4—182no—
293C2×Dic3—24no—
294C2×Dic3—22no—
295C2×Dic3—21no—
296C2×Dic3—21no—
297C4×D3—64no—
298C4×D3—64no—
299C4×D3—68no—
300C4×D3—122no—
301C4×D3—122no—
302C2×C32—24no—
303C2×C32—22no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32—21no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—41no—
311C2×C32—41no—
312C2×C32—41no—
313C2×C32—41no—
314C2×C32—41no—
315C2×C32—41no—
316C2×C32—42no—
317C2×C32—44no—
318C2×C32—82no—
319C2×C32C2×C3×C4⋊C411yescharacteristic
320C2×C32C3×#12011yescharacteristic
321C2×C32C3×#12311yescharacteristic
322C3⋊D3—22no—
323C3⋊D3C3×C4×D412yes—
324C3⋊D3C3×C22⋊Q812yes—
325C3×D3—62no—
326C3×D3—62no—
327C3×D3—62no—
328C3×D3—62no—
329C3×D3—64no—
330C3×D3—122no—
331C24—91no—
332C22×C4—61no—
333C22×C4—92no—
334C22×C4—91no—
335C22×C4—91no—
336C22×C4—92no—
337C22×C4—184no—
338C22×C4—181no—
339C4⋊C4—184no—
340C22⋊C4—61no—
341C22⋊C4—94no—
342C22⋊C4—181no—
343C22×C3—22no—
344C22×C3—22no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—42no—
354C22×C3—42no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—42no—
358C22×C3—42no—
359C22×C3—81no—
360C22×C3—81no—
361C22×C3—92no—
362C22×C3—91no—
363C22×C3—91no—
364C22×C3—92no—
365C22×C3—92no—
366C22×C3—184no—
367C22×C3—184no—
368C22×C3—181no—
369C22×C3—181no—
370C22×C3#377312yescomplemented
371C22×C3C2×GO+(4,𝔽2)11yescharacteristic
372C2×D3—62no—
373C2×D3—62no—
374C2×D3—62no—
375C2×D3—62no—
376C2×D3—61no—
377C2×D3—61no—
378C2×D3—61no—
379C2×D3—61no—
380C2×D3—62no—
381C2×D3—62no—
382C2×D3—62no—
383C2×D3—68no—
384C2×D3—68no—
385C2×D3—62no—
386C2×D3—124no—
387C2×D3—124no—
388C2×D3—121no—
389C2×D3—121no—
390C3×C4—64no—
391C3×C4—64no—
392C3×C4—124no—
393C3×C4—121no—
394C3×C4—121no—
395C3×C4—121no—
396C3×C4—121no—
397C3×C4—184no—
398C3×C4—241no—
399C3×C4—241no—
400Dic3—24no—
401Dic3—41no—
402Dic3—41no—
403C32—21no—
404C32—21no—
405C32—41no—
406C32—41no—
407C32C3×#38511yescharacteristic, complemented
408C23—91no—
409C23—91no—
410C23—92no—
411C23—92no—
412C23—182no—
413C23—182no—
414C23C3×GO+(4,𝔽2)11yescharacteristic
415C2×C4—64no—
416C2×C4—62no—
417C2×C4—61no—
418C2×C4—61no—
419C2×C4—94no—
420C2×C4—94no—
421C2×C4—182no—
422C2×C4—184no—
423C2×C4—182no—
424C2×C4—182no—
425C2×C4—182no—
426C2×C4—182no—
427C2×C3—24no—
428C2×C3—22no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—42no—
436C2×C3—44no—
437C2×C3—41no—
438C2×C3—41no—
439C2×C3—41no—
440C2×C3—41no—
441C2×C3—41no—
442C2×C3—41no—
443C2×C3—82no—
444C2×C3—92no—
445C2×C3—92no—
446C2×C3—182no—
447C2×C3#6423311yescharacteristic
448C2×C3C2×#377311yescharacteristic
449C2×C3#6424111yescharacteristic
450D3—62no—
451D3—62no—
452D3—62no—
453D3—62no—
454D3—64no—
455D3—122no—
456C22—21no—
457C22—21no—
458C22—92no—
459C22—91no—
460C22—91no—
461C22—92no—
462C22—92no—
463C22—184no—
464C22—184no—
465C22—181no—
466C22—181no—
467C22C3×#377312yescomplemented
468C22C2×C3×GO+(4,𝔽2)11yescharacteristic
469C4—64no—
470C4—121no—
471C4—121no—
472C4—184no—
473C3—21no—
474C3—21no—
475C3—41no—
476C3—41no—
477C3#1059348511yescharacteristic, direct product factor
478C2—22no—
479C2—92no—
480C2—92no—
481C2—182no—
482C2C3×#6423311yescharacteristic
483C2C2×C3×#377311yescharacteristic
484C2C3×#6424111yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutatorC2×C3⋊D3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C3⋊D3C23×C3
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#385
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
3. C2×C3×#3773C2semidirect
5. C3×#63737C2semidirect
6. #10593485C3direct (N × H)
15. C3×C22⋊#157C4semidirect
22. C2×#3773C2×C3semidirect
25. #63737C2×C3semidirect
35. C2×C3×#157C2×C4semidirect
64. C22⋊#157C3×C4semidirect
75. C22×C33C4⋊C4semidirect
81. C2×C3×C3⋊D3C4⋊C4semidirect
122. C2×#157C2×C3×C4semidirect
202. C22×C32C3×C4⋊C4semidirect
208. C2×C3⋊D3C3×C4⋊C4semidirect
249. C33#385semidirect
370. C22×C3#3773semidirect
407. C32C3×#385semidirect
467. C22C3×#3773semidirect
477. C3#10593485direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6242
C2×C3×C22⋊#1572
C2×C3×#37732
C3×#639742
C3×#637372
#105934853
C3×#3859