Finite Group Databasebeta

← C3×#10591217 #13158306

Subgroups — C3×#10591217 #13158306

Order 1728.

834 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3750C211yesmaximal, characteristic
2C2×C3×#3724C211yesmaximal, characteristic
3C32×C3⋊#95C212yesmaximal
4C3×#63579C211yesmaximal, characteristic
5C3×#63579C212yesmaximal
6C3×#4658—32nomaximal
7#10591217C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2212yes—
13C3×#3750—22no—
14C3×#3724—22no—
15C2×#3750C2×C311yescharacteristic
16C2×C3×Dic3⋊C4—32no—
17C2×C3×Dic3⋊C4—32no—
18C2×C3×Dic3⋊C4—61no—
19C2×#3724C2×C311yescharacteristic
20C32×#95—32no—
21C3×C3⋊#95—22no—
22C3×C3⋊#95—32no—
23C3×C3⋊#95—32no—
24C3×C3⋊#95—32no—
25C3×C3⋊#95D312yes—
26C3×C3⋊#95C2×C312yes—
27#63579C2×C311yescharacteristic
28#63579C2×C312yes—
29C23×C33C2311yescharacteristic
30C2×C33×C4—21no—
31C2×C33×C4—22no—
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3D412yes—
36C2×C3×C3⋊Dic3Q812yes—
37C2×C32×Dic3—22no—
38C2×C32×Dic3—22no—
39C2×C32×Dic3—22no—
40C2×C32×Dic3—22no—
41C2×C32×Dic3—22no—
42C2×C32×Dic3—22no—
43C3×#400—91no—
44#4658—32no—
45C22×C32×C4—21no—
46C22×C32×C4—22no—
47C22×C32×C4—32no—
48C22×C32×C4—32no—
49C22×C32×C4—41no—
50C22×C32×C4C2×D312yes—
51C22×C32×C4C22×C311yescharacteristic
52C22×C3⋊Dic3C22×C311yescharacteristic
53C22×C3⋊Dic3C22×C311yescharacteristic
54C22×C3×Dic3—22no—
55C22×C3×Dic3—22no—
56C22×C3×Dic3—32no—
57C22×C3×Dic3—32no—
58C22×C3×Dic3—61no—
59C22×C3×Dic3—61no—
60C22×C3×Dic3C2×D312yes—
61C22×C3×Dic3C2×D312yes—
62C22×C3×Dic3C22×C312yes—
63C22×C3×Dic3C22×C312yes—
64#3750—22no—
65C3×Dic3⋊C4—64no—
66C3×Dic3⋊C4—64no—
67C3×Dic3⋊C4—64no—
68#3724—22no—
69C22×C33C2×D411yescharacteristic
70C22×C33C2×Q811yescharacteristic
71C22×C33#4111yescharacteristic
72C22×C33#4112yes—
73C22×C33#4112yes—
74C33×C4—41no—
75C3×C3⋊Dic3—22no—
76C3×C3⋊Dic3—22no—
77C32×Dic3—42no—
78C32×Dic3—42no—
79C2×C3×C4⋊C4—91no—
80C2×C3×C4⋊C4—91no—
81C2×Dic3⋊C4—32no—
82C2×Dic3⋊C4—32no—
83C2×Dic3⋊C4—61no—
84C3×#95—32no—
85C3×#95—32no—
86C3×#95—62no—
87C3×#95—91no—
88C3×#95—92no—
89C3⋊#95—32no—
90C3⋊#95—32no—
91C3⋊#95—32no—
92C3⋊#95C3×D312yes—
93C23×C32—21no—
94C23×C32—22no—
95C23×C32—41no—
96C23×C32C22×D312yes—
97C23×C32C23×C311yescharacteristic
98C2×C32×C4—22no—
99C2×C32×C4—21no—
100C2×C32×C4—22no—
101C2×C32×C4—24no—
102C2×C32×C4—22no—
103C2×C32×C4—22no—
104C2×C32×C4—22no—
105C2×C32×C4—42no—
106C2×C32×C4—42no—
107C2×C32×C4—42no—
108C2×C32×C4—42no—
109C2×C32×C4—44no—
110C2×C32×C4—62no—
111C2×C32×C4—62no—
112C2×C32×C4—62no—
113C2×C32×C4—62no—
114C2×C32×C4—62no—
115C2×C32×C4—62no—
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3—22no—
119C2×C3⋊Dic3C3×D412yes—
120C2×C3⋊Dic3C3×Q812yes—
121C2×C3×Dic3—24no—
122C2×C3×Dic3—24no—
123C2×C3×Dic3—22no—
124C2×C3×Dic3—22no—
125C2×C3×Dic3—22no—
126C2×C3×Dic3—22no—
127C2×C3×Dic3—22no—
128C2×C3×Dic3—22no—
129C2×C3×Dic3—22no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—22no—
132C2×C3×Dic3—22no—
133C2×C3×Dic3—22no—
134C2×C3×Dic3—22no—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—34no—
137C2×C3×Dic3—42no—
138C2×C3×Dic3—42no—
139C2×C3×Dic3—42no—
140C2×C3×Dic3—42no—
141C2×C3×Dic3—62no—
142C2×C3×Dic3—62no—
143C2×C3×Dic3—62no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—122no—
150#400—91noSylow
151C2×C33C22⋊Q811yescharacteristic
152C2×C33#12311yescharacteristic
153C2×C33#12411yescharacteristic
154C2×C33#12512yes—
155C2×C33#12612yes—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—32no—
159C22×C3×C4—32no—
160C22×C3×C4—32no—
161C22×C3×C4—32no—
162C22×C3×C4—41no—
163C22×C3×C4—62no—
164C22×C3×C4—62no—
165C22×C3×C4—91no—
166C22×C3×C4—91no—
167C22×C3×C4D3211yescharacteristic
168C22×C3×C4C2×C3×D312yes—
169C22×Dic3—32no—
170C22×Dic3—32no—
171C22×Dic3—61no—
172C22×Dic3—61no—
173C22×Dic3C2×C3×D312yes—
174C22×Dic3C2×C3×D312yes—
175C3×C4⋊C4—182no—
176C3×C4⋊C4—182no—
177Dic3⋊C4—64no—
178Dic3⋊C4—64no—
179Dic3⋊C4—64no—
180C22×C32—22no—
181C22×C32—21no—
182C22×C32—21no—
183C22×C32—22no—
184C22×C32—21no—
185C22×C32—22no—
186C22×C32—22no—
187C22×C32—22no—
188C22×C32—22no—
189C22×C32—22no—
190C22×C32—22no—
191C22×C32—22no—
192C22×C32—41no—
193C22×C32—41no—
194C22×C32—42no—
195C22×C32—42no—
196C22×C32—41no—
197C22×C32#23512yes—
198C22×C32#23512yes—
199C22×C32#23512yes—
200C22×C32D3×D412yes—
201C22×C32#23712yes—
202C22×C32#23712yes—
203C22×C32D3×Q812yes—
204C22×C32C2×C3×D411yescharacteristic
205C22×C32C2×C3×Q811yescharacteristic
206C22×C32C3×#4111yescharacteristic
207C22×C32C3×#4112yes—
208C22×C32C3×#4112yes—
209C32×C4—44no—
210C32×C4—42no—
211C32×C4—44no—
212C32×C4—42no—
213C32×C4—41no—
214C32×C4—122no—
215C32×C4—122no—
216C3⋊Dic3—22no—
217C3⋊Dic3—22no—
218C3×Dic3—44no—
219C3×Dic3—44no—
220C3×Dic3—42no—
221C3×Dic3—42no—
222C3×Dic3—42no—
223C3×Dic3—42no—
224C3×Dic3—64no—
225C3×Dic3—64no—
226C3×Dic3—64no—
227C3×Dic3—122no—
228C2×C4⋊C4—91no—
229C2×C4⋊C4—91no—
230#95—32no—
231#95—91no—
232#95—92no—
233C33#40011yescharacteristic, complemented, Sylow
234C23×C3—22no—
235C23×C3—21no—
236C23×C3—41no—
237C23×C3C2×D3211yescharacteristic
238C23×C3C22×C3×D312yes—
239C2×C3×C4—22no—
240C2×C3×C4—24no—
241C2×C3×C4—22no—
242C2×C3×C4—21no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—44no—
250C2×C3×C4—42no—
251C2×C3×C4—64no—
252C2×C3×C4—64no—
253C2×C3×C4—62no—
254C2×C3×C4—62no—
255C2×C3×C4—62no—
256C2×C3×C4—62no—
257C2×C3×C4—62no—
258C2×C3×C4—62no—
259C2×C3×C4—62no—
260C2×C3×C4—62no—
261C2×C3×C4—62no—
262C2×C3×C4—62no—
263C2×C3×C4—62no—
264C2×C3×C4—62no—
265C2×C3×C4—92no—
266C2×C3×C4—92no—
267C2×C3×C4—122no—
268C2×C3×C4—122no—
269C2×C3×C4—122no—
270C2×C3×C4—122no—
271C2×C3×C4—181no—
272C2×C3×C4—182no—
273C2×C3×C4—181no—
274C2×Dic3—22no—
275C2×Dic3—22no—
276C2×Dic3—22no—
277C2×Dic3—22no—
278C2×Dic3—22no—
279C2×Dic3—22no—
280C2×Dic3—34no—
281C2×Dic3—34no—
282C2×Dic3—62no—
283C2×Dic3—62no—
284C2×Dic3—62no—
285C2×Dic3—62no—
286C2×Dic3—62no—
287C2×Dic3—62no—
288C2×Dic3—62no—
289C2×Dic3—62no—
290C2×Dic3—122no—
291C2×C32—22no—
292C2×C32—22no—
293C2×C32—22no—
294C2×C32—22no—
295C2×C32—22no—
296C2×C32—22no—
297C2×C32—22no—
298C2×C32—22no—
299C2×C32—21no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32—22no—
303C2×C32—41no—
304C2×C32—41no—
305C2×C32—41no—
306C2×C32—42no—
307C2×C32—42no—
308C2×C32#87612yes—
309C2×C32#87912yes—
310C2×C32#88312yes—
311C2×C32#88512yes—
312C2×C32#88912yes—
313C2×C32#88912yes—
314C2×C32#89612yes—
315C2×C32C3×C22⋊Q811yescharacteristic
316C2×C32C3×#12311yescharacteristic
317C2×C32C3×#12411yescharacteristic
318C2×C32C3×#12512yes—
319C2×C32C3×#12612yes—
320C22×C4—32no—
321C22×C4—32no—
322C22×C4—91no—
323C22×C4—91no—
324C22×C4C3×D3211yescharacteristic
325C4⋊C4—182no—
326C4⋊C4—182no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—21no—
331C22×C3—22no—
332C22×C3—21no—
333C22×C3—22no—
334C22×C3—22no—
335C22×C3—22no—
336C22×C3—22no—
337C22×C3—21no—
338C22×C3—22no—
339C22×C3—42no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—42no—
343C22×C3—41no—
344C22×C3#379512yes—
345C22×C3#379711yescharacteristic
346C22×C3#379811yescharacteristic
347C22×C3#380412yes—
348C22×C3#381111yescharacteristic
349C22×C3C3×#23512yes—
350C22×C3C3×#23512yes—
351C22×C3C3×#23512yes—
352C22×C3C3×D3×D412yes—
353C22×C3C3×#23712yes—
354C22×C3C3×#23712yes—
355C22×C3C3×D3×Q812yes—
356C3×C4—42no—
357C3×C4—44no—
358C3×C4—44no—
359C3×C4—42no—
360C3×C4—41no—
361C3×C4—124no—
362C3×C4—124no—
363C3×C4—122no—
364C3×C4—122no—
365C3×C4—122no—
366C3×C4—122no—
367C3×C4—182no—
368C3×C4—182no—
369Dic3—42no—
370Dic3—42no—
371Dic3—64no—
372Dic3—64no—
373Dic3—64no—
374Dic3—122no—
375C32—22no—
376C32—21no—
377C32—41no—
378C32#465812yescomplemented
379C32C3×#40011yescharacteristic, complemented
380C23C2×C3×D3211yescharacteristic
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—62no—
389C2×C4—92no—
390C2×C4—92no—
391C2×C4—181no—
392C2×C4—182no—
393C2×C4—181no—
394C2×C3—21no—
395C2×C3—22no—
396C2×C3—22no—
397C2×C3—22no—
398C2×C3—22no—
399C2×C3—22no—
400C2×C3—22no—
401C2×C3—22no—
402C2×C3—21no—
403C2×C3—22no—
404C2×C3—22no—
405C2×C3—21no—
406C2×C3—41no—
407C2×C3—42no—
408C2×C3—42no—
409C2×C3—41no—
410C2×C3—41no—
411C2×C3#6384712yes—
412C2×C3#6385912yes—
413C2×C3#6386511yescharacteristic
414C2×C3#6391311yescharacteristic
415C2×C3#6397511yescharacteristic
416C2×C3C3×#87612yes—
417C2×C3C3×#87912yes—
418C2×C3C3×#88312yes—
419C2×C3C3×#88512yes—
420C2×C3C3×#88912yes—
421C2×C3C3×#88912yes—
422C2×C3C3×#89612yes—
423C22C3×#379512yes—
424C22C3×#379711yescharacteristic
425C22C3×#379811yescharacteristic
426C22C3×#380412yes—
427C22C3×#381111yescharacteristic
428C4—41no—
429C4—122no—
430C4—122no—
431C4—182no—
432C4—182no—
433C3—22no—
434C3—21no—
435C3—41no—
436C3#1059121711yescharacteristic, direct product factor
437C3C3×#465812yescomplemented
438C2C3×#6384712yes—
439C2C3×#6385912yes—
440C2C3×#6386511yescharacteristic
441C2C3×#6391311yescharacteristic
442C2C3×#6397511yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC23×C32C23×C3
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
hypercenterC22×C3×C4D32
hypocenterC32C3×#400
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
7. #10591217C3direct (N × H)
233. C33#400semidirect
378. C32#4658semidirect
379. C32C3×#400semidirect
436. C3#10591217direct (N × H)
437. C3C3×#4658semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37502
C2×C3×#37242
C32×C3⋊#952
C3×#635792
C3×#46583
#105912173