Finite Group Databasebeta

← C3×#10590852 #13157819

Subgroups — C3×#10590852 #13157819

Order 1728.

650 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3746C211yesmaximal, characteristic, complemented
2C2×C32×D3×C8C211yesmaximal, characteristic
3C3×#63857C212yesmaximal
4C32×#818C211yesmaximal, characteristic, complemented
5C3×#63567C212yesmaximal, complemented
6C3×#5116—31nomaximal
7C3×#4887—31nomaximal
8#10590852C311yesmaximal, characteristic, direct product factor
9C2×C3×#634C2212yes—
10C2×C32×C4×D3C2211yescharacteristic
11C2×C33×C8C2211yescharacteristic
12C3×#3746—22no—
13C32×C4⋊Dic3C2212yes—
14C2×C32×C3⋊C8C2211yescharacteristic
15C32×D3×C8—22no—
16C3×Dic3⋊Dic3—22no—
17C3×(C2×D3)⋊Dic3—22no—
18C22×C32×C8—31no—
19C2×#3746C2×C311yescharacteristic, complemented
20C3×#924—32no—
21C2×C3×Dic12—31no—
22C2×C3×Dic12—31no—
23C2×C3×Dic12—61no—
24C2×C3×D3×C8—21no—
25C2×C3×D3×C8D311yescharacteristic
26C2×C3×D3×C8C2×C311yescharacteristic
27C3×#897—32no—
28#63857C2×C312yes—
29C32×#107—31no—
30C3×#818—21no—
31C3×#818D311yescharacteristic, complemented
32C3×#818C2×C311yescharacteristic, complemented
33C3×#816—32no—
34C3×#810—32no—
35#63567C2×C312yescomplemented
36C22×C32×D3D411yescharacteristic
37C2×C33×C4C2311yescharacteristic
38C2×C3×C3⋊Dic3—22no—
39C3×#634—22no—
40C3×#634—22no—
41C2×C32×Dic3—22no—
42C2×C32×Dic3D411yescharacteristic
43C32×C4×D3—21no—
44C32×C4×D3—21no—
45C33×C8D412yes—
46C32×C3⋊C8—21no—
47C3×#467—91no—
48#5116—31no—
49#4887—31no—
50C22×C32×C4—31no—
51C2×#634C22×C312yes—
52C2×C3×C4×D3—21no—
53C2×C3×C4×D3C2×D311yescharacteristic
54C2×C3×C4×D3C22×C311yescharacteristic
55C2×C3×Dic6—32no—
56C2×C3×Dic6—32no—
57C2×C3×Dic6—62no—
58C2×C32×C8—21no—
59C2×C32×C8—21no—
60C2×C32×C8—21no—
61C2×C32×C8—31no—
62C2×C32×C8—41no—
63C2×C32×C8—62no—
64C2×C32×C8C2×D311yescharacteristic
65C2×C32×C8C2×D311yescharacteristic
66C2×C32×C8C22×C311yescharacteristic
67C32×C4⋊C4—32no—
68#3746—22no—
69C3×#217—62no—
70C3×(C2×D3)⋊C4—62no—
71C3×C4⋊Dic3—22no—
72C3×C4⋊Dic3—62no—
73C3×C4⋊Dic3C2×D312yes—
74C3×C4⋊Dic3C22×C312yes—
75C3×Dic3⋊C4—62no—
76C2×C3×C3⋊C8—21no—
77C2×C3×C3⋊C8C2×D311yescharacteristic
78C2×C3×C3⋊C8C22×C311yescharacteristic
79C3×Dic12—64no—
80C3×Dic12—62no—
81C3×Dic12—62no—
82C3×D3×C8—24no—
83C3×D3×C8—22no—
84C3×D3×C8—22no—
85Dic3⋊Dic3—22no—
86(C2×D3)⋊Dic3—22no—
87C22×C33C2×D411yescharacteristic
88C2×C32×D3—21no—
89C2×C32×D3—21no—
90C2×C32×D3Q1612yescomplemented
91C33×C4C2×D411yescharacteristic
92C33×C4#4111yescharacteristic
93C3×C3⋊Dic3—42no—
94C32×Dic3—21no—
95C32×Dic3—42no—
96C2×C3×Q16—91no—
97C22×C3×C8—31no—
98C22×C3×C8—31no—
99C22×C3×C8—61no—
100C3×C22⋊Q8—92no—
101#924—32no—
102C2×Dic12—31no—
103C2×Dic12—31no—
104C2×Dic12—61no—
105C2×D3×C8C3×D311yescharacteristic
106#897—32no—
107C3×#107—31no—
108C3×#107—31no—
109C3×#107—61no—
110C3×#103—92no—
111#818C3×D311yescharacteristic, complemented
112#816—32no—
113#810—32no—
114C23×C32—31no—
115C22×C3×D3—21no—
116C22×C3×D3D1211yescharacteristic
117C22×C3×D3C3×D411yescharacteristic
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—31no—
122C2×C32×C4—41no—
123C2×C32×C4—61no—
124C2×C32×C4—61no—
125C2×C32×C4—62no—
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C22×D311yescharacteristic
128C2×C32×C4C23×C311yescharacteristic
129C2×C3⋊Dic3—22no—
130#634—22no—
131#634—22no—
132C2×C3×Dic3—22no—
133C2×C3×Dic3—22no—
134C2×C3×Dic3—21no—
135C2×C3×Dic3—42no—
136C2×C3×Dic3—62no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—122no—
139C2×C3×Dic3D1211yescharacteristic
140C2×C3×Dic3C3×D411yescharacteristic
141C3×C4×D3—22no—
142C3×C4×D3—22no—
143C3×C4×D3—21no—
144C3×C4×D3—21no—
145C3×C4×D3—21no—
146C3×C4×D3—21no—
147C3×Dic6—64no—
148C3×Dic6—62no—
149C3×Dic6—62no—
150C3×Dic6—62no—
151C3×Dic6—62no—
152C3×Dic6—122no—
153C32×C8—22no—
154C32×C8—22no—
155C32×C8—22no—
156C32×C8—42no—
157C32×C8—61no—
158C32×C8D1212yes—
159C32×C8#6712yes—
160C32×C8C3×D412yes—
161C3×C3⋊C8—22no—
162C3×C3⋊C8—21no—
163C3×C3⋊C8—21no—
164#467—91noSylow
165C2×C33#12111yescharacteristic
166C2×C33C2×Q1611yescharacteristic
167C2×C33#13511yescharacteristic
168C32×D3—22no—
169C2×C3×Q8—92no—
170C22×C3×C4—31no—
171C22×C3×C4—31no—
172C22×C3×C4—61no—
173C2×C4×D3C2×C3×D311yescharacteristic
174C2×Dic6—32no—
175C2×Dic6—32no—
176C2×Dic6—62no—
177C3×Q16—182no—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—31no—
182C2×C3×C8—31no—
183C2×C3×C8—41no—
184C2×C3×C8—61no—
185C2×C3×C8—64no—
186C2×C3×C8—62no—
187C2×C3×C8—62no—
188C2×C3×C8D3211yescharacteristic
189C2×C3×C8C2×C3×D311yescharacteristic
190C2×C3×C8C2×C3×D311yescharacteristic
191C3×C4⋊C4—32no—
192C3×C4⋊C4—32no—
193C3×C4⋊C4—62no—
194C3×C4⋊C4—182no—
195C3×C22⋊C4—182no—
196#217—62no—
197(C2×D3)⋊C4—62no—
198C4⋊Dic3—62no—
199C4⋊Dic3C2×C3×D312yes—
200Dic3⋊C4—62no—
201C2×C3⋊C8C2×C3×D311yescharacteristic
202Dic12—64no—
203Dic12—62no—
204Dic12—62no—
205D3×C8—22no—
206C22×C32—21no—
207C22×C32—21no—
208C22×C32—21no—
209C22×C32—32no—
210C22×C32—41no—
211C22×C32—61no—
212C22×C32—61no—
213C22×C32C2×D1211yescharacteristic
214C22×C32D3×D411yescharacteristic
215C22×C32C2×C3×D411yescharacteristic
216C2×C3×D3—22no—
217C2×C3×D3—22no—
218C2×C3×D3—21no—
219C2×C3×D3—21no—
220C2×C3×D3—21no—
221C2×C3×D3—21no—
222C2×C3×D3—22no—
223C2×C3×D3Dic1212yescomplemented
224C2×C3×D3C3×Q1612yescomplemented
225C32×C4—21no—
226C32×C4—21no—
227C32×C4—21no—
228C32×C4—21no—
229C32×C4—21no—
230C32×C4—21no—
231C32×C4—41no—
232C32×C4—41no—
233C32×C4—61no—
234C32×C4—122no—
235C32×C4C2×D1211yescharacteristic
236C32×C4#23511yescharacteristic
237C32×C4#23911yescharacteristic
238C32×C4C2×#6711yescharacteristic
239C32×C4C2×C3×D411yescharacteristic
240C32×C4C3×#4111yescharacteristic
241C3⋊Dic3—42no—
242C3×Dic3—22no—
243C3×Dic3—21no—
244C3×Dic3—21no—
245C3×Dic3—42no—
246C3×Dic3—42no—
247C3×Dic3—82no—
248C3×Dic3—124no—
249C3×Dic3—122no—
250C3×Dic3—122no—
251C2×Q16—91no—
252C22×C8—31no—
253C22⋊Q8—92no—
254#107—31no—
255#103—92no—
256C33#46711yescharacteristic, complemented, Sylow
257C23×C3—31no—
258C23×C3—31no—
259C23×C3—61no—
260C22×D3C3×D1211yescharacteristic
261C3×Q8—182no—
262C3×Q8—182no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—41no—
269C2×C3×C4—61no—
270C2×C3×C4—62no—
271C2×C3×C4—62no—
272C2×C3×C4—61no—
273C2×C3×C4—61no—
274C2×C3×C4—61no—
275C2×C3×C4—61no—
276C2×C3×C4—62no—
277C2×C3×C4—62no—
278C2×C3×C4—122no—
279C2×C3×C4—182no—
280C2×C3×C4C2×D3211yescharacteristic
281C2×C3×C4C22×C3×D311yescharacteristic
282C2×C3×C4C22×C3×D311yescharacteristic
283C2×Dic3—22no—
284C2×Dic3—62no—
285C2×Dic3—62no—
286C2×Dic3—122no—
287C2×Dic3C3×D1211yescharacteristic
288C4×D3—21no—
289C4×D3—21no—
290Dic6—64no—
291Dic6—62no—
292Dic6—62no—
293Dic6—62no—
294Dic6—62no—
295Dic6—122no—
296C3×C8—22no—
297C3×C8—22no—
298C3×C8—22no—
299C3×C8—42no—
300C3×C8—62no—
301C3×C8—61no—
302C3×C8—61no—
303C3×C8#62612yes—
304C3×C8C3×D1212yes—
305C3×C8C3×#6712yes—
306C3⋊C8—21no—
307C2×C32—21no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—21no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32—21no—
316C2×C32—41no—
317C2×C32—41no—
318C2×C32—41no—
319C2×C32—62no—
320C2×C32#89511yescharacteristic
321C2×C32#90411yescharacteristic
322C2×C32C2×Dic1211yescharacteristic
323C2×C32#91211yescharacteristic
324C2×C32D3×Q1611yescharacteristic
325C2×C32#93011yescharacteristic
326C2×C32C3×#12111yescharacteristic
327C2×C32C2×C3×Q1611yescharacteristic
328C2×C32C3×#13511yescharacteristic
329C3×D3—24no—
330C3×D3—22no—
331C3×D3—22no—
332C2×Q8—92no—
333C22×C4—31no—
334Q16—182no—
335C2×C8—31no—
336C2×C8—62no—
337C2×C8C3×D3211yescharacteristic
338C4⋊C4—32no—
339C4⋊C4—182no—
340C22⋊C4—182no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—32no—
345C22×C3—32no—
346C22×C3—41no—
347C22×C3—62no—
348C22×C3—62no—
349C22×C3—62no—
350C22×C3—61no—
351C22×C3—61no—
352C22×C3—61no—
353C22×C3—61no—
354C22×C3D3×D1211yescharacteristic
355C22×C3C2×C3×D1211yescharacteristic
356C22×C3C3×D3×D411yescharacteristic
357C2×D3—21no—
358C2×D3—21no—
359C2×D3C3×Dic1212yescomplemented
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—41no—
367C3×C4—41no—
368C3×C4—62no—
369C3×C4—61no—
370C3×C4—61no—
371C3×C4—122no—
372C3×C4—122no—
373C3×C4—242no—
374C3×C4—362no—
375C3×C4#379911yescharacteristic
376C3×C4C2×#62611yescharacteristic
377C3×C4C2×C3×D1211yescharacteristic
378C3×C4C3×#23511yescharacteristic
379C3×C4C3×#23911yescharacteristic
380C3×C4C2×C3×#6711yescharacteristic
381Dic3—21no—
382Dic3—42no—
383Dic3—124no—
384Dic3—122no—
385Dic3—122no—
386C32—21no—
387C32—21no—
388C32—21no—
389C32—41no—
390C32#488711yescharacteristic, complemented
391C32#511611yescharacteristic, complemented
392C32C3×#46711yescharacteristic, complemented
393C23—31no—
394Q8—182no—
395Q8—182no—
396C2×C4—31no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—62no—
400C2×C4—182no—
401C2×C4C2×C3×D3211yescharacteristic
402C8—61no—
403C8C3×#62612yes—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—41no—
414C2×C3—41no—
415C2×C3—41no—
416C2×C3—64no—
417C2×C3—62no—
418C2×C3—62no—
419C2×C3D3×Dic1211yescharacteristic
420C2×C3#6380711yescharacteristic
421C2×C3#6390911yescharacteristic
422C2×C3C3×#89511yescharacteristic
423C2×C3C3×#90411yescharacteristic
424C2×C3C2×C3×Dic1211yescharacteristic
425C2×C3C3×#91211yescharacteristic
426C2×C3C3×D3×Q1611yescharacteristic
427C2×C3C3×#93011yescharacteristic
428D3—22no—
429C22—32no—
430C22—61no—
431C22—61no—
432C22C3×D3×D1211yescharacteristic
433C4—61no—
434C4—122no—
435C4—362no—
436C4C3×#379911yescharacteristic
437C4C2×C3×#62611yescharacteristic
438C3—21no—
439C3—21no—
440C3—21no—
441C3—41no—
442C3#1059085211yescharacteristic, direct product factor
443C3C3×#488711yescharacteristic, complemented
444C3C3×#511611yescharacteristic, complemented
445C2—62no—
446C2C3×D3×Dic1211yescharacteristic
447C2C3×#6380711yescharacteristic
448C2C3×#6390911yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypercenterC2×C3×C8D32
hypocenterC32C3×#467
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3746C2semidirect
4. C32×#818C2semidirect
5. C3×#63567C2semidirect
8. #10590852C3direct (N × H)
19. C2×#3746C2×C3semidirect
31. C3×#818D3semidirect
32. C3×#818C2×C3semidirect
35. #63567C2×C3semidirect
90. C2×C32×D3Q16semidirect
111. #818C3×D3semidirect
223. C2×C3×D3Dic12semidirect
224. C2×C3×D3C3×Q16semidirect
256. C33#467semidirect
359. C2×D3C3×Dic12semidirect
390. C32#4887semidirect
391. C32#5116semidirect
392. C32C3×#467semidirect
442. C3#10590852direct (N × H)
443. C3C3×#4887semidirect
444. C3C3×#5116semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37462
C2×C32×D3×C82
C3×#638572
C32×#8182
C3×#635672
C3×#51163
C3×#48873
#105908523