Finite Group Databasebeta

← C3×#10590851 #13157818

Subgroups — C3×#10590851 #13157818

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×C32×Dic12C211yesmaximal, characteristic, complemented
2C2×C32×D3×C8C211yesmaximal, characteristic
3C3×#63856C212yesmaximal
4C3×#63645C211yesmaximal, characteristic, complemented
5C3×#63566C212yesmaximal, complemented
6C3×#5188—31nomaximal
7C3×#5116—31nomaximal
8#10590851C311yesmaximal, characteristic, direct product factor
9C2×C32×C4×D3C2211yescharacteristic
10C2×C32×Dic6C2212yes—
11C2×C33×C8C2211yescharacteristic
12C3×C4⋊(C3⋊Dic3)C2212yes—
13C2×C32×C3⋊C8C2211yescharacteristic
14C32×Dic12—22no—
15C32×D3×C8—22no—
16C3×#3724—22no—
17C3×(C2×D3)⋊Dic3—22no—
18C2×C32×Q16—31no—
19C22×C32×C8—31no—
20C3×#946—32no—
21C3×#924—32no—
22C2×C3×Dic12—21no—
23C2×C3×Dic12D311yescharacteristic, complemented
24C2×C3×Dic12C2×C311yescharacteristic, complemented
25C2×C3×D3×C8—21no—
26C2×C3×D3×C8D311yescharacteristic
27C2×C3×D3×C8C2×C311yescharacteristic
28#63856C2×C312yes—
29#63645C2×C311yescharacteristic, complemented
30C3×#835—32no—
31C3×#818—31no—
32C3×#818—31no—
33C3×#818—61no—
34C3×#816—32no—
35#63566C2×C312yescomplemented
36C22×C32×D3D411yescharacteristic
37C2×C33×C4C2311yescharacteristic
38C2×C3×C3⋊Dic3—22no—
39C2×C32×Dic3—22no—
40C2×C32×Dic3D411yescharacteristic
41C32×C4×D3—21no—
42C32×C4×D3—21no—
43C32×Dic6—22no—
44C32×Dic6—22no—
45C33×C8D412yes—
46C32×C3⋊C8—21no—
47C3×#467—91no—
48#5188—31no—
49#5116—31no—
50C2×C32×Q8—32no—
51C22×C32×C4—31no—
52C2×C3×C4×D3—21no—
53C2×C3×C4×D3C2×D311yescharacteristic
54C2×C3×C4×D3C22×C311yescharacteristic
55C2×C3×Dic6—22no—
56C2×C3×Dic6C2×D312yes—
57C2×C3×Dic6C22×C312yes—
58C32×Q16—62no—
59C2×C32×C8—21no—
60C2×C32×C8—21no—
61C2×C32×C8—21no—
62C2×C32×C8—31no—
63C2×C32×C8—41no—
64C2×C32×C8—62no—
65C2×C32×C8C2×D311yescharacteristic
66C2×C32×C8C2×D311yescharacteristic
67C2×C32×C8C22×C311yescharacteristic
68C4⋊(C3⋊Dic3)C22×C312yes—
69C3×#217—62no—
70C3×(C2×D3)⋊C4—62no—
71C3×C4⋊Dic3—32no—
72C3×C4⋊Dic3—32no—
73C3×C4⋊Dic3—62no—
74C3×Dic3⋊C4—62no—
75C3×Dic3⋊C4—62no—
76C2×C3×C3⋊C8—21no—
77C2×C3×C3⋊C8C2×D311yescharacteristic
78C2×C3×C3⋊C8C22×C311yescharacteristic
79C3×Dic12—24no—
80C3×Dic12—22no—
81C3×Dic12—22no—
82C3×D3×C8—24no—
83C3×D3×C8—22no—
84C3×D3×C8—22no—
85#3724—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—31no—
97C2×C3×Q16—31no—
98C2×C3×Q16—61no—
99C22×C3×C8—31no—
100C22×C3×C8—31no—
101C22×C3×C8—61no—
102C3×C22⋊Q8—92no—
103#946—32no—
104#924—32no—
105C2×Dic12C3×D311yescharacteristic, complemented
106C2×D3×C8C3×D311yescharacteristic
107C3×#107—91no—
108C3×#103—92no—
109#835—32no—
110#818—31no—
111#818—31no—
112#818—61no—
113#816—32no—
114C23×C32—31no—
115C22×C3×D3—21no—
116C22×C3×D3D1211yescharacteristic
117C22×C3×D3C3×D411yescharacteristic
118C32×Q8—62no—
119C32×Q8—62no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—31no—
124C2×C32×C4—41no—
125C2×C32×C4—61no—
126C2×C32×C4—61no—
127C2×C32×C4—62no—
128C2×C32×C4C22×D311yescharacteristic
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C23×C311yescharacteristic
131C2×C3⋊Dic3—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—24no—
148C3×Dic6—22no—
149C3×Dic6—22no—
150C3×Dic6—22no—
151C3×Dic6—22no—
152C3×Dic6—42no—
153C32×C8—22no—
154C32×C8—22no—
155C32×C8—22no—
156C32×C8—42no—
157C32×C8—61no—
158C32×C8#6712yes—
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—32no—
170C2×C3×Q8—32no—
171C2×C3×Q8—62no—
172C22×C3×C4—31no—
173C22×C3×C4—31no—
174C22×C3×C4—61no—
175C2×C4×D3C2×C3×D311yescharacteristic
176C2×Dic6C2×C3×D312yes—
177C3×Q16—64no—
178C3×Q16—62no—
179C3×Q16—62no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C2×C3×C8—21no—
183C2×C3×C8—31no—
184C2×C3×C8—31no—
185C2×C3×C8—41no—
186C2×C3×C8—61no—
187C2×C3×C8—64no—
188C2×C3×C8—62no—
189C2×C3×C8—62no—
190C2×C3×C8D3211yescharacteristic
191C2×C3×C8C2×C3×D311yescharacteristic
192C2×C3×C8C2×C3×D311yescharacteristic
193C3×C4⋊C4—92no—
194C3×C4⋊C4—182no—
195C3×C22⋊C4—182no—
196#217—62no—
197(C2×D3)⋊C4—62no—
198C4⋊Dic3—32no—
199C4⋊Dic3—32no—
200C4⋊Dic3—62no—
201Dic3⋊C4—62no—
202Dic3⋊C4—62no—
203C2×C3⋊C8C2×C3×D311yescharacteristic
204Dic12—22no—
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×C4#23511yescharacteristic
236C32×C4#23711yescharacteristic
237C32×C4C2×#6711yescharacteristic
238C32×C4C2×#6711yescharacteristic
239C32×C4C2×C3×D411yescharacteristic
240C32×C4C3×#4111yescharacteristic
241C3⋊Dic3—42no—
242C3×Dic3—22no—
243C3×Dic3—21no—
244C3×Dic3—21no—
245C3×Dic3—44no—
246C3×Dic3—42no—
247C3×Dic3—42no—
248C3×Dic3—122no—
249C3×Dic3—122no—
250C3×Dic3—242no—
251C2×Q16—31no—
252C22×C8—31no—
253C22⋊Q8—92no—
254#107—91no—
255#103—92no—
256C33#46711yescharacteristic, complemented, Sylow
257C23×C3—31no—
258C23×C3—31no—
259C23×C3—61no—
260C22×D3C3×D1211yescharacteristic
261C3×Q8—64no—
262C3×Q8—62no—
263C3×Q8—62no—
264C3×Q8—62no—
265C3×Q8—62no—
266C3×Q8—122no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—31no—
271C2×C3×C4—31no—
272C2×C3×C4—41no—
273C2×C3×C4—61no—
274C2×C3×C4—62no—
275C2×C3×C4—62no—
276C2×C3×C4—61no—
277C2×C3×C4—61no—
278C2×C3×C4—61no—
279C2×C3×C4—61no—
280C2×C3×C4—62no—
281C2×C3×C4—62no—
282C2×C3×C4—122no—
283C2×C3×C4—182no—
284C2×C3×C4C2×D3211yescharacteristic
285C2×C3×C4C22×C3×D311yescharacteristic
286C2×C3×C4C22×C3×D311yescharacteristic
287C2×Dic3—22no—
288C2×Dic3—62no—
289C2×Dic3—62no—
290C2×Dic3—122no—
291C2×Dic3C3×D1211yescharacteristic
292C4×D3—21no—
293C4×D3—21no—
294Dic6—22no—
295Dic6—22no—
296C3×C8—22no—
297C3×C8—22no—
298C3×C8—22no—
299C3×C8—42no—
300C3×C8—62no—
301C3×C8—61no—
302C3×C8—61no—
303C3×C8#62512yes—
304C3×C8C3×#6712yes—
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#90411yescharacteristic
321C2×C32C2×Dic1211yescharacteristic
322C2×C32D3×Q1611yescharacteristic
323C2×C32#91911yescharacteristic
324C2×C32#93011yescharacteristic
325C2×C32#93811yescharacteristic
326C2×C32C3×#12111yescharacteristic
327C2×C32C2×C3×Q1611yescharacteristic
328C2×C32C3×#13511yescharacteristic
329C3×D3—24no—
330C3×D3—22no—
331C3×D3—22no—
332C2×Q8—32no—
333C22×C4—31no—
334Q16—62no—
335C2×C8—31no—
336C2×C8—62no—
337C2×C8C3×D3211yescharacteristic
338C4⋊C4—92no—
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—124no—
372C3×C4—122no—
373C3×C4—122no—
374C3×C4—362no—
375C3×C4#379511yescharacteristic
376C3×C4C2×#62511yescharacteristic
377C3×C4C3×#23511yescharacteristic
378C3×C4C3×#23711yescharacteristic
379C3×C4C2×C3×#6711yescharacteristic
380C3×C4C2×C3×#6711yescharacteristic
381Dic3—21no—
382Dic3—42no—
383Dic3—122no—
384Dic3—122no—
385Dic3—242no—
386C32—21no—
387C32—21no—
388C32—21no—
389C32—41no—
390C32#511611yescharacteristic, complemented
391C32#518811yescharacteristic, complemented
392C32C3×#46711yescharacteristic, complemented
393C23—31no—
394Q8—62no—
395Q8—62no—
396C2×C4—31no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—62no—
400C2×C4—182no—
401C2×C4C2×C3×D3211yescharacteristic
402C8—61no—
403C8C3×#62512yes—
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#6380811yescharacteristic
421C2×C3#6390811yescharacteristic
422C2×C3C3×#90411yescharacteristic
423C2×C3C2×C3×Dic1211yescharacteristic
424C2×C3C3×D3×Q1611yescharacteristic
425C2×C3C3×#91911yescharacteristic
426C2×C3C3×#93011yescharacteristic
427C2×C3C3×#93811yescharacteristic
428D3—22no—
429C22—32no—
430C22—61no—
431C22—61no—
432C22C3×D3×D1211yescharacteristic
433C4—61no—
434C4—122no—
435C4—362no—
436C4C3×#379511yescharacteristic
437C4C2×C3×#62511yescharacteristic
438C3—21no—
439C3—21no—
440C3—21no—
441C3—41no—
442C3#1059085111yescharacteristic, direct product factor
443C3C3×#511611yescharacteristic, complemented
444C3C3×#518811yescharacteristic, complemented
445C2—62no—
446C2C3×D3×Dic1211yescharacteristic
447C2C3×#6380811yescharacteristic
448C2C3×#6390811yescharacteristic
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×C32×Dic12C2semidirect
4. C3×#63645C2semidirect
5. C3×#63566C2semidirect
8. #10590851C3direct (N × H)
23. C2×C3×Dic12D3semidirect
24. C2×C3×Dic12C2×C3semidirect
29. #63645C2×C3semidirect
35. #63566C2×C3semidirect
90. C2×C32×D3Q16semidirect
105. C2×Dic12C3×D3semidirect
223. C2×C3×D3Dic12semidirect
224. C2×C3×D3C3×Q16semidirect
256. C33#467semidirect
359. C2×D3C3×Dic12semidirect
390. C32#5116semidirect
391. C32#5188semidirect
392. C32C3×#467semidirect
442. C3#10590851direct (N × H)
443. C3C3×#5116semidirect
444. C3C3×#5188semidirect

Maximal subgroups

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

subgroupindex
C2×C32×Dic122
C2×C32×D3×C82
C3×#638562
C3×#636452
C3×#635662
C3×#51883
C3×#51163
#105908513