Finite Group Databasebeta

← C3×#10590363 #13157379

Subgroups — C3×#10590363 #13157379

Order 1728.

710 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×C4×D12C211yesmaximal, characteristic
2C32×#869C211yesmaximal, characteristic, complemented
3C2×C3×#3715C212yesmaximal
4C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
5C3×#63565C212yesmaximal, complemented
6C3×#5083—31nomaximal
7C3×#5019—31nomaximal
8#10590363C311yesmaximal, characteristic, direct product factor
9C2×C32×D12C2211yescharacteristic
10C2×C32×C4×D3—21no—
11C2×C32×Dic6C2212yes—
12C33×C42C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2212yes—
14C32×(C2×D3)⋊C4—21no—
15C32×C4⋊Dic3—21no—
16C32×C4⋊Dic3—21no—
17C32×C4⋊Dic3C2211yescharacteristic
18C3×#3715—24no—
19C32×C4⋊Q8—31no—
20C32×C4×D4—31no—
21C2×C3×#215—32no—
22C2×C3×#214—32no—
23C3×C4×D12—21no—
24C3×C4×D12D311yescharacteristic
25C3×C4×D12C2×C311yescharacteristic
26C3×#869—21no—
27C3×#869D311yescharacteristic, complemented
28C3×#869C2×C311yescharacteristic, complemented
29C2×#3715C2×C312yes—
30C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
31C3×#835—32no—
32C3×#810—32no—
33C3×C4⋊(C3⋊C8)—31no—
34C3×C4⋊(C3⋊C8)—31no—
35C3×C4⋊(C3⋊C8)—61no—
36#63565C2×C312yescomplemented
37C22×C32×D3—21no—
38C2×C33×C4D411yescharacteristic
39C2×C33×C4D411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C32×Dic3—22no—
42C2×C32×Dic3—21no—
43C32×D12—21no—
44C32×D12D412yes—
45C32×C4×D3—42no—
46C32×Dic6—22no—
47C32×Dic6—22no—
48C3×C3⋊(C3⋊C8)—22no—
49C3×#461—91no—
50#5083—31no—
51#5019—31no—
52C2×C32×Q8—32no—
53C2×C32×D4—31no—
54C22×C32×C4—61no—
55C2×C3×D12—21no—
56C2×C3×D12C2×D311yescharacteristic
57C2×C3×D12C22×C311yescharacteristic
58C2×C3×C4×D3—22no—
59C2×C3×C4×D3—21no—
60C2×C3×C4×D3—21no—
61C2×C3×Dic6—22no—
62C2×C3×Dic6C2×D312yes—
63C2×C3×Dic6C22×C312yes—
64C32×C4⋊C4—31no—
65C32×C4⋊C4—61no—
66C32×C4⋊C4—61no—
67C32×C22⋊C4—61no—
68C32×C42—21no—
69C32×C42—21no—
70C32×C42—21no—
71C32×C42—41no—
72C32×C42C2×D311yescharacteristic
73C32×C42C2×D311yescharacteristic
74C32×C42C22×C311yescharacteristic
75C2×C3⋊(C3⋊C8)C22×C312yes—
76C3×#215—64no—
77C3×#214—64no—
78C3×(C2×D3)⋊C4—22no—
79C3×(C2×D3)⋊C4—21no—
80C3×(C2×D3)⋊C4—21no—
81C3×C4⋊Dic3—22no—
82C3×C4⋊Dic3—22no—
83C3×C4⋊Dic3—21no—
84C3×C4⋊Dic3—21no—
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×C4⋊Dic3C22×C311yescharacteristic
90C2×C3×C3⋊C8—32no—
91C2×C3×C3⋊C8—32no—
92C2×C3×C3⋊C8—62no—
93#3715—24no—
94C22×C33C2×D411yescharacteristic
95C2×C32×D3—22no—
96C2×C32×D3—41no—
97C2×C32×D3—41no—
98C33×C4—21no—
99C33×C4SD1612yescomplemented
100C33×C4C2×D411yescharacteristic
101C33×C4#4111yescharacteristic
102C32×Dic3—42no—
103C32×Dic3—41no—
104C2×C3×SD16—92no—
105C3×C4⋊Q8—31no—
106C3×C4⋊Q8—31no—
107C3×C4⋊Q8—61no—
108C3×C4×D4—31no—
109C3×C4×D4—31no—
110C3×C4×D4—61no—
111C2×#215—32no—
112C2×#214—32no—
113C4×D12C3×D311yescharacteristic
114#869C3×D311yescharacteristic, complemented
115C3×C4⋊C8—91no—
116C3×#103—92no—
117#835—32no—
118#810—32no—
119C4⋊(C3⋊C8)—31no—
120C4⋊(C3⋊C8)—31no—
121C4⋊(C3⋊C8)—61no—
122C23×C32—61no—
123C22×C3×D3—22no—
124C22×C3×D3—21no—
125C22×C3×D3—21no—
126C32×Q8—62no—
127C32×Q8—62no—
128C32×D4—32no—
129C32×D4—61no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—41no—
140C2×C32×C4—41no—
141C2×C32×C4—41no—
142C2×C32×C4—61no—
143C2×C32×C4—62no—
144C2×C32×C4—122no—
145C2×C32×C4#6711yescharacteristic
146C2×C32×C4#6711yescharacteristic
147C2×C32×C4#6711yescharacteristic
148C2×C32×C4#6711yescharacteristic
149C2×C32×C4C3×D411yescharacteristic
150C2×C32×C4C3×D411yescharacteristic
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C23×C311yescharacteristic
154C2×C3×Dic3—24no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—21no—
160C3×D12—22no—
161C3×D12—21no—
162C3×D12—21no—
163C3×D12—22no—
164C3×D12D1212yes—
165C3×D12C3×D412yes—
166C3×C4×D3—44no—
167C3×C4×D3—42no—
168C3×C4×D3—42no—
169C3×Dic6—22no—
170C3×Dic6—22no—
171C3×Dic6—22no—
172C3×Dic6—22no—
173C3×Dic6—42no—
174C3×Dic6—42no—
175C3⋊(C3⋊C8)—22no—
176C3×C3⋊C8—62no—
177C3×C3⋊C8—62no—
178C3×C3⋊C8—122no—
179#461—91noSylow
180C2×C33#12111yescharacteristic
181C2×C33C2×SD1611yescharacteristic
182C2×C33#13711yescharacteristic
183C32×D3—42no—
184C2×C3×Q8—32no—
185C2×C3×Q8—32no—
186C2×C3×Q8—62no—
187C2×C3×D4—31no—
188C2×C3×D4—31no—
189C2×C3×D4—61no—
190C22×C3×C4—62no—
191C22×C3×C4—61no—
192C22×C3×C4—61no—
193C2×D12C2×C3×D311yescharacteristic
194C2×C4×D3—21no—
195C2×Dic6C2×C3×D312yes—
196C3×SD16—184no—
197C2×C3×C8—92no—
198C3×C4⋊C4—31no—
199C3×C4⋊C4—31no—
200C3×C4⋊C4—61no—
201C3×C4⋊C4—62no—
202C3×C4⋊C4—62no—
203C3×C4⋊C4—61no—
204C3×C4⋊C4—61no—
205C3×C4⋊C4—61no—
206C3×C4⋊C4—61no—
207C3×C22⋊C4—62no—
208C3×C22⋊C4—61no—
209C3×C22⋊C4—61no—
210C3×C42—21no—
211C3×C42—21no—
212C3×C42—21no—
213C3×C42—41no—
214C3×C42D3211yescharacteristic
215C3×C42C2×C3×D311yescharacteristic
216C3×C42C2×C3×D311yescharacteristic
217#215—64no—
218#214—64no—
219(C2×D3)⋊C4—21no—
220C4⋊Dic3—21no—
221C4⋊Dic3—21no—
222C4⋊Dic3C2×C3×D311yescharacteristic
223C2×C3⋊C8—32no—
224C2×C3⋊C8—32no—
225C2×C3⋊C8—62no—
226C22×C32—21no—
227C22×C32—21no—
228C22×C32—21no—
229C22×C32—41no—
230C22×C32—62no—
231C22×C32—121no—
232C22×C32—121no—
233C22×C32C2×#6711yescharacteristic
234C22×C32C2×#6711yescharacteristic
235C22×C32C2×C3×D411yescharacteristic
236C2×C3×D3—24no—
237C2×C3×D3—22no—
238C2×C3×D3—22no—
239C2×C3×D3—42no—
240C2×C3×D3—42no—
241C2×C3×D3—41no—
242C2×C3×D3—41no—
243C2×C3×D3—41no—
244C2×C3×D3—41no—
245C32×C4—21no—
246C32×C4—22no—
247C32×C4—21no—
248C32×C4—21no—
249C32×C4—22no—
250C32×C4—22no—
251C32×C4—22no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—21no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4—21no—
258C32×C4—42no—
259C32×C4—41no—
260C32×C4—41no—
261C32×C4—41no—
262C32×C4—41no—
263C32×C4—42no—
264C32×C4—122no—
265C32×C4—121no—
266C32×C4#21412yescomplemented
267C32×C4#21512yescomplemented
268C32×C4C3×SD1612yescomplemented
269C32×C4C2×D1211yescharacteristic
270C32×C4#23511yescharacteristic
271C32×C4D3×D411yescharacteristic
272C32×C4#23711yescharacteristic
273C32×C4C2×C3×D411yescharacteristic
274C32×C4C3×#4111yescharacteristic
275C3×Dic3—44no—
276C3×Dic3—42no—
277C3×Dic3—41no—
278C3×Dic3—42no—
279C3×Dic3—42no—
280C3×Dic3—41no—
281C2×SD16—92no—
282C4⋊Q8—31no—
283C4×D4—31no—
284C4⋊C8—91no—
285#103—92no—
286C33#46111yescharacteristic, complemented, Sylow
287C23×C3—62no—
288C23×C3—61no—
289C23×C3—61no—
290C22×D3—21no—
291C3×Q8—62no—
292C3×Q8—62no—
293C3×Q8—62no—
294C3×Q8—62no—
295C3×Q8—122no—
296C3×Q8—122no—
297C3×D4—32no—
298C3×D4—32no—
299C3×D4—62no—
300C3×D4—62no—
301C3×D4—61no—
302C3×D4—61no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—62no—
316C2×C3×C4—64no—
317C2×C3×C4—61no—
318C2×C3×C4—62no—
319C2×C3×C4—61no—
320C2×C3×C4—62no—
321C2×C3×C4—124no—
322C2×C3×C4—122no—
323C2×C3×C4—122no—
324C2×C3×C4#62511yescharacteristic
325C2×C3×C4#62511yescharacteristic
326C2×C3×C4C3×#6711yescharacteristic
327C2×C3×C4C3×#6711yescharacteristic
328C2×C3×C4C3×#6711yescharacteristic
329C2×C3×C4C3×#6711yescharacteristic
330C2×C3×C4C2×D3211yescharacteristic
331C2×C3×C4C22×C3×D311yescharacteristic
332C2×C3×C4C22×C3×D311yescharacteristic
333C2×Dic3—22no—
334C2×Dic3—21no—
335D12—21no—
336D12C3×D1212yes—
337C4×D3—42no—
338Dic6—22no—
339Dic6—22no—
340C3×C8—182no—
341C3⋊C8—62no—
342C3⋊C8—62no—
343C3⋊C8—122no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—21no—
353C2×C32—41no—
354C2×C32—41no—
355C2×C32—41no—
356C2×C32—122no—
357C2×C32#93011yescharacteristic
358C2×C32C2×#21411yescharacteristic
359C2×C32#93811yescharacteristic
360C2×C32C2×#21511yescharacteristic
361C2×C32#94211yescharacteristic
362C2×C32#95111yescharacteristic
363C2×C32C3×#12111yescharacteristic
364C2×C32C2×C3×SD1611yescharacteristic
365C2×C32C3×#13711yescharacteristic
366C3×D3—44no—
367C3×D3—42no—
368C3×D3—42no—
369C2×Q8—32no—
370C2×D4—31no—
371C22×C4—61no—
372SD16—184no—
373C2×C8—92no—
374C4⋊C4—31no—
375C4⋊C4—61no—
376C4⋊C4—61no—
377C22⋊C4—61no—
378C42C3×D3211yescharacteristic
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—41no—
383C22×C3—64no—
384C22×C3—62no—
385C22×C3—62no—
386C22×C3—122no—
387C22×C3—122no—
388C22×C3—121no—
389C22×C3—121no—
390C22×C3—121no—
391C22×C3—121no—
392C22×C3C2×#62511yescharacteristic
393C22×C3C2×C3×#6711yescharacteristic
394C22×C3C2×C3×#6711yescharacteristic
395C2×D3—22no—
396C2×D3—41no—
397C2×D3—41no—
398C3×C4—22no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—21no—
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—41no—
412C3×C4—41no—
413C3×C4—42no—
414C3×C4—41no—
415C3×C4—41no—
416C3×C4—42no—
417C3×C4—122no—
418C3×C4—124no—
419C3×C4—122no—
420C3×C4—121no—
421C3×C4—121no—
422C3×C4—122no—
423C3×C4#371512yescomplemented
424C3×C4C3×#21412yescomplemented
425C3×C4C3×#21512yescomplemented
426C3×C4#379511yescharacteristic
427C3×C4D3×D1211yescharacteristic
428C3×C4C2×C3×D1211yescharacteristic
429C3×C4C3×#23511yescharacteristic
430C3×C4C3×D3×D411yescharacteristic
431C3×C4C3×#23711yescharacteristic
432Dic3—41no—
433Dic3—42no—
434C32—21no—
435C32—21no—
436C32—21no—
437C32—41no—
438C32#501911yescharacteristic, complemented
439C32#508311yescharacteristic, complemented
440C32C3×#46111yescharacteristic, complemented
441C23—61no—
442Q8—62no—
443Q8—62no—
444D4—32no—
445D4—61no—
446C2×C4—61no—
447C2×C4—62no—
448C2×C4—122no—
449C2×C4C3×#62511yescharacteristic
450C2×C4C3×#62511yescharacteristic
451C2×C4C2×C3×D3211yescharacteristic
452C8—182no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—41no—
465C2×C3—124no—
466C2×C3—122no—
467C2×C3—122no—
468C2×C3C2×#371511yescharacteristic
469C2×C3#6382711yescharacteristic
470C2×C3#6390811yescharacteristic
471C2×C3C3×#93011yescharacteristic
472C2×C3C2×C3×#21411yescharacteristic
473C2×C3C3×#93811yescharacteristic
474C2×C3C2×C3×#21511yescharacteristic
475C2×C3C3×#94211yescharacteristic
476C2×C3C3×#95111yescharacteristic
477D3—42no—
478C22—62no—
479C22—121no—
480C22—121no—
481C22C2×C3×#62511yescharacteristic
482C4—21no—
483C4—121no—
484C4—122no—
485C4C3×#371512yescomplemented
486C4C3×#379511yescharacteristic
487C4C3×D3×D1211yescharacteristic
488C3—21no—
489C3—21no—
490C3—21no—
491C3—41no—
492C3#1059036311yescharacteristic, direct product factor
493C3C3×#501911yescharacteristic, complemented
494C3C3×#508311yescharacteristic, complemented
495C2—122no—
496C2C2×C3×#371511yescharacteristic
497C2C3×#6382711yescharacteristic
498C2C3×#6390811yescharacteristic
subgroupG/H
centerC22×C3C2×#625
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC3×C42D32
hypocenterC32C3×#461
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
2. C32×#869C2semidirect
4. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
5. C3×#63565C2semidirect
8. #10590363C3direct (N × H)
27. C3×#869D3semidirect
28. C3×#869C2×C3semidirect
30. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
36. #63565C2×C3semidirect
99. C33×C4SD16semidirect
114. #869C3×D3semidirect
266. C32×C4#214semidirect
267. C32×C4#215semidirect
268. C32×C4C3×SD16semidirect
286. C33#461semidirect
423. C3×C4#3715semidirect
424. C3×C4C3×#214semidirect
425. C3×C4C3×#215semidirect
438. C32#5019semidirect
439. C32#5083semidirect
440. C32C3×#461semidirect
485. C4C3×#3715semidirect
492. C3#10590363direct (N × H)
493. C3C3×#5019semidirect
494. C3C3×#5083semidirect

Maximal subgroups

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

subgroupindex
C32×C4×D122
C32×#8692
C2×C3×#37152
C3×C4⋊(C3⋊(C3⋊C8))2
C3×#635652
C3×#50833
C3×#50193
#105903633