Finite Group Databasebeta

← C3×#10590898 #13157865

Subgroups — C3×#10590898 #13157865

Order 1728.

506 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63869C211yesmaximal, characteristic
2C3×#63866C211yesmaximal, characteristic
3C3×#63643C211yesmaximal, characteristic
4C32×#816C211yesmaximal, characteristic
5C32×#815C211yesmaximal, characteristic
6C3×#63567C211yesmaximal, characteristic
7C3×#63566C211yesmaximal, characteristic
8C3×#4797—31nomaximal
9C3×#4703—31nomaximal
10#10590898C311yesmaximal, characteristic, direct product factor
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C32×C4⋊Dic3C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C3×#3724—21no—
19C3×Dic3⋊Dic3—21no—
20C3×Dic3⋊Dic3—21no—
21C3×Dic3⋊Dic3—21no—
22C3×#944—31no—
23C3×#890—31no—
24C3×#870—31no—
25C3×#869—31no—
26#63869C2×C311yescharacteristic
27#63866C2×C311yescharacteristic
28C32×#103—31no—
29C32×#97—31no—
30#63643C2×C311yescharacteristic
31C3×#835—31no—
32C3×#816—21no—
33C3×#816—31no—
34C3×#816—31no—
35C3×#816—31no—
36C3×#816—31no—
37C3×#816—61no—
38C3×#816D311yescharacteristic
39C3×#816C2×C311yescharacteristic
40C3×#815—21no—
41C3×#815D311yescharacteristic
42C3×#815C2×C311yescharacteristic
43C3×#810—31no—
44#63567C2×C311yescharacteristic
45#63566C2×C311yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3—21no—
49C3×#634—21no—
50C3×#634—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C32×Dic6—21no—
56C32×Dic6—21no—
57C33×C8—21no—
58C32×C3⋊C8—21no—
59C3×#491—91no—
60#4797—31no—
61#4703—31no—
62C2×C32×Q8—31no—
63C2×#634C22×C311yescharacteristic
64C2×C3×Dic6—21no—
65C2×C3×Dic6—31no—
66C2×C3×Dic6—31no—
67C2×C3×Dic6—61no—
68C2×C3×Dic6C2×D311yescharacteristic
69C2×C3×Dic6C22×C311yescharacteristic
70C2×C32×C8—21no—
71C2×C32×C8—21no—
72C2×C32×C8—21no—
73C2×C32×C8—31no—
74C2×C32×C8—41no—
75C2×C32×C8C2×D311yescharacteristic
76C2×C32×C8C2×D311yescharacteristic
77C2×C32×C8C22×C311yescharacteristic
78C32×C4⋊C4—31no—
79C32×C42—31no—
80C4⋊(C3⋊Dic3)C22×C311yescharacteristic
81C3×C4⋊Dic3—21no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—61no—
85C3×C4⋊Dic3—61no—
86C3×C4⋊Dic3—61no—
87C3×C4⋊Dic3—61no—
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×C4⋊Dic3C22×C311yescharacteristic
90C3×Dic3⋊C4—61no—
91C3×Dic3⋊C4—61no—
92C3×Dic3⋊C4—61no—
93C3×Dic3⋊C4—61no—
94C3×Dic3⋊C4—61no—
95C3×C4×Dic3—21no—
96C3×C4×Dic3C2×D311yescharacteristic
97C3×C4×Dic3C22×C311yescharacteristic
98C2×C3×C3⋊C8—21no—
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C22×C311yescharacteristic
101#3724—21no—
102Dic3⋊Dic3—21no—
103Dic3⋊Dic3—21no—
104Dic3⋊Dic3—21no—
105C22×C33C2×D411yescharacteristic
106C33×C4#4111yescharacteristic
107C33×C4#4111yescharacteristic
108C3×C3⋊Dic3—41no—
109C3×C3⋊Dic3—41no—
110C32×Dic3—21no—
111C32×Dic3—21no—
112C32×Dic3—41no—
113C32×Dic3—41no—
114C3×C4⋊Q8—91no—
115C3×#125—91no—
116#944—31no—
117#890—31no—
118#870—31no—
119#869—31no—
120C3×#103—31no—
121C3×#103—31no—
122C3×#103—61no—
123C3×#103—91no—
124C3×#103—91no—
125C3×#103—91no—
126C3×#97—31no—
127C3×#97—31no—
128C3×#97—61no—
129#835—31no—
130#816—31no—
131#816—31no—
132#816—31no—
133#816—31no—
134#816—61no—
135#816C3×D311yescharacteristic
136#815C3×D311yescharacteristic
137#810—31no—
138C32×Q8—61no—
139C32×Q8—61no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—31no—
144C2×C32×C4—31no—
145C2×C32×C4—41no—
146C2×C32×C4—61no—
147C2×C32×C4—61no—
148C2×C32×C4C22×D311yescharacteristic
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C23×C311yescharacteristic
151C2×C3⋊Dic3—21no—
152C2×C3⋊Dic3—21no—
153#634—21no—
154#634—21no—
155C2×C3×Dic3—21no—
156C2×C3×Dic3—21no—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—41no—
162C2×C3×Dic3—41no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—121no—
168C2×C3×Dic3—121no—
169C2×C3×Dic3D1211yescharacteristic
170C2×C3×Dic3D1211yescharacteristic
171C2×C3×Dic3C3×D411yescharacteristic
172C2×C3×Dic3C3×D411yescharacteristic
173C3×Dic6—22no—
174C3×Dic6—21no—
175C3×Dic6—21no—
176C3×Dic6—21no—
177C3×Dic6—21no—
178C3×Dic6—41no—
179C3×Dic6—62no—
180C3×Dic6—61no—
181C3×Dic6—61no—
182C3×Dic6—61no—
183C3×Dic6—61no—
184C3×Dic6—121no—
185C32×C8—21no—
186C32×C8—22no—
187C32×C8—21no—
188C32×C8—21no—
189C32×C8—41no—
190C32×C8—41no—
191C32×C8—42no—
192C32×C8—61no—
193C3×C3⋊C8—22no—
194C3×C3⋊C8—21no—
195C3×C3⋊C8—21no—
196#491—91noSylow
197C2×C33#12411yescharacteristic
198C2×C33#13711yescharacteristic
199C2×C33#13711yescharacteristic
200C2×C3×Q8—31no—
201C2×C3×Q8—31no—
202C2×C3×Q8—61no—
203C2×C3×Q8—91no—
204C2×Dic6—31no—
205C2×Dic6—31no—
206C2×Dic6—61no—
207C2×Dic6C2×C3×D311yescharacteristic
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—31no—
212C2×C3×C8—31no—
213C2×C3×C8—41no—
214C2×C3×C8—61no—
215C2×C3×C8D3211yescharacteristic
216C2×C3×C8C2×C3×D311yescharacteristic
217C2×C3×C8C2×C3×D311yescharacteristic
218C3×C4⋊C4—31no—
219C3×C4⋊C4—31no—
220C3×C4⋊C4—61no—
221C3×C4⋊C4—91no—
222C3×C4⋊C4—181no—
223C3×C4⋊C4—181no—
224C3×C4⋊C4—181no—
225C3×C4⋊C4—181no—
226C3×C42—31no—
227C3×C42—31no—
228C3×C42—61no—
229C4⋊Dic3—31no—
230C4⋊Dic3—31no—
231C4⋊Dic3—61no—
232C4⋊Dic3—61no—
233C4⋊Dic3—61no—
234C4⋊Dic3—61no—
235C4⋊Dic3C2×C3×D311yescharacteristic
236Dic3⋊C4—61no—
237Dic3⋊C4—61no—
238Dic3⋊C4—61no—
239Dic3⋊C4—61no—
240Dic3⋊C4—61no—
241C4×Dic3C2×C3×D311yescharacteristic
242C2×C3⋊C8C2×C3×D311yescharacteristic
243C22×C32—21no—
244C22×C32—21no—
245C22×C32—21no—
246C22×C32—41no—
247C22×C32C2×D1211yescharacteristic
248C22×C32D3×D411yescharacteristic
249C22×C32C2×C3×D411yescharacteristic
250C32×C4—21no—
251C32×C4—21no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—21no—
255C32×C4—21no—
256C32×C4—41no—
257C32×C4—41no—
258C32×C4—61no—
259C32×C4—61no—
260C32×C4—121no—
261C32×C4—121no—
262C32×C4#23511yescharacteristic
263C32×C4#23511yescharacteristic
264C32×C4#23511yescharacteristic
265C32×C4#23711yescharacteristic
266C32×C4C3×#4111yescharacteristic
267C32×C4C3×#4111yescharacteristic
268C3⋊Dic3—41no—
269C3⋊Dic3—41no—
270C3×Dic3—21no—
271C3×Dic3—21no—
272C3×Dic3—21no—
273C3×Dic3—21no—
274C3×Dic3—41no—
275C3×Dic3—41no—
276C3×Dic3—42no—
277C3×Dic3—41no—
278C3×Dic3—41no—
279C3×Dic3—41no—
280C3×Dic3—41no—
281C3×Dic3—81no—
282C3×Dic3—122no—
283C3×Dic3—121no—
284C3×Dic3—121no—
285C3×Dic3—121no—
286C3×Dic3—121no—
287C3×Dic3—241no—
288C4⋊Q8—91no—
289#125—91no—
290#103—31no—
291#103—91no—
292#103—91no—
293#103—91no—
294#97—31no—
295C33#49111yescharacteristic, complemented, Sylow
296C3×Q8—62no—
297C3×Q8—61no—
298C3×Q8—61no—
299C3×Q8—61no—
300C3×Q8—61no—
301C3×Q8—121no—
302C3×Q8—181no—
303C3×Q8—181no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—31no—
308C2×C3×C4—31no—
309C2×C3×C4—31no—
310C2×C3×C4—31no—
311C2×C3×C4—41no—
312C2×C3×C4—61no—
313C2×C3×C4—61no—
314C2×C3×C4—61no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—121no—
319C2×C3×C4—121no—
320C2×C3×C4—181no—
321C2×C3×C4—181no—
322C2×C3×C4C2×D3211yescharacteristic
323C2×C3×C4C22×C3×D311yescharacteristic
324C2×C3×C4C22×C3×D311yescharacteristic
325C2×Dic3—21no—
326C2×Dic3—21no—
327C2×Dic3—61no—
328C2×Dic3—61no—
329C2×Dic3—61no—
330C2×Dic3—61no—
331C2×Dic3—121no—
332C2×Dic3—121no—
333C2×Dic3C3×D1211yescharacteristic
334C2×Dic3C3×D1211yescharacteristic
335Dic6—21no—
336Dic6—21no—
337Dic6—62no—
338Dic6—61no—
339Dic6—61no—
340Dic6—61no—
341Dic6—61no—
342Dic6—121no—
343C3×C8—22no—
344C3×C8—21no—
345C3×C8—21no—
346C3×C8—21no—
347C3×C8—41no—
348C3×C8—41no—
349C3×C8—42no—
350C3×C8—62no—
351C3×C8—61no—
352C3×C8—61no—
353C3⋊C8—21no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—41no—
364C2×C32—41no—
365C2×C32—41no—
366C2×C32#87511yescharacteristic
367C2×C32#88511yescharacteristic
368C2×C32#90911yescharacteristic
369C2×C32#90911yescharacteristic
370C2×C32#91511yescharacteristic
371C2×C32#91811yescharacteristic
372C2×C32C3×#12411yescharacteristic
373C2×C32C3×#13711yescharacteristic
374C2×C32C3×#13711yescharacteristic
375C2×Q8—31no—
376C2×Q8—91no—
377C2×C8—31no—
378C2×C8C3×D3211yescharacteristic
379C4⋊C4—31no—
380C4⋊C4—91no—
381C4⋊C4—181no—
382C4⋊C4—181no—
383C4⋊C4—181no—
384C4⋊C4—181no—
385C42—31no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—41no—
390C22×C3D3×D1211yescharacteristic
391C22×C3C2×C3×D1211yescharacteristic
392C22×C3C3×D3×D411yescharacteristic
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—41no—
400C3×C4—41no—
401C3×C4—61no—
402C3×C4—61no—
403C3×C4—61no—
404C3×C4—61no—
405C3×C4—121no—
406C3×C4—121no—
407C3×C4—122no—
408C3×C4—121no—
409C3×C4—121no—
410C3×C4—121no—
411C3×C4—121no—
412C3×C4—241no—
413C3×C4—361no—
414C3×C4—361no—
415C3×C4#379811yescharacteristic
416C3×C4#380411yescharacteristic
417C3×C4C3×#23511yescharacteristic
418C3×C4C3×#23511yescharacteristic
419C3×C4C3×#23511yescharacteristic
420C3×C4C3×#23711yescharacteristic
421Dic3—21no—
422Dic3—21no—
423Dic3—41no—
424Dic3—41no—
425Dic3—122no—
426Dic3—121no—
427Dic3—121no—
428Dic3—121no—
429Dic3—121no—
430Dic3—241no—
431C32—21no—
432C32—21no—
433C32—21no—
434C32—41no—
435C32#470311yescharacteristic, complemented
436C32#479711yescharacteristic, complemented
437C32C3×#49111yescharacteristic, complemented
438Q8—61no—
439Q8—61no—
440Q8—181no—
441Q8—181no—
442C2×C4—31no—
443C2×C4—31no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—181no—
447C2×C4—181no—
448C2×C4C2×C3×D3211yescharacteristic
449C8—21no—
450C8—61no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—41no—
461C2×C3—41no—
462C2×C3—41no—
463C2×C3#6380011yescharacteristic
464C2×C3#6380111yescharacteristic
465C2×C3#6385211yescharacteristic
466C2×C3C3×#87511yescharacteristic
467C2×C3C3×#88511yescharacteristic
468C2×C3C3×#90911yescharacteristic
469C2×C3C3×#90911yescharacteristic
470C2×C3C3×#91511yescharacteristic
471C2×C3C3×#91811yescharacteristic
472C22C3×D3×D1211yescharacteristic
473C4—61no—
474C4—61no—
475C4—121no—
476C4—121no—
477C4—361no—
478C4—361no—
479C4C3×#379811yescharacteristic
480C4C3×#380411yescharacteristic
481C3—21no—
482C3—21no—
483C3—21no—
484C3—41no—
485C3#1059089811yescharacteristic, direct product factor
486C3C3×#470311yescharacteristic, complemented
487C3C3×#479711yescharacteristic, complemented
488C2C3×#6380011yescharacteristic
489C2C3×#6380111yescharacteristic
490C2C3×#6385211yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×C3×C8D32
hypocenterC32C3×#491
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
10. #10590898C3direct (N × H)
295. C33#491semidirect
435. C32#4703semidirect
436. C32#4797semidirect
437. C32C3×#491semidirect
485. C3#10590898direct (N × H)
486. C3C3×#4703semidirect
487. C3C3×#4797semidirect

Maximal subgroups

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

subgroupindex
C3×#638692
C3×#638662
C3×#636432
C32×#8162
C32×#8152
C3×#635672
C3×#635662
C3×#47973
C3×#47033
#105908983