Finite Group Databasebeta

← C3×#10590890 #13157853

Subgroups — C3×#10590890 #13157853

Order 1728.

542 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
1C3×#63918C211yesmaximal, characteristic
2C3×#63866C211yesmaximal, characteristic
3C3×#63643C211yesmaximal, characteristic
4C32×#816C211yesmaximal, characteristic
5C32×C8×Dic3C211yesmaximal, characteristic
6C3×#63567C211yesmaximal, characteristic
7C3×#63566C211yesmaximal, characteristic
8C3×#4792—31nomaximal
9C3×#4681—31nomaximal
10#10590890C311yesmaximal, characteristic, direct product factor
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C3×#627—21no—
14C2×C33×C8C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C3×Dic3⋊Dic3—21no—
20C3×Dic3⋊Dic3—21no—
21C3×Dic3⋊Dic3—21no—
22C3×#944—31no—
23C3×#888—31no—
24C3×#869—31no—
25C3×#869—31no—
26#63918C2×C311yescharacteristic
27#63866C2×C311yescharacteristic
28C32×#103—31no—
29C32×C4×C8—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×C8×Dic3—21no—
41C3×C8×Dic3D311yescharacteristic
42C3×C8×Dic3C2×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—
57C3×#627—42no—
58C33×C8—21no—
59C32×C3⋊C8—21no—
60C3×#487—91no—
61#4792—31no—
62#4681—31no—
63C2×C32×Q8—31no—
64C2×#634C22×C311yescharacteristic
65C2×C3×Dic6—21no—
66C2×C3×Dic6—31no—
67C2×C3×Dic6—31no—
68C2×C3×Dic6—61no—
69C2×C3×Dic6—61no—
70C2×C3×Dic6—61no—
71C2×C3×Dic6C2×D311yescharacteristic
72C2×C3×Dic6C22×C311yescharacteristic
73C2×#627—21no—
74C2×C32×C8—21no—
75C2×C32×C8—21no—
76C2×C32×C8—21no—
77C2×C32×C8—31no—
78C2×C32×C8—41no—
79C2×C32×C8C2×D311yescharacteristic
80C2×C32×C8C2×D311yescharacteristic
81C2×C32×C8C22×C311yescharacteristic
82C32×C4⋊C4—31no—
83C32×C42—31no—
84C4⋊(C3⋊Dic3)C22×C311yescharacteristic
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3—61no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C22×C311yescharacteristic
94C3×Dic3⋊C4—61no—
95C3×Dic3⋊C4—61no—
96C3×Dic3⋊C4—61no—
97C3×C4×Dic3—21no—
98C3×C4×Dic3C2×D311yescharacteristic
99C3×C4×Dic3C22×C311yescharacteristic
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C22×C311yescharacteristic
103Dic3⋊Dic3—21no—
104Dic3⋊Dic3—21no—
105Dic3⋊Dic3—21no—
106C22×C33C2×D411yescharacteristic
107C33×C4#4111yescharacteristic
108C33×C4#4111yescharacteristic
109C3×C3⋊Dic3—41no—
110C3×C3⋊Dic3—41no—
111C32×Dic3—41no—
112C32×Dic3—41no—
113C32×Dic3SD1612yes—
114C32×Dic3Q1612yes—
115C3×C4⋊Q8—91no—
116C3×C4⋊Q8—91no—
117#944—31no—
118#888—31no—
119#869—31no—
120#869—31no—
121C3×#103—31no—
122C3×#103—31no—
123C3×#103—61no—
124C3×#103—91no—
125C3×#103—91no—
126C3×#103—91no—
127C3×C4×C8—31no—
128C3×C4×C8—31no—
129C3×C4×C8—61no—
130#835—31no—
131#816—31no—
132#816—31no—
133#816—31no—
134#816—31no—
135#816—61no—
136#816C3×D311yescharacteristic
137C8×Dic3C3×D311yescharacteristic
138#810—31no—
139C32×Q8—61no—
140C32×Q8—61no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—31no—
145C2×C32×C4—31no—
146C2×C32×C4—41no—
147C2×C32×C4—61no—
148C2×C32×C4—61no—
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C23×C311yescharacteristic
152C2×C3⋊Dic3—21no—
153C2×C3⋊Dic3—21no—
154#634—21no—
155#634—21no—
156C2×C3×Dic3—21no—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—41no—
163C2×C3×Dic3—41no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—121no—
169C2×C3×Dic3—121no—
170C2×C3×Dic3D1211yescharacteristic
171C2×C3×Dic3D1211yescharacteristic
172C2×C3×Dic3C3×D411yescharacteristic
173C2×C3×Dic3C3×D411yescharacteristic
174C3×Dic6—22no—
175C3×Dic6—21no—
176C3×Dic6—21no—
177C3×Dic6—21no—
178C3×Dic6—21no—
179C3×Dic6—41no—
180C3×Dic6—62no—
181C3×Dic6—61no—
182C3×Dic6—61no—
183C3×Dic6—61no—
184C3×Dic6—61no—
185C3×Dic6—121no—
186C3×Dic6—122no—
187C3×Dic6—122no—
188#627—42no—
189C32×C8—21no—
190C32×C8—22no—
191C32×C8—21no—
192C32×C8—21no—
193C32×C8—41no—
194C32×C8—41no—
195C32×C8—42no—
196C32×C8—61no—
197C3×C3⋊C8—22no—
198C3×C3⋊C8—21no—
199C3×C3⋊C8—21no—
200#487—91noSylow
201C2×C33#12411yescharacteristic
202C2×C33C2×SD1611yescharacteristic
203C2×C33C2×Q1611yescharacteristic
204C2×C3×Q8—31no—
205C2×C3×Q8—31no—
206C2×C3×Q8—61no—
207C2×C3×Q8—91no—
208C2×C3×Q8—181no—
209C2×Dic6—31no—
210C2×Dic6—31no—
211C2×Dic6—61no—
212C2×Dic6—61no—
213C2×Dic6—61no—
214C2×Dic6C2×C3×D311yescharacteristic
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—31no—
219C2×C3×C8—31no—
220C2×C3×C8—41no—
221C2×C3×C8—61no—
222C2×C3×C8D3211yescharacteristic
223C2×C3×C8C2×C3×D311yescharacteristic
224C2×C3×C8C2×C3×D311yescharacteristic
225C3×C4⋊C4—31no—
226C3×C4⋊C4—31no—
227C3×C4⋊C4—61no—
228C3×C4⋊C4—91no—
229C3×C4⋊C4—181no—
230C3×C4⋊C4—181no—
231C3×C4⋊C4—181no—
232C3×C42—31no—
233C3×C42—31no—
234C3×C42—61no—
235C4⋊Dic3—31no—
236C4⋊Dic3—31no—
237C4⋊Dic3—61no—
238C4⋊Dic3—61no—
239C4⋊Dic3—61no—
240C4⋊Dic3—61no—
241C4⋊Dic3C2×C3×D311yescharacteristic
242Dic3⋊C4—61no—
243Dic3⋊C4—61no—
244Dic3⋊C4—61no—
245C4×Dic3C2×C3×D311yescharacteristic
246C2×C3⋊C8C2×C3×D311yescharacteristic
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32—41no—
251C22×C32C2×D1211yescharacteristic
252C22×C32D3×D411yescharacteristic
253C22×C32C2×C3×D411yescharacteristic
254C32×C4—21no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—32no—
261C32×C4—32no—
262C32×C4—41no—
263C32×C4—41no—
264C32×C4—121no—
265C32×C4—121no—
266C32×C4#23511yescharacteristic
267C32×C4#23511yescharacteristic
268C32×C4#23511yescharacteristic
269C32×C4#23711yescharacteristic
270C32×C4C3×#4111yescharacteristic
271C32×C4C3×#4111yescharacteristic
272C3⋊Dic3—41no—
273C3⋊Dic3—41no—
274C3×Dic3—22no—
275C3×Dic3—22no—
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—
288C3×Dic3#20412yes—
289C3×Dic3Dic1212yes—
290C3×Dic3C3×SD1612yes—
291C3×Dic3C3×Q1612yes—
292C4⋊Q8—91no—
293C4⋊Q8—91no—
294#103—31no—
295#103—91no—
296#103—91no—
297#103—91no—
298C4×C8—31no—
299C33#48711yescharacteristic, complemented, Sylow
300C3×Q8—62no—
301C3×Q8—61no—
302C3×Q8—61no—
303C3×Q8—61no—
304C3×Q8—61no—
305C3×Q8—121no—
306C3×Q8—181no—
307C3×Q8—181no—
308C3×Q8—362no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—31no—
313C2×C3×C4—31no—
314C2×C3×C4—31no—
315C2×C3×C4—31no—
316C2×C3×C4—41no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—121no—
324C2×C3×C4—121no—
325C2×C3×C4—181no—
326C2×C3×C4—181no—
327C2×C3×C4C2×D3211yescharacteristic
328C2×C3×C4C22×C3×D311yescharacteristic
329C2×C3×C4C22×C3×D311yescharacteristic
330C2×Dic3—21no—
331C2×Dic3—21no—
332C2×Dic3—61no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—61no—
336C2×Dic3—121no—
337C2×Dic3—121no—
338C2×Dic3C3×D1211yescharacteristic
339C2×Dic3C3×D1211yescharacteristic
340Dic6—21no—
341Dic6—21no—
342Dic6—62no—
343Dic6—61no—
344Dic6—61no—
345Dic6—61no—
346Dic6—61no—
347Dic6—121no—
348Dic6—122no—
349Dic6—122no—
350C3×C8—22no—
351C3×C8—21no—
352C3×C8—21no—
353C3×C8—21no—
354C3×C8—41no—
355C3×C8—41no—
356C3×C8—42no—
357C3×C8—62no—
358C3×C8—61no—
359C3×C8—61no—
360C3⋊C8—21no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—41no—
371C2×C32—41no—
372C2×C32—41no—
373C2×C32#87511yescharacteristic
374C2×C32#88511yescharacteristic
375C2×C32C2×#20411yescharacteristic
376C2×C32C2×Dic1211yescharacteristic
377C2×C32D3×SD1611yescharacteristic
378C2×C32D3×Q1611yescharacteristic
379C2×C32C3×#12411yescharacteristic
380C2×C32C2×C3×SD1611yescharacteristic
381C2×C32C2×C3×Q1611yescharacteristic
382C2×Q8—31no—
383C2×Q8—91no—
384C2×Q8—181no—
385C2×C8—31no—
386C2×C8C3×D3211yescharacteristic
387C4⋊C4—31no—
388C4⋊C4—91no—
389C4⋊C4—181no—
390C4⋊C4—181no—
391C4⋊C4—181no—
392C42—31no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—41no—
397C22×C3D3×D1211yescharacteristic
398C22×C3C2×C3×D1211yescharacteristic
399C22×C3C3×D3×D411yescharacteristic
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—32no—
407C3×C4—32no—
408C3×C4—32no—
409C3×C4—32no—
410C3×C4—41no—
411C3×C4—41no—
412C3×C4—62no—
413C3×C4—62no—
414C3×C4—122no—
415C3×C4—121no—
416C3×C4—121no—
417C3×C4—121no—
418C3×C4—121no—
419C3×C4—241no—
420C3×C4—361no—
421C3×C4—361no—
422C3×C4#379811yescharacteristic
423C3×C4#380411yescharacteristic
424C3×C4C3×#23511yescharacteristic
425C3×C4C3×#23511yescharacteristic
426C3×C4C3×#23511yescharacteristic
427C3×C4C3×#23711yescharacteristic
428Dic3—41no—
429Dic3—41no—
430Dic3—122no—
431Dic3—121no—
432Dic3—121no—
433Dic3—121no—
434Dic3—121no—
435Dic3—241no—
436Dic3C3×#20412yes—
437Dic3C3×Dic1212yes—
438C32—21no—
439C32—21no—
440C32—21no—
441C32—41no—
442C32#468111yescharacteristic, complemented
443C32#479211yescharacteristic, complemented
444C32C3×#48711yescharacteristic, complemented
445Q8—61no—
446Q8—61no—
447Q8—181no—
448Q8—181no—
449Q8—362no—
450C2×C4—31no—
451C2×C4—31no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—181no—
455C2×C4—181no—
456C2×C4C2×C3×D3211yescharacteristic
457C8—21no—
458C8—61no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—41no—
469C2×C3—41no—
470C2×C3—41no—
471C2×C3D3×#20411yescharacteristic
472C2×C3D3×Dic1211yescharacteristic
473C2×C3#6385211yescharacteristic
474C2×C3C3×#87511yescharacteristic
475C2×C3C3×#88511yescharacteristic
476C2×C3C2×C3×#20411yescharacteristic
477C2×C3C2×C3×Dic1211yescharacteristic
478C2×C3C3×D3×SD1611yescharacteristic
479C2×C3C3×D3×Q1611yescharacteristic
480C22C3×D3×D1211yescharacteristic
481C4—32no—
482C4—32no—
483C4—121no—
484C4—121no—
485C4—361no—
486C4—361no—
487C4C3×#379811yescharacteristic
488C4C3×#380411yescharacteristic
489C3—21no—
490C3—21no—
491C3—21no—
492C3—41no—
493C3#1059089011yescharacteristic, direct product factor
494C3C3×#468111yescharacteristic, complemented
495C3C3×#479211yescharacteristic, complemented
496C2C3×D3×#20411yescharacteristic
497C2C3×D3×Dic1211yescharacteristic
498C2C3×#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×#487
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. #10590890C3direct (N × H)
299. C33#487semidirect
442. C32#4681semidirect
443. C32#4792semidirect
444. C32C3×#487semidirect
493. C3#10590890direct (N × H)
494. C3C3×#4681semidirect
495. C3C3×#4792semidirect

Maximal subgroups

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

subgroupindex
C3×#639182
C3×#638662
C3×#636432
C32×#8162
C32×C8×Dic32
C3×#635672
C3×#635662
C3×#47923
C3×#46813
#105908903