Finite Group Databasebeta

← C3×#10590388 #13157403

Subgroups — C3×#10590388 #13157403

Order 1728.

638 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#634C211yesmaximal, characteristic
2C32×#869C211yesmaximal, characteristic
3C2×C3×#3719C212yesmaximal
4C32×C4⋊(C3⋊C8)C211yesmaximal, characteristic
5C3×#63566C212yesmaximal
6C3×#5090—31nomaximal
7C3×#4849—31nomaximal
8#10590388C311yesmaximal, characteristic, direct product factor
9C2×C3×#634C2211yescharacteristic
10C2×C32×Dic6C2212yes—
11C33×C42C2211yescharacteristic
12C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
13C3×#3750—21no—
14C3×C4×C3⋊Dic3—21no—
15C32×C4⋊Dic3—21no—
16C32×C4⋊Dic3—21no—
17C2×C32×C3⋊C8C2212yes—
18C3×#3719—24no—
19C32×C4⋊Q8—31no—
20C4×#634C2×C311yescharacteristic
21C2×C3×#216—32no—
22C2×C3×Dic12—32no—
23C3×#869—21no—
24C3×#869D311yescharacteristic
25C3×#869C2×C311yescharacteristic
26C3×C4×Dic6—31no—
27C3×C4×Dic6—31no—
28C3×C4×Dic6—61no—
29C2×#3719C2×C312yes—
30C32×C4⋊C8—31no—
31C3×#835—32no—
32C3×#816—32no—
33C3×C4⋊(C3⋊C8)—21no—
34C3×C4⋊(C3⋊C8)D311yescharacteristic
35C3×C4⋊(C3⋊C8)C2×C311yescharacteristic
36#63566C2×C312yes—
37C2×C33×C4D411yescharacteristic
38C2×C33×C4D411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3—21no—
42C3×#634—21no—
43C3×#634D412yes—
44C2×C32×Dic3—22no—
45C32×Dic6—22no—
46C32×Dic6—22no—
47C32×C3⋊C8—22no—
48C3×#462—91no—
49#5090—31no—
50#4849—31no—
51C2×C32×Q8—32no—
52C2×#634C22×C311yescharacteristic
53C2×C3×Dic6—22no—
54C2×C3×Dic6—31no—
55C2×C3×Dic6—31no—
56C2×C3×Dic6—61no—
57C2×C3×Dic6C2×D312yes—
58C2×C3×Dic6C22×C312yes—
59C2×C32×C8—32no—
60C32×C4⋊C4—61no—
61C32×C4⋊C4—61no—
62C32×C42—21no—
63C32×C42—21no—
64C32×C42—21no—
65C32×C42—41no—
66C32×C42C2×D311yescharacteristic
67C32×C42C2×D311yescharacteristic
68C32×C42C22×C311yescharacteristic
69C4⋊(C3⋊Dic3)C22×C311yescharacteristic
70#3750—21no—
71C4×C3⋊Dic3—21no—
72C3×#216—64no—
73C3×C4⋊Dic3—22no—
74C3×C4⋊Dic3—22no—
75C3×C4⋊Dic3—21no—
76C3×C4⋊Dic3—21no—
77C3×C4⋊Dic3—21no—
78C3×C4⋊Dic3—21no—
79C3×C4⋊Dic3—31no—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—61no—
82C3×Dic3⋊C4—62no—
83C3×Dic3⋊C4—61no—
84C3×Dic3⋊C4—61no—
85C3×C4×Dic3—62no—
86C3×C4×Dic3—61no—
87C3×C4×Dic3—61no—
88C2×C3×C3⋊C8—22no—
89C2×C3×C3⋊C8C2×D312yes—
90C2×C3×C3⋊C8C22×C312yes—
91C3×Dic12—64no—
92#3719—24no—
93C22×C33C2×D411yescharacteristic
94C33×C4—21no—
95C33×C4Q1612yescomplemented
96C33×C4C2×D411yescharacteristic
97C33×C4#4111yescharacteristic
98C3×C3⋊Dic3—22no—
99C3×C3⋊Dic3—41no—
100C32×Dic3—42no—
101C2×C3×Q16—92no—
102C3×C4⋊Q8—31no—
103C3×C4⋊Q8—31no—
104C3×C4⋊Q8—61no—
105C3×C4×Q8—91no—
106C2×#216—32no—
107C2×Dic12—32no—
108#869C3×D311yescharacteristic
109C4×Dic6—31no—
110C4×Dic6—31no—
111C4×Dic6—61no—
112C3×C4⋊C8—31no—
113C3×C4⋊C8—31no—
114C3×C4⋊C8—61no—
115C3×#103—92no—
116#835—32no—
117#816—32no—
118C4⋊(C3⋊C8)C3×D311yescharacteristic
119C32×Q8—62no—
120C32×Q8—62no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—41no—
131C2×C32×C4—41no—
132C2×C32×C4—41no—
133C2×C32×C4—62no—
134C2×C32×C4D1211yescharacteristic
135C2×C32×C4D1211yescharacteristic
136C2×C32×C4#6711yescharacteristic
137C2×C32×C4#6711yescharacteristic
138C2×C32×C4C3×D411yescharacteristic
139C2×C32×C4C3×D411yescharacteristic
140C2×C32×C4C22×D311yescharacteristic
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C23×C311yescharacteristic
143C2×C3⋊Dic3—21no—
144C2×C3⋊Dic3—21no—
145#634—21no—
146#634C3×D412yes—
147C2×C3×Dic3—24no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—62no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—62no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—61no—
156C3×Dic6—22no—
157C3×Dic6—22no—
158C3×Dic6—22no—
159C3×Dic6—22no—
160C3×Dic6—32no—
161C3×Dic6—32no—
162C3×Dic6—42no—
163C3×Dic6—42no—
164C3×Dic6—62no—
165C3×Dic6—62no—
166C3×Dic6—61no—
167C3×Dic6—61no—
168C32×C8—62no—
169C3×C3⋊C8—22no—
170C3×C3⋊C8—22no—
171C3×C3⋊C8—42no—
172#462—91noSylow
173C2×C33#12111yescharacteristic
174C2×C33C2×Q1611yescharacteristic
175C2×C33#13711yescharacteristic
176C2×C3×Q8—32no—
177C2×C3×Q8—32no—
178C2×C3×Q8—62no—
179C2×C3×Q8—91no—
180C2×Dic6—31no—
181C2×Dic6—31no—
182C2×Dic6—61no—
183C2×Dic6C2×C3×D312yes—
184C3×Q16—184no—
185C2×C3×C8—32no—
186C2×C3×C8—32no—
187C2×C3×C8—62no—
188C3×C4⋊C4—62no—
189C3×C4⋊C4—62no—
190C3×C4⋊C4—61no—
191C3×C4⋊C4—61no—
192C3×C4⋊C4—61no—
193C3×C4⋊C4—61no—
194C3×C4⋊C4—91no—
195C3×C4⋊C4—181no—
196C3×C42—21no—
197C3×C42—21no—
198C3×C42—21no—
199C3×C42—41no—
200C3×C42—181no—
201C3×C42D3211yescharacteristic
202C3×C42C2×C3×D311yescharacteristic
203C3×C42C2×C3×D311yescharacteristic
204#216—64no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207C4⋊Dic3—31no—
208C4⋊Dic3—31no—
209C4⋊Dic3—61no—
210Dic3⋊C4—62no—
211Dic3⋊C4—61no—
212Dic3⋊C4—61no—
213C4×Dic3—62no—
214C4×Dic3—61no—
215C4×Dic3—61no—
216C2×C3⋊C8C2×C3×D312yes—
217Dic12—64no—
218C22×C32—21no—
219C22×C32—21no—
220C22×C32—21no—
221C22×C32—41no—
222C22×C32C2×D1211yescharacteristic
223C22×C32C2×#6711yescharacteristic
224C22×C32C2×C3×D411yescharacteristic
225C32×C4—21no—
226C32×C4—22no—
227C32×C4—21no—
228C32×C4—21no—
229C32×C4—22no—
230C32×C4—22no—
231C32×C4—22no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4—21no—
235C32×C4—21no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4—42no—
239C32×C4—41no—
240C32×C4—41no—
241C32×C4—41no—
242C32×C4—41no—
243C32×C4—42no—
244C32×C4—122no—
245C32×C4Dic1212yescomplemented
246C32×C4#21612yescomplemented
247C32×C4C3×Q1612yescomplemented
248C32×C4D3×D411yescharacteristic
249C32×C4D3×D411yescharacteristic
250C32×C4#23711yescharacteristic
251C32×C4#23911yescharacteristic
252C32×C4C2×C3×D411yescharacteristic
253C32×C4C3×#4111yescharacteristic
254C3⋊Dic3—22no—
255C3⋊Dic3—41no—
256C3×Dic3—44no—
257C3×Dic3—42no—
258C3×Dic3—42no—
259C3×Dic3—64no—
260C3×Dic3—62no—
261C3×Dic3—62no—
262C3×Dic3—122no—
263C3×Dic3—121no—
264C3×Dic3—121no—
265C2×Q16—92no—
266C4⋊Q8—31no—
267C4×Q8—91no—
268C4⋊C8—31no—
269#103—92no—
270C33#46211yescharacteristic, complemented, Sylow
271C3×Q8—62no—
272C3×Q8—62no—
273C3×Q8—62no—
274C3×Q8—62no—
275C3×Q8—92no—
276C3×Q8—122no—
277C3×Q8—122no—
278C3×Q8—181no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—64no—
292C2×C3×C4—62no—
293C2×C3×C4—62no—
294C2×C3×C4—181no—
295C2×C3×C4—181no—
296C2×C3×C4#62611yescharacteristic
297C2×C3×C4#62611yescharacteristic
298C2×C3×C4C3×D1211yescharacteristic
299C2×C3×C4C3×D1211yescharacteristic
300C2×C3×C4C3×#6711yescharacteristic
301C2×C3×C4C3×#6711yescharacteristic
302C2×C3×C4C2×D3211yescharacteristic
303C2×C3×C4C22×C3×D311yescharacteristic
304C2×C3×C4C22×C3×D311yescharacteristic
305C2×Dic3—22no—
306C2×Dic3—62no—
307C2×Dic3—61no—
308C2×Dic3—61no—
309C2×Dic3—62no—
310C2×Dic3—61no—
311C2×Dic3—61no—
312Dic6—22no—
313Dic6—22no—
314Dic6—32no—
315Dic6—32no—
316Dic6—62no—
317Dic6—62no—
318Dic6—61no—
319Dic6—61no—
320C3×C8—62no—
321C3×C8—62no—
322C3×C8—122no—
323C3⋊C8—22no—
324C2×C32—21no—
325C2×C32—21no—
326C2×C32—21no—
327C2×C32—21no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—41no—
334C2×C32—41no—
335C2×C32—41no—
336C2×C32#89511yescharacteristic
337C2×C32C2×Dic1211yescharacteristic
338C2×C32#90911yescharacteristic
339C2×C32#93811yescharacteristic
340C2×C32#94211yescharacteristic
341C2×C32C2×#21611yescharacteristic
342C2×C32C3×#12111yescharacteristic
343C2×C32C2×C3×Q1611yescharacteristic
344C2×C32C3×#13711yescharacteristic
345C2×Q8—32no—
346C2×Q8—91no—
347Q16—184no—
348C2×C8—32no—
349C4⋊C4—61no—
350C4⋊C4—61no—
351C4⋊C4—91no—
352C4⋊C4—181no—
353C42—181no—
354C42C3×D3211yescharacteristic
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—41no—
359C22×C3C2×#62611yescharacteristic
360C22×C3C2×C3×D1211yescharacteristic
361C22×C3C2×C3×#6711yescharacteristic
362C3×C4—22no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—22no—
373C3×C4—22no—
374C3×C4—22no—
375C3×C4—41no—
376C3×C4—41no—
377C3×C4—42no—
378C3×C4—41no—
379C3×C4—41no—
380C3×C4—42no—
381C3×C4—124no—
382C3×C4—122no—
383C3×C4—122no—
384C3×C4—182no—
385C3×C4—361no—
386C3×C4#371912yescomplemented
387C3×C4C3×Dic1212yescomplemented
388C3×C4C3×#21612yescomplemented
389C3×C4#379611yescharacteristic
390C3×C4#380211yescharacteristic
391C3×C4C3×D3×D411yescharacteristic
392C3×C4C3×D3×D411yescharacteristic
393C3×C4C3×#23711yescharacteristic
394C3×C4C3×#23911yescharacteristic
395Dic3—42no—
396Dic3—64no—
397Dic3—62no—
398Dic3—62no—
399Dic3—122no—
400Dic3—121no—
401Dic3—121no—
402C32—21no—
403C32—21no—
404C32—21no—
405C32—41no—
406C32#484911yescharacteristic, complemented
407C32#509011yescharacteristic, complemented
408C32C3×#46211yescharacteristic, complemented
409Q8—62no—
410Q8—62no—
411Q8—92no—
412Q8—181no—
413C2×C4—62no—
414C2×C4—181no—
415C2×C4—181no—
416C2×C4C3×#62611yescharacteristic
417C2×C4C3×#62611yescharacteristic
418C2×C4C2×C3×D3211yescharacteristic
419C8—62no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—41no—
430C2×C3—41no—
431C2×C3—41no—
432C2×C3#6383311yescharacteristic
433C2×C3C2×#371911yescharacteristic
434C2×C3#6391611yescharacteristic
435C2×C3C3×#89511yescharacteristic
436C2×C3C2×C3×Dic1211yescharacteristic
437C2×C3C3×#90911yescharacteristic
438C2×C3C3×#93811yescharacteristic
439C2×C3C3×#94211yescharacteristic
440C2×C3C2×C3×#21611yescharacteristic
441C22C2×C3×#62611yescharacteristic
442C4—21no—
443C4—122no—
444C4—182no—
445C4—361no—
446C4C3×#371912yescomplemented
447C4C3×#379611yescharacteristic
448C4C3×#380211yescharacteristic
449C3—21no—
450C3—21no—
451C3—21no—
452C3—41no—
453C3#1059038811yescharacteristic, direct product factor
454C3C3×#484911yescharacteristic, complemented
455C3C3×#509011yescharacteristic, complemented
456C2C3×#6383311yescharacteristic
457C2C2×C3×#371911yescharacteristic
458C2C3×#6391611yescharacteristic
subgroupG/H
centerC22×C3C2×#626
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socle, prime radicalC22×C33C2×D4
hypercenterC3×C42D32
hypocenterC32C3×#462
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
8. #10590388C3direct (N × H)
95. C33×C4Q16semidirect
245. C32×C4Dic12semidirect
246. C32×C4#216semidirect
247. C32×C4C3×Q16semidirect
270. C33#462semidirect
386. C3×C4#3719semidirect
387. C3×C4C3×Dic12semidirect
388. C3×C4C3×#216semidirect
406. C32#4849semidirect
407. C32#5090semidirect
408. C32C3×#462semidirect
446. C4C3×#3719semidirect
453. C3#10590388direct (N × H)
454. C3C3×#4849semidirect
455. C3C3×#5090semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#6342
C32×#8692
C2×C3×#37192
C32×C4⋊(C3⋊C8)2
C3×#635662
C3×#50903
C3×#48493
#105903883