Finite Group Databasebeta

← C3×#10591070 #13158096

Subgroups — C3×#10591070 #13158096

Order 1728.

994 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#235C211yesmaximal, characteristic
2C2×C3×#3748C211yesmaximal, characteristic, complemented
3C22×C32×C3⋊C8C211yesmaximal, characteristic, complemented
4C3×(C2×D3)⋊(C3⋊C8)C214yesmaximal, complemented
5C3×#5234—31nomaximal
6C3×#5110—31nomaximal
7#10591070C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C2×C32×#67C412yes—
10C32×#235—22no—
11C32×#235—22no—
12C2×C32×D12C411yescharacteristic
13C2×C32×C4×D3C2212yes—
14C2×C32×Dic6C411yescharacteristic
15C3×#3748—22no—
16C2×C3×C3⋊(C3⋊C8)C2212yes—
17C2×C32×C3⋊C8—22no—
18C2×C32×C3⋊C8C2212yes—
19C2×C32×#41—31no—
20C2×C3×#235—21no—
21C2×C3×#235D311yescharacteristic
22C2×C3×#235C2×C311yescharacteristic
23C22×C32×C8—31no—
24C2×#3748C2×C311yescharacteristic, complemented
25C2×C3×#208—31no—
26C2×C3×#208—31no—
27C2×C3×#208—61no—
28C22×C3×C3⋊C8—21no—
29C22×C3×C3⋊C8D311yescharacteristic, complemented
30C22×C3×C3⋊C8C2×C311yescharacteristic, complemented
31C3×#830—34no—
32C3×(C2×D3)⋊C8—34no—
33(C2×D3)⋊(C3⋊C8)C2×C314yescomplemented
34C23×C33C2×C411yescharacteristic
35C22×C32×D3C2×C412yes—
36C2×C33×C4C2×C411yescharacteristic
37C2×C33×C4D412yes—
38C2×C33×C4D412yes—
39C2×C33×C4C2311yescharacteristic
40C32×#67—24no—
41C2×C32×Dic3C2×C412yes—
42C32×D12—22no—
43C32×C4×D3—22no—
44C32×C4×D3—22no—
45C32×Dic6—22no—
46C3×C3⋊(C3⋊C8)—22no—
47C32×C3⋊C8—22no—
48C3×#408—91no—
49#5234—31no—
50#5110—31no—
51C32×#41—62no—
52C32×#41—62no—
53C2×C32×Q8—31no—
54C2×C32×D4—32no—
55C2×C32×D4—31no—
56C22×C32×C4—21no—
57C22×C32×C4—21no—
58C22×C32×C4—21no—
59C22×C32×C4—32no—
60C22×C32×C4—41no—
61C22×C32×C4C2×D311yescharacteristic
62C22×C32×C4C2×D311yescharacteristic
63C22×C32×C4C22×C311yescharacteristic
64C2×C3×#67—22no—
65C2×C3×#67Dic312yes—
66C2×C3×#67C3×C412yes—
67C3×#235—24no—
68C3×#235—24no—
69C3×#235—22no—
70C3×#235—22no—
71C3×#235—22no—
72C3×#235—22no—
73C2×C3×D12—21no—
74C2×C3×D12Dic311yescharacteristic
75C2×C3×D12C3×C411yescharacteristic
76C2×C3×C4×D3—22no—
77C2×C3×C4×D3C2×D312yes—
78C2×C3×C4×D3C22×C312yes—
79C2×C3×Dic6—21no—
80C2×C3×Dic6Dic311yescharacteristic
81C2×C3×Dic6C3×C411yescharacteristic
82C2×C32×C8—32no—
83C2×C32×C8—62no—
84#3748—22no—
85C2×C3⋊(C3⋊C8)C22×C312yes—
86C3×#208—64no—
87C3×#208—62no—
88C3×#208—62no—
89C2×C3×C3⋊C8—24no—
90C2×C3×C3⋊C8—22no—
91C2×C3×C3⋊C8—22no—
92C2×C3×C3⋊C8—22no—
93C2×C3×C3⋊C8—32no—
94C2×C3×C3⋊C8—32no—
95C2×C3×C3⋊C8—62no—
96C2×C3×C3⋊C8C2×D312yes—
97C2×C3×C3⋊C8C22×C312yes—
98C22×C33—22no—
99C22×C33C22⋊C412yes—
100C22×C33C22×C411yescharacteristic
101C2×C32×D3—22no—
102C2×C32×D3—24no—
103C33×C4C22⋊C412yes—
104C33×C4C2×D411yescharacteristic
105C33×C4C2×D411yescharacteristic
106C32×Dic3—22no—
107C2×C3×#41—31no—
108C2×C3×#41—31no—
109C2×C3×#41—61no—
110C2×#235C3×D311yescharacteristic
111C2×C3×M16—91no—
112C22×C3×C8—31no—
113C22×C3×C8—31no—
114C22×C3×C8—61no—
115C2×#208—31no—
116C2×#208—31no—
117C2×#208—61no—
118C22×C3⋊C8C3×D311yescharacteristic, complemented
119C3×C22⋊C8—94no—
120#830—34no—
121(C2×D3)⋊C8—34no—
122C23×C32—21no—
123C23×C32—21no—
124C23×C32—21no—
125C23×C32—32no—
126C23×C32—41no—
127C23×C32C4×D311yescharacteristic
128C23×C32C2×Dic311yescharacteristic
129C23×C32C2×C3×C411yescharacteristic
130C22×C3×D3—22no—
131C22×C3×D3C2×Dic312yes—
132C22×C3×D3C2×C3×C412yes—
133C32×Q8—62no—
134C32×D4—64no—
135C32×D4—62no—
136C2×C32×C4—21no—
137C2×C32×C4—22no—
138C2×C32×C4—22no—
139C2×C32×C4—21no—
140C2×C32×C4—22no—
141C2×C32×C4—21no—
142C2×C32×C4—22no—
143C2×C32×C4—22no—
144C2×C32×C4—22no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—32no—
149C2×C32×C4—41no—
150C2×C32×C4—42no—
151C2×C32×C4—42no—
152C2×C32×C4—41no—
153C2×C32×C4—62no—
154C2×C32×C4—62no—
155C2×C32×C4C4×D311yescharacteristic
156C2×C32×C4D1212yes—
157C2×C32×C4C2×Dic311yescharacteristic
158C2×C32×C4#6712yes—
159C2×C32×C4#6712yes—
160C2×C32×C4#6712yes—
161C2×C32×C4C2×C3×C411yescharacteristic
162C2×C32×C4C3×D412yes—
163C2×C32×C4C3×D412yes—
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C23×C311yescharacteristic
167C3×#67—28no—
168C3×#67—24no—
169C3×#67—24no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3C2×Dic312yes—
172C2×C3×Dic3C2×C3×C412yes—
173C3×D12—24no—
174C3×D12—22no—
175C3×D12—22no—
176C3×C4×D3—24no—
177C3×C4×D3—24no—
178C3×C4×D3—22no—
179C3×C4×D3—22no—
180C3×C4×D3—22no—
181C3×C4×D3—22no—
182C3×Dic6—24no—
183C3×Dic6—22no—
184C3×Dic6—22no—
185C32×C8—62no—
186C3⋊(C3⋊C8)—22no—
187C3×C3⋊C8—24no—
188C3×C3⋊C8—22no—
189C3×C3⋊C8—22no—
190C3×C3⋊C8—64no—
191C3×C3⋊C8—62no—
192C3×C3⋊C8—62no—
193#408—91noSylow
194C2×C33—22no—
195C2×C33C2×C22⋊C411yescharacteristic
196C2×C33#13112yes—
197C32×D3—42no—
198C3×#41—64no—
199C3×#41—64no—
200C3×#41—62no—
201C3×#41—62no—
202C3×#41—62no—
203C3×#41—62no—
204C2×C3×Q8—31no—
205C2×C3×Q8—31no—
206C2×C3×Q8—61no—
207C2×C3×D4—32no—
208C2×C3×D4—32no—
209C2×C3×D4—31no—
210C2×C3×D4—31no—
211C2×C3×D4—61no—
212C2×C3×D4—62no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—32no—
217C22×C3×C4—32no—
218C22×C3×C4—41no—
219C22×C3×C4—62no—
220C22×C3×C4D3211yescharacteristic
221C22×C3×C4C2×C3×D311yescharacteristic
222C22×C3×C4C2×C3×D311yescharacteristic
223C2×#67C3×Dic312yes—
224#235—22no—
225#235—22no—
226C2×D12C3×Dic311yescharacteristic
227C2×C4×D3C2×C3×D312yes—
228C2×Dic6C3×Dic311yescharacteristic
229C3×M16—182no—
230C2×C3×C8—32no—
231C2×C3×C8—32no—
232C2×C3×C8—62no—
233C2×C3×C8—64no—
234C2×C3×C8—62no—
235C2×C3×C8—62no—
236C2×C3×C8—92no—
237#208—64no—
238#208—62no—
239#208—62no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—32no—
242C2×C3⋊C8—32no—
243C2×C3⋊C8—62no—
244C2×C3⋊C8C2×C3×D312yes—
245C22×C32—22no—
246C22×C32—24no—
247C22×C32—22no—
248C22×C32—22no—
249C22×C32—22no—
250C22×C32—22no—
251C22×C32—22no—
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—42no—
256C22×C32—41no—
257C22×C32—42no—
258C22×C32—42no—
259C22×C32—44no—
260C22×C32—62no—
261C22×C32—64no—
262C22×C32(C2×D3)⋊C412yes—
263C22×C32#21712yes—
264C22×C32C3×C22⋊C412yes—
265C22×C32C2×C4×D311yescharacteristic
266C22×C32C22×Dic311yescharacteristic
267C22×C32C22×C3×C411yescharacteristic
268C2×C3×D3—24no—
269C2×C3×D3—24no—
270C2×C3×D3—22no—
271C2×C3×D3—22no—
272C2×C3×D3—24no—
273C2×C3×D3—44no—
274C32×C4—22no—
275C32×C4—22no—
276C32×C4—22no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—42no—
284C32×C4—41no—
285C32×C4—41no—
286C32×C4—62no—
287C32×C4(C2×D3)⋊C412yes—
288C32×C4#21712yes—
289C32×C4C3×C22⋊C412yes—
290C32×C4C2×D1211yescharacteristic
291C32×C4C2×#6711yescharacteristic
292C32×C4C2×#6711yescharacteristic
293C32×C4C2×#6711yescharacteristic
294C32×C4C2×C3×D411yescharacteristic
295C32×C4C2×C3×D411yescharacteristic
296C3×Dic3—24no—
297C3×Dic3—22no—
298C3×Dic3—22no—
299C2×#41—31no—
300C2×M16—91no—
301C22×C8—31no—
302C22⋊C8—94no—
303C33#40811yescharacteristic, complemented, Sylow
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—32no—
308C23×C3—32no—
309C23×C3—41no—
310C23×C3—62no—
311C23×C3D3×Dic311yescharacteristic
312C23×C3C3×C4×D311yescharacteristic
313C23×C3C2×C3×Dic311yescharacteristic
314C22×D3C2×C3×Dic312yes—
315C3×Q8—64no—
316C3×Q8—62no—
317C3×Q8—62no—
318C3×D4—64no—
319C3×D4—68no—
320C3×D4—64no—
321C3×D4—64no—
322C3×D4—62no—
323C3×D4—62no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—32no—
337C2×C3×C4—32no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—41no—
342C2×C3×C4—62no—
343C2×C3×C4—64no—
344C2×C3×C4—64no—
345C2×C3×C4—62no—
346C2×C3×C4—62no—
347C2×C3×C4—62no—
348C2×C3×C4—62no—
349C2×C3×C4D3×Dic311yescharacteristic
350C2×C3×C4#62512yes—
351C2×C3×C4#62612yes—
352C2×C3×C4C3×C4×D311yescharacteristic
353C2×C3×C4C3×D1212yes—
354C2×C3×C4C2×C3×Dic311yescharacteristic
355C2×C3×C4C3×#6712yes—
356C2×C3×C4C3×#6712yes—
357C2×C3×C4C3×#6712yes—
358C2×C3×C4C2×D3211yescharacteristic
359C2×C3×C4C22×C3×D311yescharacteristic
360C2×C3×C4C22×C3×D311yescharacteristic
361#67—24no—
362C2×Dic3C2×C3×Dic312yes—
363D12—22no—
364C4×D3—22no—
365C4×D3—22no—
366Dic6—22no—
367C3×C8—64no—
368C3×C8—62no—
369C3×C8—62no—
370C3×C8—182no—
371C3⋊C8—22no—
372C3⋊C8—64no—
373C3⋊C8—62no—
374C3⋊C8—62no—
375C2×C32—22no—
376C2×C32—24no—
377C2×C32—22no—
378C2×C32—22no—
379C2×C32—22no—
380C2×C32—21no—
381C2×C32—22no—
382C2×C32—22no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—41no—
386C2×C32—42no—
387C2×C32—42no—
388C2×C32—42no—
389C2×C32—44no—
390C2×C32—122no—
391C2×C32#90112yes—
392C2×C32C2×(C2×D3)⋊C411yescharacteristic
393C2×C32#94812yes—
394C2×C32C2×#21711yescharacteristic
395C2×C32C2×C3×C22⋊C411yescharacteristic
396C2×C32C3×#13112yes—
397C3×D3—44no—
398C3×D3—42no—
399C3×D3—42no—
400#41—62no—
401#41—62no—
402C2×Q8—31no—
403C2×D4—32no—
404C2×D4—31no—
405C22×C4—32no—
406C22×C4C3×D3211yescharacteristic
407M16—182no—
408C2×C8—32no—
409C2×C8—62no—
410C2×C8—92no—
411C22×C3—24no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—44no—
424C22×C3—41no—
425C22×C3—42no—
426C22×C3—64no—
427C22×C3—64no—
428C22×C3—62no—
429C22×C3—64no—
430C22×C3—62no—
431C22×C3—124no—
432C22×C3(C2×D3)⋊Dic312yes—
433C22×C3C3×(C2×D3)⋊C412yes—
434C22×C3C3×#21712yes—
435C22×C3C2×D3×Dic311yescharacteristic
436C22×C3C2×C3×C4×D311yescharacteristic
437C22×C3C22×C3×Dic311yescharacteristic
438C2×D3—24no—
439C2×D3—22no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—22no—
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—41no—
450C3×C4—41no—
451C3×C4—42no—
452C3×C4—64no—
453C3×C4—62no—
454C3×C4—62no—
455C3×C4(C2×D3)⋊Dic312yes—
456C3×C4C3×(C2×D3)⋊C412yes—
457C3×C4C3×#21712yes—
458C3×C4C2×#62511yescharacteristic
459C3×C4C2×#62611yescharacteristic
460C3×C4C2×C3×D1211yescharacteristic
461C3×C4C2×C3×#6711yescharacteristic
462C3×C4C2×C3×#6711yescharacteristic
463C3×C4C2×C3×#6711yescharacteristic
464Dic3—22no—
465C32—21no—
466C32—21no—
467C32—21no—
468C32—41no—
469C32#511011yescharacteristic, complemented
470C32#523411yescharacteristic, complemented
471C32C3×#40811yescharacteristic, complemented
472C23—32no—
473C23C3×D3×Dic311yescharacteristic
474Q8—62no—
475D4—64no—
476D4—62no—
477C2×C4—32no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4C3×D3×Dic311yescharacteristic
481C2×C4C3×#62512yes—
482C2×C4C3×#62612yes—
483C2×C4C2×C3×D3211yescharacteristic
484C8—62no—
485C8—182no—
486C2×C3—24no—
487C2×C3—22no—
488C2×C3—22no—
489C2×C3—22no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—22no—
494C2×C3—22no—
495C2×C3—22no—
496C2×C3—42no—
497C2×C3—42no—
498C2×C3—44no—
499C2×C3—41no—
500C2×C3—42no—
501C2×C3—124no—
502C2×C3—122no—
503C2×C3—122no—
504C2×C3#6381412yes—
505C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
506C2×C3C3×#90112yes—
507C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
508C2×C3C3×#94812yes—
509C2×C3C2×C3×#21711yescharacteristic
510D3—42no—
511C22—22no—
512C22—62no—
513C22—64no—
514C22C3×(C2×D3)⋊Dic312yes—
515C22C2×C3×D3×Dic311yescharacteristic
516C4—62no—
517C4C3×(C2×D3)⋊Dic312yes—
518C4C2×C3×#62511yescharacteristic
519C4C2×C3×#62611yescharacteristic
520C3—21no—
521C3—21no—
522C3—21no—
523C3—41no—
524C3#1059107011yescharacteristic, direct product factor
525C3C3×#511011yescharacteristic, complemented
526C3C3×#523411yescharacteristic, complemented
527C2—22no—
528C2—122no—
529C2C3×#6381412yes—
530C2C2×C3×(C2×D3)⋊Dic311yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C22×C4
prime radicalC2×C32×#235C2
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#408
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3748C2semidirect
3. C22×C32×C3⋊C8C2semidirect
4. C3×(C2×D3)⋊(C3⋊C8)C2semidirect
7. #10591070C3direct (N × H)
24. C2×#3748C2×C3semidirect
29. C22×C3×C3⋊C8D3semidirect
30. C22×C3×C3⋊C8C2×C3semidirect
33. (C2×D3)⋊(C3⋊C8)C2×C3semidirect
118. C22×C3⋊C8C3×D3semidirect
303. C33#408semidirect
469. C32#5110semidirect
470. C32#5234semidirect
471. C32C3×#408semidirect
524. C3#10591070direct (N × H)
525. C3C3×#5110semidirect
526. C3C3×#5234semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2352
C2×C3×#37482
C22×C32×C3⋊C82
C3×(C2×D3)⋊(C3⋊C8)2
C3×#52343
C3×#51103
#105910703