Finite Group Databasebeta

← C3×#71674 #11775292

Subgroups — C3×#71674 #11775292

Order 1152.

865 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#121C211yesmaximal, characteristic
2C3×#5211C211yesmaximal, characteristic, complemented
3C3×#5207C211yesmaximal, characteristic, complemented
4C3×#4537C214yesmaximal, complemented
5C3×#1510—31nomaximal
6#71674C311yesmaximal, characteristic, direct product factor
7C22×C32×D4—21no—
8C22×C32×D4C411yescharacteristic, complemented
9C23×C32×C4C2211yescharacteristic
10C32×#121—21no—
11C32×#121—21no—
12C32×#121C412yescomplemented
13C32×#121C2212yes—
14C2×C32×C4⋊C4C411yescharacteristic, complemented
15C2×C32×C22⋊C4—21no—
16C2×C3×#217—21no—
17C2×C3×#208—21no—
18C3×C3⋊#95—21no—
19C3×C3⋊#95C2212yescomplemented
20C3×#830—21no—
21C3×#830C2212yescomplemented
22C2×C3×#121—21no—
23C2×C3×#121D311yescharacteristic
24C2×C3×#121C2×C311yescharacteristic
25C3×#407—31no—
26C3×#381—31no—
27#5211C2×C311yescharacteristic, complemented
28#5207C2×C311yescharacteristic, complemented
29C3×#327—34no—
30#4537C2×C314yescomplemented
31C24×C32—21no—
32C24×C32D411yescharacteristic
33C2×C32×D4—22no—
34C2×C32×D4—22no—
35C2×C32×D4—21no—
36C2×C32×D4—42no—
37C2×C32×D4—44no—
38C2×C32×D4C2×C412yes—
39C22×C32×C4—21no—
40C22×C32×C4—42no—
41C22×C32×C4C2×C411yescharacteristic
42C22×C32×C4C2×C411yescharacteristic
43C22×C32×C4D411yescharacteristic
44C22×C32×C4D411yescharacteristic
45C22×C32×C4D411yescharacteristic
46C22×C32×C4C2311yescharacteristic
47C22×C3×Dic3—22no—
48C32×C4⋊C4—21no—
49C32×C4⋊C4C2×C412yes—
50C32×C22⋊C4—22no—
51C32×C22⋊C4—22no—
52C3×#217—42no—
53C3×#208—42no—
54C2×C3×C3⋊C8—22no—
55#1510—31noSylow
56C22×C3×D4—22no—
57C22×C3×D4—21no—
58C22×C3×D4—21no—
59C22×C3×D4—21no—
60C22×C3×D4Dic311yescharacteristic, complemented
61C22×C3×D4C3×C411yescharacteristic, complemented
62C23×C3×C4—21no—
63C23×C3×C4C2×D311yescharacteristic
64C23×C3×C4C22×C311yescharacteristic
65C2×C3×M16—61no—
66C3×#121—22no—
67C3×#121—22no—
68C3×#121—21no—
69C3×#121—21no—
70C3×#121—21no—
71C3×#121—21no—
72C3×#121—22no—
73C3×#121—22no—
74C3×#121Dic312yescomplemented
75C3×#121C3×C412yescomplemented
76C3×#121C2×D312yes—
77C3×#121C22×C312yes—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4Dic311yescharacteristic, complemented
80C2×C3×C4⋊C4C3×C411yescharacteristic, complemented
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—61no—
85C2×#217—21no—
86C2×#208—21no—
87C3×C22⋊C8—32no—
88C3×C22⋊C8—61no—
89C3×#95—32no—
90C3×#95—61no—
91C3⋊#95—21no—
92C3⋊#95C22×C312yescomplemented
93#830—21no—
94#830C22×C312yescomplemented
95C23×C32—22no—
96C23×C32—22no—
97C23×C32—22no—
98C23×C32—41no—
99C23×C32—42no—
100C23×C32—42no—
101C23×C32—42no—
102C23×C32C22⋊C411yescharacteristic
103C23×C32C22⋊C411yescharacteristic
104C23×C32C2×D411yescharacteristic
105C32×D4—42no—
106C32×D4—42no—
107C32×D4—44no—
108C32×D4—44no—
109C2×C32×C4—22no—
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4—41no—
113C2×C32×C4—41no—
114C2×C32×C4—41no—
115C2×C32×C4—41no—
116C2×C32×C4—42no—
117C2×C32×C4C22⋊C411yescharacteristic
118C2×C32×C4C22⋊C411yescharacteristic
119C2×C32×C4C22×C411yescharacteristic
120C2×C32×C4C2×D411yescharacteristic
121C2×C3×Dic3—42no—
122C2×C3×Dic3—42no—
123C2×C3×Dic3—42no—
124C3×C3⋊C8—42no—
125C2×#121C3×D311yescharacteristic
126#407—31no—
127#381—31no—
128#327—34no—
129C24×C3—22no—
130C24×C3—21no—
131C24×C3—21no—
132C24×C3—21no—
133C24×C3#6711yescharacteristic
134C24×C3C3×D411yescharacteristic
135C2×C3×D4—24no—
136C2×C3×D4—24no—
137C2×C3×D4—22no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—22no—
145C2×C3×D4—42no—
146C2×C3×D4—42no—
147C2×C3×D4—44no—
148C2×C3×D4—48no—
149C2×C3×D4—44no—
150C2×C3×D4—44no—
151C2×C3×D4C2×Dic312yes—
152C2×C3×D4C2×C3×C412yes—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—42no—
163C22×C3×C4—42no—
164C22×C3×C4—44no—
165C22×C3×C4—62no—
166C22×C3×C4C2×Dic311yescharacteristic
167C22×C3×C4C2×Dic311yescharacteristic
168C22×C3×C4#6711yescharacteristic
169C22×C3×C4#6711yescharacteristic
170C22×C3×C4#6711yescharacteristic
171C22×C3×C4C2×C3×C411yescharacteristic
172C22×C3×C4C2×C3×C411yescharacteristic
173C22×C3×C4C3×D411yescharacteristic
174C22×C3×C4C3×D411yescharacteristic
175C22×C3×C4C3×D411yescharacteristic
176C22×C3×C4C22×D311yescharacteristic
177C22×C3×C4C23×C311yescharacteristic
178C22×Dic3—22no—
179C3×M16—122no—
180C2×C3×C8—62no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4C2×Dic312yes—
186C3×C4⋊C4C2×C3×C412yes—
187C3×C22⋊C4—24no—
188C3×C22⋊C4—24no—
189C3×C22⋊C4—22no—
190C3×C22⋊C4—22no—
191C3×C22⋊C4—22no—
192C3×C22⋊C4—22no—
193C3×C22⋊C4—122no—
194#217—42no—
195#208—42no—
196C2×C3⋊C8—22no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—21no—
201C22×C32—21no—
202C22×C32—21no—
203C22×C32—21no—
204C22×C32—44no—
205C22×C32—42no—
206C22×C32—42no—
207C22×C32—42no—
208C22×C32—84no—
209C22×C32#9912yes—
210C22×C32C2×C22⋊C411yescharacteristic
211C22×C32AGL(1,ℤ/8)12yes—
212C32×C4—21no—
213C32×C4—21no—
214C32×C4—21no—
215C32×C4—21no—
216C32×C4—42no—
217C3×Dic3—82no—
218C22×D4—21no—
219C22×D4C3×Dic311yescharacteristic, complemented
220C23×C4C2×C3×D311yescharacteristic
221C2×M16—61no—
222#121—21no—
223#121—21no—
224#121C3×Dic312yescomplemented
225#121C2×C3×D312yes—
226C2×C4⋊C4C3×Dic311yescharacteristic, complemented
227C2×C22⋊C4—21no—
228C2×C22⋊C4—61no—
229C22⋊C8—32no—
230C22⋊C8—61no—
231#95—32no—
232#95—61no—
233C23×C3—24no—
234C23×C3—22no—
235C23×C3—22no—
236C23×C3—22no—
237C23×C3—22no—
238C23×C3—24no—
239C23×C3—22no—
240C23×C3—22no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—42no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3—42no—
248C23×C3—44no—
249C23×C3—42no—
250C23×C3—42no—
251C23×C3—44no—
252C23×C3—44no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3#21711yescharacteristic
258C23×C3#21711yescharacteristic
259C23×C3C3×C22⋊C411yescharacteristic
260C23×C3C3×C22⋊C411yescharacteristic
261C23×C3C2×#6711yescharacteristic
262C23×C3C2×C3×D411yescharacteristic
263C3×D4—42no—
264C3×D4—42no—
265C3×D4—44no—
266C3×D4—44no—
267C3×D4—42no—
268C3×D4—42no—
269C3×D4—48no—
270C3×D4—44no—
271C3×D4—44no—
272C3×D4—48no—
273C3×D4—44no—
274C3×D4—44no—
275C2×C3×C4—24no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—24no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—42no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—44no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—122no—
304C2×C3×C4—122no—
305C2×C3×C4—122no—
306C2×C3×C4#21711yescharacteristic
307C2×C3×C4#21711yescharacteristic
308C2×C3×C4C3×C22⋊C411yescharacteristic
309C2×C3×C4C3×C22⋊C411yescharacteristic
310C2×C3×C4C22×Dic311yescharacteristic
311C2×C3×C4C2×#6711yescharacteristic
312C2×C3×C4C22×C3×C411yescharacteristic
313C2×C3×C4C2×C3×D411yescharacteristic
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C3×C8—122no—
318C3⋊C8—42no—
319C2×C32—22no—
320C2×C32—22no—
321C2×C32—41no—
322C2×C32—82no—
323C2×C32C2×#9911yescharacteristic
324C2×C32#41811yescharacteristic
325C2×C32#42111yescharacteristic
326C24—21no—
327C24C3×#6711yescharacteristic
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—21no—
331C2×D4—42no—
332C2×D4—44no—
333C2×D4C2×C3×Dic312yes—
334C22×C4—21no—
335C22×C4—42no—
336C22×C4—62no—
337C22×C4C2×C3×Dic311yescharacteristic
338C22×C4C2×C3×Dic311yescharacteristic
339C22×C4C3×#6711yescharacteristic
340C22×C4C3×#6711yescharacteristic
341C22×C4C3×#6711yescharacteristic
342C22×C4C22×C3×D311yescharacteristic
343M16—122no—
344C2×C8—62no—
345C4⋊C4—21no—
346C4⋊C4C2×C3×Dic312yes—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—122no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—22no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—22no—
368C22×C3—21no—
369C22×C3—22no—
370C22×C3—41no—
371C22×C3—41no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—44no—
375C22×C3—44no—
376C22×C3—48no—
377C22×C3—44no—
378C22×C3—44no—
379C22×C3—44no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—88no—
387C22×C3—84no—
388C22×C3—84no—
389C22×C3#83412yes—
390C22×C3C3×#9912yes—
391C22×C3#93212yes—
392C22×C3C2×#21711yescharacteristic
393C22×C3C2×C3×C22⋊C411yescharacteristic
394C22×C3C3×AGL(1,ℤ/8)12yes—
395C3×C4—22no—
396C3×C4—22no—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—41no—
406C3×C4—41no—
407C3×C4—44no—
408C3×C4—42no—
409C3×C4—42no—
410C3×C4—242no—
411Dic3—82no—
412C32#151011yescharacteristic, complemented, Sylow
413C23—22no—
414C23—22no—
415C23—22no—
416C23—41no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23C3×#21711yescharacteristic
421C23C3×#21711yescharacteristic
422C23C2×C3×#6711yescharacteristic
423D4—42no—
424D4—42no—
425D4—44no—
426D4—44no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—122no—
436C2×C4—122no—
437C2×C4—122no—
438C2×C4C3×#21711yescharacteristic
439C2×C4C3×#21711yescharacteristic
440C2×C4C22×C3×Dic311yescharacteristic
441C2×C4C2×C3×#6711yescharacteristic
442C8—122no—
443C2×C3—24no—
444C2×C3—22no—
445C2×C3—22no—
446C2×C3—22no—
447C2×C3—22no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—42no—
452C2×C3—42no—
453C2×C3—41no—
454C2×C3—41no—
455C2×C3—84no—
456C2×C3—82no—
457C2×C3—82no—
458C2×C3#521511yescharacteristic
459C2×C3C2×#83411yescharacteristic
460C2×C3#523611yescharacteristic
461C2×C3C2×C3×#9911yescharacteristic
462C2×C3C3×#41811yescharacteristic
463C2×C3C3×#42111yescharacteristic
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—44no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—84no—
476C22C3×#83412yes—
477C22C3×#93212yes—
478C22C2×C3×#21711yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—42no—
484C4—242no—
485C3—21no—
486C3#7167411yescharacteristic, direct product factor
487C3C3×#151011yescharacteristic, complemented
488C2—22no—
489C2—22no—
490C2—41no—
491C2—82no—
492C2C3×#521511yescharacteristic
493C2C2×C3×#83411yescharacteristic
494C2C3×#523611yescharacteristic
subgroupG/H
centerC22×C3C2×#217
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
Fitting, prime radicalC2×C32×#121C2
socleC22×C32C2×C22⋊C4
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×#121D3
hypocenterC3C3×#1510
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C2×C3×#121

Complements

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

NH ≅ G/Ntype
2. C3×#5211C2semidirect
3. C3×#5207C2semidirect
4. C3×#4537C2semidirect
6. #71674C3direct (N × H)
8. C22×C32×D4C4semidirect
12. C32×#121C4semidirect
14. C2×C32×C4⋊C4C4semidirect
19. C3×C3⋊#95C22semidirect
21. C3×#830C22semidirect
27. #5211C2×C3semidirect
28. #5207C2×C3semidirect
30. #4537C2×C3semidirect
60. C22×C3×D4Dic3semidirect
61. C22×C3×D4C3×C4semidirect
74. C3×#121Dic3semidirect
75. C3×#121C3×C4semidirect
79. C2×C3×C4⋊C4Dic3semidirect
80. C2×C3×C4⋊C4C3×C4semidirect
92. C3⋊#95C22×C3semidirect
94. #830C22×C3semidirect
219. C22×D4C3×Dic3semidirect
224. #121C3×Dic3semidirect
226. C2×C4⋊C4C3×Dic3semidirect
412. C32#1510semidirect
486. C3#71674direct (N × H)
487. C3C3×#1510semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1212
C3×#52112
C3×#52072
C3×#45372
C3×#15103
#716743