Finite Group Databasebeta

← C3×#71682 #11788135

Subgroups — C3×#71682 #11788135

Order 1152.

1037 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
1C2×C32×#121C211yesmaximal, characteristic
2C22×C3×#208C211yesmaximal, characteristic, complemented
3C2×C3×#832C212yesmaximal, complemented
4C3×#5212C211yesmaximal, characteristic, complemented
5C3×#4529C212yesmaximal, complemented
6C3×#1895—31nomaximal
7#71682C311yesmaximal, characteristic, direct product factor
8C22×C32×D4—21no—
9C22×C32×D4C2211yescharacteristic
10C23×C32×C4C2211yescharacteristic
11C32×#121—22no—
12C32×#121C414yescomplemented
13C2×C32×C4⋊C4C2211yescharacteristic
14C2×C32×C22⋊C4—21no—
15C2×C3×#217—21no—
16C2×C3×C4⋊Dic3C2212yes—
17C2×C3×#208—22no—
18C2×C3×#208—24no—
19C22×C3×C3⋊C8C2212yes—
20C3×#832—24no—
21C3×C3⋊#95—21no—
22C22×C3×M16—31no—
23C2×C3×#121—21no—
24C2×C3×#121D311yescharacteristic
25C2×C3×#121C2×C311yescharacteristic
26C22×#208C2×C311yescharacteristic, complemented
27C2×C3×#102—32no—
28C3×#380—31no—
29C2×#832C2×C312yescomplemented
30#5212C2×C311yescharacteristic, complemented
31C3×#340—32no—
32#4529C2×C312yescomplemented
33C24×C32—21no—
34C24×C32D411yescharacteristic
35C2×C32×D4—24no—
36C2×C32×D4—22no—
37C2×C32×D4—21no—
38C2×C32×D4—22no—
39C2×C32×D4—44no—
40C2×C32×D4C2×C412yes—
41C22×C32×C4—22no—
42C22×C32×C4—22no—
43C22×C32×C4—21no—
44C22×C32×C4C2×C412yes—
45C22×C32×C4D411yescharacteristic
46C22×C32×C4D412yes—
47C22×C32×C4C2311yescharacteristic
48C22×C3×Dic3—22no—
49C32×C4⋊C4—21no—
50C32×C4⋊C4C2×C412yes—
51C32×C22⋊C4—24no—
52C3×#217—42no—
53C3×C4⋊Dic3—22no—
54C3×C4⋊Dic3—22no—
55C3×#208—28no—
56C2×C3×C3⋊C8—22no—
57C2×C3×C3⋊C8—22no—
58C2×C3×C3⋊C8D414yescomplemented
59#1895—31noSylow
60C22×C3×D4—22no—
61C22×C3×D4—21no—
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4C2×D311yescharacteristic
65C22×C3×D4C22×C311yescharacteristic
66C23×C3×C4—21no—
67C23×C3×C4C2×D311yescharacteristic
68C23×C3×C4C22×C311yescharacteristic
69C2×C3×M16—64no—
70C2×C3×M16—62no—
71C22×C3×C8—32no—
72C3×#121—24no—
73C3×#121—22no—
74C3×#121—22no—
75C3×#121—24no—
76C3×#121Dic314yescomplemented
77C3×#121C3×C414yescomplemented
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—32no—
80C2×C3×C4⋊C4C2×D311yescharacteristic
81C2×C3×C4⋊C4C22×C311yescharacteristic
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—61no—
86C2×#217—21no—
87C2×C4⋊Dic3C22×C312yes—
88C2×#208—22no—
89C2×#208—24no—
90C22×C3⋊C8C22×C312yes—
91C3×#102—64no—
92C3×#95—61no—
93#832—24no—
94C3⋊#95—21no—
95C23×C32—22no—
96C23×C32—22no—
97C23×C32—22no—
98C23×C32—21no—
99C23×C32—21no—
100C23×C32—42no—
101C23×C32—41no—
102C23×C32—42no—
103C23×C32—41no—
104C23×C32C22⋊C412yes—
105C23×C32C2×D411yescharacteristic
106C32×D4—24no—
107C32×D4—48no—
108C32×D4—41no—
109C32×D4—41no—
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4—24no—
113C2×C32×C4—22no—
114C2×C32×C4—22no—
115C2×C32×C4—42no—
116C2×C32×C4C22⋊C412yes—
117C2×C32×C4C22×C411yescharacteristic
118C2×C32×C4C2×D412yes—
119C2×C32×C4C2×D411yescharacteristic
120C2×C32×C4#4112yes—
121C2×C3×Dic3—42no—
122C2×C3×Dic3—42no—
123C2×C3×Dic3—42no—
124C3×C3⋊C8—24no—
125C22×M16—31no—
126C2×#121C3×D311yescharacteristic
127C2×#102—32no—
128#380—31no—
129#340—32no—
130C24×C3—22no—
131C24×C3—21no—
132C24×C3—21no—
133C24×C3—21no—
134C24×C3#6711yescharacteristic
135C24×C3C3×D411yescharacteristic
136C2×C3×D4—28no—
137C2×C3×D4—22no—
138C2×C3×D4—24no—
139C2×C3×D4—24no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—21no—
143C2×C3×D4—22no—
144C2×C3×D4—21no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—42no—
148C2×C3×D4—42no—
149C2×C3×D4—48no—
150C2×C3×D4—44no—
151C2×C3×D4—44no—
152C2×C3×D4C2×Dic312yes—
153C2×C3×D4C2×C3×C412yes—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—42no—
166C22×C3×C4—42no—
167C22×C3×C4—62no—
168C22×C3×C4C2×Dic312yes—
169C22×C3×C4#6711yescharacteristic
170C22×C3×C4#6712yes—
171C22×C3×C4C2×C3×C412yes—
172C22×C3×C4C3×D411yescharacteristic
173C22×C3×C4C3×D412yes—
174C22×C3×C4C22×D311yescharacteristic
175C22×C3×C4C23×C311yescharacteristic
176C22×Dic3—22no—
177C3×M16—68no—
178C2×C3×C8—34no—
179C2×C3×C8—62no—
180C2×C3×C8—62no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—62no—
186C3×C4⋊C4—62no—
187C3×C4⋊C4C2×Dic312yes—
188C3×C4⋊C4C2×C3×C412yes—
189C3×C22⋊C4—28no—
190C3×C22⋊C4—24no—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—122no—
193#217—42no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196#208—28no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—22no—
199C2×C3⋊C8C3×D414yescomplemented
200C22×C32—22no—
201C22×C32—22no—
202C22×C32—22no—
203C22×C32—42no—
204C22×C32—42no—
205C22×C32—42no—
206C22×C32—42no—
207C22×C32—42no—
208C22×C32—82no—
209C22×C32—82no—
210C22×C32C2×C22⋊C411yescharacteristic
211C22×C32#12112yes—
212C22×C32AGL(1,ℤ/8)12yes—
213C22×C32AGL(1,ℤ/8)11yescharacteristic
214C22×C32#13711yescharacteristic
215C32×C4—22no—
216C32×C4—42no—
217C32×C4C4×D412yes—
218C32×C4#12011yescharacteristic
219C32×C4#12311yescharacteristic
220C3×Dic3—82no—
221C22×D4—21no—
222C22×D4C2×C3×D311yescharacteristic
223C23×C4C2×C3×D311yescharacteristic
224C2×M16—64no—
225C2×M16—62no—
226C22×C8—32no—
227#121—22no—
228#121C3×Dic314yescomplemented
229C2×C4⋊C4—32no—
230C2×C4⋊C4C2×C3×D311yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—61no—
233#102—64no—
234#95—61no—
235C23×C3—24no—
236C23×C3—22no—
237C23×C3—22no—
238C23×C3—21no—
239C23×C3—21no—
240C23×C3—22no—
241C23×C3—21no—
242C23×C3—22no—
243C23×C3—21no—
244C23×C3—24no—
245C23×C3—22no—
246C23×C3—22no—
247C23×C3—22no—
248C23×C3—21no—
249C23×C3—42no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—44no—
253C23×C3—44no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3#21712yes—
265C23×C3C3×C22⋊C412yes—
266C23×C3C2×#6711yescharacteristic
267C23×C3C2×C3×D411yescharacteristic
268C3×D4—24no—
269C3×D4—24no—
270C3×D4—44no—
271C3×D4—42no—
272C3×D4—42no—
273C3×D4—416no—
274C3×D4—48no—
275C3×D4—48no—
276C3×D4—41no—
277C3×D4—41no—
278C3×D4—41no—
279C3×D4—41no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—24no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—24no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—24no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—44no—
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#21712yes—
307C2×C3×C4C3×C22⋊C412yes—
308C2×C3×C4D3×D412yes—
309C2×C3×C4#23712yes—
310C2×C3×C4C22×Dic311yescharacteristic
311C2×C3×C4C2×#6711yescharacteristic
312C2×C3×C4C22×C3×C411yescharacteristic
313C2×C3×C4C2×C3×D412yes—
314C2×C3×C4C2×C3×D411yescharacteristic
315C2×C3×C4C3×#4112yes—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C3×C8—64no—
320C3⋊C8—24no—
321C2×C32—42no—
322C2×C32—82no—
323C2×C32#38611yescharacteristic
324C2×C32#41711yescharacteristic
325C2×C32#41811yescharacteristic
326C2×C32#46812yes—
327C2×C32#46912yes—
328C24—21no—
329C24C3×#6711yescharacteristic
330C2×D4—24no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—21no—
334C2×D4—44no—
335C2×D4C2×C3×Dic312yes—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—62no—
340C22×C4C2×C3×Dic312yes—
341C22×C4C3×#6711yescharacteristic
342C22×C4C3×#6712yes—
343C22×C4C22×C3×D311yescharacteristic
344M16—68no—
345C2×C8—34no—
346C2×C8—62no—
347C2×C8—62no—
348C4⋊C4—21no—
349C4⋊C4—62no—
350C4⋊C4—62no—
351C4⋊C4C2×C3×Dic312yes—
352C22⋊C4—24no—
353C22⋊C4—122no—
354C22×C3—24no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—42no—
367C22×C3—42no—
368C22×C3—44no—
369C22×C3—44no—
370C22×C3—44no—
371C22×C3—44no—
372C22×C3—44no—
373C22×C3—42no—
374C22×C3—42no—
375C22×C3—42no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—42no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—42no—
383C22×C3—84no—
384C22×C3—84no—
385C22×C3—82no—
386C22×C3—82no—
387C22×C3—82no—
388C22×C3—82no—
389C22×C3#93212yes—
390C22×C3#93912yes—
391C22×C3#94911yescharacteristic
392C22×C3#95111yescharacteristic
393C22×C3C2×#21711yescharacteristic
394C22×C3C2×C3×C22⋊C411yescharacteristic
395C22×C3C3×#12112yes—
396C22×C3C3×AGL(1,ℤ/8)12yes—
397C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
398C22×C3C3×#13711yescharacteristic
399C3×C4—22no—
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—42no—
405C3×C4—44no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4—242no—
409C3×C4D4×Dic312yes—
410C3×C4#93511yescharacteristic
411C3×C4#93711yescharacteristic
412C3×C4C3×C4×D412yes—
413C3×C4C3×#12011yescharacteristic
414C3×C4C3×#12311yescharacteristic
415Dic3—82no—
416C32#189511yescharacteristic, complemented, Sylow
417C23—22no—
418C23—22no—
419C23—21no—
420C23—21no—
421C23—22no—
422C23—42no—
423C23—42no—
424C23—41no—
425C23—41no—
426C23C3×#21712yes—
427C23C2×C3×#6711yescharacteristic
428D4—24no—
429D4—48no—
430D4—41no—
431D4—41no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—42no—
438C2×C4—122no—
439C2×C4—122no—
440C2×C4—122no—
441C2×C4C3×#21712yes—
442C2×C4C3×D3×D412yes—
443C2×C4C3×#23712yes—
444C2×C4C22×C3×Dic311yescharacteristic
445C2×C4C2×C3×#6711yescharacteristic
446C8—64no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—22no—
451C2×C3—22no—
452C2×C3—44no—
453C2×C3—42no—
454C2×C3—42no—
455C2×C3—84no—
456C2×C3—82no—
457C2×C3—82no—
458C2×C3#503912yes—
459C2×C3#504212yes—
460C2×C3#521511yescharacteristic
461C2×C3#522011yescharacteristic
462C2×C3#523211yescharacteristic
463C2×C3C3×#38611yescharacteristic
464C2×C3C3×#41711yescharacteristic
465C2×C3C3×#41811yescharacteristic
466C2×C3C3×#46812yes—
467C2×C3C3×#46912yes—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—82no—
477C22—82no—
478C22C3×#93212yes—
479C22C3×#93912yes—
480C22C3×#94911yescharacteristic
481C22C3×#95111yescharacteristic
482C22C2×C3×#21711yescharacteristic
483C4—22no—
484C4—42no—
485C4—242no—
486C4C3×D4×Dic312yes—
487C4C3×#93511yescharacteristic
488C4C3×#93711yescharacteristic
489C3—21no—
490C3#7168211yescharacteristic, direct product factor
491C3C3×#189511yescharacteristic, complemented
492C2—42no—
493C2—82no—
494C2C3×#503912yes—
495C2C3×#504212yes—
496C2C3×#521511yescharacteristic
497C2C3×#522011yescharacteristic
498C2C3×#523211yescharacteristic
subgroupG/H
centerC23×C3C2×#67
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
Fitting, prime radicalC2×C32×#121C2
socleC23×C32C2×D4
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×#121D3
hypocenterC3C3×#1895
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C2×C3×#121

Complements

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

NH ≅ G/Ntype
2. C22×C3×#208C2semidirect
3. C2×C3×#832C2semidirect
4. C3×#5212C2semidirect
5. C3×#4529C2semidirect
7. #71682C3direct (N × H)
12. C32×#121C4semidirect
26. C22×#208C2×C3semidirect
29. C2×#832C2×C3semidirect
30. #5212C2×C3semidirect
32. #4529C2×C3semidirect
58. C2×C3×C3⋊C8D4semidirect
76. C3×#121Dic3semidirect
77. C3×#121C3×C4semidirect
199. C2×C3⋊C8C3×D4semidirect
228. #121C3×Dic3semidirect
416. C32#1895semidirect
490. C3#71682direct (N × H)
491. C3C3×#1895semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1212
C22×C3×#2082
C2×C3×#8322
C3×#52122
C3×#45292
C3×#18953
#716823