Finite Group Databasebeta

← C3×#69679 #11791138

Subgroups — C3×#69679 #11791138

Order 1152.

773 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#215C211yesmaximal, characteristic
2C32×#391C211yesmaximal, characteristic, complemented
3C2×C3×#835C211yesmaximal, characteristic, complemented
4C2×C3×#809C211yesmaximal, characteristic
5C2×C3×#807C211yesmaximal, characteristic, complemented
6C3×#4939C211yesmaximal, characteristic
7C2×C3×#803C211yesmaximal, characteristic, complemented
8C3×#1973—31nomaximal
9#69679C311yesmaximal, characteristic, direct product factor
10C22×C32×Q8C2211yescharacteristic
11C22×C3×D12C2211yescharacteristic
12C2×C32×C4⋊C4C2211yescharacteristic
13C2×C32×C42C2211yescharacteristic
14C2×C3×#215—22no—
15C2×C3×#215—22no—
16C2×C3×#215C414yescomplemented
17C2×C3×(C2×D3)⋊C4—22no—
18C2×C3×C4⋊Dic3C2211yescharacteristic
19C22×C3×C3⋊C8C2211yescharacteristic
20C22×C3×C3⋊C8C2211yescharacteristic
21C32×#95—22no—
22C3×#835—22no—
23C3×#809—22no—
24C3×#807—22no—
25C3×#803—22no—
26C22×C3×SD16—31no—
27C22×#215C2×C311yescharacteristic
28C2×C3×#107—31no—
29C2×C3×#103—31no—
30C2×C3×#102—31no—
31C2×C3×#97—31no—
32C3×#391—21no—
33C3×#391D311yescharacteristic, complemented
34C3×#391C2×C311yescharacteristic, complemented
35C3×#390—31no—
36C2×#835C2×C311yescharacteristic, complemented
37C2×#809C2×C311yescharacteristic
38C2×#807C2×C311yescharacteristic, complemented
39#4939C2×C311yescharacteristic
40C2×#803C2×C311yescharacteristic, complemented
41C23×C3×D3—21no—
42C2×C32×Q8—21no—
43C2×C32×Q8—21no—
44C2×C32×Q8—21no—
45C2×C32×Q8—21no—
46C2×C32×Q8—21no—
47C2×C32×Q8C2×C412yes—
48C22×C32×C4—21no—
49C22×C32×C4—21no—
50C22×C32×C4D412yes—
51C22×C32×C4C2311yescharacteristic
52C22×C3×Dic3—21no—
53C2×C3×D12—21no—
54C2×C3×D12—21no—
55C2×C3×D12—21no—
56C2×C3×D12—21no—
57C2×C3×D12—21no—
58C2×C3×D12C2×C412yes—
59C32×C4⋊C4—21no—
60C32×C4⋊C4—21no—
61C32×C42—21no—
62C32×C42—21no—
63C3×#215—28no—
64C3×(C2×D3)⋊C4—44no—
65C3×C4⋊Dic3—21no—
66C3×C4⋊Dic3—21no—
67C2×C3×C3⋊C8—21no—
68C2×C3×C3⋊C8—21no—
69C2×C3×C3⋊C8C2×C412yes—
70C2×C3×C3⋊C8D412yes—
71C2×C3×C3⋊C8D412yes—
72C2×C3×C3⋊C8D412yes—
73#1973—31noSylow
74C22×C3×Q8—21no—
75C22×C3×Q8C2×D311yescharacteristic
76C22×C3×Q8C22×C311yescharacteristic
77C22×C3×D4—31no—
78C22×D12C22×C311yescharacteristic
79C2×C3×SD16—34no—
80C2×C3×SD16—62no—
81C2×C3×SD16—62no—
82C22×C3×C8—31no—
83C22×C3×C8—31no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—31no—
86C2×C3×C4⋊C4C2×D311yescharacteristic
87C2×C3×C4⋊C4C22×C311yescharacteristic
88C2×C3×C22⋊C4—62no—
89C2×C3×C42—21no—
90C2×C3×C42C2×D311yescharacteristic
91C2×C3×C42C22×C311yescharacteristic
92C2×#215—22no—
93C2×#215—22no—
94C2×#215C3×C414yescomplemented
95C2×(C2×D3)⋊C4—22no—
96C2×C4⋊Dic3C22×C311yescharacteristic
97C22×C3⋊C8C22×C311yescharacteristic
98C22×C3⋊C8C22×C311yescharacteristic
99C3×#107—62no—
100C3×#103—62no—
101C3×#102—62no—
102C3×#97—62no—
103C3×#95—24no—
104C3×#95—22no—
105C3×#95—22no—
106#835—22no—
107#809—22no—
108#807—22no—
109#803—22no—
110C23×C32C2×D411yescharacteristic
111C22×C3×D3—22no—
112C22×C3×D3—41no—
113C22×C3×D3—41no—
114C22×C3×D3—42no—
115C22×C3×D3—42no—
116C32×Q8—22no—
117C32×Q8—22no—
118C32×Q8—41no—
119C32×Q8—41no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—22no—
123C2×C32×C4—22no—
124C2×C32×C4—41no—
125C2×C32×C4—41no—
126C2×C32×C4—41no—
127C2×C32×C4—42no—
128C2×C32×C4C22×C411yescharacteristic
129C2×C32×C4C2×D411yescharacteristic
130C2×C32×C4C2×D411yescharacteristic
131C2×C32×C4C2×D411yescharacteristic
132C2×C32×C4#4111yescharacteristic
133C2×C32×C4#4111yescharacteristic
134C2×C3×Dic3—41no—
135C2×C3×Dic3—42no—
136C3×D12—22no—
137C3×D12—22no—
138C3×D12—41no—
139C3×D12—41no—
140C3×C3⋊C8—22no—
141C3×C3⋊C8C22⋊C414yescomplemented
142C22×SD16—31no—
143C2×#107—31no—
144C2×#103—31no—
145C2×#102—31no—
146C2×#97—31no—
147#391C3×D311yescharacteristic, complemented
148#390—31no—
149C24×C3—61no—
150C23×D3—21no—
151C2×C3×Q8—22no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—21no—
162C2×C3×Q8—22no—
163C2×C3×Q8—41no—
164C2×C3×Q8—41no—
165C2×C3×Q8—41no—
166C2×C3×Q8—41no—
167C2×C3×Q8C4×D312yes—
168C2×C3×Q8C2×C3×C412yes—
169C2×C3×D4—32no—
170C2×C3×D4—61no—
171C2×C3×D4—61no—
172C2×C3×D4—61no—
173C2×C3×D4—61no—
174C2×C3×D4—61no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4—21no—
183C22×C3×C4—61no—
184C22×C3×C4#6712yes—
185C22×C3×C4C3×D412yes—
186C22×C3×C4C22×D311yescharacteristic
187C22×C3×C4C23×C311yescharacteristic
188C22×Dic3—21no—
189C2×D12—21no—
190C2×D12—21no—
191C2×D12—21no—
192C2×D12—21no—
193C2×D12—21no—
194C2×D12C2×C3×C412yes—
195C3×SD16—68no—
196C2×C3×C8—32no—
197C2×C3×C8—32no—
198C2×C3×C8—32no—
199C2×C3×C8—32no—
200C2×C3×C8—61no—
201C2×C3×C8—61no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—41no—
207C3×C4⋊C4—41no—
208C3×C4⋊C4—61no—
209C3×C4⋊C4—61no—
210C3×C22⋊C4—124no—
211C3×C42—21no—
212C3×C42—21no—
213C3×C42—21no—
214C3×C42—21no—
215C3×C42—41no—
216C3×C42—41no—
217#215—28no—
218(C2×D3)⋊C4—44no—
219C4⋊Dic3—21no—
220C4⋊Dic3—21no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8C2×C3×C412yes—
224C2×C3⋊C8C3×D412yes—
225C2×C3⋊C8C3×D412yes—
226C2×C3⋊C8C3×D412yes—
227C22×C32C4×D411yescharacteristic
228C22×C32#12111yescharacteristic
229C22×C32#12711yescharacteristic
230C22×C32AGL(1,ℤ/8)12yes—
231C22×C32#13712yes—
232C2×C3×D3—42no—
233C2×C3×D3—44no—
234C2×C3×D3—82no—
235C2×C3×D3—81no—
236C2×C3×D3—81no—
237C32×C4—42no—
238C32×C4—42no—
239C32×C4—81no—
240C32×C4C2×C22⋊C411yescharacteristic
241C32×C4C4×D411yescharacteristic
242C32×C4#12111yescharacteristic
243C32×C4#12411yescharacteristic
244C3×Dic3—81no—
245C22×Q8C2×C3×D311yescharacteristic
246C22×D4—31no—
247C2×SD16—34no—
248C2×SD16—62no—
249C2×SD16—62no—
250C22×C8—31no—
251C22×C8—31no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4C2×C3×D311yescharacteristic
254C2×C22⋊C4—62no—
255C2×C42C2×C3×D311yescharacteristic
256#107—62no—
257#103—62no—
258#102—62no—
259#97—62no—
260#95—22no—
261C23×C3—21no—
262C23×C3—62no—
263C23×C3—121no—
264C23×C3—121no—
265C23×C3—122no—
266C23×C3—122no—
267C23×C3C2×#6711yescharacteristic
268C23×C3C2×C3×D411yescharacteristic
269C22×D3—22no—
270C22×D3—41no—
271C22×D3—41no—
272C22×D3—42no—
273C22×D3—42no—
274C3×Q8—22no—
275C3×Q8—22no—
276C3×Q8—22no—
277C3×Q8—22no—
278C3×Q8—42no—
279C3×Q8—42no—
280C3×Q8—42no—
281C3×Q8—42no—
282C3×Q8—41no—
283C3×Q8—41no—
284C3×Q8—41no—
285C3×Q8—41no—
286C3×D4—62no—
287C3×D4—62no—
288C3×D4—121no—
289C3×D4—121no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—42no—
315C2×C3×C4—44no—
316C2×C3×C4—41no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4—42no—
320C2×C3×C4—121no—
321C2×C3×C4—122no—
322C2×C3×C4C2×C4×D311yescharacteristic
323C2×C3×C4#23511yescharacteristic
324C2×C3×C4D3×D411yescharacteristic
325C2×C3×C4D3×D411yescharacteristic
326C2×C3×C4D3×D411yescharacteristic
327C2×C3×C4#23711yescharacteristic
328C2×C3×C4C22×C3×C411yescharacteristic
329C2×C3×C4C2×C3×D411yescharacteristic
330C2×C3×C4C2×C3×D411yescharacteristic
331C2×C3×C4C2×C3×D411yescharacteristic
332C2×C3×C4C3×#4111yescharacteristic
333C2×C3×C4C3×#4111yescharacteristic
334C2×Dic3—41no—
335C2×Dic3—42no—
336D12—22no—
337D12—22no—
338D12—41no—
339D12—41no—
340C3×C8—34no—
341C3×C8—62no—
342C3⋊C8—22no—
343C3⋊C8C3×C22⋊C414yescomplemented
344C2×C32#39011yescharacteristic
345C2×C32#44012yes—
346C2×C32#46812yes—
347C2×C32#49611yescharacteristic
348C2×C32#49711yescharacteristic
349C3×D3—82no—
350C24—61no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8C3×C4×D312yes—
357C2×D4—32no—
358C2×D4—61no—
359C2×D4—61no—
360C2×D4—61no—
361C2×D4—61no—
362C2×D4—61no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—61no—
366C22×C4C3×#6712yes—
367C22×C4C22×C3×D311yescharacteristic
368SD16—68no—
369C2×C8—32no—
370C2×C8—32no—
371C2×C8—32no—
372C2×C8—32no—
373C2×C8—61no—
374C2×C8—61no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—61no—
378C4⋊C4—61no—
379C22⋊C4—124no—
380C42—21no—
381C42—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—122no—
388C22×C3—124no—
389C22×C3—242no—
390C22×C3—241no—
391C22×C3—241no—
392C22×C3C4×#6711yescharacteristic
393C22×C3#93911yescharacteristic
394C22×C3#94011yescharacteristic
395C22×C3#94212yes—
396C22×C3#94912yes—
397C22×C3C3×C4×D411yescharacteristic
398C22×C3C3×#12111yescharacteristic
399C22×C3C3×#12711yescharacteristic
400C22×C3C3×AGL(1,ℤ/8)12yes—
401C22×C3C3×#13712yes—
402C2×D3—42no—
403C2×D3—44no—
404C2×D3—82no—
405C2×D3—81no—
406C2×D3—81no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—44no—
412C3×C4—44no—
413C3×C4—42no—
414C3×C4—42no—
415C3×C4—42no—
416C3×C4—42no—
417C3×C4—82no—
418C3×C4—81no—
419C3×C4—81no—
420C3×C4—241no—
421C3×C4D3×C22⋊C411yescharacteristic
422C3×C4#67⋊C411yescharacteristic
423C3×C4#88411yescharacteristic
424C3×C4#88511yescharacteristic
425C3×C4C2×C3×C22⋊C411yescharacteristic
426C3×C4C3×C4×D411yescharacteristic
427C3×C4C3×#12111yescharacteristic
428C3×C4C3×#12411yescharacteristic
429Dic3—81no—
430C32#197311yescharacteristic, complemented, Sylow
431C23—62no—
432C23—121no—
433C23—121no—
434C23—122no—
435C23—122no—
436C23C2×C3×#6711yescharacteristic
437Q8—22no—
438Q8—22no—
439Q8—41no—
440Q8—41no—
441D4—62no—
442D4—62no—
443D4—121no—
444D4—121no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—121no—
454C2×C4—122no—
455C2×C4C2×C3×C4×D311yescharacteristic
456C2×C4C3×#23511yescharacteristic
457C2×C4C3×D3×D411yescharacteristic
458C2×C4C3×D3×D411yescharacteristic
459C2×C4C3×D3×D411yescharacteristic
460C2×C4C3×#23711yescharacteristic
461C8—34no—
462C8—62no—
463C2×C3—21no—
464C2×C3—22no—
465C2×C3—22no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—242no—
469C2×C3#495711yescharacteristic
470C2×C3#502612yes—
471C2×C3#504912yes—
472C2×C3#505811yescharacteristic
473C2×C3#508611yescharacteristic
474C2×C3C3×#39011yescharacteristic
475C2×C3C3×#44012yes—
476C2×C3C3×#46812yes—
477C2×C3C3×#49611yescharacteristic
478C2×C3C3×#49711yescharacteristic
479D3—82no—
480C22—122no—
481C22—124no—
482C22—242no—
483C22—241no—
484C22—241no—
485C22C3×C4×#6711yescharacteristic
486C22C3×#93911yescharacteristic
487C22C3×#94011yescharacteristic
488C22C3×#94212yes—
489C22C3×#94912yes—
490C4—42no—
491C4—42no—
492C4—81no—
493C4—241no—
494C4C3×D3×C22⋊C411yescharacteristic
495C4C3×#67⋊C411yescharacteristic
496C4C3×#88411yescharacteristic
497C4C3×#88511yescharacteristic
498C3—21no—
499C3#6967911yescharacteristic, direct product factor
500C3C3×#197311yescharacteristic, complemented
501C2—242no—
502C2C3×#495711yescharacteristic
503C2C3×#502612yes—
504C2C3×#504912yes—
505C2C3×#505811yescharacteristic
506C2C3×#508611yescharacteristic
subgroupG/H
centerC23×C3C2×#67
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#391C2
socleC23×C32C2×D4
prime radicalC22×C3×D12C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#391D3
hypocenterC3C3×#1973
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C3×#391

Complements

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

NH ≅ G/Ntype
2. C32×#391C2semidirect
3. C2×C3×#835C2semidirect
5. C2×C3×#807C2semidirect
7. C2×C3×#803C2semidirect
9. #69679C3direct (N × H)
16. C2×C3×#215C4semidirect
33. C3×#391D3semidirect
34. C3×#391C2×C3semidirect
36. C2×#835C2×C3semidirect
38. C2×#807C2×C3semidirect
40. C2×#803C2×C3semidirect
94. C2×#215C3×C4semidirect
141. C3×C3⋊C8C22⋊C4semidirect
147. #391C3×D3semidirect
343. C3⋊C8C3×C22⋊C4semidirect
430. C32#1973semidirect
499. C3#69679direct (N × H)
500. C3C3×#1973semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#2152
C32×#3912
C2×C3×#8352
C2×C3×#8092
C2×C3×#8072
C3×#49392
C2×C3×#8032
C3×#19733
#696793