Finite Group Databasebeta

← C3×#70148 #11796492

Subgroups — C3×#70148 #11796492

Order 1152.

575 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5832C211yesmaximal, characteristic
2C32×#413C211yesmaximal, characteristic, complemented
3C3×#5237C211yesmaximal, characteristic, complemented
4C3×#4478C212yesmaximal, complemented
5C3×#4477C212yesmaximal, complemented
6C3×#2132—31nomaximal
7#70148C311yesmaximal, characteristic, direct product factor
8C2×C32×#41C2211yescharacteristic
9C2×C3×#235C411yescharacteristic
10C2×C32×M16—21no—
11C3×#947C2211yescharacteristic
12C3×#944C2211yescharacteristic
13C3×#940C411yescharacteristic
14C3×#939—21no—
15C3×#939—21no—
16C3×#936C411yescharacteristic
17C3×C4×#67—21no—
18C3×C4×#67—21no—
19C2×C3×C4×Dic3C411yescharacteristic
20C2×C3×#208—21no—
21C32×#101C2212yes—
22C32×#100—21no—
23C3×#836C2212yes—
24C3×#833—21no—
25C3×#532—31no—
26#5832C2×C311yescharacteristic
27C3×#413—21no—
28C3×#413—31no—
29C3×#413D311yescharacteristic, complemented
30C3×#413C2×C311yescharacteristic, complemented
31#5237C2×C311yescharacteristic, complemented
32C3×#356—32no—
33C3×#355—32no—
34#4478C2×C312yescomplemented
35#4477C2×C312yescomplemented
36C32×#41—41no—
37C32×#41—41no—
38C2×C32×Q8C2311yescharacteristic
39C2×C32×D4C2×C411yescharacteristic
40C2×C32×D4D411yescharacteristic
41C2×C32×D4D411yescharacteristic
42C22×C32×C4C2×C411yescharacteristic
43C22×C32×C4D411yescharacteristic
44C22×C32×C4D411yescharacteristic
45C2×C3×#67—21no—
46C2×C3×#67—21no—
47C2×C3×#67—21no—
48C22×C3×Dic3—21no—
49C3×#235—41no—
50C3×#235—41no—
51C2×C3×D12C2×C411yescharacteristic
52C2×C3×C4×D3—21no—
53C2×C3×Dic6C2×C411yescharacteristic
54C32×M16—22no—
55C32×M16—41no—
56C2×C32×C8—41no—
57C3×#217—21no—
58C3×#217—21no—
59C3×(C2×D3)⋊C4—21no—
60C3×(C2×D3)⋊C4—21no—
61C3×Dic3⋊C4—21no—
62C3×Dic3⋊C4—21no—
63C3×C4×Dic3—21no—
64C3×C4×Dic3C2×C411yescharacteristic
65C3×C4×Dic3C2×C411yescharacteristic
66C3×#208—22no—
67C3×#208—41no—
68C2×C3×C3⋊C8—41no—
69#2132—31noSylow
70C2×C3×#41—21no—
71C2×C3×#41—31no—
72C2×C3×#41C2×D311yescharacteristic
73C2×C3×#41C22×C311yescharacteristic
74C2×#235C3×C411yescharacteristic
75C2×C3×M16—22no—
76C2×C3×M16—21no—
77C2×C3×M16—21no—
78C2×C3×M16—61no—
79C3×C4⋊Q8—31no—
80C3×#127—31no—
81C3×#124—31no—
82C3×#124—31no—
83C3×#121—61no—
84C3×#121—61no—
85C3×C4×D4—61no—
86C3×C4×D4—61no—
87C2×C3×C42—31no—
88#947C22×C311yescharacteristic
89#944C22×C311yescharacteristic
90#940C3×C411yescharacteristic
91#939—21no—
92#939—21no—
93#936C3×C411yescharacteristic
94C4×#67—21no—
95C4×#67—21no—
96C2×C4×Dic3C3×C411yescharacteristic
97C2×#208—21no—
98C3×#101—22no—
99C3×#101—32no—
100C3×#101C2×D312yes—
101C3×#101C22×C312yes—
102C3×#100—22no—
103C3×#100—21no—
104C3×#100—21no—
105C3×#100—61no—
106#836C22×C312yes—
107#833—21no—
108C23×C32C22⋊C411yescharacteristic
109C23×C32C22⋊C411yescharacteristic
110C23×C32C22⋊C411yescharacteristic
111C22×C3×D3—21no—
112C32×Q8—41no—
113C32×D4—41no—
114C32×D4—41no—
115C32×D4—41no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4C22⋊C411yescharacteristic
123C2×C32×C4C22×C411yescharacteristic
124C2×C32×C4C2×D411yescharacteristic
125C2×C32×C4C2×D411yescharacteristic
126C3×#67—42no—
127C3×#67—42no—
128C3×#67—42no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—42no—
133C3×D12—41no—
134C3×C4×D3—41no—
135C3×C4×D3—41no—
136C3×Dic6—41no—
137C32×C8—42no—
138C3×C3⋊C8—42no—
139#532—31no—
140#413—31no—
141#413C3×D311yescharacteristic, complemented
142#356—32no—
143#355—32no—
144C3×#41—41no—
145C3×#41—41no—
146C3×#41—41no—
147C3×#41—41no—
148C3×#41—42no—
149C3×#41—42no—
150C3×#41—121no—
151C3×#41—121no—
152C2×C3×Q8—21no—
153C2×C3×Q8—31no—
154C2×C3×Q8C22×D311yescharacteristic
155C2×C3×Q8C23×C311yescharacteristic
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—31no—
160C2×C3×D4—61no—
161C2×C3×D4—61no—
162C2×C3×D4—61no—
163C2×C3×D4C4×D311yescharacteristic
164C2×C3×D4D1211yescharacteristic
165C2×C3×D4#6711yescharacteristic
166C2×C3×D4C2×C3×C411yescharacteristic
167C2×C3×D4C3×D411yescharacteristic
168C2×C3×D4C3×D411yescharacteristic
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—61no—
173C22×C3×C4—61no—
174C22×C3×C4C4×D311yescharacteristic
175C22×C3×C4D1211yescharacteristic
176C22×C3×C4#6711yescharacteristic
177C22×C3×C4C2×C3×C411yescharacteristic
178C22×C3×C4C3×D411yescharacteristic
179C22×C3×C4C3×D411yescharacteristic
180C2×#67—21no—
181C2×#67—21no—
182C2×#67—21no—
183C22×Dic3—21no—
184#235—41no—
185#235—41no—
186C2×D12C2×C3×C411yescharacteristic
187C2×C4×D3—21no—
188C2×Dic6C2×C3×C411yescharacteristic
189C3×M16—24no—
190C3×M16—22no—
191C3×M16—22no—
192C3×M16—42no—
193C3×M16—41no—
194C3×M16—41no—
195C3×M16—62no—
196C3×M16—121no—
197C2×C3×C8—42no—
198C2×C3×C8—41no—
199C2×C3×C8—41no—
200C2×C3×C8—121no—
201C3×C4⋊C4—61no—
202C3×C4⋊C4—61no—
203C3×C22⋊C4—61no—
204C3×C22⋊C4—61no—
205C3×C22⋊C4—61no—
206C3×C22⋊C4—61no—
207C3×C42—31no—
208C3×C42—31no—
209C3×C42—61no—
210#217—21no—
211#217—21no—
212(C2×D3)⋊C4—21no—
213(C2×D3)⋊C4—21no—
214Dic3⋊C4—21no—
215Dic3⋊C4—21no—
216C4×Dic3—21no—
217C4×Dic3C2×C3×C411yescharacteristic
218C4×Dic3C2×C3×C411yescharacteristic
219#208—22no—
220#208—41no—
221C2×C3⋊C8—41no—
222C22×C32—21no—
223C22×C32—21no—
224C22×C32—21no—
225C22×C32—41no—
226C22×C32—41no—
227C22×C32—41no—
228C22×C32C2×C22⋊C411yescharacteristic
229C2×C3×D3—41no—
230C2×C3×D3—41no—
231C2×C3×D3—41no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4—21no—
235C32×C4#9911yescharacteristic
236C32×C4#9911yescharacteristic
237C3×Dic3—22no—
238C3×Dic3—41no—
239C3×Dic3—41no—
240C2×#41—31no—
241C2×#41C2×C3×D311yescharacteristic
242C2×M16—21no—
243C2×M16—61no—
244C4⋊Q8—31no—
245#127—31no—
246#124—31no—
247#124—31no—
248#121—61no—
249#121—61no—
250C4×D4—61no—
251C4×D4—61no—
252C2×C42—31no—
253#101—32no—
254#101C2×C3×D312yes—
255#100—21no—
256#100—61no—
257C23×C3—21no—
258C23×C3—21no—
259C23×C3—21no—
260C23×C3—61no—
261C23×C3(C2×D3)⋊C411yescharacteristic
262C23×C3(C2×D3)⋊C411yescharacteristic
263C23×C3(C2×D3)⋊C411yescharacteristic
264C23×C3C3×C22⋊C411yescharacteristic
265C23×C3C3×C22⋊C411yescharacteristic
266C23×C3C3×C22⋊C411yescharacteristic
267C22×D3—21no—
268C3×Q8—41no—
269C3×Q8—41no—
270C3×Q8—42no—
271C3×Q8—121no—
272C3×D4—41no—
273C3×D4—41no—
274C3×D4—41no—
275C3×D4—41no—
276C3×D4—41no—
277C3×D4—41no—
278C3×D4—42no—
279C3×D4—42no—
280C3×D4—42no—
281C3×D4—121no—
282C3×D4—122no—
283C3×D4—122no—
284C3×D4—122no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—61no—
310C2×C3×C4—122no—
311C2×C3×C4—121no—
312C2×C3×C4—121no—
313C2×C3×C4(C2×D3)⋊C411yescharacteristic
314C2×C3×C4C3×C22⋊C411yescharacteristic
315C2×C3×C4C2×C4×D311yescharacteristic
316C2×C3×C4C2×D1211yescharacteristic
317C2×C3×C4C2×#6711yescharacteristic
318C2×C3×C4C22×C3×C411yescharacteristic
319C2×C3×C4C2×C3×D411yescharacteristic
320C2×C3×C4C2×C3×D411yescharacteristic
321#67—42no—
322#67—42no—
323#67—42no—
324C2×Dic3—21no—
325C2×Dic3—21no—
326C2×Dic3—21no—
327C2×Dic3—42no—
328D12—41no—
329C4×D3—41no—
330C4×D3—41no—
331Dic6—41no—
332C3×C8—44no—
333C3×C8—42no—
334C3×C8—42no—
335C3×C8—122no—
336C3⋊C8—42no—
337C2×C32—21no—
338C2×C32—41no—
339C2×C32—41no—
340C2×C32—41no—
341C2×C32C2×#9911yescharacteristic
342C3×D3—81no—
343#41—41no—
344#41—41no—
345#41—121no—
346#41—121no—
347C2×Q8—31no—
348C2×Q8C22×C3×D311yescharacteristic
349C2×D4—31no—
350C2×D4—61no—
351C2×D4—61no—
352C2×D4—61no—
353C2×D4C3×C4×D311yescharacteristic
354C2×D4C3×D1211yescharacteristic
355C2×D4C3×#6711yescharacteristic
356C22×C4—61no—
357C22×C4—61no—
358C22×C4C3×C4×D311yescharacteristic
359C22×C4C3×D1211yescharacteristic
360C22×C4C3×#6711yescharacteristic
361M16—22no—
362M16—41no—
363M16—62no—
364M16—121no—
365C2×C8—41no—
366C2×C8—121no—
367C4⋊C4—61no—
368C4⋊C4—61no—
369C22⋊C4—61no—
370C22⋊C4—61no—
371C22⋊C4—61no—
372C22⋊C4—61no—
373C42—31no—
374C42—31no—
375C42—61no—
376C22×C3—22no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—41no—
385C22×C3—41no—
386C22×C3—41no—
387C22×C3—41no—
388C22×C3—41no—
389C22×C3—41no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3—81no—
395C22×C3—121no—
396C22×C3—121no—
397C22×C3—121no—
398C22×C3C2×(C2×D3)⋊C411yescharacteristic
399C22×C3C2×C3×C22⋊C411yescharacteristic
400C2×D3—41no—
401C2×D3—41no—
402C2×D3—41no—
403C3×C4—22no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—41no—
413C3×C4—41no—
414C3×C4—62no—
415C3×C4—121no—
416C3×C4—121no—
417C3×C4#80611yescharacteristic
418C3×C4#80611yescharacteristic
419C3×C4C3×#9911yescharacteristic
420C3×C4C3×#9911yescharacteristic
421Dic3—22no—
422Dic3—41no—
423Dic3—41no—
424C32#213211yescharacteristic, complemented, Sylow
425C23—61no—
426C23C3×(C2×D3)⋊C411yescharacteristic
427C23C3×(C2×D3)⋊C411yescharacteristic
428C23C3×(C2×D3)⋊C411yescharacteristic
429Q8—41no—
430Q8—121no—
431D4—41no—
432D4—41no—
433D4—41no—
434D4—121no—
435D4—122no—
436D4—122no—
437D4—122no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—122no—
448C2×C4—121no—
449C2×C4—121no—
450C2×C4C3×(C2×D3)⋊C411yescharacteristic
451C2×C4C2×C3×C4×D311yescharacteristic
452C2×C4C2×C3×D1211yescharacteristic
453C2×C4C2×C3×#6711yescharacteristic
454C8—42no—
455C8—122no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—41no—
460C2×C3—42no—
461C2×C3—41no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—41no—
465C2×C3—41no—
466C2×C3—41no—
467C2×C3—81no—
468C2×C3—81no—
469C2×C3—241no—
470C2×C3C2×#80611yescharacteristic
471C2×C3C2×C3×#9911yescharacteristic
472D3—81no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—121no—
480C22—121no—
481C22—121no—
482C22C2×C3×(C2×D3)⋊C411yescharacteristic
483C4—21no—
484C4—21no—
485C4—21no—
486C4—62no—
487C4—121no—
488C4—121no—
489C4C3×#80611yescharacteristic
490C4C3×#80611yescharacteristic
491C3—21no—
492C3#7014811yescharacteristic, direct product factor
493C3C3×#213211yescharacteristic, complemented
494C2—21no—
495C2—41no—
496C2—41no—
497C2—41no—
498C2—241no—
499C2C2×C3×#80611yescharacteristic
subgroupG/H
centerC3×C4#806
commutatorC2×C3×C4C22×C3×C4
FrattiniC2×Q8C22×C3×D3
FittingC32×#413C2
socleC2×C32C2×#99
prime radicalC3×#5832C2
cosocleC2×C3×Q8C23×C3
hypercenterC3×#413D3
hypocenterC3C3×#2132
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×#41 ◁ C3×#413

Complements

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

NH ≅ G/Ntype
2. C32×#413C2semidirect
3. C3×#5237C2semidirect
4. C3×#4478C2semidirect
5. C3×#4477C2semidirect
7. #70148C3direct (N × H)
29. C3×#413D3semidirect
30. C3×#413C2×C3semidirect
31. #5237C2×C3semidirect
34. #4478C2×C3semidirect
35. #4477C2×C3semidirect
141. #413C3×D3semidirect
424. C32#2132semidirect
492. C3#70148direct (N × H)
493. C3C3×#2132semidirect

Maximal subgroups

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

subgroupindex
C3×#58322
C32×#4132
C3×#52372
C3×#44782
C3×#44772
C3×#21323
#701483