Finite Group Databasebeta

← C3×#70052 #11788524

Subgroups — C3×#70052 #11788524

Order 1152.

699 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#951C211yesmaximal, characteristic
2C3×#5516C211yesmaximal, characteristic, complemented
3C32×#410C211yesmaximal, characteristic
4C3×#5236C211yesmaximal, characteristic
5C3×#5004C211yesmaximal, characteristic, complemented
6C3×#4533C211yesmaximal, characteristic, complemented
7C3×#4531C211yesmaximal, characteristic, complemented
8C3×#1905—31nomaximal
9#70052C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C32×#117C2211yescharacteristic
13C32×#117C2211yescharacteristic
14C3×#951—22no—
15C3×#951—22no—
16C2×C3×#216C412yescomplemented
17C2×C3×#214C412yescomplemented
18C3×#926C2211yescharacteristic
19C2×C3×#208C2211yescharacteristic
20C2×C3×#208C2211yescharacteristic
21C3×#887—22no—
22C3×#877—22no—
23C3×C4×Dic6—22no—
24C32×#99—22no—
25C3×#837—22no—
26C3×#810—22no—
27C2×C3×#137—31no—
28C3×#519—31no—
29C2×#951C2×C311yescharacteristic
30#5516C2×C311yescharacteristic, complemented
31C3×#421—31no—
32C3×#419—31no—
33C3×#410—21no—
34C3×#410D311yescharacteristic
35C3×#410C2×C311yescharacteristic
36#5236C2×C311yescharacteristic
37#5004C2×C311yescharacteristic, complemented
38C3×#344—31no—
39C3×H(3,ℤ/4)—31no—
40#4533C2×C311yescharacteristic, complemented
41#4531C2×C311yescharacteristic, complemented
42C32×#41—21no—
43C32×#41—21no—
44C32×#41—21no—
45C32×#41—21no—
46C2×C32×Q8C2×C411yescharacteristic
47C2×C32×D4—21no—
48C2×C32×D4C2×C411yescharacteristic
49C22×C32×C4—21no—
50C22×C32×C4C2311yescharacteristic
51C22×C3×Dic3—21no—
52C2×C3×Dic6—21no—
53C2×C3×Dic6—21no—
54C2×C3×Dic6—21no—
55C2×C3×Dic6C2×C412yes—
56C2×C3×Dic6D412yes—
57C2×C3×Dic6D412yes—
58C32×C4⋊C4—21no—
59C32×C4⋊C4—21no—
60C32×C22⋊C4—21no—
61C32×C22⋊C4D412yes—
62C32×C42—21no—
63C32×C42—21no—
64C3×#217—21no—
65C3×#216—24no—
66C3×#214—24no—
67C3×C4⋊Dic3—21no—
68C3×Dic3⋊C4—24no—
69C3×C4×Dic3—24no—
70C3×C4×Dic3D412yes—
71C3×#208—21no—
72C3×#208—21no—
73C3×#208—21no—
74C3×#208—21no—
75C2×C3×C3⋊C8—21no—
76C2×C3×C3⋊C8C2×C412yes—
77#1905—31noSylow
78C2×C3×#41—21no—
79C2×C3×#41C2×D311yescharacteristic
80C2×C3×#41C22×C311yescharacteristic
81C22×C3×Q8—31no—
82C22×Dic6C22×C311yescharacteristic
83C3×#137—62no—
84C3×#137—62no—
85C2×C3×Q16—32no—
86C2×C3×SD16—32no—
87C2×C3×M16—31no—
88C2×C3×M16—31no—
89C3×C4×Q8—62no—
90C3×C4×Q8—62no—
91C3×#117—21no—
92C3×#117—21no—
93C3×#117—31no—
94C3×#117—62no—
95C3×#117C2×D311yescharacteristic
96C3×#117C2×D311yescharacteristic
97C3×#117C22×C311yescharacteristic
98C3×#117C22×C311yescharacteristic
99#951—22no—
100#951—22no—
101C2×#216C3×C412yescomplemented
102C2×#214C3×C412yescomplemented
103#926C22×C311yescharacteristic
104C2×#208C22×C311yescharacteristic
105C2×#208C22×C311yescharacteristic
106#887—22no—
107#877—22no—
108C4×Dic6—22no—
109C3×#104—62no—
110C3×#103—62no—
111C3×#99—24no—
112C3×#99—22no—
113C3×#99—22no—
114#837—22no—
115#810—22no—
116C23×C32—21no—
117C23×C32C2×D411yescharacteristic
118C32×Q8—21no—
119C32×Q8—21no—
120C32×D4—21no—
121C32×D4—21no—
122C32×D4—41no—
123C32×D4—41no—
124C2×C32×C4—21no—
125C2×C32×C4—22no—
126C2×C32×C4—41no—
127C2×C32×C4—41no—
128C2×C32×C4—41no—
129C2×C32×C4C22×C411yescharacteristic
130C2×C32×C4C2×D411yescharacteristic
131C2×C32×C4C2×D411yescharacteristic
132C2×C32×C4C2×D411yescharacteristic
133C2×C32×C4#4111yescharacteristic
134C2×C32×C4#4111yescharacteristic
135C2×C3×Dic3—22no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—22no—
138C2×C3×Dic3—22no—
139C3×Dic6—22no—
140C3×Dic6—22no—
141C3×Dic6—22no—
142C3×Dic6C22⋊C414yes—
143C3×C3⋊C8—22no—
144C3×C3⋊C8—41no—
145C2×#137—31no—
146#519—31no—
147#421—31no—
148#419—31no—
149#410C3×D311yescharacteristic
150#344—31no—
151H(3,ℤ/4)—31no—
152C3×#41—21no—
153C3×#41—21no—
154C3×#41—21no—
155C3×#41—21no—
156C3×#41—21no—
157C3×#41—21no—
158C3×#41—21no—
159C3×#41—21no—
160C3×#41—41no—
161C3×#41—41no—
162C3×#41—41no—
163C3×#41—41no—
164C2×C3×Q8—21no—
165C2×C3×Q8—32no—
166C2×C3×Q8—32no—
167C2×C3×Q8—32no—
168C2×C3×Q8—61no—
169C2×C3×Q8—61no—
170C2×C3×Q8—61no—
171C2×C3×Q8C4×D311yescharacteristic
172C2×C3×Q8C2×C3×C411yescharacteristic
173C2×C3×D4—22no—
174C2×C3×D4—21no—
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4C4×D311yescharacteristic
178C2×C3×D4C2×C3×C411yescharacteristic
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—61no—
184C22×C3×C4C22×D311yescharacteristic
185C22×C3×C4C23×C311yescharacteristic
186C22×Dic3—21no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6—21no—
190C2×Dic6C2×C3×C412yes—
191C2×Dic6C3×D412yes—
192C2×Dic6C3×D412yes—
193C3×Q16—64no—
194C3×SD16—64no—
195C3×M16—61no—
196C3×M16—61no—
197C3×M16—61no—
198C3×M16—61no—
199C2×C3×C8—32no—
200C2×C3×C8—61no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—41no—
206C3×C4⋊C4—41no—
207C3×C4⋊C4—64no—
208C3×C4⋊C4—61no—
209C3×C22⋊C4—22no—
210C3×C22⋊C4—21no—
211C3×C22⋊C4—21no—
212C3×C22⋊C4—22no—
213C3×C22⋊C4—61no—
214C3×C22⋊C4#6712yes—
215C3×C22⋊C4C3×D412yes—
216C3×C42—21no—
217C3×C42—21no—
218C3×C42—21no—
219C3×C42—21no—
220C3×C42—32no—
221C3×C42—41no—
222C3×C42—41no—
223C3×C42—64no—
224#217—21no—
225#216—24no—
226#214—24no—
227C4⋊Dic3—21no—
228Dic3⋊C4—24no—
229C4×Dic3—24no—
230C4×Dic3C3×D412yes—
231#208—21no—
232#208—21no—
233#208—21no—
234#208—21no—
235C2×C3⋊C8—21no—
236C2×C3⋊C8C2×C3×C412yes—
237C22×C32—41no—
238C22×C32—41no—
239C22×C32—81no—
240C22×C32C4×D411yescharacteristic
241C22×C32#12011yescharacteristic
242C22×C32#12111yescharacteristic
243C32×C4—41no—
244C32×C4—42no—
245C32×C4—81no—
246C32×C4C2×C22⋊C411yescharacteristic
247C32×C4C4×D411yescharacteristic
248C32×C4#12111yescharacteristic
249C32×C4#12311yescharacteristic
250C3×Dic3—24no—
251C3×Dic3—42no—
252C2×#41C2×C3×D311yescharacteristic
253C22×Q8—31no—
254#137—62no—
255#137—62no—
256C2×Q16—32no—
257C2×SD16—32no—
258C2×M16—31no—
259C2×M16—31no—
260C4×Q8—62no—
261C4×Q8—62no—
262#117—31no—
263#117—62no—
264#117C2×C3×D311yescharacteristic
265#117C2×C3×D311yescharacteristic
266#104—62no—
267#103—62no—
268#99—22no—
269C23×C3—22no—
270C23×C3—21no—
271C23×C3—21no—
272C23×C3—21no—
273C23×C3C2×#6711yescharacteristic
274C23×C3C2×C3×D411yescharacteristic
275C3×Q8—21no—
276C3×Q8—21no—
277C3×Q8—21no—
278C3×Q8—21no—
279C3×Q8—34no—
280C3×Q8—41no—
281C3×Q8—41no—
282C3×Q8—62no—
283C3×Q8—62no—
284C3×Q8—62no—
285C3×D4—21no—
286C3×D4—21no—
287C3×D4—21no—
288C3×D4—21no—
289C3×D4—41no—
290C3×D4—41no—
291C3×D4—42no—
292C3×D4—42no—
293C3×D4—41no—
294C3×D4—41no—
295C3×D4—41no—
296C3×D4—41no—
297C2×C3×C4—22no—
298C2×C3×C4—21no—
299C2×C3×C4—22no—
300C2×C3×C4—21no—
301C2×C3×C4—22no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—42no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—62no—
319C2×C3×C4—62no—
320C2×C3×C4—62no—
321C2×C3×C4—62no—
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—22no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338Dic6—22no—
339Dic6—22no—
340Dic6—22no—
341Dic6C3×C22⋊C414yes—
342C3×C8—62no—
343C3×C8—121no—
344C3⋊C8—22no—
345C3⋊C8—41no—
346C2×C32—21no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—81no—
350C2×C32#38611yescharacteristic
351#41—21no—
352#41—21no—
353#41—21no—
354#41—21no—
355C2×Q8—32no—
356C2×Q8—32no—
357C2×Q8—32no—
358C2×Q8—61no—
359C2×Q8—61no—
360C2×Q8—61no—
361C2×Q8C3×C4×D311yescharacteristic
362C2×D4—21no—
363C2×D4C3×C4×D311yescharacteristic
364C22×C4—21no—
365C22×C4—61no—
366C22×C4C22×C3×D311yescharacteristic
367Q16—64no—
368SD16—64no—
369M16—61no—
370M16—61no—
371M16—61no—
372M16—61no—
373C2×C8—32no—
374C2×C8—61no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—64no—
378C4⋊C4—61no—
379C22⋊C4—21no—
380C22⋊C4—61no—
381C22⋊C4C3×#6712yes—
382C42—21no—
383C42—21no—
384C42—32no—
385C42—64no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—41no—
390C22×C3—41no—
391C22×C3—42no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—81no—
395C22×C3—81no—
396C22×C3—81no—
397C22×C3—82no—
398C22×C3C4×#6711yescharacteristic
399C22×C3#93711yescharacteristic
400C22×C3#93811yescharacteristic
401C22×C3C3×C4×D411yescharacteristic
402C22×C3C3×#12011yescharacteristic
403C22×C3C3×#12111yescharacteristic
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—21no—
408C3×C4—42no—
409C3×C4—42no—
410C3×C4—42no—
411C3×C4—41no—
412C3×C4—41no—
413C3×C4—64no—
414C3×C4—82no—
415C3×C4—81no—
416C3×C4—81no—
417C3×C4—82no—
418C3×C4—122no—
419C3×C4D3×C22⋊C411yescharacteristic
420C3×C4#67⋊C411yescharacteristic
421C3×C4#88311yescharacteristic
422C3×C4#88411yescharacteristic
423C3×C4C2×C3×C22⋊C411yescharacteristic
424C3×C4C3×C4×D411yescharacteristic
425C3×C4C3×#12111yescharacteristic
426C3×C4C3×#12311yescharacteristic
427Dic3—24no—
428Dic3—42no—
429C32#190511yescharacteristic, complemented, Sylow
430C23—21no—
431C23C2×C3×#6711yescharacteristic
432Q8—21no—
433Q8—21no—
434Q8—34no—
435Q8—62no—
436Q8—62no—
437Q8—62no—
438D4—21no—
439D4—21no—
440D4—41no—
441D4—41no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4C2×C3×C4×D311yescharacteristic
452C2×C4C3×#23511yescharacteristic
453C2×C4C3×D3×D411yescharacteristic
454C2×C4C3×D3×D411yescharacteristic
455C2×C4C3×D3×D411yescharacteristic
456C2×C4C3×#23711yescharacteristic
457C8—62no—
458C8—121no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—41no—
467C2×C3—41no—
468C2×C3—41no—
469C2×C3—82no—
470C2×C3—81no—
471C2×C3—81no—
472C2×C3#495611yescharacteristic
473C2×C3C3×#38611yescharacteristic
474C22—41no—
475C22—41no—
476C22—81no—
477C22C3×C4×#6711yescharacteristic
478C22C3×#93711yescharacteristic
479C22C3×#93811yescharacteristic
480C4—42no—
481C4—41no—
482C4—64no—
483C4—81no—
484C4—122no—
485C4C3×D3×C22⋊C411yescharacteristic
486C4C3×#67⋊C411yescharacteristic
487C4C3×#88311yescharacteristic
488C4C3×#88411yescharacteristic
489C3—21no—
490C3#7005211yescharacteristic, direct product factor
491C3C3×#190511yescharacteristic, complemented
492C2—21no—
493C2—21no—
494C2—21no—
495C2—81no—
496C2C3×#495611yescharacteristic
subgroupG/H
centerC2×C3#4956
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#410C2
socleC2×C32#386
prime radicalC2×C32×#41C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#410D3
hypocenterC3C3×#1905
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#410

Complements

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

NH ≅ G/Ntype
2. C3×#5516C2semidirect
5. C3×#5004C2semidirect
6. C3×#4533C2semidirect
7. C3×#4531C2semidirect
9. #70052C3direct (N × H)
16. C2×C3×#216C4semidirect
17. C2×C3×#214C4semidirect
30. #5516C2×C3semidirect
37. #5004C2×C3semidirect
40. #4533C2×C3semidirect
41. #4531C2×C3semidirect
101. C2×#216C3×C4semidirect
102. C2×#214C3×C4semidirect
429. C32#1905semidirect
490. C3#70052direct (N × H)
491. C3C3×#1905semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9512
C3×#55162
C32×#4102
C3×#52362
C3×#50042
C3×#45332
C3×#45312
C3×#19053
#700523