Finite Group Databasebeta

← C3×#72101 #11788521

Subgroups — C3×#72101 #11788521

Order 1152.

749 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#137C211yesmaximal, characteristic
2C3×#5812C211yesmaximal, characteristic, complemented
3C3×#5240C211yesmaximal, characteristic
4C3×#5236C211yesmaximal, characteristic
5C3×#5225C211yesmaximal, characteristic, complemented
6C3×#4557C211yesmaximal, characteristic, complemented
7C3×#4556C211yesmaximal, characteristic, complemented
8C3×#1905—31nomaximal
9#72101C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C22×C32×Q8C2211yescharacteristic
12C32×#137—22no—
13C32×#137—22no—
14C2×C32×Q16C412yescomplemented
15C2×C32×SD16C412yescomplemented
16C2×C32×M16C2211yescharacteristic
17C3×Q8×Dic3—22no—
18C3×Q8×Dic3—22no—
19C3×#926—22no—
20C3×#926C2211yescharacteristic
21C3×#926C2211yescharacteristic
22C3×#926C2211yescharacteristic
23C2×C3×#208C2211yescharacteristic
24C3×#837—22no—
25C3×#835—22no—
26C3×#834—22no—
27C2×C3×#137—21no—
28C2×C3×#137D311yescharacteristic
29C2×C3×#137C2×C311yescharacteristic
30C3×#519—31no—
31#5812C2×C311yescharacteristic, complemented
32C3×#421—31no—
33C3×#419—31no—
34C3×#410—31no—
35#5240C2×C311yescharacteristic
36#5236C2×C311yescharacteristic
37#5225C2×C311yescharacteristic, complemented
38C3×#344—31no—
39C3×H(3,ℤ/4)—31no—
40#4557C2×C311yescharacteristic, complemented
41#4556C2×C311yescharacteristic, complemented
42C32×#41—21no—
43C32×#41—21no—
44C32×#41—21no—
45C32×#41—21no—
46C2×C32×Q8—21no—
47C2×C32×Q8—21no—
48C2×C32×Q8—21no—
49C2×C32×Q8C2×C411yescharacteristic
50C2×C32×Q8C2×C412yes—
51C2×C32×Q8D412yes—
52C2×C32×Q8D412yes—
53C2×C32×D4—21no—
54C2×C32×D4C2×C411yescharacteristic
55C22×C32×C4—21no—
56C22×C32×C4—21no—
57C22×C32×C4C2311yescharacteristic
58C32×Q16—24no—
59C32×SD16—24no—
60C32×M16—21no—
61C32×M16—21no—
62C2×C32×C8C2×C412yes—
63C3×#217—21no—
64C3×#217—21no—
65C3×#217D412yes—
66C3×C4⋊Dic3—24no—
67C3×C4⋊Dic3—21no—
68C3×C4⋊Dic3—21no—
69C3×C4⋊Dic3—21no—
70C3×C4×Dic3—24no—
71C3×C4×Dic3—21no—
72C3×C4×Dic3—21no—
73C3×C4×Dic3D412yes—
74C3×#208—21no—
75C3×#208—21no—
76C2×C3×C3⋊C8—21no—
77#1905—31noSylow
78C2×C3×#41—21no—
79C2×C3×#41C2×D311yescharacteristic
80C2×C3×#41C22×C311yescharacteristic
81C22×C3×Q8—21no—
82C22×C3×Q8C2×D311yescharacteristic
83C22×C3×Q8C22×C311yescharacteristic
84C3×#137—24no—
85C3×#137—24no—
86C3×#137—22no—
87C3×#137—22no—
88C3×#137—22no—
89C3×#137—22no—
90C2×C3×Q16—22no—
91C2×C3×Q16Dic312yescomplemented
92C2×C3×Q16C3×C412yescomplemented
93C2×C3×SD16—22no—
94C2×C3×SD16Dic312yescomplemented
95C2×C3×SD16C3×C412yescomplemented
96C2×C3×M16—21no—
97C2×C3×M16—31no—
98C2×C3×M16C2×D311yescharacteristic
99C2×C3×M16C22×C311yescharacteristic
100C3×C4×Q8—62no—
101C3×C4×Q8—62no—
102C3×#117—31no—
103C3×#117—31no—
104C3×#117—31no—
105C3×#117—62no—
106Q8×Dic3—22no—
107Q8×Dic3—22no—
108#926—22no—
109#926C22×C311yescharacteristic
110#926C22×C311yescharacteristic
111#926C22×C311yescharacteristic
112C2×#208C22×C311yescharacteristic
113C3×#104—62no—
114C3×#103—62no—
115C3×#99—62no—
116#837—22no—
117#835—22no—
118#834—22no—
119C23×C32—21no—
120C23×C32C2×D411yescharacteristic
121C32×Q8—21no—
122C32×Q8—22no—
123C32×Q8—22no—
124C32×Q8—22no—
125C32×Q8—21no—
126C32×Q8C22⋊C414yes—
127C32×D4—21no—
128C32×D4—21no—
129C32×D4—41no—
130C32×D4—41no—
131C2×C32×C4—22no—
132C2×C32×C4—22no—
133C2×C32×C4—22no—
134C2×C32×C4—21no—
135C2×C32×C4—41no—
136C2×C32×C4—41no—
137C2×C32×C4C22×C411yescharacteristic
138C2×C32×C4C2×D411yescharacteristic
139C2×C32×C4C2×D411yescharacteristic
140C2×C32×C4C2×D411yescharacteristic
141C2×C32×C4#4111yescharacteristic
142C2×C32×C4#4111yescharacteristic
143C2×C3×Dic3—22no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—41no—
146C32×C8—22no—
147C3×C3⋊C8—41no—
148C2×#137C3×D311yescharacteristic
149#519—31no—
150#421—31no—
151#419—31no—
152#410—31no—
153#344—31no—
154H(3,ℤ/4)—31no—
155C3×#41—22no—
156C3×#41—22no—
157C3×#41—21no—
158C3×#41—21no—
159C3×#41—21no—
160C3×#41—21no—
161C3×#41—21no—
162C3×#41—21no—
163C3×#41—21no—
164C3×#41—21no—
165C3×#41—41no—
166C3×#41—41no—
167C2×C3×Q8—21no—
168C2×C3×Q8—21no—
169C2×C3×Q8—21no—
170C2×C3×Q8—21no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—22no—
174C2×C3×Q8—21no—
175C2×C3×Q8—22no—
176C2×C3×Q8—22no—
177C2×C3×Q8—41no—
178C2×C3×Q8—41no—
179C2×C3×Q8—41no—
180C2×C3×Q8C2×Dic311yescharacteristic
181C2×C3×Q8C2×Dic312yes—
182C2×C3×Q8#6712yes—
183C2×C3×Q8#6712yes—
184C2×C3×Q8C2×C3×C411yescharacteristic
185C2×C3×Q8C2×C3×C412yes—
186C2×C3×Q8C3×D412yes—
187C2×C3×Q8C3×D412yes—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—41no—
192C2×C3×D4C2×Dic311yescharacteristic
193C2×C3×D4C2×C3×C411yescharacteristic
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—41no—
200C22×C3×C4—41no—
201C22×C3×C4C22×D311yescharacteristic
202C22×C3×C4C23×C311yescharacteristic
203C3×Q16—28no—
204C3×Q16—24no—
205C3×Q16—24no—
206C3×SD16—28no—
207C3×SD16—24no—
208C3×SD16—24no—
209C3×M16—22no—
210C3×M16—22no—
211C3×M16—21no—
212C3×M16—21no—
213C3×M16—21no—
214C3×M16—21no—
215C3×M16—61no—
216C3×M16—61no—
217C2×C3×C8—22no—
218C2×C3×C8—61no—
219C2×C3×C8C2×Dic312yes—
220C2×C3×C8C2×C3×C412yes—
221C3×C4⋊C4—64no—
222C3×C4⋊C4—61no—
223C3×C4⋊C4—61no—
224C3×C4⋊C4—61no—
225C3×C22⋊C4—32no—
226C3×C22⋊C4—61no—
227C3×C22⋊C4—61no—
228C3×C42—32no—
229C3×C42—64no—
230C3×C42—61no—
231C3×C42—61no—
232#217—21no—
233#217—21no—
234#217C3×D412yes—
235C4⋊Dic3—24no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4×Dic3—24no—
240C4×Dic3—21no—
241C4×Dic3—21no—
242C4×Dic3C3×D412yes—
243#208—21no—
244#208—21no—
245C2×C3⋊C8—21no—
246C22×C32—41no—
247C22×C32—41no—
248C22×C32—81no—
249C22×C32C4×D411yescharacteristic
250C22×C32#12011yescharacteristic
251C22×C32#12111yescharacteristic
252C32×C4—24no—
253C32×C4—41no—
254C32×C4C2×C22⋊C411yescharacteristic
255C32×C4C4×D411yescharacteristic
256C32×C4#12111yescharacteristic
257C32×C4#12311yescharacteristic
258C3×Dic3—42no—
259C3×Dic3—42no—
260C3×Dic3—81no—
261C2×#41C2×C3×D311yescharacteristic
262C22×Q8C2×C3×D311yescharacteristic
263#137—22no—
264#137—22no—
265C2×Q16C3×Dic312yescomplemented
266C2×SD16C3×Dic312yescomplemented
267C2×M16—31no—
268C2×M16C2×C3×D311yescharacteristic
269C4×Q8—62no—
270C4×Q8—62no—
271#117—31no—
272#117—31no—
273#117—31no—
274#117—62no—
275#104—62no—
276#103—62no—
277#99—62no—
278C23×C3—21no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—41no—
282C23×C3D3×D411yescharacteristic
283C23×C3C2×C3×D411yescharacteristic
284C3×Q8—22no—
285C3×Q8—22no—
286C3×Q8—22no—
287C3×Q8—22no—
288C3×Q8—22no—
289C3×Q8—22no—
290C3×Q8—22no—
291C3×Q8—21no—
292C3×Q8—21no—
293C3×Q8—21no—
294C3×Q8—21no—
295C3×Q8—24no—
296C3×Q8—42no—
297C3×Q8—42no—
298C3×Q8—42no—
299C3×Q8—41no—
300C3×Q8#21714yes—
301C3×Q8C3×C22⋊C414yes—
302C3×D4—22no—
303C3×D4—21no—
304C3×D4—21no—
305C3×D4—21no—
306C3×D4—21no—
307C3×D4—41no—
308C3×D4—42no—
309C3×D4—42no—
310C3×D4—41no—
311C3×D4—41no—
312C3×D4—41no—
313C3×D4—41no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—41no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—62no—
339C2×C3×C4—62no—
340C2×C3×C4—121no—
341C2×C3×C4D3×D411yescharacteristic
342C2×C3×C4#23711yescharacteristic
343C2×C3×C4#23711yescharacteristic
344C2×C3×C4C22×Dic311yescharacteristic
345C2×C3×C4C2×#6711yescharacteristic
346C2×C3×C4C2×#6711yescharacteristic
347C2×C3×C4C22×C3×C411yescharacteristic
348C2×C3×C4C2×C3×D411yescharacteristic
349C2×C3×C4C2×C3×D411yescharacteristic
350C2×C3×C4C2×C3×D411yescharacteristic
351C2×C3×C4C3×#4111yescharacteristic
352C2×C3×C4C3×#4111yescharacteristic
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—41no—
356C3×C8—24no—
357C3×C8—22no—
358C3×C8—22no—
359C3×C8—121no—
360C3⋊C8—41no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—81no—
365C2×C32#38611yescharacteristic
366#41—21no—
367#41—21no—
368#41—21no—
369#41—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8C2×C3×Dic311yescharacteristic
374C2×Q8C2×C3×Dic312yes—
375C2×Q8C3×#6712yes—
376C2×Q8C3×#6712yes—
377C2×D4—21no—
378C2×D4C2×C3×Dic311yescharacteristic
379C22×C4—21no—
380C22×C4—21no—
381C22×C4C22×C3×D311yescharacteristic
382Q16—24no—
383SD16—24no—
384M16—21no—
385M16—21no—
386M16—61no—
387M16—61no—
388C2×C8—61no—
389C2×C8C2×C3×Dic312yes—
390C4⋊C4—64no—
391C4⋊C4—61no—
392C4⋊C4—61no—
393C4⋊C4—61no—
394C22⋊C4—32no—
395C22⋊C4—61no—
396C22⋊C4—61no—
397C42—32no—
398C42—64no—
399C42—61no—
400C42—61no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—41no—
405C22×C3—41no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3—41no—
409C22×C3—41no—
410C22×C3—81no—
411C22×C3—81no—
412C22×C3—82no—
413C22×C3D4×Dic311yescharacteristic
414C22×C3#93711yescharacteristic
415C22×C3#93911yescharacteristic
416C22×C3C3×C4×D411yescharacteristic
417C22×C3C3×#12011yescharacteristic
418C22×C3C3×#12111yescharacteristic
419C3×C4—24no—
420C3×C4—24no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—44no—
426C3×C4—42no—
427C3×C4—41no—
428C3×C4—41no—
429C3×C4—122no—
430C3×C4—122no—
431C3×C4—241no—
432C3×C4D4×Dic311yescharacteristic
433C3×C4#93511yescharacteristic
434C3×C4#93911yescharacteristic
435C3×C4C2×#21711yescharacteristic
436C3×C4C2×C3×C22⋊C411yescharacteristic
437C3×C4C3×C4×D411yescharacteristic
438C3×C4C3×#12111yescharacteristic
439C3×C4C3×#12311yescharacteristic
440Dic3—42no—
441Dic3—42no—
442Dic3—81no—
443C32#190511yescharacteristic, complemented, Sylow
444C23—21no—
445C23C3×D3×D411yescharacteristic
446Q8—22no—
447Q8—22no—
448Q8—22no—
449Q8—21no—
450Q8—21no—
451Q8C3×#21714yes—
452D4—21no—
453D4—21no—
454D4—41no—
455D4—41no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—121no—
465C2×C4C3×D3×D411yescharacteristic
466C2×C4C3×#23711yescharacteristic
467C2×C4C3×#23711yescharacteristic
468C2×C4C22×C3×Dic311yescharacteristic
469C2×C4C2×C3×#6711yescharacteristic
470C2×C4C2×C3×#6711yescharacteristic
471C8—22no—
472C8—121no—
473C2×C3—22no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—41no—
482C2×C3—41no—
483C2×C3—82no—
484C2×C3—81no—
485C2×C3—81no—
486C2×C3#522011yescharacteristic
487C2×C3C3×#38611yescharacteristic
488C22—41no—
489C22—41no—
490C22—81no—
491C22C3×D4×Dic311yescharacteristic
492C22C3×#93711yescharacteristic
493C22C3×#93911yescharacteristic
494C4—24no—
495C4—41no—
496C4—122no—
497C4—122no—
498C4—241no—
499C4C3×D4×Dic311yescharacteristic
500C4C3×#93511yescharacteristic
501C4C3×#93911yescharacteristic
502C4C2×C3×#21711yescharacteristic
503C3—21no—
504C3#7210111yescharacteristic, direct product factor
505C3C3×#190511yescharacteristic, complemented
506C2—21no—
507C2—21no—
508C2—21no—
509C2—81no—
510C2C3×#522011yescharacteristic
subgroupG/H
centerC2×C3#5220
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#137C2
socleC2×C32#386
prime radicalC2×C32×#41C22
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×#137D3
hypocenterC3C3×#1905
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#137

Complements

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

NH ≅ G/Ntype
2. C3×#5812C2semidirect
5. C3×#5225C2semidirect
6. C3×#4557C2semidirect
7. C3×#4556C2semidirect
9. #72101C3direct (N × H)
14. C2×C32×Q16C4semidirect
15. C2×C32×SD16C4semidirect
31. #5812C2×C3semidirect
37. #5225C2×C3semidirect
40. #4557C2×C3semidirect
41. #4556C2×C3semidirect
91. C2×C3×Q16Dic3semidirect
92. C2×C3×Q16C3×C4semidirect
94. C2×C3×SD16Dic3semidirect
95. C2×C3×SD16C3×C4semidirect
265. C2×Q16C3×Dic3semidirect
266. C2×SD16C3×Dic3semidirect
443. C32#1905semidirect
504. C3#72101direct (N × H)
505. C3C3×#1905semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1372
C3×#58122
C3×#52402
C3×#52362
C3×#52252
C3×#45572
C3×#45562
C3×#19053
#721013