Finite Group Databasebeta

← C3×#72100 #11787676

Subgroups — C3×#72100 #11787676

Order 1152.

797 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#137C211yesmaximal, characteristic
2C3×#5827C211yesmaximal, characteristic
3C2×C3×#837C211yesmaximal, characteristic
4C3×#5229C211yesmaximal, characteristic, complemented
5C3×#5225C211yesmaximal, characteristic, complemented
6C3×#4555C211yesmaximal, characteristic, complemented
7C3×#4554C211yesmaximal, characteristic, complemented
8C3×#1882—31nomaximal
9#72100C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C22×C32×Q8C2211yescharacteristic
12C32×#137—22no—
13C32×#137C414yescomplemented
14C2×C32×Q16—21no—
15C2×C32×SD16—21no—
16C2×C32×M16C2211yescharacteristic
17C3×Q8×Dic3—21no—
18C3×D4×Dic3—21no—
19C3×D4×Dic3—22no—
20C3×#926—21no—
21C3×#926C2211yescharacteristic
22C2×C3×C4⋊Dic3—21no—
23C2×C3×C4⋊Dic3C2211yescharacteristic
24C2×C3×C4×Dic3C2211yescharacteristic
25C2×C3×#208C2211yescharacteristic
26C3×#837—22no—
27C3×#835—21no—
28C3×#835—21no—
29C3×C3⋊#95—22no—
30C2×C3×#137—21no—
31C2×C3×#137D311yescharacteristic
32C2×C3×#137C2×C311yescharacteristic
33C3×#520—31no—
34#5827C2×C311yescharacteristic
35C2×C3×#104—31no—
36C3×#419—31no—
37C3×#391—31no—
38C2×#837C2×C311yescharacteristic
39#5229C2×C311yescharacteristic, complemented
40#5225C2×C311yescharacteristic, complemented
41C3×#343—31no—
42C3×#339—31no—
43#4555C2×C311yescharacteristic, complemented
44#4554C2×C311yescharacteristic, complemented
45C32×#41—21no—
46C32×#41—21no—
47C32×#41C2×C412yes—
48C32×#41D412yes—
49C2×C32×Q8—21no—
50C2×C32×Q8—21no—
51C2×C32×Q8—21no—
52C2×C32×Q8—21no—
53C2×C32×Q8—21no—
54C2×C32×Q8C2×C412yescomplemented
55C2×C32×Q8D411yescharacteristic
56C2×C32×D4—21no—
57C2×C32×D4D411yescharacteristic
58C22×C32×C4—21no—
59C22×C32×C4—21no—
60C22×C32×C4C2311yescharacteristic
61C22×C3×Dic3—21no—
62C22×C3×Dic3—22no—
63C22×C3×Dic3D412yes—
64C32×Q16—24no—
65C32×SD16—24no—
66C32×M16—21no—
67C32×M16C2×C412yescomplemented
68C2×C32×C8—21no—
69C3×#217—22no—
70C3×#217—21no—
71C3×C4⋊Dic3—22no—
72C3×C4⋊Dic3—21no—
73C3×C4⋊Dic3—21no—
74C3×C4⋊Dic3D411yescharacteristic
75C3×C4⋊Dic3D411yescharacteristic
76C3×C4×Dic3—22no—
77C3×C4×Dic3—21no—
78C3×C4×Dic3—21no—
79C3×C4×Dic3—21no—
80C3×#208—21no—
81C3×#208—21no—
82C2×C3×C3⋊C8—21no—
83#1882—31noSylow
84C2×C3×#41—21no—
85C2×C3×#41C2×D311yescharacteristic
86C2×C3×#41C22×C311yescharacteristic
87C22×C3×Q8—21no—
88C22×C3×Q8C2×D311yescharacteristic
89C22×C3×Q8C22×C311yescharacteristic
90C3×#137—24no—
91C3×#137—22no—
92C3×#137—22no—
93C3×#137—24no—
94C3×#137Dic314yescomplemented
95C3×#137C3×C414yescomplemented
96C2×C3×Q16—22no—
97C2×C3×Q16—21no—
98C2×C3×Q16—21no—
99C2×C3×SD16—22no—
100C2×C3×SD16—21no—
101C2×C3×SD16—21no—
102C2×C3×M16—21no—
103C2×C3×M16—31no—
104C2×C3×M16C2×D311yescharacteristic
105C2×C3×M16C22×C311yescharacteristic
106C3×C4×Q8—61no—
107C3×C4×D4—62no—
108C3×C4×D4—61no—
109C3×#117—31no—
110C3×#117—61no—
111C2×C3×C4⋊C4—31no—
112C2×C3×C4⋊C4—61no—
113C2×C3×C42—31no—
114Q8×Dic3—21no—
115D4×Dic3—21no—
116D4×Dic3—22no—
117#926—21no—
118#926C22×C311yescharacteristic
119C2×C4⋊Dic3—21no—
120C2×C4⋊Dic3C22×C311yescharacteristic
121C2×C4×Dic3C22×C311yescharacteristic
122C2×#208C22×C311yescharacteristic
123C3×#104—62no—
124C3×#103—61no—
125C3×#103—61no—
126C3×#95—62no—
127#837—22no—
128#835—21no—
129#835—21no—
130C3⋊#95—22no—
131C23×C32—21no—
132C23×C32C2×D411yescharacteristic
133C32×Q8—22no—
134C32×Q8—21no—
135C32×Q8—22no—
136C32×Q8—41no—
137C32×Q8—41no—
138C32×Q8C22⋊C412yes—
139C32×D4—22no—
140C32×D4—22no—
141C32×D4—21no—
142C32×D4C22⋊C412yes—
143C2×C32×C4—21no—
144C2×C32×C4—22no—
145C2×C32×C4—22no—
146C2×C32×C4—22no—
147C2×C32×C4—41no—
148C2×C32×C4—41no—
149C2×C32×C4C22×C411yescharacteristic
150C2×C32×C4C2×D411yescharacteristic
151C2×C32×C4C2×D411yescharacteristic
152C2×C32×C4C2×D411yescharacteristic
153C2×C32×C4#4111yescharacteristic
154C2×C32×C4#4111yescharacteristic
155C2×C3×Dic3—22no—
156C2×C3×Dic3—21no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—41no—
160C2×C3×Dic3—41no—
161C2×C3×Dic3—42no—
162C2×C3×Dic3—82no—
163C32×C8—22no—
164C3×C3⋊C8—41no—
165C2×#137C3×D311yescharacteristic
166#520—31no—
167C2×#104—31no—
168#419—31no—
169#391—31no—
170#343—31no—
171#339—31no—
172C3×#41—21no—
173C3×#41—21no—
174C3×#41—21no—
175C3×#41—21no—
176C3×#41—22no—
177C3×#41—22no—
178C3×#41—41no—
179C3×#41—41no—
180C3×#41C2×Dic312yes—
181C3×#41#6712yes—
182C3×#41C2×C3×C412yes—
183C3×#41C3×D412yes—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—22no—
187C2×C3×Q8—22no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—22no—
197C2×C3×Q8—21no—
198C2×C3×Q8—41no—
199C2×C3×Q8—41no—
200C2×C3×Q8—41no—
201C2×C3×Q8C2×Dic312yescomplemented
202C2×C3×Q8#6711yescharacteristic
203C2×C3×Q8C2×C3×C412yescomplemented
204C2×C3×Q8C3×D411yescharacteristic
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—41no—
209C2×C3×D4#6711yescharacteristic
210C2×C3×D4C3×D411yescharacteristic
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—32no—
217C22×C3×C4—41no—
218C22×C3×C4—41no—
219C22×C3×C4—61no—
220C22×C3×C4—62no—
221C22×C3×C4C22×D311yescharacteristic
222C22×C3×C4C23×C311yescharacteristic
223C22×Dic3—21no—
224C22×Dic3—22no—
225C22×Dic3C3×D412yes—
226C3×Q16—28no—
227C3×Q16—24no—
228C3×Q16—24no—
229C3×SD16—28no—
230C3×SD16—24no—
231C3×SD16—24no—
232C3×M16—22no—
233C3×M16—21no—
234C3×M16—21no—
235C3×M16—22no—
236C3×M16—61no—
237C3×M16—61no—
238C3×M16C2×Dic312yescomplemented
239C3×M16C2×C3×C412yescomplemented
240C2×C3×C8—22no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—61no—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—31no—
246C3×C4⋊C4—62no—
247C3×C4⋊C4—61no—
248C3×C4⋊C4—61no—
249C3×C22⋊C4—62no—
250C3×C22⋊C4—61no—
251C3×C42—62no—
252C3×C42—61no—
253C3×C42—61no—
254C3×C42—61no—
255#217—22no—
256#217—21no—
257C4⋊Dic3—22no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3C3×D411yescharacteristic
261C4⋊Dic3C3×D411yescharacteristic
262C4×Dic3—22no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266#208—21no—
267#208—21no—
268C2×C3⋊C8—21no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—22no—
272C22×C32—41no—
273C22×C32—41no—
274C22×C32C4×D411yescharacteristic
275C22×C32#12011yescharacteristic
276C22×C32#12111yescharacteristic
277C32×C4—22no—
278C32×C4—42no—
279C32×C4C2×C22⋊C411yescharacteristic
280C32×C4C4×D411yescharacteristic
281C32×C4#12111yescharacteristic
282C32×C4#12311yescharacteristic
283C3×Dic3—42no—
284C3×Dic3—42no—
285C3×Dic3—41no—
286C3×Dic3—41no—
287C2×#41C2×C3×D311yescharacteristic
288C22×Q8C2×C3×D311yescharacteristic
289#137—22no—
290#137C3×Dic314yescomplemented
291C2×Q16—21no—
292C2×SD16—21no—
293C2×M16—31no—
294C2×M16C2×C3×D311yescharacteristic
295C4×Q8—61no—
296C4×D4—62no—
297C4×D4—61no—
298#117—31no—
299#117—61no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—61no—
302C2×C42—31no—
303#104—62no—
304#103—61no—
305#103—61no—
306#95—62no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—41no—
311C23×C3D3×D411yescharacteristic
312C23×C3C2×C3×D411yescharacteristic
313C3×Q8—21no—
314C3×Q8—21no—
315C3×Q8—24no—
316C3×Q8—22no—
317C3×Q8—22no—
318C3×Q8—22no—
319C3×Q8—22no—
320C3×Q8—22no—
321C3×Q8—42no—
322C3×Q8—41no—
323C3×Q8—42no—
324C3×Q8—42no—
325C3×Q8—41no—
326C3×Q8—41no—
327C3×Q8—41no—
328C3×Q8—41no—
329C3×Q8#21712yes—
330C3×Q8C3×C22⋊C412yes—
331C3×D4—22no—
332C3×D4—22no—
333C3×D4—22no—
334C3×D4—22no—
335C3×D4—21no—
336C3×D4—21no—
337C3×D4—22no—
338C3×D4—41no—
339C3×D4—42no—
340C3×D4—42no—
341C3×D4#21712yes—
342C3×D4C3×C22⋊C412yes—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—41no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—62no—
368C2×C3×C4—61no—
369C2×C3×C4—61no—
370C2×C3×C4—62no—
371C2×C3×C4—121no—
372C2×C3×C4—121no—
373C2×C3×C4—122no—
374C2×C3×C4—242no—
375C2×C3×C4D3×D411yescharacteristic
376C2×C3×C4#23711yescharacteristic
377C2×C3×C4#23711yescharacteristic
378C2×C3×C4C22×Dic311yescharacteristic
379C2×C3×C4C2×#6711yescharacteristic
380C2×C3×C4C2×#6711yescharacteristic
381C2×C3×C4C22×C3×C411yescharacteristic
382C2×C3×C4C2×C3×D411yescharacteristic
383C2×C3×C4C2×C3×D411yescharacteristic
384C2×C3×C4C2×C3×D411yescharacteristic
385C2×C3×C4C3×#4111yescharacteristic
386C2×C3×C4C3×#4111yescharacteristic
387C2×Dic3—22no—
388C2×Dic3—21no—
389C2×Dic3—22no—
390C2×Dic3—21no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—42no—
394C2×Dic3—82no—
395C3×C8—24no—
396C3×C8—22no—
397C3×C8—22no—
398C3×C8—121no—
399C3⋊C8—41no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—42no—
403C2×C32#38611yescharacteristic
404C2×C32#45511yescharacteristic
405C2×C32#45611yescharacteristic
406#41—21no—
407#41—21no—
408#41C2×C3×Dic312yes—
409#41C3×#6712yes—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8C2×C3×Dic312yescomplemented
416C2×Q8C3×#6711yescharacteristic
417C2×D4—21no—
418C2×D4C3×#6711yescharacteristic
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—32no—
422C22×C4—61no—
423C22×C4—62no—
424C22×C4C22×C3×D311yescharacteristic
425Q16—24no—
426SD16—24no—
427M16—21no—
428M16—61no—
429M16—61no—
430M16C2×C3×Dic312yescomplemented
431C2×C8—21no—
432C2×C8—61no—
433C4⋊C4—31no—
434C4⋊C4—31no—
435C4⋊C4—62no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C22⋊C4—62no—
439C22⋊C4—61no—
440C42—62no—
441C42—61no—
442C42—61no—
443C42—61no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—41no—
454C22×C3—41no—
455C22×C3—42no—
456C22×C3—41no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—81no—
461C22×C3—81no—
462C22×C3D4×Dic311yescharacteristic
463C22×C3#93711yescharacteristic
464C22×C3#93911yescharacteristic
465C22×C3C3×C4×D411yescharacteristic
466C22×C3C3×#12011yescharacteristic
467C22×C3C3×#12111yescharacteristic
468C3×C4—22no—
469C3×C4—22no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—42no—
475C3×C4—44no—
476C3×C4—42no—
477C3×C4—42no—
478C3×C4—122no—
479C3×C4—122no—
480C3×C4—121no—
481C3×C4—121no—
482C3×C4D4×Dic311yescharacteristic
483C3×C4#93511yescharacteristic
484C3×C4#93911yescharacteristic
485C3×C4C2×#21711yescharacteristic
486C3×C4C2×C3×C22⋊C411yescharacteristic
487C3×C4C3×C4×D411yescharacteristic
488C3×C4C3×#12111yescharacteristic
489C3×C4C3×#12311yescharacteristic
490Dic3—42no—
491Dic3—42no—
492Dic3—41no—
493Dic3—41no—
494C32#188211yescharacteristic, complemented, Sylow
495C23—21no—
496C23C3×D3×D411yescharacteristic
497Q8—21no—
498Q8—22no—
499Q8—22no—
500Q8—41no—
501Q8—41no—
502Q8C3×#21712yes—
503D4—22no—
504D4—22no—
505D4—21no—
506D4C3×#21712yes—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—62no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—62no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—122no—
520C2×C4—242no—
521C2×C4C3×D3×D411yescharacteristic
522C2×C4C3×#23711yescharacteristic
523C2×C4C3×#23711yescharacteristic
524C2×C4C22×C3×Dic311yescharacteristic
525C2×C4C2×C3×#6711yescharacteristic
526C2×C4C2×C3×#6711yescharacteristic
527C8—22no—
528C8—121no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—41no—
537C2×C3—41no—
538C2×C3—42no—
539C2×C3—42no—
540C2×C3—82no—
541C2×C3#520211yescharacteristic
542C2×C3#520511yescharacteristic
543C2×C3#522011yescharacteristic
544C2×C3C3×#38611yescharacteristic
545C2×C3C3×#45511yescharacteristic
546C2×C3C3×#45611yescharacteristic
547C22—21no—
548C22—21no—
549C22—22no—
550C22—41no—
551C22—41no—
552C22C3×D4×Dic311yescharacteristic
553C22C3×#93711yescharacteristic
554C22C3×#93911yescharacteristic
555C4—22no—
556C4—42no—
557C4—122no—
558C4—122no—
559C4—121no—
560C4—121no—
561C4C3×D4×Dic311yescharacteristic
562C4C3×#93511yescharacteristic
563C4C3×#93911yescharacteristic
564C4C2×C3×#21711yescharacteristic
565C3—21no—
566C3#7210011yescharacteristic, direct product factor
567C3C3×#188211yescharacteristic, complemented
568C2—21no—
569C2—21no—
570C2—42no—
571C2C3×#520211yescharacteristic
572C2C3×#520511yescharacteristic
573C2C3×#522011yescharacteristic
subgroupG/H
centerC22×C3#937
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#137C2
socleC22×C32#120
prime radicalC2×C32×#41C22
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×#137D3
hypocenterC3C3×#1882
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#137

Complements

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

NH ≅ G/Ntype
4. C3×#5229C2semidirect
5. C3×#5225C2semidirect
6. C3×#4555C2semidirect
7. C3×#4554C2semidirect
9. #72100C3direct (N × H)
13. C32×#137C4semidirect
39. #5229C2×C3semidirect
40. #5225C2×C3semidirect
43. #4555C2×C3semidirect
44. #4554C2×C3semidirect
54. C2×C32×Q8C2×C4semidirect
67. C32×M16C2×C4semidirect
94. C3×#137Dic3semidirect
95. C3×#137C3×C4semidirect
201. C2×C3×Q8C2×Dic3semidirect
203. C2×C3×Q8C2×C3×C4semidirect
238. C3×M16C2×Dic3semidirect
239. C3×M16C2×C3×C4semidirect
290. #137C3×Dic3semidirect
415. C2×Q8C2×C3×Dic3semidirect
430. M16C2×C3×Dic3semidirect
494. C32#1882semidirect
566. C3#72100direct (N × H)
567. C3C3×#1882semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1372
C3×#58272
C2×C3×#8372
C3×#52292
C3×#52252
C3×#45552
C3×#45542
C3×#18823
#721003