Finite Group Databasebeta

← C3×#72108 #11792179

Subgroups — C3×#72108 #11792179

Order 1152.

853 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#137C211yesmaximal, characteristic
2C3×#5825C211yesmaximal, characteristic
3C2×C3×#944C211yesmaximal, characteristic, complemented
4C2×C3×#837C211yesmaximal, characteristic
5C3×#5225C211yesmaximal, characteristic, complemented
6C2×C3×#824C211yesmaximal, characteristic, complemented
7C3×#4555C211yesmaximal, characteristic, complemented
8C3×#2005—31nomaximal
9#72108C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C22×C32×Q8C2211yescharacteristic
12C22×C3×Dic6C2211yescharacteristic
13C32×#137—22no—
14C32×#137—22no—
15C2×C32×Q16—21no—
16C2×C32×SD16—21no—
17C2×C32×M16C2211yescharacteristic
18C3×#944—22no—
19C3×#944—21no—
20C3×#944—22no—
21C3×#936—21no—
22C3×#935—22no—
23C3×#926C2211yescharacteristic
24C3×#924—22no—
25C2×C3×Dic3⋊C4—22no—
26C2×C3×C4×Dic3C2211yescharacteristic
27C2×C3×#208C2211yescharacteristic
28C3×#837—22no—
29C3×#835—22no—
30C3×#824—22no—
31C2×C3×#137—21no—
32C2×C3×#137D311yescharacteristic
33C2×C3×#137C2×C311yescharacteristic
34C3×#536—31no—
35C2×C3×C4⋊Q8—31no—
36#5825C2×C311yescharacteristic
37C2×#944C2×C311yescharacteristic, complemented
38C2×C3×#104—31no—
39C3×#419—31no—
40C2×C3×#101—31no—
41C2×#837C2×C311yescharacteristic
42#5225C2×C311yescharacteristic, complemented
43C2×#824C2×C311yescharacteristic, complemented
44C3×#339—31no—
45#4555C2×C311yescharacteristic, complemented
46C32×#41—21no—
47C32×#41—21no—
48C32×#41—21no—
49C32×#41—21no—
50C2×C32×Q8—21no—
51C2×C32×Q8—21no—
52C2×C32×Q8—21no—
53C2×C32×Q8—21no—
54C2×C32×Q8—21no—
55C2×C32×Q8D412yes—
56C2×C32×Q8D411yescharacteristic
57C2×C32×D4—21no—
58C2×C32×D4D411yescharacteristic
59C22×C32×C4—21no—
60C22×C32×C4—21no—
61C22×C32×C4C2311yescharacteristic
62C22×C3×Dic3—21no—
63C22×C3×Dic3D412yes—
64C2×C3×Dic6—21no—
65C2×C3×Dic6—21no—
66C2×C3×Dic6—21no—
67C2×C3×Dic6—21no—
68C2×C3×Dic6—21no—
69C2×C3×Dic6—21no—
70C32×Q16—24no—
71C32×SD16—24no—
72C32×M16—21no—
73C32×M16D412yes—
74C2×C32×C8—21no—
75C3×#217—24no—
76C3×#217—21no—
77C3×C4⋊Dic3D412yes—
78C3×Dic3⋊C4—28no—
79C3×Dic3⋊C4—24no—
80C3×C4×Dic3—21no—
81C3×C4×Dic3—21no—
82C3×C4×Dic3D412yes—
83C3×#208—21no—
84C3×#208—21no—
85C2×C3×C3⋊C8—21no—
86#2005—31noSylow
87C2×C3×#41—21no—
88C2×C3×#41C2×D311yescharacteristic
89C2×C3×#41C22×C311yescharacteristic
90C22×C3×Q8—21no—
91C22×C3×Q8—31no—
92C22×C3×Q8C2×D311yescharacteristic
93C22×C3×Q8C22×C311yescharacteristic
94C22×Dic6C22×C311yescharacteristic
95C3×#137—24no—
96C3×#137—24no—
97C3×#137—22no—
98C3×#137—22no—
99C3×#137—22no—
100C3×#137—22no—
101C2×C3×Q16—22no—
102C2×C3×Q16—21no—
103C2×C3×Q16—21no—
104C2×C3×SD16—22no—
105C2×C3×SD16—21no—
106C2×C3×SD16—21no—
107C2×C3×M16—21no—
108C2×C3×M16—31no—
109C2×C3×M16C2×D311yescharacteristic
110C2×C3×M16C22×C311yescharacteristic
111C3×C4⋊Q8—62no—
112C3×C4⋊Q8—62no—
113C3×C4⋊Q8—61no—
114C3×#124—61no—
115C3×#123—62no—
116C3×C22⋊Q8—62no—
117C3×#117—31no—
118C2×C3×C4⋊C4—62no—
119C2×C3×C42—31no—
120#944—22no—
121#944—21no—
122#944—22no—
123#936—21no—
124#935—22no—
125#926C22×C311yescharacteristic
126#924—22no—
127C2×Dic3⋊C4—22no—
128C2×C4×Dic3C22×C311yescharacteristic
129C2×#208C22×C311yescharacteristic
130C3×#104—62no—
131C3×#103—62no—
132C3×#101—62no—
133#837—22no—
134#835—22no—
135#824—22no—
136C23×C32—21no—
137C23×C32C2×D411yescharacteristic
138C32×Q8—22no—
139C32×Q8—21no—
140C32×Q8—22no—
141C32×Q8—21no—
142C32×Q8—41no—
143C32×Q8—41no—
144C32×D4—21no—
145C32×D4—21no—
146C32×D4—41no—
147C32×D4—41no—
148C2×C32×C4—21no—
149C2×C32×C4—22no—
150C2×C32×C4—41no—
151C2×C32×C4—41no—
152C2×C32×C4—41no—
153C2×C32×C4—41no—
154C2×C32×C4C2×D411yescharacteristic
155C2×C32×C4C2×D411yescharacteristic
156C2×C32×C4C2×D411yescharacteristic
157C2×C32×C4C2×D411yescharacteristic
158C2×C32×C4C2×D411yescharacteristic
159C2×C32×C4#4111yescharacteristic
160C2×C3×Dic3—24no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—22no—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—22no—
165C2×C3×Dic3—22no—
166C3×Dic6—41no—
167C3×Dic6—41no—
168C3×Dic6—41no—
169C3×Dic6—41no—
170C32×C8—22no—
171C3×C3⋊C8—41no—
172C2×#137C3×D311yescharacteristic
173#536—31no—
174C2×C4⋊Q8—31no—
175C2×#104—31no—
176#419—31no—
177C2×#101—31no—
178#339—31no—
179C3×#41—22no—
180C3×#41—22no—
181C3×#41—21no—
182C3×#41—21no—
183C3×#41—21no—
184C3×#41—21no—
185C3×#41—21no—
186C3×#41—21no—
187C3×#41—21no—
188C3×#41—21no—
189C3×#41—41no—
190C3×#41—41no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—22no—
194C2×C3×Q8—22no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—22no—
205C2×C3×Q8—41no—
206C2×C3×Q8—41no—
207C2×C3×Q8—41no—
208C2×C3×Q8—61no—
209C2×C3×Q8—61no—
210C2×C3×Q8—61no—
211C2×C3×Q8—61no—
212C2×C3×Q8—61no—
213C2×C3×Q8—61no—
214C2×C3×Q8#6711yescharacteristic
215C2×C3×Q8#6712yes—
216C2×C3×Q8C3×D411yescharacteristic
217C2×C3×Q8C3×D412yes—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—21no—
221C2×C3×D4—41no—
222C2×C3×D4#6711yescharacteristic
223C2×C3×D4C3×D411yescharacteristic
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4—32no—
230C22×C3×C4—41no—
231C22×C3×C4—41no—
232C22×C3×C4—61no—
233C22×C3×C4C22×D311yescharacteristic
234C22×C3×C4C23×C311yescharacteristic
235C22×Dic3—21no—
236C22×Dic3C3×D412yes—
237C2×Dic6—21no—
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6—21no—
241C2×Dic6—21no—
242C2×Dic6—21no—
243C3×Q16—28no—
244C3×Q16—24no—
245C3×Q16—24no—
246C3×SD16—28no—
247C3×SD16—24no—
248C3×SD16—24no—
249C3×M16—22no—
250C3×M16—21no—
251C3×M16—21no—
252C3×M16—22no—
253C3×M16—61no—
254C3×M16—61no—
255C3×M16#6712yes—
256C3×M16C3×D412yes—
257C2×C3×C8—22no—
258C2×C3×C8—21no—
259C2×C3×C8—21no—
260C2×C3×C8—61no—
261C3×C4⋊C4—32no—
262C3×C4⋊C4—68no—
263C3×C4⋊C4—64no—
264C3×C22⋊C4—64no—
265C3×C22⋊C4—61no—
266C3×C42—32no—
267C3×C42—61no—
268C3×C42—61no—
269#217—24no—
270#217—21no—
271C4⋊Dic3C3×D412yes—
272Dic3⋊C4—28no—
273Dic3⋊C4—24no—
274C4×Dic3—21no—
275C4×Dic3—21no—
276C4×Dic3C3×D412yes—
277#208—21no—
278#208—21no—
279C2×C3⋊C8—21no—
280C22×C32—22no—
281C22×C32—41no—
282C22×C32—42no—
283C22×C32#12011yescharacteristic
284C22×C32#12011yescharacteristic
285C22×C32#12111yescharacteristic
286C32×C4—41no—
287C32×C4—42no—
288C32×C4#12011yescharacteristic
289C32×C4#12111yescharacteristic
290C32×C4#12111yescharacteristic
291C32×C4#12711yescharacteristic
292C3×Dic3—24no—
293C3×Dic3—42no—
294C3×Dic3—42no—
295C2×#41C2×C3×D311yescharacteristic
296C22×Q8—31no—
297C22×Q8C2×C3×D311yescharacteristic
298#137—22no—
299#137—22no—
300C2×Q16—21no—
301C2×SD16—21no—
302C2×M16—31no—
303C2×M16C2×C3×D311yescharacteristic
304C4⋊Q8—62no—
305C4⋊Q8—62no—
306C4⋊Q8—61no—
307#124—61no—
308#123—62no—
309C22⋊Q8—62no—
310#117—31no—
311C2×C4⋊C4—62no—
312C2×C42—31no—
313#104—62no—
314#103—62no—
315#101—62no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—41no—
320C23×C3D3×D411yescharacteristic
321C23×C3C2×C3×D411yescharacteristic
322C3×Q8—24no—
323C3×Q8—22no—
324C3×Q8—22no—
325C3×Q8—21no—
326C3×Q8—22no—
327C3×Q8—21no—
328C3×Q8—22no—
329C3×Q8—21no—
330C3×Q8—22no—
331C3×Q8—21no—
332C3×Q8—42no—
333C3×Q8—41no—
334C3×Q8—42no—
335C3×Q8—42no—
336C3×Q8—41no—
337C3×Q8—41no—
338C3×Q8—41no—
339C3×Q8—41no—
340C3×Q8—121no—
341C3×Q8—121no—
342C3×Q8—121no—
343C3×Q8—121no—
344C3×D4—22no—
345C3×D4—21no—
346C3×D4—21no—
347C3×D4—21no—
348C3×D4—21no—
349C3×D4—41no—
350C3×D4—42no—
351C3×D4—42no—
352C3×D4—41no—
353C3×D4—41no—
354C3×D4—41no—
355C3×D4—41no—
356C2×C3×C4—22no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—22no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—41no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—64no—
381C2×C3×C4—62no—
382C2×C3×C4—62no—
383C2×C3×C4—62no—
384C2×C3×C4—62no—
385C2×C3×C4—62no—
386C2×C3×C4D3×D411yescharacteristic
387C2×C3×C4D3×D411yescharacteristic
388C2×C3×C4#23711yescharacteristic
389C2×C3×C4C2×#6711yescharacteristic
390C2×C3×C4C2×#6711yescharacteristic
391C2×C3×C4C2×#6711yescharacteristic
392C2×C3×C4C2×C3×D411yescharacteristic
393C2×C3×C4C2×C3×D411yescharacteristic
394C2×C3×C4C2×C3×D411yescharacteristic
395C2×C3×C4C2×C3×D411yescharacteristic
396C2×C3×C4C2×C3×D411yescharacteristic
397C2×C3×C4C3×#4111yescharacteristic
398C2×Dic3—24no—
399C2×Dic3—22no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3—22no—
403C2×Dic3—22no—
404Dic6—41no—
405Dic6—41no—
406Dic6—41no—
407Dic6—41no—
408C3×C8—24no—
409C3×C8—22no—
410C3×C8—22no—
411C3×C8—121no—
412C3⋊C8—41no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—81no—
416C2×C32#39211yescharacteristic
417C2×C32#45612yes—
418#41—21no—
419#41—21no—
420#41—21no—
421#41—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8C3×#6711yescharacteristic
434C2×Q8C3×#6712yes—
435C2×D4—21no—
436C2×D4C3×#6711yescharacteristic
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—32no—
440C22×C4—61no—
441C22×C4C22×C3×D311yescharacteristic
442Q16—24no—
443SD16—24no—
444M16—21no—
445M16—61no—
446M16—61no—
447M16C3×#6712yes—
448C2×C8—21no—
449C2×C8—61no—
450C4⋊C4—32no—
451C4⋊C4—68no—
452C4⋊C4—64no—
453C22⋊C4—64no—
454C22⋊C4—61no—
455C42—32no—
456C42—61no—
457C42—61no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—41no—
467C22×C3—41no—
468C22×C3—42no—
469C22×C3—82no—
470C22×C3#93711yescharacteristic
471C22×C3#93711yescharacteristic
472C22×C3#93811yescharacteristic
473C22×C3C3×#12011yescharacteristic
474C22×C3C3×#12011yescharacteristic
475C22×C3C3×#12111yescharacteristic
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—42no—
481C3×C4—44no—
482C3×C4—41no—
483C3×C4—42no—
484C3×C4—41no—
485C3×C4—42no—
486C3×C4—64no—
487C3×C4—122no—
488C3×C4—122no—
489C3×C4#93911yescharacteristic
490C3×C4#93911yescharacteristic
491C3×C4#94011yescharacteristic
492C3×C4#95311yescharacteristic
493C3×C4C3×#12011yescharacteristic
494C3×C4C3×#12111yescharacteristic
495C3×C4C3×#12111yescharacteristic
496C3×C4C3×#12711yescharacteristic
497Dic3—24no—
498Dic3—42no—
499Dic3—42no—
500C32#200511yescharacteristic, complemented, Sylow
501C23—21no—
502C23C3×D3×D411yescharacteristic
503Q8—22no—
504Q8—21no—
505Q8—22no—
506Q8—21no—
507Q8—41no—
508Q8—41no—
509Q8—121no—
510Q8—121no—
511Q8—121no—
512Q8—121no—
513D4—21no—
514D4—21no—
515D4—41no—
516D4—41no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—64no—
524C2×C4—62no—
525C2×C4—62no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4C3×D3×D411yescharacteristic
530C2×C4C3×D3×D411yescharacteristic
531C2×C4C3×#23711yescharacteristic
532C2×C4C2×C3×#6711yescharacteristic
533C2×C4C2×C3×#6711yescharacteristic
534C2×C4C2×C3×#6711yescharacteristic
535C8—22no—
536C8—121no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—22no—
542C2×C3—21no—
543C2×C3—41no—
544C2×C3—41no—
545C2×C3—82no—
546C2×C3—81no—
547C2×C3—81no—
548C2×C3#520512yes—
549C2×C3#522311yescharacteristic
550C2×C3C3×#39211yescharacteristic
551C2×C3C3×#45612yes—
552C22—22no—
553C22—42no—
554C22—41no—
555C22C3×#93711yescharacteristic
556C22C3×#93711yescharacteristic
557C22C3×#93811yescharacteristic
558C4—41no—
559C4—42no—
560C4—64no—
561C4—122no—
562C4—122no—
563C4C3×#93911yescharacteristic
564C4C3×#93911yescharacteristic
565C4C3×#94011yescharacteristic
566C4C3×#95311yescharacteristic
567C3—21no—
568C3#7210811yescharacteristic, direct product factor
569C3C3×#200511yescharacteristic, complemented
570C2—21no—
571C2—21no—
572C2—81no—
573C2C3×#520512yes—
574C2C3×#522311yescharacteristic
subgroupG/H
centerC22×C3#937
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#137C2
socleC22×C32#120
prime radicalC2×C32×#41C22
hypercenterC2×C3×#137D3
hypocenterC3C3×#2005
LCSG ▷ C22×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
3. C2×C3×#944C2semidirect
5. C3×#5225C2semidirect
6. C2×C3×#824C2semidirect
7. C3×#4555C2semidirect
9. #72108C3direct (N × H)
37. C2×#944C2×C3semidirect
42. #5225C2×C3semidirect
43. C2×#824C2×C3semidirect
45. #4555C2×C3semidirect
500. C32#2005semidirect
568. C3#72108direct (N × H)
569. C3C3×#2005semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1372
C3×#58252
C2×C3×#9442
C2×C3×#8372
C3×#52252
C2×C3×#8242
C3×#45552
C3×#20053
#721083