Finite Group Databasebeta

← C3×#72072 #11792574

Subgroups — C3×#72072 #11792574

Order 1152.

909 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×Q16C211yesmaximal, characteristic, complemented
2C2×C3×#947C211yesmaximal, characteristic
3C22×C3×#215C211yesmaximal, characteristic
4C3×#5229C211yesmaximal, characteristic, complemented
5C2×C3×#835C211yesmaximal, characteristic, complemented
6C2×C3×#821C211yesmaximal, characteristic
7C3×#4550C211yesmaximal, characteristic, complemented
8C3×#2015—31nomaximal
9#72072C311yesmaximal, characteristic, direct product factor
10C22×C32×Q8C2211yescharacteristic
11C22×C32×Q8C2211yescharacteristic
12C22×C3×D12C2211yescharacteristic
13C2×C32×Q16—22no—
14C2×C32×Q16—22no—
15C2×C32×Q16—22no—
16C22×C32×C8C2211yescharacteristic
17C3×#947—22no—
18C3×#947—22no—
19C2×C3×#215—22no—
20C2×C3×#215—22no—
21C2×C3×#215—22no—
22C2×C3×(C2×D3)⋊C4—22no—
23C2×C3×C4⋊Dic3C2211yescharacteristic
24C2×C3×C4×Dic3C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26C3×#835—22no—
27C3×C3⋊#95—22no—
28C3×#821—22no—
29C22×C3×Q16—21no—
30C22×C3×Q16D311yescharacteristic, complemented
31C22×C3×Q16C2×C311yescharacteristic, complemented
32C22×C3×SD16—31no—
33C2×C3×#124—31no—
34C2×#947C2×C311yescharacteristic
35C22×#215C2×C311yescharacteristic
36C2×C3×#103—31no—
37C2×C3×#102—31no—
38C3×#391—31no—
39#5229C2×C311yescharacteristic, complemented
40C2×#835C2×C311yescharacteristic, complemented
41C2×#821C2×C311yescharacteristic
42C3×#336—31no—
43#4550C2×C311yescharacteristic, complemented
44C23×C3×D3—21no—
45C2×C32×Q8—21no—
46C2×C32×Q8—21no—
47C2×C32×Q8—21no—
48C2×C32×Q8—21no—
49C2×C32×Q8—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×Q8D412yes—
57C22×C32×C4—21no—
58C22×C32×C4—21no—
59C22×C32×C4C2311yescharacteristic
60C22×C3×Dic3—21no—
61C22×C3×Dic3D412yes—
62C2×C3×D12—21no—
63C2×C3×D12—21no—
64C2×C3×D12—21no—
65C2×C3×D12—21no—
66C2×C3×D12—21no—
67C2×C3×D12D412yes—
68C32×Q16—28no—
69C2×C32×C8—21no—
70C2×C32×C8—21no—
71C2×C32×C8D412yes—
72C3×#215—28no—
73C3×(C2×D3)⋊C4—28no—
74C3×C4⋊Dic3—21no—
75C3×C4⋊Dic3—21no—
76C3×C4×Dic3—21no—
77C3×C4×Dic3—21no—
78C2×C3×C3⋊C8—21no—
79C2×C3×C3⋊C8—21no—
80C2×C3×C3⋊C8D412yes—
81#2015—31noSylow
82C22×C3×Q8—21no—
83C22×C3×Q8—21no—
84C22×C3×Q8C2×D311yescharacteristic
85C22×C3×Q8C2×D311yescharacteristic
86C22×C3×Q8C22×C311yescharacteristic
87C22×C3×Q8C22×C311yescharacteristic
88C22×C3×D4—31no—
89C22×D12C22×C311yescharacteristic
90C2×C3×Q16—24no—
91C2×C3×Q16—24no—
92C2×C3×Q16—24no—
93C2×C3×Q16—22no—
94C2×C3×Q16—22no—
95C2×C3×Q16—22no—
96C2×C3×Q16—22no—
97C2×C3×Q16—22no—
98C2×C3×Q16—22no—
99C2×C3×SD16—62no—
100C2×C3×SD16—62no—
101C2×C3×SD16—62no—
102C22×C3×C8—21no—
103C22×C3×C8—31no—
104C22×C3×C8C2×D311yescharacteristic
105C22×C3×C8C22×C311yescharacteristic
106C3×#124—62no—
107C3×#124—62no—
108C2×C3×C4⋊C4—31no—
109C2×C3×C22⋊C4—62no—
110C2×C3×C42—31no—
111#947—22no—
112#947—22no—
113C2×#215—22no—
114C2×#215—22no—
115C2×#215—22no—
116C2×(C2×D3)⋊C4—22no—
117C2×C4⋊Dic3C22×C311yescharacteristic
118C2×C4×Dic3C22×C311yescharacteristic
119C22×C3⋊C8C22×C311yescharacteristic
120C3×#103—62no—
121C3×#102—62no—
122C3×#95—62no—
123#835—22no—
124C3⋊#95—22no—
125#821—22no—
126C23×C32C2×D411yescharacteristic
127C22×C3×D3—22no—
128C22×C3×D3—41no—
129C22×C3×D3—41no—
130C22×C3×D3—42no—
131C22×C3×D3—42no—
132C32×Q8—22no—
133C32×Q8—22no—
134C32×Q8—22no—
135C32×Q8—22no—
136C32×Q8—41no—
137C32×Q8—41no—
138C32×Q8—41no—
139C32×Q8—41no—
140C2×C32×C4—22no—
141C2×C32×C4—22no—
142C2×C32×C4—41no—
143C2×C32×C4—41no—
144C2×C32×C4—41no—
145C2×C32×C4—41no—
146C2×C32×C4C2×D411yescharacteristic
147C2×C32×C4C2×D411yescharacteristic
148C2×C32×C4C2×D411yescharacteristic
149C2×C32×C4C2×D411yescharacteristic
150C2×C32×C4C2×D411yescharacteristic
151C2×C32×C4#4111yescharacteristic
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—41no—
156C2×C3×Dic3—42no—
157C3×D12—22no—
158C3×D12—22no—
159C3×D12—41no—
160C3×D12—41no—
161C32×C8—22no—
162C3×C3⋊C8—22no—
163C22×Q16C3×D311yescharacteristic, complemented
164C22×SD16—31no—
165C2×#124—31no—
166C2×#103—31no—
167C2×#102—31no—
168#391—31no—
169#336—31no—
170C24×C3—61no—
171C23×D3—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8—21no—
176C2×C3×Q8—22no—
177C2×C3×Q8—22no—
178C2×C3×Q8—22no—
179C2×C3×Q8—22no—
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
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—22no—
198C2×C3×Q8—41no—
199C2×C3×Q8—41no—
200C2×C3×Q8—41no—
201C2×C3×Q8—41no—
202C2×C3×Q8—41no—
203C2×C3×Q8—41no—
204C2×C3×Q8#6712yes—
205C2×C3×Q8#6712yes—
206C2×C3×Q8C3×D412yes—
207C2×C3×Q8C3×D412yes—
208C2×C3×D4—32no—
209C2×C3×D4—61no—
210C2×C3×D4—61no—
211C2×C3×D4—61no—
212C2×C3×D4—61no—
213C2×C3×D4—61no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—32no—
220C22×C3×C4—41no—
221C22×C3×C4—41no—
222C22×C3×C4—61no—
223C22×C3×C4C22×D311yescharacteristic
224C22×C3×C4C23×C311yescharacteristic
225C22×Dic3—21no—
226C22×Dic3C3×D412yes—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12C3×D412yes—
233C3×Q16—216no—
234C3×Q16—28no—
235C3×Q16—28no—
236C3×SD16—68no—
237C2×C3×C8—22no—
238C2×C3×C8—22no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—22no—
244C2×C3×C8—32no—
245C2×C3×C8—61no—
246C2×C3×C8—61no—
247C2×C3×C8#6712yes—
248C2×C3×C8C3×D412yes—
249C3×C4⋊C4—61no—
250C3×C4⋊C4—61no—
251C3×C22⋊C4—68no—
252C3×C42—61no—
253C3×C42—61no—
254#215—28no—
255(C2×D3)⋊C4—28no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8C3×D412yes—
263C22×C32#12011yescharacteristic
264C22×C32#12111yescharacteristic
265C22×C32#12711yescharacteristic
266C22×C32#13512yes—
267C22×C32#13712yes—
268C2×C3×D3—42no—
269C2×C3×D3—44no—
270C2×C3×D3—82no—
271C2×C3×D3—81no—
272C2×C3×D3—81no—
273C32×C4—42no—
274C32×C4—42no—
275C32×C4#12011yescharacteristic
276C32×C4#12011yescharacteristic
277C32×C4#12111yescharacteristic
278C32×C4#12111yescharacteristic
279C3×Dic3—42no—
280C3×Dic3—81no—
281C22×Q8C2×C3×D311yescharacteristic
282C22×Q8C2×C3×D311yescharacteristic
283C22×D4—31no—
284C2×Q16—22no—
285C2×Q16—22no—
286C2×Q16—22no—
287C2×SD16—62no—
288C2×SD16—62no—
289C2×SD16—62no—
290C22×C8—31no—
291C22×C8C2×C3×D311yescharacteristic
292#124—62no—
293#124—62no—
294C2×C4⋊C4—31no—
295C2×C22⋊C4—62no—
296C2×C42—31no—
297#103—62no—
298#102—62no—
299#95—62no—
300C23×C3—21no—
301C23×C3—62no—
302C23×C3—122no—
303C23×C3—122no—
304C23×C3—121no—
305C23×C3—121no—
306C23×C3D3×D411yescharacteristic
307C23×C3C2×C3×D411yescharacteristic
308C22×D3—22no—
309C22×D3—41no—
310C22×D3—41no—
311C22×D3—42no—
312C22×D3—42no—
313C3×Q8—24no—
314C3×Q8—24no—
315C3×Q8—22no—
316C3×Q8—22no—
317C3×Q8—22no—
318C3×Q8—22no—
319C3×Q8—22no—
320C3×Q8—22no—
321C3×Q8—22no—
322C3×Q8—22no—
323C3×Q8—42no—
324C3×Q8—42no—
325C3×Q8—42no—
326C3×Q8—42no—
327C3×Q8—42no—
328C3×Q8—42no—
329C3×Q8—41no—
330C3×Q8—41no—
331C3×Q8—41no—
332C3×Q8—41no—
333C3×Q8—41no—
334C3×Q8—41no—
335C3×Q8—41no—
336C3×Q8—41no—
337C3×D4—62no—
338C3×D4—62no—
339C3×D4—121no—
340C3×D4—121no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—62no—
366C2×C3×C4—62no—
367C2×C3×C4—62no—
368C2×C3×C4—122no—
369C2×C3×C4—121no—
370C2×C3×C4D3×D411yescharacteristic
371C2×C3×C4D3×D411yescharacteristic
372C2×C3×C4#23711yescharacteristic
373C2×C3×C4C2×#6711yescharacteristic
374C2×C3×C4C2×#6711yescharacteristic
375C2×C3×C4C2×#6711yescharacteristic
376C2×C3×C4C2×C3×D411yescharacteristic
377C2×C3×C4C2×C3×D411yescharacteristic
378C2×C3×C4C2×C3×D411yescharacteristic
379C2×C3×C4C2×C3×D411yescharacteristic
380C2×C3×C4C2×C3×D411yescharacteristic
381C2×C3×C4C3×#4111yescharacteristic
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—41no—
386C2×Dic3—42no—
387D12—22no—
388D12—22no—
389D12—41no—
390D12—41no—
391C3×C8—24no—
392C3×C8—22no—
393C3×C8—22no—
394C3×C8—62no—
395C3⋊C8—22no—
396C2×C32#39211yescharacteristic
397C2×C32#45212yes—
398C2×C32#46412yes—
399C2×C32#49511yescharacteristic
400C2×C32#49711yescharacteristic
401C3×D3—82no—
402C24—61no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8C3×#6712yes—
414C2×Q8C3×#6712yes—
415C2×D4—32no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—61no—
420C2×D4—61no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—32no—
424C22×C4—61no—
425C22×C4C22×C3×D311yescharacteristic
426Q16—28no—
427SD16—68no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—32no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8C3×#6712yes—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C22⋊C4—68no—
437C42—61no—
438C42—61no—
439C22×C3—22no—
440C22×C3—22no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—124no—
445C22×C3—122no—
446C22×C3—242no—
447C22×C3—241no—
448C22×C3—241no—
449C22×C3#91812yes—
450C22×C3#91912yes—
451C22×C3#93711yescharacteristic
452C22×C3#93911yescharacteristic
453C22×C3#94011yescharacteristic
454C22×C3C3×#12011yescharacteristic
455C22×C3C3×#12111yescharacteristic
456C22×C3C3×#12711yescharacteristic
457C22×C3C3×#13512yes—
458C22×C3C3×#13712yes—
459C2×D3—42no—
460C2×D3—44no—
461C2×D3—82no—
462C2×D3—81no—
463C2×D3—81no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—44no—
469C3×C4—44no—
470C3×C4—42no—
471C3×C4—42no—
472C3×C4—42no—
473C3×C4—42no—
474C3×C4—122no—
475C3×C4—241no—
476C3×C4#93711yescharacteristic
477C3×C4#93811yescharacteristic
478C3×C4#93911yescharacteristic
479C3×C4#95311yescharacteristic
480C3×C4C3×#12011yescharacteristic
481C3×C4C3×#12011yescharacteristic
482C3×C4C3×#12111yescharacteristic
483C3×C4C3×#12111yescharacteristic
484Dic3—42no—
485Dic3—81no—
486C32#201511yescharacteristic, complemented, Sylow
487C23—62no—
488C23—122no—
489C23—122no—
490C23—121no—
491C23—121no—
492C23C3×D3×D411yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—22no—
496Q8—22no—
497Q8—41no—
498Q8—41no—
499Q8—41no—
500Q8—41no—
501D4—62no—
502D4—62no—
503D4—121no—
504D4—121no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—62no—
514C2×C4—122no—
515C2×C4—121no—
516C2×C4C3×D3×D411yescharacteristic
517C2×C4C3×D3×D411yescharacteristic
518C2×C4C3×#23711yescharacteristic
519C2×C4C2×C3×#6711yescharacteristic
520C2×C4C2×C3×#6711yescharacteristic
521C2×C4C2×C3×#6711yescharacteristic
522C8—22no—
523C8—62no—
524C2×C3—22no—
525C2×C3—22no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—242no—
530C2×C3#518512yes—
531C2×C3#518712yes—
532C2×C3#519011yescharacteristic
533C2×C3#519111yescharacteristic
534C2×C3#522311yescharacteristic
535C2×C3C3×#39211yescharacteristic
536C2×C3C3×#45212yes—
537C2×C3C3×#46412yes—
538C2×C3C3×#49511yescharacteristic
539C2×C3C3×#49711yescharacteristic
540D3—82no—
541C22—124no—
542C22—122no—
543C22—242no—
544C22—241no—
545C22—241no—
546C22C3×#91812yes—
547C22C3×#91912yes—
548C22C3×#93711yescharacteristic
549C22C3×#93911yescharacteristic
550C22C3×#94011yescharacteristic
551C4—42no—
552C4—42no—
553C4—122no—
554C4—241no—
555C4C3×#93711yescharacteristic
556C4C3×#93811yescharacteristic
557C4C3×#93911yescharacteristic
558C4C3×#95311yescharacteristic
559C3—21no—
560C3#7207211yescharacteristic, direct product factor
561C3C3×#201511yescharacteristic, complemented
562C2—242no—
563C2C3×#518512yes—
564C2C3×#518712yes—
565C2C3×#519011yescharacteristic
566C2C3×#519111yescharacteristic
567C2C3×#522311yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC22×C32×Q16C2
socleC23×C32C2×D4
prime radicalC22×C3×D12C22
hypercenterC22×C3×Q16D3
hypocenterC3C3×#2015
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C22×C3×Q16

Complements

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

NH ≅ G/Ntype
1. C22×C32×Q16C2semidirect
4. C3×#5229C2semidirect
5. C2×C3×#835C2semidirect
7. C3×#4550C2semidirect
9. #72072C3direct (N × H)
30. C22×C3×Q16D3semidirect
31. C22×C3×Q16C2×C3semidirect
39. #5229C2×C3semidirect
40. C2×#835C2×C3semidirect
43. #4550C2×C3semidirect
163. C22×Q16C3×D3semidirect
486. C32#2015semidirect
560. C3#72072direct (N × H)
561. C3C3×#2015semidirect

Maximal subgroups

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

subgroupindex
C22×C32×Q162
C2×C3×#9472
C22×C3×#2152
C3×#52292
C2×C3×#8352
C2×C3×#8212
C3×#45502
C3×#20153
#720723