Finite Group Databasebeta

← C3×#72095 #11792672

Subgroups — C3×#72095 #11792672

Order 1152.

913 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2C2×C3×#949C211yesmaximal, characteristic, complemented
3C22×C3×#213C211yesmaximal, characteristic, complemented
4C3×#5234C211yesmaximal, characteristic, complemented
5C2×C3×#833C211yesmaximal, characteristic, complemented
6C2×C3×#823C211yesmaximal, characteristic, complemented
7C3×#4558C211yesmaximal, characteristic, complemented
8C3×#2018—31nomaximal
9#72095C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C22×C32×D4C2211yescharacteristic
12C22×C3×D12C2211yescharacteristic
13C32×AGL(1,ℤ/8)—22no—
14C32×AGL(1,ℤ/8)—22no—
15C2×C32×SD16—21no—
16C2×C32×D8—21no—
17C2×C32×M16C2211yescharacteristic
18C3×#949—22no—
19C3×#949—22no—
20C2×C3×#215—21no—
21C2×C3×#213—22no—
22C2×C3×#213—21no—
23C2×C3×#213—22no—
24C2×C3×#213—22no—
25C2×C3×#208C2211yescharacteristic
26C2×C3×#208C2211yescharacteristic
27C22×C3×C3⋊C8C2211yescharacteristic
28C3×#833—22no—
29C3×#830—22no—
30C3×#823—22no—
31C2×C3×AGL(1,ℤ/8)—21no—
32C2×C3×AGL(1,ℤ/8)—31no—
33C2×C3×AGL(1,ℤ/8)D311yescharacteristic, complemented
34C2×C3×AGL(1,ℤ/8)C2×C311yescharacteristic, complemented
35C22×C3×D8—31no—
36C2×#949C2×C311yescharacteristic, complemented
37C22×#213C2×C311yescharacteristic, complemented
38C2×C3×#100—31no—
39C2×C3×#100—31no—
40C3×#408—31no—
41#5234C2×C311yescharacteristic, complemented
42C2×#833C2×C311yescharacteristic, complemented
43C2×#823C2×C311yescharacteristic, complemented
44C3×#341—31no—
45#4558C2×C311yescharacteristic, complemented
46C24×C32—21no—
47C23×C3×D3—21no—
48C32×#41—21no—
49C32×#41—21no—
50C32×#41—21no—
51C32×#41—21no—
52C2×C32×Q8D411yescharacteristic
53C2×C32×D4—21no—
54C2×C32×D4—21no—
55C2×C32×D4—21no—
56C2×C32×D4—21no—
57C2×C32×D4—21no—
58C2×C32×D4—21no—
59C2×C32×D4D412yes—
60C2×C32×D4D411yescharacteristic
61C22×C32×C4—21no—
62C22×C32×C4C2311yescharacteristic
63C2×C3×D12—21no—
64C2×C3×D12—21no—
65C2×C3×D12—21no—
66C2×C3×D12—21no—
67C2×C3×D12—21no—
68C2×C3×D12D412yes—
69C32×SD16—24no—
70C32×D8—24no—
71C32×M16—21no—
72C32×M16D412yes—
73C2×C32×C8—21no—
74C3×#215—24no—
75C3×#213—28no—
76C3×#213—24no—
77C3×#208—21no—
78C3×#208—21no—
79C3×#208—21no—
80C3×#208D412yes—
81C2×C3×C3⋊C8—21no—
82C2×C3×C3⋊C8—21no—
83C2×C3×C3⋊C8—21no—
84C2×C3×C3⋊C8—21no—
85C2×C3×C3⋊C8D412yes—
86#2018—31noSylow
87C2×C3×#41—21no—
88C2×C3×#41C2×D311yescharacteristic
89C2×C3×#41C22×C311yescharacteristic
90C22×C3×D4—21no—
91C22×C3×D4—31no—
92C22×C3×D4C2×D311yescharacteristic
93C22×C3×D4C22×C311yescharacteristic
94C22×D12C22×C311yescharacteristic
95C3×AGL(1,ℤ/8)—24no—
96C3×AGL(1,ℤ/8)—24no—
97C3×AGL(1,ℤ/8)—22no—
98C3×AGL(1,ℤ/8)—22no—
99C3×AGL(1,ℤ/8)—22no—
100C3×AGL(1,ℤ/8)—22no—
101C3×AGL(1,ℤ/8)—62no—
102C3×AGL(1,ℤ/8)—62no—
103C2×C3×SD16—22no—
104C2×C3×SD16—21no—
105C2×C3×SD16—21no—
106C2×C3×SD16—61no—
107C2×C3×D8—22no—
108C2×C3×D8—21no—
109C2×C3×D8—21no—
110C2×C3×D8—62no—
111C2×C3×D8—62no—
112C2×C3×D8—62no—
113C2×C3×D8—61no—
114C2×C3×M16—21no—
115C2×C3×M16—31no—
116C2×C3×M16—31no—
117C2×C3×M16C2×D311yescharacteristic
118C2×C3×M16C22×C311yescharacteristic
119C22×C3×C8—31no—
120#949—22no—
121#949—22no—
122C2×#215—21no—
123C2×#213—22no—
124C2×#213—21no—
125C2×#213—22no—
126C2×#213—22no—
127C2×#208C22×C311yescharacteristic
128C2×#208C22×C311yescharacteristic
129C22×C3⋊C8C22×C311yescharacteristic
130C3×#100—62no—
131C3×#100—62no—
132C3×C22⋊C8—62no—
133#833—22no—
134#830—22no—
135#823—22no—
136C23×C32—21no—
137C23×C32—22no—
138C23×C32—41no—
139C23×C32—41no—
140C23×C32—82no—
141C23×C32#4111yescharacteristic
142C22×C3×D3—22no—
143C22×C3×D3—41no—
144C22×C3×D3—41no—
145C22×C3×D3—82no—
146C32×Q8—21no—
147C32×Q8—21no—
148C32×D4—21no—
149C32×D4—22no—
150C32×D4—22no—
151C32×D4—21no—
152C32×D4—41no—
153C32×D4—41no—
154C32×D4—41no—
155C32×D4—41no—
156C2×C32×C4—21no—
157C2×C32×C4—41no—
158C2×C32×C4—41no—
159C2×C32×C4C2×D411yescharacteristic
160C2×C32×C4C2×D411yescharacteristic
161C2×C32×C4C2×D411yescharacteristic
162C2×C32×C4C2×D411yescharacteristic
163C2×C32×C4C2×D411yescharacteristic
164C2×C32×C4C2×D411yescharacteristic
165C3×D12—22no—
166C3×D12—22no—
167C3×D12—41no—
168C3×D12—41no—
169C32×C8—22no—
170C3×C3⋊C8—22no—
171C3×C3⋊C8—22no—
172C3×C3⋊C8—41no—
173C2×AGL(1,ℤ/8)—31no—
174C2×AGL(1,ℤ/8)C3×D311yescharacteristic, complemented
175C22×D8—31no—
176C2×#100—31no—
177C2×#100—31no—
178#408—31no—
179#341—31no—
180C24×C3—21no—
181C24×C3—21no—
182C24×C3—41no—
183C24×C3—61no—
184C23×D3—21no—
185C3×#41—22no—
186C3×#41—22no—
187C3×#41—21no—
188C3×#41—21no—
189C3×#41—21no—
190C3×#41—21no—
191C3×#41—21no—
192C3×#41—21no—
193C3×#41—21no—
194C3×#41—21no—
195C3×#41—41no—
196C3×#41—41no—
197C2×C3×Q8—21no—
198C2×C3×Q8#6711yescharacteristic
199C2×C3×Q8C3×D411yescharacteristic
200C2×C3×D4—22no—
201C2×C3×D4—22no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—22no—
216C2×C3×D4—32no—
217C2×C3×D4—41no—
218C2×C3×D4—41no—
219C2×C3×D4—41no—
220C2×C3×D4—41no—
221C2×C3×D4—61no—
222C2×C3×D4—61no—
223C2×C3×D4—61no—
224C2×C3×D4—61no—
225C2×C3×D4—61no—
226C2×C3×D4#6711yescharacteristic
227C2×C3×D4#6712yes—
228C2×C3×D4C3×D411yescharacteristic
229C2×C3×D4C3×D412yes—
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4—21no—
233C22×C3×C4—41no—
234C22×C3×C4C22×D311yescharacteristic
235C22×C3×C4C23×C311yescharacteristic
236C2×D12—21no—
237C2×D12—21no—
238C2×D12—21no—
239C2×D12—21no—
240C2×D12—21no—
241C2×D12C3×D412yes—
242C3×SD16—28no—
243C3×SD16—24no—
244C3×SD16—24no—
245C3×SD16—64no—
246C3×D8—28no—
247C3×D8—24no—
248C3×D8—24no—
249C3×D8—68no—
250C3×D8—64no—
251C3×M16—22no—
252C3×M16—21no—
253C3×M16—21no—
254C3×M16—22no—
255C3×M16—32no—
256C3×M16—61no—
257C3×M16—61no—
258C3×M16—61no—
259C3×M16#6712yes—
260C3×M16C3×D412yes—
261C2×C3×C8—22no—
262C2×C3×C8—21no—
263C2×C3×C8—21no—
264C2×C3×C8—32no—
265C2×C3×C8—61no—
266C2×C3×C8—61no—
267C2×C3×C8—61no—
268C2×C3×C8—61no—
269#215—24no—
270#213—28no—
271#213—24no—
272#208—21no—
273#208—21no—
274#208—21no—
275#208C3×D412yes—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8C3×D412yes—
281C22×C32—22no—
282C22×C32—41no—
283C22×C32—42no—
284C22×C32—42no—
285C22×C32—82no—
286C22×C32—82no—
287C22×C32—82no—
288C22×C32#12111yescharacteristic
289C22×C32#12111yescharacteristic
290C22×C32#12111yescharacteristic
291C2×C3×D3—42no—
292C2×C3×D3—82no—
293C2×C3×D3—82no—
294C2×C3×D3—82no—
295C32×C4—41no—
296C32×C4#12011yescharacteristic
297C32×C4#12011yescharacteristic
298C32×C4#12011yescharacteristic
299C32×C4#12711yescharacteristic
300C2×#41C2×C3×D311yescharacteristic
301C22×D4—31no—
302C22×D4C2×C3×D311yescharacteristic
303AGL(1,ℤ/8)—22no—
304AGL(1,ℤ/8)—22no—
305AGL(1,ℤ/8)—62no—
306AGL(1,ℤ/8)—62no—
307C2×SD16—21no—
308C2×SD16—61no—
309C2×D8—21no—
310C2×D8—62no—
311C2×D8—62no—
312C2×D8—62no—
313C2×D8—61no—
314C2×M16—31no—
315C2×M16—31no—
316C2×M16C2×C3×D311yescharacteristic
317C22×C8—31no—
318#100—62no—
319#100—62no—
320C22⋊C8—62no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—21no—
326C23×C3—41no—
327C23×C3—42no—
328C23×C3—42no—
329C23×C3—42no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—62no—
335C23×C3—82no—
336C23×C3—82no—
337C23×C3—84no—
338C23×C3—121no—
339C23×C3—121no—
340C23×C3—242no—
341C23×C3#23711yescharacteristic
342C23×C3C3×#4111yescharacteristic
343C22×D3—22no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—82no—
347C3×Q8—22no—
348C3×Q8—21no—
349C3×Q8—21no—
350C3×Q8—21no—
351C3×Q8—21no—
352C3×Q8—41no—
353C3×D4—22no—
354C3×D4—24no—
355C3×D4—22no—
356C3×D4—21no—
357C3×D4—21no—
358C3×D4—22no—
359C3×D4—22no—
360C3×D4—21no—
361C3×D4—21no—
362C3×D4—22no—
363C3×D4—42no—
364C3×D4—41no—
365C3×D4—42no—
366C3×D4—42no—
367C3×D4—42no—
368C3×D4—42no—
369C3×D4—41no—
370C3×D4—41no—
371C3×D4—41no—
372C3×D4—41no—
373C3×D4—41no—
374C3×D4—41no—
375C3×D4—41no—
376C3×D4—41no—
377C3×D4—62no—
378C3×D4—62no—
379C3×D4—121no—
380C3×D4—121no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—41no—
390C2×C3×C4—42no—
391C2×C3×C4—42no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4D3×D411yescharacteristic
397C2×C3×C4D3×D411yescharacteristic
398C2×C3×C4D3×D411yescharacteristic
399C2×C3×C4C2×#6711yescharacteristic
400C2×C3×C4C2×#6711yescharacteristic
401C2×C3×C4C2×#6711yescharacteristic
402C2×C3×C4C2×C3×D411yescharacteristic
403C2×C3×C4C2×C3×D411yescharacteristic
404C2×C3×C4C2×C3×D411yescharacteristic
405C2×C3×C4C2×C3×D411yescharacteristic
406C2×C3×C4C2×C3×D411yescharacteristic
407C2×C3×C4C2×C3×D411yescharacteristic
408D12—22no—
409D12—22no—
410D12—41no—
411D12—41no—
412C3×C8—24no—
413C3×C8—22no—
414C3×C8—22no—
415C3×C8—62no—
416C3×C8—62no—
417C3×C8—121no—
418C3⋊C8—22no—
419C3⋊C8—22no—
420C3⋊C8—41no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—81no—
424C2×C32—82no—
425C2×C32#39211yescharacteristic
426C2×C32#47212yes—
427C3×D3—82no—
428C24—21no—
429C24—61no—
430#41—21no—
431#41—21no—
432#41—21no—
433#41—21no—
434C2×Q8C3×#6711yescharacteristic
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—32no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4C3×#6711yescharacteristic
448C2×D4C3×#6712yes—
449C22×C4—21no—
450C22×C4C22×C3×D311yescharacteristic
451SD16—24no—
452SD16—64no—
453D8—24no—
454D8—68no—
455D8—64no—
456M16—21no—
457M16—32no—
458M16—61no—
459M16—61no—
460M16—61no—
461M16C3×#6712yes—
462C2×C8—21no—
463C2×C8—32no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—44no—
476C22×C3—42no—
477C22×C3—41no—
478C22×C3—42no—
479C22×C3—41no—
480C22×C3—42no—
481C22×C3—42no—
482C22×C3—82no—
483C22×C3—82no—
484C22×C3—82no—
485C22×C3—82no—
486C22×C3—82no—
487C22×C3—84no—
488C22×C3—84no—
489C22×C3—84no—
490C22×C3—82no—
491C22×C3—82no—
492C22×C3—122no—
493C22×C3—242no—
494C22×C3—242no—
495C22×C3—242no—
496C22×C3#93811yescharacteristic
497C22×C3#93911yescharacteristic
498C22×C3#93911yescharacteristic
499C22×C3C3×#12111yescharacteristic
500C22×C3C3×#12111yescharacteristic
501C22×C3C3×#12111yescharacteristic
502C2×D3—42no—
503C2×D3—82no—
504C2×D3—82no—
505C2×D3—82no—
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—21no—
510C3×C4—42no—
511C3×C4—41no—
512C3×C4—41no—
513C3×C4#93711yescharacteristic
514C3×C4#93711yescharacteristic
515C3×C4#94011yescharacteristic
516C3×C4#95311yescharacteristic
517C3×C4C3×#12011yescharacteristic
518C3×C4C3×#12011yescharacteristic
519C3×C4C3×#12011yescharacteristic
520C3×C4C3×#12711yescharacteristic
521C32#201811yescharacteristic, complemented, Sylow
522C23—21no—
523C23—22no—
524C23—41no—
525C23—41no—
526C23—62no—
527C23—82no—
528C23—121no—
529C23—121no—
530C23—242no—
531C23C3×#23711yescharacteristic
532Q8—21no—
533Q8—21no—
534D4—22no—
535D4—21no—
536D4—21no—
537D4—22no—
538D4—41no—
539D4—41no—
540D4—41no—
541D4—41no—
542D4—62no—
543D4—62no—
544D4—121no—
545D4—121no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4C3×D3×D411yescharacteristic
550C2×C4C3×D3×D411yescharacteristic
551C2×C4C3×D3×D411yescharacteristic
552C2×C4C2×C3×#6711yescharacteristic
553C2×C4C2×C3×#6711yescharacteristic
554C2×C4C2×C3×#6711yescharacteristic
555C8—22no—
556C8—62no—
557C8—62no—
558C8—121no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—22no—
564C2×C3—21no—
565C2×C3—41no—
566C2×C3—41no—
567C2×C3—84no—
568C2×C3—82no—
569C2×C3—81no—
570C2×C3—82no—
571C2×C3—81no—
572C2×C3—82no—
573C2×C3—242no—
574C2×C3#519912yes—
575C2×C3#522311yescharacteristic
576C2×C3C3×#39211yescharacteristic
577C2×C3C3×#47212yes—
578D3—82no—
579C22—22no—
580C22—42no—
581C22—41no—
582C22—42no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—122no—
587C22—242no—
588C22—242no—
589C22—242no—
590C22C3×#93811yescharacteristic
591C22C3×#93911yescharacteristic
592C22C3×#93911yescharacteristic
593C4—41no—
594C4C3×#93711yescharacteristic
595C4C3×#93711yescharacteristic
596C4C3×#94011yescharacteristic
597C4C3×#95311yescharacteristic
598C3—21no—
599C3#7209511yescharacteristic, direct product factor
600C3C3×#201811yescharacteristic, complemented
601C2—21no—
602C2—21no—
603C2—81no—
604C2—82no—
605C2—242no—
606C2C3×#519912yes—
607C2C3×#522311yescharacteristic
subgroupG/H
centerC22×C3#939
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC2×C32×AGL(1,ℤ/8)C2
socleC22×C32#121
hypercenterC2×C3×AGL(1,ℤ/8)D3
hypocenterC3C3×#2018
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×C32×AGL(1,ℤ/8)C2semidirect
2. C2×C3×#949C2semidirect
3. C22×C3×#213C2semidirect
4. C3×#5234C2semidirect
5. C2×C3×#833C2semidirect
6. C2×C3×#823C2semidirect
7. C3×#4558C2semidirect
9. #72095C3direct (N × H)
33. C2×C3×AGL(1,ℤ/8)D3semidirect
34. C2×C3×AGL(1,ℤ/8)C2×C3semidirect
36. C2×#949C2×C3semidirect
37. C22×#213C2×C3semidirect
41. #5234C2×C3semidirect
42. C2×#833C2×C3semidirect
43. C2×#823C2×C3semidirect
45. #4558C2×C3semidirect
174. C2×AGL(1,ℤ/8)C3×D3semidirect
521. C32#2018semidirect
599. C3#72095direct (N × H)
600. C3C3×#2018semidirect

Maximal subgroups

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

subgroupindex
C2×C32×AGL(1,ℤ/8)2
C2×C3×#9492
C22×C3×#2132
C3×#52342
C2×C3×#8332
C2×C3×#8232
C3×#45582
C3×#20183
#720953