Finite Group Databasebeta

← C3×#70209 #11792437

Subgroups — C3×#70209 #11792437

Order 1152.

1121 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#940C211yesmaximal, characteristic, complemented
2C22×C3×#213C211yesmaximal, characteristic, complemented
3C22×C3×D24C211yesmaximal, characteristic, complemented
4C2×C32×#102C211yesmaximal, characteristic, complemented
5C3×#4988C211yesmaximal, characteristic, complemented
6C2×C3×#809C211yesmaximal, characteristic, complemented
7C3×#4550C211yesmaximal, characteristic, complemented
8C3×#2011—31nomaximal
9#70209C311yesmaximal, characteristic, direct product factor
10C22×C32×D4C2211yescharacteristic
11C22×C3×#67—22no—
12C22×C3×D12C2211yescharacteristic
13C22×C3×D12C2211yescharacteristic
14C22×C32×C8C2211yescharacteristic
15C2×C32×C4⋊C4C2211yescharacteristic
16C3×#940—22no—
17C3×#940—22no—
18C2×C3×#213—22no—
19C2×C3×#213—22no—
20C2×C3×#213—22no—
21C2×C3×(C2×D3)⋊C4—22no—
22C2×C3×C4×Dic3C2211yescharacteristic
23C22×C3×C3⋊C8C2211yescharacteristic
24C2×C3×D24—22no—
25C2×C3×D24—22no—
26C2×C3×D24—22no—
27C32×#102—22no—
28C3×#809—22no—
29C22×C3×D8—31no—
30C22×C3×D8—31no—
31C2×C3×#127—31no—
32C2×#940C2×C311yescharacteristic, complemented
33C22×#213C2×C311yescharacteristic, complemented
34C22×D24C2×C311yescharacteristic, complemented
35C2×C3×#102—21no—
36C2×C3×#102—31no—
37C2×C3×#102D311yescharacteristic, complemented
38C2×C3×#102C2×C311yescharacteristic, complemented
39C3×#390—31no—
40#4988C2×C311yescharacteristic, complemented
41C2×#809C2×C311yescharacteristic, complemented
42C3×#336—31no—
43#4550C2×C311yescharacteristic, complemented
44C24×C32—21no—
45C23×C3×D3—21no—
46C23×C3×D3—21no—
47C2×C32×D4—21no—
48C2×C32×D4—21no—
49C2×C32×D4—21no—
50C2×C32×D4—21no—
51C2×C32×D4—21no—
52C2×C32×D4D412yes—
53C22×C32×C4—21no—
54C22×C32×C4C2311yescharacteristic
55C2×C3×#67—28no—
56C2×C3×#67—44no—
57C2×C3×#67—44no—
58C22×C3×Dic3D412yes—
59C2×C3×D12—21no—
60C2×C3×D12—21no—
61C2×C3×D12—21no—
62C2×C3×D12—21no—
63C2×C3×D12—21no—
64C2×C3×D12—21no—
65C2×C3×D12—21no—
66C2×C3×D12—21no—
67C2×C3×D12—21no—
68C2×C3×D12—21no—
69C2×C3×D12D412yes—
70C2×C3×D12D412yescomplemented
71C2×C32×C8—21no—
72C2×C32×C8—21no—
73C2×C32×C8D412yescomplemented
74C32×C4⋊C4—21no—
75C32×C4⋊C4—21no—
76C3×#213—28no—
77C3×(C2×D3)⋊C4—44no—
78C3×C4×Dic3—21no—
79C3×C4×Dic3—21no—
80C2×C3×C3⋊C8—21no—
81C2×C3×C3⋊C8—21no—
82C2×C3×C3⋊C8D412yescomplemented
83C3×D24—28no—
84#2011—31noSylow
85C22×C3×D4—21no—
86C22×C3×D4—31no—
87C22×C3×D4—31no—
88C22×C3×D4—62no—
89C22×C3×D4C2×D311yescharacteristic
90C22×C3×D4C22×C311yescharacteristic
91C22×#67—22no—
92C22×D12C22×C311yescharacteristic
93C22×D12C22×C311yescharacteristic
94C2×C3×D8—62no—
95C2×C3×D8—62no—
96C2×C3×D8—62no—
97C2×C3×D8—62no—
98C2×C3×D8—62no—
99C2×C3×D8—62no—
100C22×C3×C8—21no—
101C22×C3×C8—31no—
102C22×C3×C8C2×D311yescharacteristic
103C22×C3×C8C22×C311yescharacteristic
104C3×#127—62no—
105C3×#127—62no—
106C2×C3×C4⋊C4—21no—
107C2×C3×C4⋊C4C2×D311yescharacteristic
108C2×C3×C4⋊C4C22×C311yescharacteristic
109C2×C3×C22⋊C4—62no—
110C2×C3×C42—31no—
111#940—22no—
112#940—22no—
113C2×#213—22no—
114C2×#213—22no—
115C2×#213—22no—
116C2×(C2×D3)⋊C4—22no—
117C2×C4×Dic3C22×C311yescharacteristic
118C22×C3⋊C8C22×C311yescharacteristic
119C2×D24—22no—
120C2×D24—22no—
121C2×D24—22no—
122C3×#102—24no—
123C3×#102—22no—
124C3×#102—22no—
125C3×#102—62no—
126#809—22no—
127C23×C32—22no—
128C23×C32—41no—
129C23×C32—41no—
130C23×C32—42no—
131C23×C32—42no—
132C23×C32C2×D411yescharacteristic
133C22×C3×D3—22no—
134C22×C3×D3—22no—
135C22×C3×D3—41no—
136C22×C3×D3—41no—
137C22×C3×D3—41no—
138C22×C3×D3—41no—
139C22×C3×D3—42no—
140C22×C3×D3—42no—
141C22×C3×D3—42no—
142C22×C3×D3—42no—
143C32×D4—22no—
144C32×D4—22no—
145C32×D4—41no—
146C32×D4—41no—
147C2×C32×C4—41no—
148C2×C32×C4—42no—
149C2×C32×C4C2×D411yescharacteristic
150C2×C32×C4C2×D411yescharacteristic
151C2×C32×C4C2×D411yescharacteristic
152C2×C32×C4C2×D411yescharacteristic
153C2×C32×C4C2×D411yescharacteristic
154C2×C32×C4#4111yescharacteristic
155C3×#67—416no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3D814yescomplemented
159C3×D12—22no—
160C3×D12—22no—
161C3×D12—22no—
162C3×D12—22no—
163C3×D12—41no—
164C3×D12—41no—
165C3×D12—41no—
166C3×D12—41no—
167C32×C8—22no—
168C3×C3⋊C8—22no—
169C22×D8—31no—
170C22×D8—31no—
171C2×#127—31no—
172C2×#102—31no—
173C2×#102C3×D311yescharacteristic, complemented
174#390—31no—
175#336—31no—
176C24×C3—21no—
177C24×C3—21no—
178C24×C3—41no—
179C24×C3—61no—
180C24×C3—61no—
181C23×D3—21no—
182C23×D3—21no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—22no—
196C2×C3×D4—32no—
197C2×C3×D4—32no—
198C2×C3×D4—41no—
199C2×C3×D4—41no—
200C2×C3×D4—41no—
201C2×C3×D4—68no—
202C2×C3×D4—61no—
203C2×C3×D4—61no—
204C2×C3×D4—61no—
205C2×C3×D4—61no—
206C2×C3×D4—61no—
207C2×C3×D4—61no—
208C2×C3×D4—61no—
209C2×C3×D4—61no—
210C2×C3×D4—61no—
211C2×C3×D4—61no—
212C2×C3×D4—124no—
213C2×C3×D4—124no—
214C2×C3×D4D1212yes—
215C2×C3×D4C3×D412yes—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—32no—
220C22×C3×C4—41no—
221C22×C3×C4C22×D311yescharacteristic
222C22×C3×C4C23×C311yescharacteristic
223C2×#67—28no—
224C2×#67—44no—
225C2×#67—44no—
226C22×Dic3C3×D412yes—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12C3×D412yes—
238C2×D12C3×D412yescomplemented
239C3×D8—68no—
240C3×D8—68no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—22no—
246C2×C3×C8—32no—
247C2×C3×C8—41no—
248C2×C3×C8—41no—
249C2×C3×C8—61no—
250C2×C3×C8—61no—
251C2×C3×C8#6712yescomplemented
252C2×C3×C8C3×D412yescomplemented
253C3×C4⋊C4—22no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—41no—
259C3×C22⋊C4—124no—
260C3×C42—61no—
261C3×C42—61no—
262#213—28no—
263(C2×D3)⋊C4—44no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8C3×D412yescomplemented
269D24—28no—
270C22×C32—42no—
271C22×C32—44no—
272C22×C32—82no—
273C22×C32—81no—
274C22×C32—81no—
275C22×C32#12011yescharacteristic
276C22×C32#12111yescharacteristic
277C22×C32#12711yescharacteristic
278C22×C32C2×D812yes—
279C22×C32AGL(1,ℤ/8)12yes—
280C2×C3×D3—42no—
281C2×C3×D3—42no—
282C2×C3×D3—44no—
283C2×C3×D3—44no—
284C2×C3×D3—82no—
285C2×C3×D3—82no—
286C2×C3×D3—81no—
287C2×C3×D3—81no—
288C2×C3×D3—81no—
289C2×C3×D3—81no—
290C32×C4—81no—
291C32×C4#12011yescharacteristic
292C32×C4#12011yescharacteristic
293C32×C4#12111yescharacteristic
294C32×C4#12111yescharacteristic
295C3×Dic3—24no—
296C22×D4—31no—
297C22×D4—31no—
298C22×D4—62no—
299C22×D4C2×C3×D311yescharacteristic
300C2×D8—62no—
301C2×D8—62no—
302C2×D8—62no—
303C2×D8—62no—
304C2×D8—62no—
305C2×D8—62no—
306C22×C8—31no—
307C22×C8C2×C3×D311yescharacteristic
308#127—62no—
309#127—62no—
310C2×C4⋊C4C2×C3×D311yescharacteristic
311C2×C22⋊C4—62no—
312C2×C42—31no—
313#102—22no—
314#102—62no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—21no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—41no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—42no—
328C23×C3—42no—
329C23×C3—62no—
330C23×C3—62no—
331C23×C3—82no—
332C23×C3—82no—
333C23×C3—121no—
334C23×C3—121no—
335C23×C3—122no—
336C23×C3—122no—
337C23×C3—122no—
338C23×C3—122no—
339C23×C3—121no—
340C23×C3—121no—
341C23×C3D3×D411yescharacteristic
342C23×C3C2×C3×D411yescharacteristic
343C22×D3—22no—
344C22×D3—22no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—42no—
350C22×D3—42no—
351C22×D3—42no—
352C22×D3—42no—
353C3×D4—24no—
354C3×D4—22no—
355C3×D4—22no—
356C3×D4—22no—
357C3×D4—22no—
358C3×D4—42no—
359C3×D4—42no—
360C3×D4—42no—
361C3×D4—41no—
362C3×D4—41no—
363C3×D4—41no—
364C3×D4—41no—
365C3×D4—62no—
366C3×D4—62no—
367C3×D4—62no—
368C3×D4—62no—
369C3×D4—1216no—
370C3×D4—121no—
371C3×D4—121no—
372C3×D4—121no—
373C3×D4—121no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—34no—
381C2×C3×C4—42no—
382C2×C3×C4—42no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—42no—
386C2×C3×C4—62no—
387C2×C3×C4—62no—
388C2×C3×C4—82no—
389C2×C3×C4C2×D1211yescharacteristic
390C2×C3×C4#23511yescharacteristic
391C2×C3×C4D3×D411yescharacteristic
392C2×C3×C4D3×D411yescharacteristic
393C2×C3×C4D3×D411yescharacteristic
394C2×C3×C4C2×#6711yescharacteristic
395C2×C3×C4C2×C3×D411yescharacteristic
396C2×C3×C4C2×C3×D411yescharacteristic
397C2×C3×C4C2×C3×D411yescharacteristic
398C2×C3×C4C2×C3×D411yescharacteristic
399C2×C3×C4C2×C3×D411yescharacteristic
400C2×C3×C4C3×#4111yescharacteristic
401#67—416no—
402C2×Dic3—22no—
403C2×Dic3—22no—
404C2×Dic3C3×D814yescomplemented
405D12—22no—
406D12—22no—
407D12—22no—
408D12—22no—
409D12—41no—
410D12—41no—
411D12—41no—
412D12—41no—
413C3×C8—22no—
414C3×C8—22no—
415C3×C8—42no—
416C3×C8—62no—
417C3⋊C8—22no—
418C2×C32—82no—
419C2×C32#39211yescharacteristic
420C2×C32#44712yes—
421C2×C32#45912yes—
422C2×C32#49311yescharacteristic
423C2×C32#49611yescharacteristic
424C3×D3—82no—
425C3×D3—82no—
426C24—21no—
427C24—61no—
428C24—61no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—32no—
435C2×D4—32no—
436C2×D4—68no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—124no—
448C2×D4—124no—
449C2×D4C3×D1212yes—
450C22×C4—21no—
451C22×C4—32no—
452C22×C4C22×C3×D311yescharacteristic
453D8—68no—
454D8—68no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—32no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8C3×#6712yescomplemented
461C4⋊C4—21no—
462C4⋊C4—21no—
463C22⋊C4—124no—
464C42—61no—
465C42—61no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—21no—
471C22×C3—44no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—44no—
475C22×C3—44no—
476C22×C3—84no—
477C22×C3—82no—
478C22×C3—84no—
479C22×C3—82no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3—82no—
483C22×C3—82no—
484C22×C3—81no—
485C22×C3—81no—
486C22×C3—122no—
487C22×C3—122no—
488C22×C3—124no—
489C22×C3—124no—
490C22×C3—241no—
491C22×C3—241no—
492C22×C3—241no—
493C22×C3—241no—
494C22×C3—242no—
495C22×C3—242no—
496C22×C3#88211yescharacteristic
497C22×C3#88411yescharacteristic
498C22×C3D3×D812yes—
499C22×C3#91412yes—
500C22×C3#94011yescharacteristic
501C22×C3C3×#12011yescharacteristic
502C22×C3C3×#12111yescharacteristic
503C22×C3C3×#12711yescharacteristic
504C22×C3C2×C3×D812yes—
505C22×C3C3×AGL(1,ℤ/8)12yes—
506C2×D3—42no—
507C2×D3—42no—
508C2×D3—44no—
509C2×D3—44no—
510C2×D3—82no—
511C2×D3—82no—
512C2×D3—81no—
513C2×D3—81no—
514C2×D3—81no—
515C2×D3—81no—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—21no—
520C3×C4—64no—
521C3×C4—82no—
522C3×C4—81no—
523C3×C4—81no—
524C3×C4#88211yescharacteristic
525C3×C4#88411yescharacteristic
526C3×C4#93011yescharacteristic
527C3×C4#93711yescharacteristic
528C3×C4C3×#12011yescharacteristic
529C3×C4C3×#12011yescharacteristic
530C3×C4C3×#12111yescharacteristic
531C3×C4C3×#12111yescharacteristic
532Dic3—24no—
533C32#201111yescharacteristic, complemented, Sylow
534C23—22no—
535C23—41no—
536C23—42no—
537C23—42no—
538C23—41no—
539C23—62no—
540C23—62no—
541C23—121no—
542C23—121no—
543C23—122no—
544C23—122no—
545C23—122no—
546C23—122no—
547C23—121no—
548C23—121no—
549C23C3×D3×D411yescharacteristic
550D4—22no—
551D4—22no—
552D4—41no—
553D4—41no—
554D4—62no—
555D4—62no—
556D4—62no—
557D4—62no—
558D4—1216no—
559D4—121no—
560D4—121no—
561D4—121no—
562D4—121no—
563C2×C4—34no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4C2×C3×D1211yescharacteristic
569C2×C4C3×#23511yescharacteristic
570C2×C4C3×D3×D411yescharacteristic
571C2×C4C3×D3×D411yescharacteristic
572C2×C4C3×D3×D411yescharacteristic
573C2×C4C2×C3×#6711yescharacteristic
574C8—22no—
575C8—62no—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3—22no—
579C2×C3—22no—
580C2×C3—21no—
581C2×C3—84no—
582C2×C3—82no—
583C2×C3—82no—
584C2×C3—242no—
585C2×C3—242no—
586C2×C3#477312yes—
587C2×C3#478612yes—
588C2×C3#496011yescharacteristic
589C2×C3#515111yescharacteristic
590C2×C3#517611yescharacteristic
591C2×C3C3×#39211yescharacteristic
592C2×C3C3×#44712yes—
593C2×C3C3×#45912yes—
594C2×C3C3×#49311yescharacteristic
595C2×C3C3×#49611yescharacteristic
596D3—82no—
597D3—82no—
598C22—42no—
599C22—44no—
600C22—81no—
601C22—82no—
602C22—81no—
603C22—122no—
604C22—122no—
605C22—124no—
606C22—124no—
607C22—241no—
608C22—241no—
609C22—242no—
610C22—242no—
611C22—241no—
612C22—241no—
613C22C3×#88211yescharacteristic
614C22C3×#88411yescharacteristic
615C22C3×D3×D812yes—
616C22C3×#91412yes—
617C22C3×#94011yescharacteristic
618C4—64no—
619C4—81no—
620C4C3×#88211yescharacteristic
621C4C3×#88411yescharacteristic
622C4C3×#93011yescharacteristic
623C4C3×#93711yescharacteristic
624C3—21no—
625C3#7020911yescharacteristic, direct product factor
626C3C3×#201111yescharacteristic, complemented
627C2—82no—
628C2—242no—
629C2—242no—
630C2C3×#477312yes—
631C2C3×#478612yes—
632C2C3×#496011yescharacteristic
633C2C3×#515111yescharacteristic
634C2C3×#517611yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#102C2
socleC23×C32C2×D4
hypercenterC2×C3×#102D3
hypocenterC3C3×#2011
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#102

Complements

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

NH ≅ G/Ntype
1. C2×C3×#940C2semidirect
2. C22×C3×#213C2semidirect
3. C22×C3×D24C2semidirect
4. C2×C32×#102C2semidirect
5. C3×#4988C2semidirect
6. C2×C3×#809C2semidirect
7. C3×#4550C2semidirect
9. #70209C3direct (N × H)
32. C2×#940C2×C3semidirect
33. C22×#213C2×C3semidirect
34. C22×D24C2×C3semidirect
37. C2×C3×#102D3semidirect
38. C2×C3×#102C2×C3semidirect
40. #4988C2×C3semidirect
41. C2×#809C2×C3semidirect
43. #4550C2×C3semidirect
70. C2×C3×D12D4semidirect
73. C2×C32×C8D4semidirect
82. C2×C3×C3⋊C8D4semidirect
158. C2×C3×Dic3D8semidirect
173. C2×#102C3×D3semidirect
238. C2×D12C3×D4semidirect
251. C2×C3×C8#67semidirect
252. C2×C3×C8C3×D4semidirect
268. C2×C3⋊C8C3×D4semidirect
404. C2×Dic3C3×D8semidirect
460. C2×C8C3×#67semidirect
533. C32#2011semidirect
625. C3#70209direct (N × H)
626. C3C3×#2011semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9402
C22×C3×#2132
C22×C3×D242
C2×C32×#1022
C3×#49882
C2×C3×#8092
C3×#45502
C3×#20113
#702093