Finite Group Databasebeta

← C2×#11722494 #13164726

Subgroups — C2×#11722494 #13164726

Order 1728.

2546 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88742C211yesmaximal, characteristic, complemented
2C2×C9×#237C211yesmaximal, characteristic, complemented
3C2×C3×#3699C211yesmaximal, characteristic, complemented
4C2×#88663C212yesmaximal, complemented
5C2×#88662C212yesmaximal, complemented
6C2×#88658C212yesmaximal, complemented
7C22×#6189C212yesmaximal, complemented
8#11722494C2116yesmaximal, direct product factor
9C2×#88640C211yesmaximal, characteristic, complemented
10C2×D3×Dic18C211yesmaximal, characteristic, complemented
11C2×(C3⋊D9)⋊Q8C211yesmaximal, characteristic, complemented
12C2×D9×Dic6C211yesmaximal, characteristic, complemented
13C2×#64309—31nomaximal
14C2×#63716—31nomaximal
15C2×C3×D4×C9C2211yescharacteristic
16C2×#6233C2212yes—
17C22×C3⋊Dic9C2212yes—
18#88742C2218yescomplemented
19C2×C4×C3⋊D9C2211yescharacteristic
20C2×#6226C2211yescharacteristic
21C2×C9×#67C2212yes—
22C2×C3×#611C2212yes—
23C22×C9×Dic3C2212yes—
24C22×C3×Dic9C2212yes—
25C9×#237C2218yescomplemented
26C3×#3699C2218yescomplemented
27C2×C4×D3×C9C2211yescharacteristic
28C2×C3×C4×D9C2211yescharacteristic
29C2×C9×Dic6C2211yescharacteristic
30C2×C3×Dic18C2211yescharacteristic
31C2×#6195C2211yescharacteristic
32C2×#6194C2211yescharacteristic
33#88663C22116yescomplemented
34#88662C22116yescomplemented
35C2×D3×Dic9C2212yes—
36C2×#6192C2211yescharacteristic
37C2×(C3⋊D9)⋊C4C2212yes—
38#88658C22116yescomplemented
39C2×Dic3×D9C2212yes—
40C2×#6189C2211yescharacteristic
41C2×#6189C2212yes—
42C2×#6189C2212yes—
43C2×#6189C2212yes—
44C2×#6189C22116yescomplemented
45#88640C2218yescomplemented
46D3×Dic18C2218yescomplemented
47(C3⋊D9)⋊Q8C2218yescomplemented
48D9×Dic6C2218yescomplemented
49C2×#3830—31no—
50C2×C3×#237—31no—
51C2×C3×#237D311yescharacteristic
52C22×#627—32no—
53C2×#3805—32no—
54C2×#3804—32no—
55C2×#3804—32no—
56#64309—316no—
57C2×#3798—31no—
58C2×#3797—31no—
59C2×D3×Dic6—31no—
60C2×D3×Dic6—31no—
61C2×C9×#41—31no—
62#63716—316no—
63C2×Q8×D9—31no—
64C2×Q8×D9—31no—
65C2×#3699—31no—
66C2×#3699—32no—
67C2×#3699—32no—
68C2×#3699—61no—
69C2×#3699D311yescharacteristic, complemented
70C2×#3697—32no—
71C2×#3697—31no—
72C22×Dic18—32no—
73C22×Dic18—31no—
74C23×C3×C9C2312yes—
75C22×C3⋊D9C2311yescharacteristic
76C22×D3×C9C2311yescharacteristic
77C22×C3×D9C2311yescharacteristic
78C3×D4×C9C2314yes—
79C2×C3×C4×C9C2311yescharacteristic
80#6233C2318yes—
81C2×C3⋊Dic9C2311yescharacteristic
82C2×C3⋊Dic9C2312yes—
83C2×C3⋊Dic9C2318yes—
84C4×C3⋊D9C2314yes—
85#6226C2314yes—
86C9×#67C2318yes—
87C3×#611C2318yes—
88C2×C9×Dic3C2311yescharacteristic
89C2×C9×Dic3C2312yes—
90C2×C9×Dic3C2318yes—
91C2×C3×Dic9C2311yescharacteristic
92C2×C3×Dic9C2312yes—
93C2×C3×Dic9C2318yes—
94C4×D3×C9C2314yes—
95C3×C4×D9C2314yes—
96C9×Dic6C2314yes—
97C3×Dic18C2314yes—
98#6195C2314yes—
99#6194C2314yes—
100D3×Dic9C2318yes—
101#6192C2314yes—
102(C3⋊D9)⋊C4C2318yes—
103Dic3×D9C2318yes—
104#6189C2318yes—
105#6189C2318yes—
106#6189C2318yes—
107#6189C2314yes—
108C2×#1010—91no—
109C2×#1010—91no—
110C2×C32×D4C2×D311yescharacteristic
111C2×#638—32no—
112C22×C3⋊Dic3—32no—
113#3830—38no—
114C2×C4×C3⋊D3—31no—
115C2×#634—31no—
116C2×C3×#67—32no—
117C2×C3×#67C2×D312yes—
118C22×C3×Dic3—32no—
119C22×C3×Dic3C2×D312yes—
120C3×#237—38no—
121C3×#237C2×D318yes—
122C2×C3×C4×D3—31no—
123C2×C3×C4×D3C2×D311yescharacteristic
124C2×C3×Dic6—31no—
125C2×C3×Dic6C2×D311yescharacteristic
126C2×#627—32no—
127C2×#627—32no—
128C2×#627—32no—
129C2×#627—31no—
130C2×#627—316no—
131C2×#626—31no—
132C2×#626—31no—
133C2×#625—31no—
134C2×#624—32no—
135#3805—316no—
136#3804—316no—
137#3804—316no—
138C2×D3×Dic3—32no—
139C2×D3×Dic3—32no—
140#3798—38no—
141#3797—38no—
142D3×Dic6—38no—
143D3×Dic6—38no—
144C9×#41—38no—
145C2×Q8×C9—31no—
146C2×D4×C9—21no—
147C2×D4×C9—32no—
148C2×D4×C9C2×D311yescharacteristic
149C22×C4×C9—32no—
150C22×C4×C9—31no—
151C2×#611—32no—
152C2×#611—31no—
153C2×#611—31no—
154C2×#611—62no—
155C2×#611C2×D312yes—
156C22×Dic9—32no—
157C22×Dic9—32no—
158C22×Dic9—62no—
159C22×Dic9C2×D312yes—
160Q8×D9—38no—
161Q8×D9—38no—
162#3699—38no—
163#3699—316no—
164#3699—316no—
165#3699—68no—
166#3699C2×D318yescomplemented
167#3697—316no—
168#3697—38no—
169C2×D36—31no—
170C2×C4×D9—32no—
171C2×C4×D9—31no—
172C2×C4×D9—32no—
173C2×C4×D9—61no—
174C2×C4×D9C2×D311yescharacteristic
175C2×Dic18—31no—
176C2×Dic18—31no—
177C2×Dic18—32no—
178C2×Dic18—32no—
179C2×Dic18—32no—
180C2×Dic18—316no—
181C2×Dic18—38no—
182C2×Dic18—61no—
183C2×Dic18C2×D311yescharacteristic
184C22×C3×C9—24no—
185C22×C3×C9C2414yes—
186C22×C3×C9C2411yescharacteristic
187C2×C3⋊D9—22no—
188C2×C3⋊D9C2412yes—
189C2×D3×C9—22no—
190C2×D3×C9C2412yes—
191C2×C3×D9—22no—
192C2×C3×D9C2412yes—
193C3×C4×C9C2412yes—
194C3⋊Dic9C2412yes—
195C3⋊Dic9C2414yes—
196C9×Dic3C2412yes—
197C9×Dic3C2414yes—
198C3×Dic9C2412yes—
199C3×Dic9C2414yes—
200C2×C3×#41—31no—
201C2×C3×#41—91no—
202#1010—916no—
203#1010—916no—
204C2×D3×Q8—91no—
205C2×D3×Q8—91no—
206C2×D3×Q8—91no—
207C2×D3×Q8—91no—
208C2×#237—31no—
209C2×#237—92no—
210C2×#237—92no—
211C2×#237—91no—
212C2×#237—91no—
213C2×#237—92no—
214C2×#237—92no—
215C2×#237—181no—
216C2×#237D911yescharacteristic, complemented
217C2×#235—91no—
218C2×#235—92no—
219C2×#235—92no—
220C2×#235—91no—
221C22×Dic6—91no—
222C22×Dic6—92no—
223C22×Dic6—92no—
224C22×Dic6—91no—
225C23×C32C22×D312yes—
226C22×C3⋊D3—31no—
227C22×C3×D3—31no—
228C22×C3×D3C22×D311yescharacteristic
229C32×D4C22×D314yes—
230C2×C32×C4C22×D311yescharacteristic
231#638—38no—
232C2×C3⋊Dic3—32no—
233C2×C3⋊Dic3—38no—
234C2×C3⋊Dic3—31no—
235C4×C3⋊D3—34no—
236#634—34no—
237C3×#67—38no—
238C3×#67C22×D318yes—
239C2×C3×Dic3—32no—
240C2×C3×Dic3—38no—
241C2×C3×Dic3—31no—
242C2×C3×Dic3C22×D312yes—
243C2×C3×Dic3C22×D318yes—
244C2×C3×Dic3C22×D311yescharacteristic
245C3×C4×D3—34no—
246C3×C4×D3C22×D314yes—
247C3×Dic6—34no—
248C3×Dic6C22×D314yes—
249#627—38no—
250#627—38no—
251#627—38no—
252#627—34no—
253#626—34no—
254#626—34no—
255#625—34no—
256#624—38no—
257D3×Dic3—38no—
258D3×Dic3—38no—
259C23×C9—22no—
260C23×C9—31no—
261C23×C9C22×D312yes—
262C22×D9—31no—
263C22×D9—61no—
264C22×D9C22×D311yescharacteristic
265Q8×C9—34no—
266D4×C9—24no—
267D4×C9—38no—
268D4×C9C22×D314yes—
269C2×C4×C9—21no—
270C2×C4×C9—31no—
271C2×C4×C9—32no—
272C2×C4×C9—38no—
273C2×C4×C9—34no—
274C2×C4×C9C22×D311yescharacteristic
275#611—38no—
276#611—34no—
277#611—34no—
278#611—68no—
279#611C22×D318yes—
280C2×Dic9—31no—
281C2×Dic9—32no—
282C2×Dic9—38no—
283C2×Dic9—38no—
284C2×Dic9—62no—
285C2×Dic9—61no—
286C2×Dic9—68no—
287C2×Dic9C22×D311yescharacteristic
288C2×Dic9C22×D312yes—
289C2×Dic9C22×D318yes—
290D36—34no—
291C4×D9—38no—
292C4×D9—34no—
293C4×D9—38no—
294C4×D9—64no—
295C4×D9C22×D314yes—
296Dic18—34no—
297Dic18—34no—
298Dic18—38no—
299Dic18—38no—
300Dic18—38no—
301Dic18—64no—
302Dic18C22×D314yes—
303C2×2−1+4—271noSylow
304C2×C3×C9—24no—
305C2×C3×C92−1+412yescomplemented
306C2×C3×C9C2511yescharacteristic
307C3⋊D9—22no—
308D3×C9—22no—
309C3×D9—22no—
310C3×#41—38no—
311C3×#41—98no—
312C2×C3×Q8—31no—
313C2×C3×Q8—91no—
314C2×C3×D4—21no—
315C2×C3×D4—32no—
316C2×C3×D4—92no—
317C2×C3×D4C2×D911yescharacteristic
318C2×C3×D4D3211yescharacteristic
319C22×C3×C4—32no—
320C22×C3×C4—31no—
321C22×C3×C4—92no—
322C22×C3×C4—91no—
323C2×#67—32no—
324C2×#67—91no—
325C2×#67—91no—
326C2×#67—92no—
327C2×#67—92no—
328C2×#67—91no—
329C2×#67—91no—
330C2×#67—182no—
331C2×#67C2×D912yes—
332C22×Dic3—32no—
333C22×Dic3—92no—
334C22×Dic3—92no—
335C22×Dic3—92no—
336C22×Dic3—92no—
337C22×Dic3—182no—
338C22×Dic3C2×D912yes—
339D3×Q8—98no—
340D3×Q8—98no—
341D3×Q8—98no—
342D3×Q8—98no—
343#237—38no—
344#237—916no—
345#237—916no—
346#237—98no—
347#237—98no—
348#237—916no—
349#237—916no—
350#237—188no—
351#237C2×D918yescomplemented
352#235—98no—
353#235—916no—
354#235—916no—
355#235—98no—
356C2×D12—91no—
357C2×D12—91no—
358C2×C4×D3—31no—
359C2×C4×D3—92no—
360C2×C4×D3—92no—
361C2×C4×D3—92no—
362C2×C4×D3—91no—
363C2×C4×D3—91no—
364C2×C4×D3—92no—
365C2×C4×D3—181no—
366C2×C4×D3C2×D911yescharacteristic
367C2×Dic6—31no—
368C2×Dic6—98no—
369C2×Dic6—92no—
370C2×Dic6—91no—
371C2×Dic6—92no—
372C2×Dic6—92no—
373C2×Dic6—916no—
374C2×Dic6—91no—
375C2×Dic6—91no—
376C2×Dic6—91no—
377C2×Dic6—92no—
378C2×Dic6—92no—
379C2×Dic6—92no—
380C2×Dic6—916no—
381C2×Dic6—98no—
382C2×Dic6—181no—
383C2×Dic6C2×D911yescharacteristic
384C22×C32—24no—
385C22×C32C23×D314yes—
386C22×C32C23×D311yescharacteristic
387C2×C3⋊D3—32no—
388C2×C3⋊D3—62no—
389C2×C3×D3—22no—
390C2×C3×D3—32no—
391C2×C3×D3—62no—
392C2×C3×D3C23×D312yes—
393C32×C4C23×D312yes—
394C3⋊Dic3—34no—
395C3⋊Dic3—32no—
396C3×Dic3—34no—
397C3×Dic3—32no—
398C3×Dic3C23×D314yes—
399C3×Dic3C23×D312yes—
400C22×C9—24no—
401C22×C9—21no—
402C22×C9—24no—
403C22×C9—32no—
404C22×C9—44no—
405C22×C9—62no—
406C22×C9C23×D314yes—
407C22×C9C23×D311yescharacteristic
408C2×D9—22no—
409C2×D9—32no—
410C2×D9—62no—
411C2×D9—62no—
412C2×D9—122no—
413C2×D9C23×D312yes—
414C4×C9—22no—
415C4×C9—32no—
416C4×C9—34no—
417C4×C9C23×D312yes—
418Dic9—32no—
419Dic9—34no—
420Dic9—64no—
421Dic9—62no—
422Dic9C23×D312yes—
423Dic9C23×D314yes—
4242−1+4—2716no—
425C2×#41—31no—
426C2×#41—91no—
427C2×#41—271no—
428C2×#41—272no—
429C2×#41—272no—
430C2×#41—272no—
431C2×#41—271no—
432C22×Q8—272no—
433C22×Q8—271no—
434C22×Q8—271no—
435C22×Q8—271no—
436C3×C9C2×2−1+411yescharacteristic, complemented, Sylow
437C23×C3—22no—
438C23×C3—31no—
439C23×C3—91no—
440C23×C3C22×D912yes—
441C23×C3C2×D3212yes—
442C22×D3—31no—
443C22×D3—91no—
444C22×D3—91no—
445C22×D3—181no—
446C22×D3C22×D911yescharacteristic
447C3×Q8—34no—
448C3×Q8—94no—
449C3×D4—24no—
450C3×D4—38no—
451C3×D4—98no—
452C3×D4C22×D914yes—
453C3×D4C2×D3214yes—
454C2×C3×C4—21no—
455C2×C3×C4—31no—
456C2×C3×C4—32no—
457C2×C3×C4—38no—
458C2×C3×C4—34no—
459C2×C3×C4—91no—
460C2×C3×C4—92no—
461C2×C3×C4—98no—
462C2×C3×C4—94no—
463C2×C3×C4C22×D911yescharacteristic
464C2×C3×C4C2×D3211yescharacteristic
465#67—38no—
466#67—94no—
467#67—94no—
468#67—98no—
469#67—98no—
470#67—94no—
471#67—94no—
472#67—188no—
473#67C22×D918yes—
474C2×Dic3—32no—
475C2×Dic3—31no—
476C2×Dic3—38no—
477C2×Dic3—98no—
478C2×Dic3—92no—
479C2×Dic3—91no—
480C2×Dic3—98no—
481C2×Dic3—92no—
482C2×Dic3—91no—
483C2×Dic3—98no—
484C2×Dic3—98no—
485C2×Dic3—182no—
486C2×Dic3—181no—
487C2×Dic3—188no—
488C2×Dic3C22×D911yescharacteristic
489C2×Dic3C22×D912yes—
490C2×Dic3C22×D918yes—
491D12—94no—
492D12—94no—
493C4×D3—34no—
494C4×D3—98no—
495C4×D3—98no—
496C4×D3—98no—
497C4×D3—94no—
498C4×D3—94no—
499C4×D3—98no—
500C4×D3—184no—
501C4×D3C22×D914yes—
502Dic6—34no—
503Dic6—98no—
504Dic6—94no—
505Dic6—98no—
506Dic6—98no—
507Dic6—94no—
508Dic6—94no—
509Dic6—94no—
510Dic6—98no—
511Dic6—98no—
512Dic6—98no—
513Dic6—184no—
514Dic6C22×D914yes—
515C2×C32—24no—
516C2×C32#101012yes—
517C2×C32C24×D311yescharacteristic
518C3⋊D3—62no—
519C3×D3—22no—
520C3×D3—62no—
521C2×C9—24no—
522C2×C9—22no—
523C2×C9—21no—
524C2×C9—44no—
525C2×C9—62no—
526C2×C9#101012yescomplemented
527C2×C9C24×D311yescharacteristic
528D9—22no—
529D9—62no—
530D9—122no—
531#41—38no—
532#41—98no—
533#41—278no—
534#41—2716no—
535#41—2716no—
536#41—2716no—
537#41—278no—
538C2×Q8—31no—
539C2×Q8—91no—
540C2×Q8—271no—
541C2×Q8—272no—
542C2×Q8—272no—
543C2×Q8—272no—
544C2×Q8—2716no—
545C2×Q8—278no—
546C2×Q8—278no—
547C2×Q8—278no—
548C2×Q8—271no—
549C2×D4—32no—
550C2×D4—92no—
551C2×D4—271no—
552C2×D4—271no—
553C2×D4—271no—
554C2×D4—272no—
555C2×D4D3×D911yescharacteristic
556C22×C4—32no—
557C22×C4—31no—
558C22×C4—92no—
559C22×C4—91no—
560C22×C4—272no—
561C22×C4—272no—
562C22×C4—272no—
563C22×C4—272no—
564C22×C4—271no—
565C22×C3—24no—
566C22×C3—24no—
567C22×C3—24no—
568C22×C3—21no—
569C22×C3—32no—
570C22×C3—44no—
571C22×C3—62no—
572C22×C3—92no—
573C22×C3—182no—
574C22×C3C23×D914yes—
575C22×C3C23×D911yescharacteristic
576C22×C3C22×D3214yes—
577C22×C3C22×D3211yescharacteristic
578C2×D3—22no—
579C2×D3—32no—
580C2×D3—62no—
581C2×D3—92no—
582C2×D3—92no—
583C2×D3—182no—
584C2×D3—182no—
585C2×D3—182no—
586C2×D3—362no—
587C2×D3C23×D912yes—
588C3×C4—22no—
589C3×C4—32no—
590C3×C4—34no—
591C3×C4—92no—
592C3×C4—94no—
593C3×C4C23×D912yes—
594C3×C4C22×D3212yes—
595Dic3—34no—
596Dic3—32no—
597Dic3—92no—
598Dic3—94no—
599Dic3—94no—
600Dic3—92no—
601Dic3—184no—
602Dic3—182no—
603Dic3C23×D912yes—
604Dic3C23×D914yes—
605C32C2×#101011yescharacteristic
606C9—21no—
607C9C2×#101011yescharacteristic, complemented
608C23—31no—
609C23—91no—
610C23—271no—
611C23C2×D3×D912yes—
612Q8—34no—
613Q8—94no—
614Q8—274no—
615Q8—278no—
616Q8—278no—
617Q8—278no—
618Q8—274no—
619D4—38no—
620D4—98no—
621D4—274no—
622D4—274no—
623D4—274no—
624D4—278no—
625D4C2×D3×D914yes—
626C2×C4—38no—
627C2×C4—31no—
628C2×C4—32no—
629C2×C4—34no—
630C2×C4—98no—
631C2×C4—91no—
632C2×C4—92no—
633C2×C4—94no—
634C2×C4—278no—
635C2×C4—278no—
636C2×C4—278no—
637C2×C4—278no—
638C2×C4—272no—
639C2×C4—271no—
640C2×C4—274no—
641C2×C4C2×D3×D911yescharacteristic
642C2×C3—24no—
643C2×C3—24no—
644C2×C3—21no—
645C2×C3—22no—
646C2×C3—44no—
647C2×C3—62no—
648C2×C3—182no—
649C2×C3#6371612yescomplemented
650C2×C3C24×D911yescharacteristic
651C2×C3#6430912yes—
652C2×C3C23×D3211yescharacteristic
653D3—22no—
654D3—62no—
655D3—182no—
656D3—182no—
657D3—362no—
658C22—24no—
659C22—32no—
660C22—62no—
661C22—92no—
662C22—182no—
663C22—272no—
664C22—542no—
665C22C22×D3×D914yes—
666C22C22×D3×D911yescharacteristic
667C4—32no—
668C4—34no—
669C4—92no—
670C4—94no—
671C4—274no—
672C4—272no—
673C4C22×D3×D912yes—
674C3—21no—
675C3C2×#6371611yescharacteristic, complemented
676C3C2×#6430911yescharacteristic
677C2—24no—
678C2—62no—
679C2—182no—
680C2—542no—
681C2#1172249412yesdirect product factor
682C2C23×D3×D911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC2×C3×C9C25
FrattiniC2×C3C23×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9C2×2−1+4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88742C2semidirect
2. C2×C9×#237C2semidirect
3. C2×C3×#3699C2semidirect
4. C2×#88663C2semidirect
5. C2×#88662C2semidirect
6. C2×#88658C2semidirect
7. C22×#6189C2semidirect
8. #11722494C2direct (N × H)
9. C2×#88640C2semidirect
10. C2×D3×Dic18C2semidirect
11. C2×(C3⋊D9)⋊Q8C2semidirect
12. C2×D9×Dic6C2semidirect
18. #88742C22semidirect
25. C9×#237C22semidirect
26. C3×#3699C22semidirect
33. #88663C22semidirect
34. #88662C22semidirect
38. #88658C22semidirect
44. C2×#6189C22semidirect
45. #88640C22semidirect
46. D3×Dic18C22semidirect
47. (C3⋊D9)⋊Q8C22semidirect
48. D9×Dic6C22semidirect
69. C2×#3699D3semidirect
166. #3699C2×D3semidirect
216. C2×#237D9semidirect
305. C2×C3×C92−1+4semidirect
351. #237C2×D9semidirect
436. C3×C9C2×2−1+4semidirect
526. C2×C9#1010semidirect
607. C9C2×#1010semidirect
649. C2×C3#63716semidirect
675. C3C2×#63716semidirect
681. C2#11722494direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#887422
C2×C9×#2372
C2×C3×#36992
C2×#886632
C2×#886622
C2×#886582
C22×#61892
#117224942
C2×#886402
C2×D3×Dic182
C2×(C3⋊D9)⋊Q82
C2×D9×Dic62
C2×#643093
C2×#637163