Finite Group Databasebeta

← C2×#11723633 #13178082

Subgroups — C2×#11723633 #13178082

Order 1728.

3932 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6312C211yesmaximal, characteristic, complemented
2C22×C3×#638C211yesmaximal, characteristic, complemented
3C22×C32×#67C211yesmaximal, characteristic, complemented
4C22×#6292C211yesmaximal, characteristic, complemented
5C22×#6291C211yesmaximal, characteristic, complemented
6C22×#6290C211yesmaximal, characteristic, complemented
7#11723633C218yesmaximal, direct product factor
8C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
9C2×#64149—31nomaximal
10C2×#63979—34nomaximal
11C24×C33C2211yescharacteristic
12C23×C3⋊(C3⋊D3)C2211yescharacteristic
13C23×C3×C3⋊D3C2211yescharacteristic
14C23×C32×D3C2211yescharacteristic
15C2×#6312—24no—
16C2×#6312C2214yescomplemented
17C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×#638—24no—
19C2×C3×#638C2214yescomplemented
20C22×C3×C3⋊Dic3C2211yescharacteristic
21C2×C32×#67—24no—
22C2×C32×#67C2214yescomplemented
23C22×C32×Dic3C2211yescharacteristic
24C2×#6292—24no—
25C2×#6292C2214yescomplemented
26C2×#6291—24no—
27C2×#6291C2214yescomplemented
28C2×#6290—24no—
29C2×#6290C2214yescomplemented
30Dic3×C3⋊Dic3C2214yescomplemented
31C22×C32×D4—31no—
32C22×#638—31no—
33C22×#638—31no—
34C22×#638—34no—
35C22×#638—31no—
36C22×#638—64no—
37C22×#638D311yescharacteristic, complemented
38C22×C4⋊(C3⋊D3)—31no—
39C22×C3×#67—34no—
40C22×C3×#67D314yescomplemented
41C22×#626—34no—
42C22×#626—34no—
43C22×#625—34no—
44#64149—38no—
45C2×C4×C3⋊Dic3—31no—
46#63979—332no—
47C2×Dic32—34no—
48C23×C33—22no—
49C23×C33—22no—
50C23×C33—22no—
51C23×C33C2312yes—
52C23×C33C2311yescharacteristic
53C22×C3⋊(C3⋊D3)—22no—
54C22×C3⋊(C3⋊D3)—22no—
55C22×C3⋊(C3⋊D3)—22no—
56C22×C3⋊(C3⋊D3)C2312yes—
57C22×C3×C3⋊D3—22no—
58C22×C3×C3⋊D3—22no—
59C22×C3×C3⋊D3—22no—
60C22×C3×C3⋊D3C2312yes—
61C22×C32×D3—22no—
62C22×C32×D3—22no—
63C22×C32×D3—22no—
64C22×C32×D3C2312yes—
65#6312—28no—
66C2×C3⋊(C3⋊Dic3)D414yescomplemented
67C2×C3⋊(C3⋊Dic3)C2312yes—
68C3×#638—28no—
69C2×C3×C3⋊Dic3D414yescomplemented
70C2×C3×C3⋊Dic3C2312yes—
71C32×#67—28no—
72C2×C32×Dic3D414yescomplemented
73C2×C32×Dic3C2312yes—
74#6292—28no—
75#6291—28no—
76#6290—28no—
77C2×#940—94no—
78C2×#940—91no—
79C24×C32—24no—
80C24×C32—31no—
81C24×C32C2×D314yes—
82C24×C32C2×D311yescharacteristic
83C23×C3⋊D3—31no—
84C23×C3⋊D3—34no—
85C23×C3⋊D3—64no—
86C23×C3⋊D3C2×D311yescharacteristic
87C23×C3×D3—34no—
88C23×C3×D3C2×D314yes—
89C2×C32×D4—34no—
90C2×C32×D4—64no—
91C22×C32×C4—31no—
92C2×#638—24no—
93C2×#638—34no—
94C2×#638—316no—
95C2×#638—34no—
96C2×#638—34no—
97C2×#638—64no—
98C2×#638—64no—
99C2×#638—616no—
100C2×#638—616no—
101C2×#638—632no—
102C2×#638—64no—
103C2×#638C2×D314yescomplemented
104C22×C3⋊Dic3—31no—
105C22×C3⋊Dic3—34no—
106C22×C3⋊Dic3—64no—
107C22×C3⋊Dic3C2×D311yescharacteristic
108C2×C4⋊(C3⋊D3)—34no—
109C2×C4⋊(C3⋊D3)—64no—
110C2×C3×#67—216no—
111C2×C3×#67—316no—
112C2×C3×#67—616no—
113C2×C3×#67C2×D3116yescomplemented
114C22×C3×Dic3—34no—
115C22×C3×Dic3C2×D314yes—
116C2×#626—316no—
117C2×#626—316no—
118C2×#626—616no—
119C2×#626—616no—
120C2×#625—316no—
121C2×#625—616no—
122C4×C3⋊Dic3—34no—
123Dic32—316no—
124C22×C33—22no—
125C22×C33—22no—
126C22×C33—22no—
127C22×C33—44no—
128C22×C33C2×D412yes—
129C22×C33C2×D412yes—
130C22×C33C2×D412yes—
131C22×C33C2411yescharacteristic
132C2×C3⋊(C3⋊D3)—22no—
133C2×C3⋊(C3⋊D3)—22no—
134C2×C3⋊(C3⋊D3)—22no—
135C2×C3⋊(C3⋊D3)—44no—
136C2×C3×C3⋊D3—22no—
137C2×C3×C3⋊D3—22no—
138C2×C3×C3⋊D3—22no—
139C2×C3×C3⋊D3—44no—
140C2×C32×D3—22no—
141C2×C32×D3—22no—
142C2×C32×D3—22no—
143C2×C32×D3—44no—
144C3⋊(C3⋊Dic3)C2×D414yescomplemented
145C3×C3⋊Dic3C2×D414yescomplemented
146C32×Dic3C2×D414yescomplemented
147C22×C3×D4—34no—
148C22×C3×D4—91no—
149C22×#67—34no—
150C22×#67—94no—
151C22×#67—94no—
152C22×#67—94no—
153C22×#67—91no—
154C22×#67—91no—
155C22×#67—91no—
156C22×#67—184no—
157C22×#67C3⋊D311yescharacteristic, complemented
158C22×D12—94no—
159C22×D12—91no—
160#940—932no—
161#940—98no—
162C2×C4×Dic3—94no—
163C2×C4×Dic3—91no—
164C23×C32—216no—
165C23×C32—28no—
166C23×C32—22no—
167C23×C32—22no—
168C23×C32—22no—
169C23×C32—24no—
170C23×C32—28no—
171C23×C32—28no—
172C23×C32—28no—
173C23×C32—32no—
174C23×C32—48no—
175C23×C32—48no—
176C23×C32—62no—
177C23×C32—62no—
178C23×C32—62no—
179C23×C32C22×D318yes—
180C23×C32C22×D314yes—
181C23×C32C22×D312yes—
182C23×C32C22×D311yescharacteristic
183C22×C3⋊D3—22no—
184C22×C3⋊D3—22no—
185C22×C3⋊D3—22no—
186C22×C3⋊D3—32no—
187C22×C3⋊D3—38no—
188C22×C3⋊D3—68no—
189C22×C3⋊D3—616no—
190C22×C3⋊D3—62no—
191C22×C3⋊D3—68no—
192C22×C3⋊D3—62no—
193C22×C3⋊D3—62no—
194C22×C3⋊D3—68no—
195C22×C3⋊D3—68no—
196C22×C3⋊D3—128no—
197C22×C3⋊D3—128no—
198C22×C3⋊D3C22×D312yes—
199C22×C3×D3—28no—
200C22×C3×D3—28no—
201C22×C3×D3—28no—
202C22×C3×D3—38no—
203C22×C3×D3—68no—
204C22×C3×D3—68no—
205C22×C3×D3—68no—
206C22×C3×D3C22×D318yes—
207C32×D4—68no—
208C2×C32×C4—34no—
209C2×C32×C4—32no—
210#638—28no—
211#638—68no—
212#638—68no—
213#638—632no—
214#638—664no—
215#638—68no—
216C2×C3⋊Dic3—34no—
217C2×C3⋊Dic3—32no—
218C2×C3⋊Dic3—316no—
219C2×C3⋊Dic3—38no—
220C2×C3⋊Dic3—616no—
221C2×C3⋊Dic3—68no—
222C2×C3⋊Dic3#6714yescomplemented
223C2×C3⋊Dic3C22×D312yes—
224C4⋊(C3⋊D3)—68no—
225C3×#67—232no—
226C3×#67—632no—
227C2×C3×Dic3—316no—
228C2×C3×Dic3—38no—
229C2×C3×Dic3#67116yescomplemented
230C2×C3×Dic3C22×D318yes—
231#626—632no—
232#626—632no—
233#625—632no—
234C2×#127—271noSylow
235C2×C33—42no—
236C2×C33#12714yescomplemented
237C2×C33C22×D411yescharacteristic
238C2×C33C22×D411yescharacteristic
239C2×C33C22×D411yescharacteristic
240C3⋊(C3⋊D3)—42no—
241C3×C3⋊D3—42no—
242C32×D3—42no—
243C24×C3—24no—
244C24×C3—34no—
245C24×C3—91no—
246C24×C3D3214yes—
247C24×C3C2×C3⋊D311yescharacteristic
248C23×D3—34no—
249C23×D3—94no—
250C23×D3—91no—
251C23×D3—184no—
252C23×D3C2×C3⋊D311yescharacteristic
253C2×C3×D4—316no—
254C2×C3×D4—616no—
255C2×C3×D4—94no—
256C2×C3×D4—184no—
257C22×C3×C4—34no—
258C22×C3×C4—91no—
259C2×#67—24no—
260C2×#67—316no—
261C2×#67—616no—
262C2×#67—916no—
263C2×#67—916no—
264C2×#67—916no—
265C2×#67—94no—
266C2×#67—94no—
267C2×#67—94no—
268C2×#67—1816no—
269C2×#67—1816no—
270C2×#67—184no—
271C2×#67—1816no—
272C2×#67—1816no—
273C2×#67—184no—
274C2×#67—1832no—
275C2×#67—184no—
276C2×#67C2×C3⋊D314yescomplemented
277C22×Dic3—34no—
278C22×Dic3—94no—
279C22×Dic3—91no—
280C22×Dic3—184no—
281C22×Dic3C2×C3⋊D311yescharacteristic
282C2×D12—916no—
283C2×D12—94no—
284C2×D12—1816no—
285C2×D12—184no—
286C4×Dic3—916no—
287C4×Dic3—94no—
288C22×C32—216no—
289C22×C32—28no—
290C22×C32—22no—
291C22×C32—22no—
292C22×C32—22no—
293C22×C32—24no—
294C22×C32—28no—
295C22×C32—28no—
296C22×C32—28no—
297C22×C32—28no—
298C22×C32—28no—
299C22×C32—48no—
300C22×C32—48no—
301C22×C32—432no—
302C22×C32—416no—
303C22×C32—44no—
304C22×C32—62no—
305C22×C32—62no—
306C22×C32—62no—
307C22×C32—124no—
308C22×C32D3×D418yes—
309C22×C32D3×D418yes—
310C22×C32D3×D412yes—
311C22×C32D3×D412yes—
312C22×C32C2×#6718yes—
313C22×C32C2×#6712yes—
314C22×C32C23×D314yes—
315C22×C32C23×D311yescharacteristic
316C2×C3⋊D3—22no—
317C2×C3⋊D3—22no—
318C2×C3⋊D3—22no—
319C2×C3⋊D3—44no—
320C2×C3⋊D3—616no—
321C2×C3⋊D3—62no—
322C2×C3⋊D3—68no—
323C2×C3⋊D3—62no—
324C2×C3⋊D3—62no—
325C2×C3⋊D3—68no—
326C2×C3⋊D3—68no—
327C2×C3⋊D3—128no—
328C2×C3⋊D3—128no—
329C2×C3⋊D3—1232no—
330C2×C3⋊D3—1216no—
331C2×C3⋊D3—124no—
332C2×C3×D3—28no—
333C2×C3×D3—28no—
334C2×C3×D3—28no—
335C2×C3×D3—416no—
336C2×C3×D3—68no—
337C2×C3×D3—68no—
338C2×C3×D3—68no—
339C2×C3×D3—1216no—
340C32×C4—34no—
341C3⋊Dic3—34no—
342C3⋊Dic3—316no—
343C3⋊Dic3—616no—
344C3⋊Dic3C2×#6714yescomplemented
345C3×Dic3—316no—
346C3×Dic3C2×#67116yescomplemented
347C22×D4—31no—
348C22×D4—91no—
349C22×D4—271no—
350C22×D4—271no—
351C22×D4—271no—
352C22×D4—271no—
353#127—278no—
354C2×C42—271no—
355C33C2×#12711yescharacteristic, complemented, Sylow
356C23×C3—216no—
357C23×C3—28no—
358C23×C3—28no—
359C23×C3—28no—
360C23×C3—22no—
361C23×C3—22no—
362C23×C3—22no—
363C23×C3—28no—
364C23×C3—24no—
365C23×C3—38no—
366C23×C3—48no—
367C23×C3—48no—
368C23×C3—68no—
369C23×C3—68no—
370C23×C3—68no—
371C23×C3—92no—
372C23×C3—182no—
373C23×C3—182no—
374C23×C3—182no—
375C23×C3C2×D3218yes—
376C23×C3C2×D3214yes—
377C23×C3C22×C3⋊D312yes—
378C23×C3C22×C3⋊D311yescharacteristic
379C22×D3—22no—
380C22×D3—22no—
381C22×D3—22no—
382C22×D3—38no—
383C22×D3—68no—
384C22×D3—68no—
385C22×D3—68no—
386C22×D3—98no—
387C22×D3—92no—
388C22×D3—188no—
389C22×D3—1816no—
390C22×D3—188no—
391C22×D3—188no—
392C22×D3—188no—
393C22×D3—182no—
394C22×D3—182no—
395C22×D3—182no—
396C22×D3—368no—
397C22×D3—368no—
398C22×D3C22×C3⋊D312yes—
399C3×D4—632no—
400C3×D4—188no—
401C2×C3×C4—38no—
402C2×C3×C4—316no—
403C2×C3×C4—92no—
404C2×C3×C4—94no—
405#67—28no—
406#67—632no—
407#67—1832no—
408#67—188no—
409#67—1832no—
410#67—1832no—
411#67—188no—
412#67—1864no—
413#67—188no—
414C2×Dic3—38no—
415C2×Dic3—316no—
416C2×Dic3—98no—
417C2×Dic3—92no—
418C2×Dic3—916no—
419C2×Dic3—94no—
420C2×Dic3—188no—
421C2×Dic3—1816no—
422C2×Dic3#63814yescomplemented
423C2×Dic3C22×C3⋊D312yes—
424D12—1832no—
425D12—188no—
426C2×C32—24no—
427C2×C32—24no—
428C2×C32—24no—
429C2×C32—216no—
430C2×C32—416no—
431C2×C32—48no—
432C2×C32—42no—
433C2×C32—122no—
434C2×C32#940116yescomplemented
435C2×C32#94014yescomplemented
436C2×C32C2×D3×D414yes—
437C2×C32C2×D3×D414yes—
438C2×C32C2×D3×D411yescharacteristic
439C2×C32C2×D3×D411yescharacteristic
440C2×C32C22×#6714yes—
441C2×C32C22×#6711yescharacteristic
442C3⋊D3—42no—
443C3⋊D3—1216no—
444C3⋊D3—128no—
445C3⋊D3—122no—
446C3×D3—48no—
447C3×D3—128no—
448C24—31no—
449C24—91no—
450C24—271no—
451C24D3×C3⋊D311yescharacteristic
452C2×D4—34no—
453C2×D4—64no—
454C2×D4—94no—
455C2×D4—184no—
456C2×D4—274no—
457C2×D4—274no—
458C2×D4—274no—
459C2×D4—274no—
460C2×D4—544no—
461C2×D4—544no—
462C2×D4—544no—
463C2×D4—544no—
464C22×C4—31no—
465C22×C4—91no—
466C22×C4—271no—
467C42—274no—
468C22×C3—216no—
469C22×C3—28no—
470C22×C3—28no—
471C22×C3—28no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—24no—
476C22×C3—28no—
477C22×C3—28no—
478C22×C3—28no—
479C22×C3—48no—
480C22×C3—48no—
481C22×C3—432no—
482C22×C3—44no—
483C22×C3—416no—
484C22×C3—68no—
485C22×C3—68no—
486C22×C3—68no—
487C22×C3—1216no—
488C22×C3—182no—
489C22×C3—182no—
490C22×C3—182no—
491C22×C3—364no—
492C22×C3D3×#6718yes—
493C22×C3D3×#6718yes—
494C22×C3#381118yes—
495C22×C3D4×C3⋊D312yes—
496C22×C3D4×C3⋊D312yes—
497C22×C3C2×#63812yes—
498C22×C3C22×D3214yes—
499C22×C3C23×C3⋊D311yescharacteristic
500C2×D3—22no—
501C2×D3—22no—
502C2×D3—22no—
503C2×D3—44no—
504C2×D3—68no—
505C2×D3—68no—
506C2×D3—68no—
507C2×D3—1216no—
508C2×D3—1816no—
509C2×D3—188no—
510C2×D3—188no—
511C2×D3—188no—
512C2×D3—182no—
513C2×D3—182no—
514C2×D3—182no—
515C2×D3—368no—
516C2×D3—368no—
517C2×D3—3632no—
518C2×D3—364no—
519C2×D3—3616no—
520C3×C4—316no—
521C3×C4—94no—
522Dic3—316no—
523Dic3—916no—
524Dic3—94no—
525Dic3—1816no—
526Dic3C2×#63814yescomplemented
527C32—24no—
528C32C2×#94014yescomplemented
529C32C2×#94011yescharacteristic, complemented
530C23—22no—
531C23—22no—
532C23—22no—
533C23—32no—
534C23—62no—
535C23—62no—
536C23—62no—
537C23—92no—
538C23—182no—
539C23—182no—
540C23—182no—
541C23—272no—
542C23—542no—
543C23—542no—
544C23—542no—
545C23C2×D3×C3⋊D312yes—
546C23C2×D3×C3⋊D311yescharacteristic
547D4—68no—
548D4—188no—
549D4—548no—
550D4—548no—
551D4—548no—
552D4—548no—
553C2×C4—32no—
554C2×C4—34no—
555C2×C4—92no—
556C2×C4—94no—
557C2×C4—272no—
558C2×C4—274no—
559C2×C3—24no—
560C2×C3—24no—
561C2×C3—24no—
562C2×C3—216no—
563C2×C3—416no—
564C2×C3—42no—
565C2×C3—48no—
566C2×C3—128no—
567C2×C3—362no—
568C2×C3#63979116yescomplemented
569C2×C3#6414914yescomplemented
570C2×C3C2×D3×#6714yes—
571C2×C3C2×D3×#6714yes—
572C2×C3C2×#381114yes—
573C2×C3C2×D4×C3⋊D311yescharacteristic
574C2×C3C2×D4×C3⋊D311yescharacteristic
575C2×C3C22×#63811yescharacteristic
576D3—42no—
577D3—128no—
578D3—3616no—
579D3—362no—
580D3—368no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—44no—
585C22—62no—
586C22—62no—
587C22—62no—
588C22—124no—
589C22—182no—
590C22—182no—
591C22—182no—
592C22—364no—
593C22—542no—
594C22—542no—
595C22—542no—
596C22—1084no—
597C22D3×#63812yes—
598C22C3⋊D3×#6712yes—
599C22#8903912yes—
600C22C22×D3×C3⋊D311yescharacteristic
601C4—34no—
602C4—94no—
603C4—274no—
604C3—24no—
605C3C2×#6397914yescomplemented
606C3C2×#6414911yescharacteristic, complemented
607C2—42no—
608C2—122no—
609C2—362no—
610C2—1082no—
611C2#1172363314yesdirect product factor
612C2C2×D3×#63811yescharacteristic
613C2C2×C3⋊D3×#6711yescharacteristic
614C2C2×#8903911yescharacteristic
subgroupG/H
centerC23C2×D3×C3⋊D3
commutator, cosocleC22×C33C24
FrattiniC22C22×D3×C3⋊D3
FittingC24×C33C22
socleC23×C33C23
hypercenterC24D3×C3⋊D3
hypocenterC33C2×#127
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×#6312C2semidirect
2. C22×C3×#638C2semidirect
3. C22×C32×#67C2semidirect
4. C22×#6292C2semidirect
5. C22×#6291C2semidirect
6. C22×#6290C2semidirect
7. #11723633C2direct (N × H)
8. C2×Dic3×C3⋊Dic3C2semidirect
16. C2×#6312C22semidirect
19. C2×C3×#638C22semidirect
22. C2×C32×#67C22semidirect
25. C2×#6292C22semidirect
27. C2×#6291C22semidirect
29. C2×#6290C22semidirect
30. Dic3×C3⋊Dic3C22semidirect
37. C22×#638D3semidirect
40. C22×C3×#67D3semidirect
66. C2×C3⋊(C3⋊Dic3)D4semidirect
69. C2×C3×C3⋊Dic3D4semidirect
72. C2×C32×Dic3D4semidirect
103. C2×#638C2×D3semidirect
113. C2×C3×#67C2×D3semidirect
144. C3⋊(C3⋊Dic3)C2×D4semidirect
145. C3×C3⋊Dic3C2×D4semidirect
146. C32×Dic3C2×D4semidirect
157. C22×#67C3⋊D3semidirect
222. C2×C3⋊Dic3#67semidirect
229. C2×C3×Dic3#67semidirect
236. C2×C33#127semidirect
276. C2×#67C2×C3⋊D3semidirect
344. C3⋊Dic3C2×#67semidirect
346. C3×Dic3C2×#67semidirect
355. C33C2×#127semidirect
422. C2×Dic3#638semidirect
434. C2×C32#940semidirect
435. C2×C32#940semidirect
526. Dic3C2×#638semidirect
528. C32C2×#940semidirect
529. C32C2×#940semidirect
568. C2×C3#63979semidirect
569. C2×C3#64149semidirect
605. C3C2×#63979semidirect
606. C3C2×#64149semidirect
611. C2#11723633direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#63122
C22×C3×#6382
C22×C32×#672
C22×#62922
C22×#62912
C22×#62902
#117236332
C2×Dic3×C3⋊Dic32
C2×#641493
C2×#639793