Finite Group Databasebeta

← C2×#11722943 #13172269

Subgroups — C2×#11722943 #13172269

Order 1728.

1602 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6293C211yesmaximal, characteristic
2C2×#88813C211yesmaximal, characteristic
3C2×#88806C211yesmaximal, characteristic
4C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic
5C2×C3×#3724C211yesmaximal, characteristic
6C2×C3×Dic3⋊Dic3C211yesmaximal, characteristic
7C2×C3×Dic32C211yesmaximal, characteristic
8#11722943C218yesmaximal, direct product factor
9C2×#63844—31nomaximal
10C2×#63838—31nomaximal
11C2×#63837—31nomaximal
12C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17C22×C32×Dic3C2211yescharacteristic
18C22×C32×Dic3C2211yescharacteristic
19C2×#6293C418yescomplemented
20C2×#6293C2214yes—
21#88813C2214yes—
22#88806C2214yes—
23Dic3×C3⋊Dic3C2214yes—
24C3×#3724C2214yes—
25C3×Dic3⋊Dic3C2214yes—
26C3×Dic32C2214yes—
27C22×#634—31no—
28C22×#627—31no—
29C22×#627—31no—
30C22×#627—61no—
31C2×#3750—31no—
32C2×C4×C3⋊Dic3—31no—
33C2×C3×C4⋊Dic3—31no—
34C2×C3×Dic3⋊C4—31no—
35C2×C3×Dic3⋊C4—31no—
36C2×C3×Dic3⋊C4—31no—
37C2×C3×C4×Dic3—31no—
38C2×C3×C4×Dic3—31no—
39C2×#3724—31no—
40C2×#3724—31no—
41C2×#3724—61no—
42C2×#3724D311yescharacteristic
43C2×Dic3⋊Dic3D311yescharacteristic
44C2×Dic32—31no—
45C2×Dic32—31no—
46C2×Dic32—31no—
47C2×Dic32—31no—
48C2×Dic32—31no—
49C2×Dic32—61no—
50C2×Dic32D311yescharacteristic
51#63844—38no—
52#63838—38no—
53#63837—38no—
54C23×C33C2311yescharacteristic
55C2×C3⋊(C3⋊Dic3)C2×C414yes—
56C2×C3⋊(C3⋊Dic3)C2312yes—
57C2×C3×C3⋊Dic3—22no—
58C2×C3×C3⋊Dic3C2×C414yes—
59C2×C3×C3⋊Dic3Q814yescomplemented
60C2×C3×C3⋊Dic3C2312yes—
61C2×C3×C3⋊Dic3C2312yes—
62C2×C3×C3⋊Dic3C2312yes—
63C2×C32×Dic3—22no—
64C2×C32×Dic3—22no—
65C2×C32×Dic3C2×C414yes—
66C2×C32×Dic3C2312yes—
67C2×C32×Dic3C2312yes—
68C2×C32×Dic3C2312yes—
69#6293C2×C4116yescomplemented
70C2×#887—91no—
71C2×#887—91no—
72C2×C4×Dic6—91no—
73C22×C32×C4—31no—
74C22×C32×C4—31no—
75C22×C32×C4—31no—
76C22×C3⋊Dic3—31no—
77C22×C3⋊Dic3—31no—
78C22×C3⋊Dic3—31no—
79C22×C3⋊Dic3—61no—
80C22×C3⋊Dic3—61no—
81C22×C3⋊Dic3—61no—
82C22×C3⋊Dic3—121no—
83C22×C3⋊Dic3C2×D311yescharacteristic
84C22×C3⋊Dic3C2×D311yescharacteristic
85C22×C3⋊Dic3C2×D311yescharacteristic
86C2×#634—38no—
87C2×#634—34no—
88C22×C3×Dic3—21no—
89C22×C3×Dic3—21no—
90C22×C3×Dic3—21no—
91C22×C3×Dic3—31no—
92C22×C3×Dic3—31no—
93C22×C3×Dic3—31no—
94C22×C3×Dic3—31no—
95C22×C3×Dic3—31no—
96C22×C3×Dic3—31no—
97C22×C3×Dic3—61no—
98C22×C3×Dic3—61no—
99C22×C3×Dic3—61no—
100C22×C3×Dic3C2×D311yescharacteristic
101C22×C3×Dic3C2×D311yescharacteristic
102C22×C3×Dic3C2×D311yescharacteristic
103C22×C3×Dic3C2×D311yescharacteristic
104C22×C3×Dic3C2×D311yescharacteristic
105C22×C3×Dic3C2×D311yescharacteristic
106C2×#627—38no—
107C2×#627—38no—
108C2×#627—34no—
109C2×#627—34no—
110C2×#627—68no—
111C2×#627—64no—
112#3750—34no—
113C4×C3⋊Dic3—34no—
114C3×C4⋊Dic3—34no—
115C3×Dic3⋊C4—34no—
116C3×Dic3⋊C4—34no—
117C3×Dic3⋊C4—34no—
118C3×C4×Dic3—34no—
119C3×C4×Dic3—34no—
120#3724—34no—
121#3724—34no—
122#3724—64no—
123#3724C2×D314yes—
124Dic3⋊Dic3C2×D314yes—
125Dic32—34no—
126Dic32—34no—
127Dic32—34no—
128Dic32—34no—
129Dic32—34no—
130Dic32—64no—
131Dic32C2×D314yes—
132C22×C33C22×C412yes—
133C22×C33C2×Q812yes—
134C22×C33#4112yes—
135C22×C33C2411yescharacteristic
136C3⋊(C3⋊Dic3)C22×C414yes—
137C3×C3⋊Dic3—22no—
138C3×C3⋊Dic3C22×C414yes—
139C3×C3⋊Dic3C2×Q814yescomplemented
140C32×Dic3—22no—
141C32×Dic3—22no—
142C32×Dic3C22×C414yes—
143C22×Dic6—91no—
144C22×Dic6—91no—
145C22×Dic6—91no—
146C22×Dic6—181no—
147C2×C3×C4⋊C4—91no—
148C2×C3×C4⋊C4—91no—
149C2×C3×C42—91no—
150C2×C4⋊Dic3—31no—
151C2×Dic3⋊C4—31no—
152C2×Dic3⋊C4—31no—
153C2×Dic3⋊C4—31no—
154C2×Dic3⋊C4—91no—
155C2×Dic3⋊C4—91no—
156C2×Dic3⋊C4—91no—
157C2×Dic3⋊C4—181no—
158C2×C4×Dic3—31no—
159C2×C4×Dic3—31no—
160C2×C4×Dic3—91no—
161C2×C4×Dic3—91no—
162C2×C4×Dic3—91no—
163C2×C4×Dic3—91no—
164C2×C4×Dic3—91no—
165C2×C4×Dic3—91no—
166C2×C4×Dic3—181no—
167#887—98no—
168#887—98no—
169C4×Dic6—98no—
170C23×C32—21no—
171C23×C32—21no—
172C23×C32—21no—
173C23×C32—41no—
174C23×C32C22×D311yescharacteristic
175C23×C32C22×D311yescharacteristic
176C23×C32C22×D311yescharacteristic
177C2×C32×C4—34no—
178C2×C32×C4—32no—
179C2×C32×C4—32no—
180C2×C32×C4—32no—
181C2×C32×C4—62no—
182C2×C32×C4—62no—
183C2×C3⋊Dic3—22no—
184C2×C3⋊Dic3—34no—
185C2×C3⋊Dic3—34no—
186C2×C3⋊Dic3—34no—
187C2×C3⋊Dic3—32no—
188C2×C3⋊Dic3—32no—
189C2×C3⋊Dic3—32no—
190C2×C3⋊Dic3—62no—
191C2×C3⋊Dic3—62no—
192C2×C3⋊Dic3—64no—
193C2×C3⋊Dic3—64no—
194C2×C3⋊Dic3—64no—
195C2×C3⋊Dic3—62no—
196C2×C3⋊Dic3—124no—
197C2×C3⋊Dic3—122no—
198C2×C3⋊Dic3Dic614yescomplemented
199C2×C3⋊Dic3C4×D314yes—
200C2×C3⋊Dic3C22×D312yes—
201C2×C3⋊Dic3C22×D312yes—
202C2×C3⋊Dic3C22×D312yes—
203#634—316no—
204C2×C3×Dic3—24no—
205C2×C3×Dic3—22no—
206C2×C3×Dic3—22no—
207C2×C3×Dic3—22no—
208C2×C3×Dic3—22no—
209C2×C3×Dic3—24no—
210C2×C3×Dic3—22no—
211C2×C3×Dic3—22no—
212C2×C3×Dic3—22no—
213C2×C3×Dic3—34no—
214C2×C3×Dic3—34no—
215C2×C3×Dic3—34no—
216C2×C3×Dic3—34no—
217C2×C3×Dic3—32no—
218C2×C3×Dic3—32no—
219C2×C3×Dic3—32no—
220C2×C3×Dic3—32no—
221C2×C3×Dic3—32no—
222C2×C3×Dic3—32no—
223C2×C3×Dic3—42no—
224C2×C3×Dic3—62no—
225C2×C3×Dic3—64no—
226C2×C3×Dic3—64no—
227C2×C3×Dic3—62no—
228C2×C3×Dic3—62no—
229C2×C3×Dic3—62no—
230C2×C3×Dic3—62no—
231C2×C3×Dic3—122no—
232C2×C3×Dic3C4×D314yes—
233C2×C3×Dic3C4×D314yes—
234C2×C3×Dic3C22×D312yes—
235C2×C3×Dic3C22×D312yes—
236C2×C3×Dic3C22×D312yes—
237C2×C3×Dic3C22×D312yes—
238C2×C3×Dic3C22×D312yes—
239C2×C3×Dic3C22×D312yes—
240#627—316no—
241#627—316no—
242#627—616no—
243C2×C4×Q8—271noSylow
244C2×C33C4×Q814yescomplemented
245C2×C33C23×C411yescharacteristic
246C2×C33C22×Q811yescharacteristic
247C2×C33C2×#4111yescharacteristic
248C22×C3×C4—31no—
249C22×C3×C4—31no—
250C22×C3×C4—31no—
251C22×C3×C4—31no—
252C22×C3×C4—31no—
253C22×C3×C4—31no—
254C22×C3×C4—61no—
255C22×C3×C4—61no—
256C22×C3×C4—61no—
257C22×C3×C4—91no—
258C22×C3×C4—91no—
259C22×C3×C4—91no—
260C22×Dic3—31no—
261C22×Dic3—31no—
262C22×Dic3—31no—
263C22×Dic3—31no—
264C22×Dic3—31no—
265C22×Dic3—31no—
266C22×Dic3—61no—
267C22×Dic3—61no—
268C22×Dic3—61no—
269C22×Dic3—91no—
270C22×Dic3—91no—
271C22×Dic3—91no—
272C22×Dic3—181no—
273C22×Dic3—181no—
274C22×Dic3—181no—
275C22×Dic3—361no—
276C22×Dic3D3211yescharacteristic
277C22×Dic3D3211yescharacteristic
278C22×Dic3D3211yescharacteristic
279C2×Dic6—94no—
280C2×Dic6—94no—
281C2×Dic6—98no—
282C2×Dic6—98no—
283C2×Dic6—98no—
284C2×Dic6—94no—
285C2×Dic6—184no—
286C2×Dic6—188no—
287C3×C4⋊C4—94no—
288C3×C4⋊C4—94no—
289C3×C42—94no—
290C4⋊Dic3—34no—
291Dic3⋊C4—34no—
292Dic3⋊C4—34no—
293Dic3⋊C4—34no—
294Dic3⋊C4—94no—
295Dic3⋊C4—94no—
296Dic3⋊C4—94no—
297Dic3⋊C4—184no—
298C4×Dic3—34no—
299C4×Dic3—34no—
300C4×Dic3—94no—
301C4×Dic3—94no—
302C4×Dic3—94no—
303C4×Dic3—94no—
304C4×Dic3—94no—
305C4×Dic3—94no—
306C4×Dic3—184no—
307C22×C32—21no—
308C22×C32—21no—
309C22×C32—21no—
310C22×C32—22no—
311C22×C32—22no—
312C22×C32—22no—
313C22×C32—22no—
314C22×C32—22no—
315C22×C32—22no—
316C22×C32—22no—
317C22×C32—22no—
318C22×C32—22no—
319C22×C32—42no—
320C22×C32—42no—
321C22×C32—42no—
322C22×C32—41no—
323C22×C32C2×Dic612yes—
324C22×C32C2×C4×D312yes—
325C22×C32C2×C4×D312yes—
326C22×C32C2×C4×D312yes—
327C22×C32#23512yes—
328C22×C32#23712yes—
329C22×C32#23712yes—
330C22×C32D3×Q812yes—
331C22×C32D3×Q812yes—
332C22×C32C23×D311yescharacteristic
333C22×C32C23×D311yescharacteristic
334C22×C32C23×D311yescharacteristic
335C32×C4—34no—
336C32×C4—62no—
337C32×C4—62no—
338C3⋊Dic3—22no—
339C3⋊Dic3—34no—
340C3⋊Dic3—34no—
341C3⋊Dic3—34no—
342C3⋊Dic3—64no—
343C3⋊Dic3—64no—
344C3⋊Dic3—64no—
345C3⋊Dic3—124no—
346C3⋊Dic3C2×Dic614yescomplemented
347C3⋊Dic3C2×C4×D314yes—
348C3×Dic3—24no—
349C3×Dic3—22no—
350C3×Dic3—22no—
351C3×Dic3—22no—
352C3×Dic3—22no—
353C3×Dic3—24no—
354C3×Dic3—34no—
355C3×Dic3—34no—
356C3×Dic3—34no—
357C3×Dic3—34no—
358C3×Dic3—42no—
359C3×Dic3—64no—
360C3×Dic3—64no—
361C3×Dic3—62no—
362C3×Dic3—62no—
363C3×Dic3—122no—
364C3×Dic3C2×C4×D314yes—
365C3×Dic3C2×C4×D314yes—
366C22×Q8—271no—
367C4×Q8—278no—
368C2×C4⋊C4—91no—
369C2×C4⋊C4—91no—
370C2×C4⋊C4—271no—
371C2×C42—91no—
372C2×C42—271no—
373C2×C42—271no—
374C33C2×C4×Q811yescharacteristic, complemented, Sylow
375C23×C3—21no—
376C23×C3—21no—
377C23×C3—21no—
378C23×C3—41no—
379C23×C3C2×D3211yescharacteristic
380C23×C3C2×D3211yescharacteristic
381C23×C3C2×D3211yescharacteristic
382C2×C3×C4—32no—
383C2×C3×C4—32no—
384C2×C3×C4—34no—
385C2×C3×C4—34no—
386C2×C3×C4—32no—
387C2×C3×C4—32no—
388C2×C3×C4—32no—
389C2×C3×C4—32no—
390C2×C3×C4—62no—
391C2×C3×C4—62no—
392C2×C3×C4—64no—
393C2×C3×C4—62no—
394C2×C3×C4—64no—
395C2×C3×C4—62no—
396C2×C3×C4—62no—
397C2×C3×C4—62no—
398C2×C3×C4—62no—
399C2×C3×C4—92no—
400C2×C3×C4—92no—
401C2×C3×C4—94no—
402C2×C3×C4—94no—
403C2×C3×C4—92no—
404C2×C3×C4—122no—
405C2×C3×C4—182no—
406C2×Dic3—22no—
407C2×Dic3—22no—
408C2×Dic3—32no—
409C2×Dic3—32no—
410C2×Dic3—32no—
411C2×Dic3—32no—
412C2×Dic3—34no—
413C2×Dic3—34no—
414C2×Dic3—34no—
415C2×Dic3—34no—
416C2×Dic3—32no—
417C2×Dic3—32no—
418C2×Dic3—64no—
419C2×Dic3—64no—
420C2×Dic3—62no—
421C2×Dic3—62no—
422C2×Dic3—62no—
423C2×Dic3—62no—
424C2×Dic3—62no—
425C2×Dic3—92no—
426C2×Dic3—92no—
427C2×Dic3—92no—
428C2×Dic3—94no—
429C2×Dic3—94no—
430C2×Dic3—94no—
431C2×Dic3—122no—
432C2×Dic3—182no—
433C2×Dic3—182no—
434C2×Dic3—182no—
435C2×Dic3—184no—
436C2×Dic3—184no—
437C2×Dic3—184no—
438C2×Dic3—362no—
439C2×Dic3—364no—
440C2×Dic3#62414yes—
441C2×Dic3C2×D3212yes—
442C2×Dic3C2×D3212yes—
443C2×Dic3C2×D3212yes—
444Dic6—916no—
445Dic6—916no—
446Dic6—916no—
447Dic6—1816no—
448C2×C32—21no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32—21no—
452C2×C32—21no—
453C2×C32—21no—
454C2×C32—21no—
455C2×C32—21no—
456C2×C32—21no—
457C2×C32—24no—
458C2×C32—24no—
459C2×C32—24no—
460C2×C32—44no—
461C2×C32—41no—
462C2×C32—41no—
463C2×C32—41no—
464C2×C32C4×Dic614yescomplemented
465C2×C32#88714yescomplemented
466C2×C32#88714yescomplemented
467C2×C32C22×Dic611yescharacteristic
468C2×C32C22×C4×D311yescharacteristic
469C2×C32C22×C4×D311yescharacteristic
470C2×C32C22×C4×D311yescharacteristic
471C2×C32C2×#23511yescharacteristic
472C2×C32C2×#23711yescharacteristic
473C2×C32C2×#23711yescharacteristic
474C2×C32C2×D3×Q811yescharacteristic
475C2×C32C2×D3×Q811yescharacteristic
476C2×Q8—274no—
477C2×Q8—278no—
478C22×C4—31no—
479C22×C4—31no—
480C22×C4—31no—
481C22×C4—91no—
482C22×C4—91no—
483C22×C4—91no—
484C22×C4—271no—
485C4⋊C4—94no—
486C4⋊C4—94no—
487C4⋊C4—274no—
488C42—94no—
489C42—274no—
490C42—274no—
491C22×C3—21no—
492C22×C3—21no—
493C22×C3—21no—
494C22×C3—22no—
495C22×C3—22no—
496C22×C3—22no—
497C22×C3—22no—
498C22×C3—22no—
499C22×C3—22no—
500C22×C3—22no—
501C22×C3—22no—
502C22×C3—22no—
503C22×C3—41no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—42no—
507C22×C3D3×Dic612yes—
508C22×C3D3×Dic612yes—
509C22×C3#379512yes—
510C22×C3#379712yes—
511C22×C3C4×D3212yes—
512C22×C3C4×D3212yes—
513C22×C3#380412yes—
514C22×C3#380512yes—
515C22×C3C2×#62412yes—
516C22×C3C22×D3211yescharacteristic
517C22×C3C22×D3211yescharacteristic
518C22×C3C22×D3211yescharacteristic
519C3×C4—34no—
520C3×C4—34no—
521C3×C4—64no—
522C3×C4—64no—
523C3×C4—62no—
524C3×C4—62no—
525C3×C4—62no—
526C3×C4—62no—
527C3×C4—94no—
528C3×C4—94no—
529C3×C4—122no—
530C3×C4—182no—
531Dic3—22no—
532Dic3—22no—
533Dic3—34no—
534Dic3—34no—
535Dic3—34no—
536Dic3—34no—
537Dic3—64no—
538Dic3—64no—
539Dic3—62no—
540Dic3—62no—
541Dic3—94no—
542Dic3—94no—
543Dic3—94no—
544Dic3—122no—
545Dic3—184no—
546Dic3—184no—
547Dic3—184no—
548Dic3—364no—
549Dic3C2×#62414yes—
550C32—21no—
551C32—21no—
552C32—21no—
553C32—41no—
554C32C2×C4×Dic611yescharacteristic, complemented
555C32C2×#88711yescharacteristic, complemented
556C32C2×#88711yescharacteristic, complemented
557C23D3311yescharacteristic
558Q8—2716no—
559C2×C4—32no—
560C2×C4—34no—
561C2×C4—32no—
562C2×C4—32no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—92no—
566C2×C4—92no—
567C2×C4—94no—
568C2×C4—94no—
569C2×C4—92no—
570C2×C4—182no—
571C2×C4—272no—
572C2×C4—274no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—21no—
581C2×C3—21no—
582C2×C3—24no—
583C2×C3—24no—
584C2×C3—24no—
585C2×C3—41no—
586C2×C3—41no—
587C2×C3—41no—
588C2×C3—44no—
589C2×C3#6383714yescomplemented
590C2×C3#6383814yescomplemented
591C2×C3#6384414yescomplemented
592C2×C3C2×D3×Dic611yescharacteristic
593C2×C3C2×D3×Dic611yescharacteristic
594C2×C3C2×#379511yescharacteristic
595C2×C3C2×#379711yescharacteristic
596C2×C3C2×C4×D3211yescharacteristic
597C2×C3C2×C4×D3211yescharacteristic
598C2×C3C2×#380411yescharacteristic
599C2×C3C2×#380511yescharacteristic
600C2×C3C22×#62411yescharacteristic
601C22D3×#62412yes—
602C22#8895712yes—
603C22#8896112yes—
604C22C2×D3311yescharacteristic
605C4—34no—
606C4—62no—
607C4—62no—
608C4—94no—
609C4—94no—
610C4—182no—
611C4—274no—
612C3—21no—
613C3—21no—
614C3—21no—
615C3—41no—
616C3C2×#6383711yescharacteristic, complemented
617C3C2×#6383811yescharacteristic, complemented
618C3C2×#6384411yescharacteristic, complemented
619C2#1172294314yesdirect product factor
620C2C2×D3×#62411yescharacteristic
621C2C2×#8895711yescharacteristic
622C2C2×#8896111yescharacteristic
subgroupG/H
center, hypercenterC23D33
commutatorC2×C33C23×C4
FrattiniC22C2×D33
Fitting, socle, prime radicalC23×C33C23
cosocleC22×C33C24
hypocenterC33C2×C4×Q8
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
8. #11722943C2direct (N × H)
19. C2×#6293C4semidirect
59. C2×C3×C3⋊Dic3Q8semidirect
69. #6293C2×C4semidirect
139. C3×C3⋊Dic3C2×Q8semidirect
198. C2×C3⋊Dic3Dic6semidirect
244. C2×C33C4×Q8semidirect
346. C3⋊Dic3C2×Dic6semidirect
374. C33C2×C4×Q8semidirect
464. C2×C32C4×Dic6semidirect
465. C2×C32#887semidirect
466. C2×C32#887semidirect
554. C32C2×C4×Dic6semidirect
555. C32C2×#887semidirect
556. C32C2×#887semidirect
589. C2×C3#63837semidirect
590. C2×C3#63838semidirect
591. C2×C3#63844semidirect
616. C3C2×#63837semidirect
617. C3C2×#63838semidirect
618. C3C2×#63844semidirect
619. C2#11722943direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62932
C2×#888132
C2×#888062
C2×Dic3×C3⋊Dic32
C2×C3×#37242
C2×C3×Dic3⋊Dic32
C2×C3×Dic322
#117229432
C2×#638443
C2×#638383
C2×#638373