Finite Group Databasebeta

← C2×#11720333 #13141145

Subgroups — C2×#11720333 #13141145

Order 1728.

867 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#638⋊C3C211yesmaximal, characteristic
2C2×#88552C211yesmaximal, characteristic, complemented
3C2×#3757⋊C3C211yesmaximal, characteristic, complemented
4#11720333C214yesmaximal, direct product factor
5C2×#63581—31nomaximal
6C2×#63580—31nomaximal
7C24×H(3,𝔽3)C2211yescharacteristic
8C23×(C3⋊D3)⋊C3C411yescharacteristic, complemented
9C22×#1084—21no—
10C2×#638⋊C3—21no—
11C2×#638⋊C3—21no—
12C2×#638⋊C3—21no—
13C2×#638⋊C3—21no—
14C2×#638⋊C3C412yescomplemented
15C2×#638⋊C3C2212yes—
16C22×(C3⋊Dic3)⋊C3—21no—
17C22×(C3⋊Dic3)⋊C3C411yescharacteristic, complemented
18#88552—21no—
19#88552C2212yescomplemented
20#3757⋊C3—21no—
21#3757⋊C3C2212yescomplemented
22C22×#638D311yescharacteristic
23C22×C3×#67—31no—
24C2×#3757D311yescharacteristic, complemented
25C2×C3×#217—31no—
26C2×C3×#217—31no—
27C2×C3×#217—31no—
28C2×C3×#217—61no—
29#63581—34no—
30#63580—34no—
31C23×H(3,𝔽3)—42no—
32C23×H(3,𝔽3)C2×C411yescharacteristic
33C23×H(3,𝔽3)C2×C411yescharacteristic
34C23×H(3,𝔽3)D411yescharacteristic
35C23×H(3,𝔽3)D411yescharacteristic
36C23×H(3,𝔽3)D411yescharacteristic
37C23×H(3,𝔽3)D411yescharacteristic
38C23×H(3,𝔽3)C2311yescharacteristic
39C22×(C3⋊D3)⋊C3—21no—
40C22×(C3⋊D3)⋊C3—21no—
41C22×(C3⋊D3)⋊C3—42no—
42C22×(C3⋊D3)⋊C3C2×C412yescomplemented
43C2×#1084—22no—
44C2×#1084—42no—
45#638⋊C3—22no—
46#638⋊C3—22no—
47#638⋊C3—22no—
48#638⋊C3—22no—
49C2×(C3⋊Dic3)⋊C3—21no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C2×(C3⋊Dic3)⋊C3—22no—
52C2×(C3⋊Dic3)⋊C3—42no—
53C2×(C3⋊Dic3)⋊C3C2×C412yescomplemented
54C2×#834—91no—
55C2×#806—91no—
56C2×#806—91no—
57C24×C32—21no—
58C24×C32C2×D311yescharacteristic
59C24×C32C2×D311yescharacteristic
60C23×C3⋊D3Dic311yescharacteristic, complemented
61C23×C3×D3—31no—
62C2×#638—21no—
63C2×#638—21no—
64C2×#638—21no—
65C2×#638—21no—
66C2×#638Dic312yescomplemented
67C2×#638C2×D312yes—
68C22×C3⋊Dic3—21no—
69C22×C3⋊Dic3Dic311yescharacteristic, complemented
70C2×C3×#67—32no—
71C2×C3×#67—32no—
72C2×C3×#67—61no—
73C2×C3×#67—61no—
74C2×C3×#67—61no—
75C2×C3×#67—61no—
76C22×C3×Dic3—31no—
77C22×C3×Dic3—62no—
78C22×C3×Dic3—61no—
79C22×C3×Dic3—61no—
80C22×C3×Dic3—61no—
81#3757—21no—
82#3757C2×D312yescomplemented
83C3×#217—32no—
84C3×#217—32no—
85C3×#217—32no—
86C3×#217—62no—
87C3×#217—62no—
88C3×#217—61no—
89C3×#217—61no—
90C3×#217—61no—
91C22×H(3,𝔽3)—22no—
92C22×H(3,𝔽3)—22no—
93C22×H(3,𝔽3)—22no—
94C22×H(3,𝔽3)—41no—
95C22×H(3,𝔽3)—41no—
96C22×H(3,𝔽3)—41no—
97C22×H(3,𝔽3)—41no—
98C22×H(3,𝔽3)C22⋊C411yescharacteristic
99C22×H(3,𝔽3)C22⋊C411yescharacteristic
100C22×H(3,𝔽3)C22⋊C411yescharacteristic
101C22×H(3,𝔽3)C22⋊C411yescharacteristic
102C22×H(3,𝔽3)C22×C411yescharacteristic
103C22×H(3,𝔽3)C2×D411yescharacteristic
104C22×H(3,𝔽3)C2×D411yescharacteristic
105C2×(C3⋊D3)⋊C3—22no—
106C2×(C3⋊D3)⋊C3—42no—
107C2×(C3⋊D3)⋊C3—42no—
108C2×(C3⋊D3)⋊C3—42no—
109#1084—42no—
110(C3⋊Dic3)⋊C3—22no—
111(C3⋊Dic3)⋊C3—42no—
112C22×C3×D4—91no—
113C22×#67—31no—
114C22×#67—91no—
115C2×C3×C22⋊C4—91no—
116C2×C3×C22⋊C4—91no—
117C2×#217—31no—
118C2×#217—31no—
119C2×#217—31no—
120C2×#217—61no—
121C2×#217—91no—
122#834—94no—
123#806—94no—
124#806—94no—
125C23×C32—21no—
126C23×C32—21no—
127C23×C32—21no—
128C23×C32—21no—
129C23×C32—21no—
130C23×C32—21no—
131C23×C32—21no—
132C23×C32—42no—
133C23×C32—42no—
134C23×C32—44no—
135C23×C32C4×D311yescharacteristic
136C23×C32C4×D311yescharacteristic
137C23×C32D1211yescharacteristic
138C23×C32D1211yescharacteristic
139C23×C32C2×Dic311yescharacteristic
140C23×C32C2×Dic311yescharacteristic
141C23×C32#6711yescharacteristic
142C23×C32#6711yescharacteristic
143C23×C32#6711yescharacteristic
144C23×C32#6711yescharacteristic
145C23×C32#6711yescharacteristic
146C23×C32#6711yescharacteristic
147C23×C32C22×D311yescharacteristic
148C23×C32C22×D311yescharacteristic
149C22×C3⋊D3—21no—
150C22×C3⋊D3—21no—
151C22×C3⋊D3—42no—
152C22×C3⋊D3C2×Dic312yescomplemented
153C22×C3×D3—32no—
154C22×C3×D3—61no—
155C22×C3×D3—61no—
156C22×C3×D3—122no—
157#638—22no—
158#638—22no—
159#638—22no—
160#638—22no—
161C2×C3⋊Dic3—21no—
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3—22no—
164C2×C3⋊Dic3—42no—
165C2×C3⋊Dic3C2×Dic312yescomplemented
166C3×#67—62no—
167C3×#67—62no—
168C3×#67—62no—
169C3×#67—62no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—64no—
174C2×C3×Dic3—62no—
175C2×C3×Dic3—62no—
176C2×C3×Dic3—62no—
177C2×C3×Dic3—122no—
178C2×C3×Dic3—124no—
179C2×C3×Dic3—122no—
180C2×C3×Dic3—122no—
181C2×#99—271noSylow
182C2×H(3,𝔽3)—21no—
183C2×H(3,𝔽3)—21no—
184C2×H(3,𝔽3)—21no—
185C2×H(3,𝔽3)—21no—
186C2×H(3,𝔽3)—21no—
187C2×H(3,𝔽3)—21no—
188C2×H(3,𝔽3)#9912yescomplemented
189C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
190(C3⋊D3)⋊C3—42no—
191C24×C3—31no—
192C24×C3—31no—
193C24×C3—61no—
194C24×C3—91no—
195C24×C3D3211yescharacteristic
196C23×D3—31no—
197C23×D3—91no—
198C2×C3×D4—92no—
199C2×C3×D4—92no—
200C2×C3×D4—181no—
201C2×C3×D4—181no—
202C2×C3×D4—181no—
203C2×C3×D4—181no—
204C22×C3×C4—91no—
205C22×C3×C4—181no—
206C22×C3×C4—181no—
207C2×#67—32no—
208C2×#67—32no—
209C2×#67—61no—
210C2×#67—61no—
211C2×#67—61no—
212C2×#67—61no—
213C2×#67—92no—
214C2×#67—92no—
215C2×#67—181no—
216C2×#67—181no—
217C2×#67—181no—
218C2×#67—181no—
219C22×Dic3—31no—
220C22×Dic3—61no—
221C22×Dic3—61no—
222C22×Dic3—62no—
223C22×Dic3—61no—
224C22×Dic3—91no—
225C22×Dic3—181no—
226C3×C22⋊C4—92no—
227C3×C22⋊C4—92no—
228C3×C22⋊C4—181no—
229C3×C22⋊C4—181no—
230#217—32no—
231#217—32no—
232#217—32no—
233#217—62no—
234#217—61no—
235#217—61no—
236#217—62no—
237#217—61no—
238#217—92no—
239#217—181no—
240C22×C32—24no—
241C22×C32—22no—
242C22×C32—22no—
243C22×C32—22no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—22no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—42no—
255C22×C32—42no—
256C22×C32—41no—
257C22×C32—41no—
258C22×C32—41no—
259C22×C32—41no—
260C22×C32—41no—
261C22×C32—41no—
262C22×C32—41no—
263C22×C32—41no—
264C22×C32—42no—
265C22×C32—42no—
266C22×C32—42no—
267C22×C32—42no—
268C22×C32(C2×D3)⋊C411yescharacteristic
269C22×C32(C2×D3)⋊C411yescharacteristic
270C22×C32(C2×D3)⋊C411yescharacteristic
271C22×C32(C2×D3)⋊C411yescharacteristic
272C22×C32#21711yescharacteristic
273C22×C32#21711yescharacteristic
274C22×C32#21711yescharacteristic
275C22×C32#21711yescharacteristic
276C22×C32C2×C4×D311yescharacteristic
277C22×C32C2×D1211yescharacteristic
278C22×C32C22×Dic311yescharacteristic
279C22×C32C2×#6711yescharacteristic
280C22×C32C2×#6711yescharacteristic
281C22×C32C2×#6711yescharacteristic
282C2×C3⋊D3—22no—
283C2×C3⋊D3—42no—
284C2×C3⋊D3—42no—
285C2×C3⋊D3—42no—
286C2×C3×D3—62no—
287C2×C3×D3—122no—
288C2×C3×D3—122no—
289C2×C3×D3—122no—
290C3⋊Dic3—22no—
291C3⋊Dic3—42no—
292C3×Dic3—62no—
293C3×Dic3—122no—
294C3×Dic3—124no—
295C3×Dic3—122no—
296C3×Dic3—122no—
297C22×D4—91no—
298C2×C22⋊C4—91no—
299C2×C22⋊C4—91no—
300#99—274no—
301H(3,𝔽3)C2×#9911yescharacteristic, complemented, Sylow
302C23×C3—31no—
303C23×C3—31no—
304C23×C3—31no—
305C23×C3—31no—
306C23×C3—31no—
307C23×C3—31no—
308C23×C3—31no—
309C23×C3—31no—
310C23×C3—31no—
311C23×C3—31no—
312C23×C3—31no—
313C23×C3—31no—
314C23×C3—31no—
315C23×C3—31no—
316C23×C3—42no—
317C23×C3—61no—
318C23×C3—61no—
319C23×C3—61no—
320C23×C3—61no—
321C23×C3—61no—
322C23×C3—61no—
323C23×C3—61no—
324C23×C3—92no—
325C23×C3—122no—
326C23×C3—122no—
327C23×C3—124no—
328C23×C3—181no—
329C23×C3—181no—
330C23×C3—362no—
331C23×C3D3×Dic311yescharacteristic
332C23×C3D3×Dic311yescharacteristic
333C23×C3#62511yescharacteristic
334C23×C3#62511yescharacteristic
335C23×C3#62611yescharacteristic
336C23×C3#62611yescharacteristic
337C23×C3C2×D3211yescharacteristic
338C22×D3—32no—
339C22×D3—61no—
340C22×D3—61no—
341C22×D3—92no—
342C22×D3—122no—
343C22×D3—181no—
344C22×D3—181no—
345C22×D3—362no—
346C3×D4—182no—
347C3×D4—182no—
348C3×D4—182no—
349C3×D4—182no—
350C2×C3×C4—92no—
351C2×C3×C4—181no—
352C2×C3×C4—181no—
353C2×C3×C4—182no—
354C2×C3×C4—182no—
355C2×C3×C4—362no—
356C2×C3×C4—362no—
357#67—62no—
358#67—62no—
359#67—62no—
360#67—62no—
361#67—182no—
362#67—182no—
363#67—182no—
364#67—182no—
365C2×Dic3—32no—
366C2×Dic3—61no—
367C2×Dic3—61no—
368C2×Dic3—62no—
369C2×Dic3—62no—
370C2×Dic3—64no—
371C2×Dic3—62no—
372C2×Dic3—92no—
373C2×Dic3—122no—
374C2×Dic3—122no—
375C2×Dic3—122no—
376C2×Dic3—124no—
377C2×Dic3—181no—
378C2×Dic3—181no—
379C2×Dic3—182no—
380C2×Dic3—362no—
381C2×C32—22no—
382C2×C32—22no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—22no—
396C2×C32—21no—
397C2×C32—41no—
398C2×C32—41no—
399C2×C32—41no—
400C2×C32—41no—
401C2×C32#80612yescomplemented
402C2×C32#83412yescomplemented
403C2×C32C2×(C2×D3)⋊C411yescharacteristic
404C2×C32C2×#21711yescharacteristic
405C3⋊D3—42no—
406C3×D3—122no—
407C24—91no—
408C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
409C2×D4—92no—
410C2×D4—92no—
411C2×D4—181no—
412C2×D4—181no—
413C2×D4—181no—
414C2×D4—181no—
415C22×C4—91no—
416C22×C4—181no—
417C22×C4—181no—
418C22⋊C4—92no—
419C22⋊C4—92no—
420C22⋊C4—181no—
421C22⋊C4—181no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—31no—
426C22×C3—31no—
427C22×C3—31no—
428C22×C3—31no—
429C22×C3—31no—
430C22×C3—31no—
431C22×C3—31no—
432C22×C3—31no—
433C22×C3—31no—
434C22×C3—31no—
435C22×C3—31no—
436C22×C3—31no—
437C22×C3—31no—
438C22×C3—31no—
439C22×C3—41no—
440C22×C3—41no—
441C22×C3—41no—
442C22×C3—41no—
443C22×C3—62no—
444C22×C3—62no—
445C22×C3—62no—
446C22×C3—62no—
447C22×C3—62no—
448C22×C3—62no—
449C22×C3—64no—
450C22×C3—61no—
451C22×C3—61no—
452C22×C3—61no—
453C22×C3—61no—
454C22×C3—61no—
455C22×C3—61no—
456C22×C3—61no—
457C22×C3—122no—
458C22×C3—122no—
459C22×C3—121no—
460C22×C3—121no—
461C22×C3—121no—
462C22×C3—121no—
463C22×C3—121no—
464C22×C3—121no—
465C22×C3—121no—
466C22×C3—121no—
467C22×C3—122no—
468C22×C3—122no—
469C22×C3—122no—
470C22×C3—122no—
471C22×C3—182no—
472C22×C3—362no—
473C22×C3—362no—
474C22×C3—362no—
475C22×C3(C2×D3)⋊Dic311yescharacteristic
476C22×C3(C2×D3)⋊Dic311yescharacteristic
477C22×C3(C2×D3)⋊Dic311yescharacteristic
478C22×C3(C2×D3)⋊Dic311yescharacteristic
479C22×C3C2×D3×Dic311yescharacteristic
480C22×C3C2×#62511yescharacteristic
481C22×C3C2×#62611yescharacteristic
482C2×D3—62no—
483C2×D3—122no—
484C2×D3—122no—
485C2×D3—122no—
486C2×D3—182no—
487C2×D3—362no—
488C2×D3—362no—
489C2×D3—362no—
490C3×C4—182no—
491C3×C4—362no—
492C3×C4—362no—
493Dic3—62no—
494Dic3—122no—
495Dic3—122no—
496Dic3—122no—
497Dic3—124no—
498Dic3—182no—
499Dic3—362no—
500C32—21no—
501C32C2×#80611yescharacteristic, complemented
502C32C2×#83411yescharacteristic, complemented
503C23—42no—
504C23—92no—
505C23—181no—
506C23—181no—
507C23—362no—
508C23((C3⋊D3)⋊C3)⋊C411yescharacteristic
509C23((C3⋊D3)⋊C3)⋊C411yescharacteristic
510C23#619811yescharacteristic
511C23#619811yescharacteristic
512C23#620011yescharacteristic
513C23#620011yescharacteristic
514C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
515D4—182no—
516D4—182no—
517D4—182no—
518D4—182no—
519C2×C4—92no—
520C2×C4—181no—
521C2×C4—181no—
522C2×C4—182no—
523C2×C4—182no—
524C2×C4—362no—
525C2×C4—362no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—31no—
533C2×C3—31no—
534C2×C3—32no—
535C2×C3—32no—
536C2×C3—61no—
537C2×C3—61no—
538C2×C3—61no—
539C2×C3—61no—
540C2×C3—61no—
541C2×C3—61no—
542C2×C3—61no—
543C2×C3—61no—
544C2×C3—61no—
545C2×C3—61no—
546C2×C3—61no—
547C2×C3—61no—
548C2×C3—62no—
549C2×C3—62no—
550C2×C3—61no—
551C2×C3—62no—
552C2×C3—121no—
553C2×C3—121no—
554C2×C3—121no—
555C2×C3—121no—
556C2×C3—362no—
557C2×C3#6358012yes—
558C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
559D3—122no—
560D3—362no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—182no—
569C22—362no—
570C22—362no—
571C22—362no—
572C22#8845011yescharacteristic
573C22#8845011yescharacteristic
574C22#8845011yescharacteristic
575C22#8845011yescharacteristic
576C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
577C22C2×#619811yescharacteristic
578C22C2×#620011yescharacteristic
579C4—182no—
580C4—362no—
581C4—362no—
582C3—31no—
583C3—31no—
584C3—61no—
585C3C2×#6358011yescharacteristic
586C2—21no—
587C2—21no—
588C2—21no—
589C2—21no—
590C2—21no—
591C2—21no—
592C2—362no—
593C2#1172033312yesdirect product factor
594C2C2×#8845011yescharacteristic
subgroupG/H
centerC22#88450
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC23×C3C2×D32
FittingC24×H(3,𝔽3)C22
socleC22×C3(C2×D3)⋊Dic3
prime radicalC22×#638⋊C3C2
cosocleC23×H(3,𝔽3)C23
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#99
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×#88552C2semidirect
3. C2×#3757⋊C3C2semidirect
4. #11720333C2direct (N × H)
8. C23×(C3⋊D3)⋊C3C4semidirect
14. C2×#638⋊C3C4semidirect
17. C22×(C3⋊Dic3)⋊C3C4semidirect
19. #88552C22semidirect
21. #3757⋊C3C22semidirect
24. C2×#3757D3semidirect
42. C22×(C3⋊D3)⋊C3C2×C4semidirect
53. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
60. C23×C3⋊D3Dic3semidirect
66. C2×#638Dic3semidirect
69. C22×C3⋊Dic3Dic3semidirect
82. #3757C2×D3semidirect
152. C22×C3⋊D3C2×Dic3semidirect
165. C2×C3⋊Dic3C2×Dic3semidirect
188. C2×H(3,𝔽3)#99semidirect
301. H(3,𝔽3)C2×#99semidirect
401. C2×C32#806semidirect
402. C2×C32#834semidirect
501. C32C2×#806semidirect
502. C32C2×#834semidirect
593. C2#11720333direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#638⋊C32
C2×#885522
C2×#3757⋊C32
#117203332
C2×#635813
C2×#635803