Finite Group Databasebeta

← C2×#11721314 #13152074

Subgroups — C2×#11721314 #13152074

Order 1728.

1623 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#638⋊C3C211yesmaximal, characteristic
2C23×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3C22×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
4C2×#88552C211yesmaximal, characteristic, complemented
5#11721314C218yesmaximal, direct product factor
6C2×#88450C211yesmaximal, characteristic, complemented
7C2×#88448C211yesmaximal, characteristic, complemented
8C2×#88447C211yesmaximal, characteristic, complemented
9C2×#67⋊Dic3—31nomaximal
10C2×#63952—31nomaximal
11C24×H(3,𝔽3)C2211yescharacteristic
12C23×(C3⋊D3)⋊C3C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C2×#638⋊C3C418yescomplemented
16C2×#638⋊C3C2214yes—
17C22×(C3⋊Dic3)⋊C3—24no—
18C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C22×(C3⋊Dic3)⋊C3C2214yescomplemented
22C2×((C3⋊D3)⋊C3)⋊C4—24no—
23C2×((C3⋊D3)⋊C3)⋊C4C2214yescomplemented
24#88552C2214yescomplemented
25#88450C2214yescomplemented
26#88448C2214yescomplemented
27#88447C2214yescomplemented
28C22×#638D311yescharacteristic
29C23×C3⋊Dic3D311yescharacteristic, complemented
30C22×C3×#67—31no—
31C23×C3×Dic3—31no—
32C22×#624—31no—
33C22×D3×Dic3—31no—
34C2×C3×#217—31no—
35C2×C3×#217—31no—
36C2×C3×#217—61no—
37#67⋊Dic3—38no—
38C2×#3724—31no—
39C2×Dic3⋊Dic3—31no—
40C2×#3722—31no—
41C2×(C2×D3)⋊Dic3—31no—
42C2×Dic32—31no—
43C2×Dic32—31no—
44#63952—38no—
45C23×H(3,𝔽3)—22no—
46C23×H(3,𝔽3)—22no—
47C23×H(3,𝔽3)C2×C414yescomplemented
48C23×H(3,𝔽3)C2312yes—
49C23×H(3,𝔽3)C2311yescharacteristic
50C22×(C3⋊D3)⋊C3—22no—
51C22×(C3⋊D3)⋊C3—22no—
52C22×(C3⋊D3)⋊C3C2×C414yescomplemented
53C22×(C3⋊D3)⋊C3C2312yes—
54C2×#1084—22no—
55C2×#1084—22no—
56C2×#1084C2312yes—
57C2×#1084C2312yes—
58#638⋊C3C2×C4116yescomplemented
59C2×(C3⋊Dic3)⋊C3—22no—
60C2×(C3⋊Dic3)⋊C3—28no—
61C2×(C3⋊Dic3)⋊C3C2×C414yescomplemented
62C2×(C3⋊Dic3)⋊C3D414yescomplemented
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66((C3⋊D3)⋊C3)⋊C4—28no—
67C2×D4×Dic3—91no—
68C2×C4×#67—91no—
69C2×#67⋊C4—91no—
70C24×C32—21no—
71C24×C32C2×D311yescharacteristic
72C24×C32C2×D311yescharacteristic
73C23×C3⋊D3C2×D311yescharacteristic
74C23×C3×D3—31no—
75C2×#638Dic318yescomplemented
76C2×#638C2×D314yes—
77C22×C3⋊Dic3—24no—
78C22×C3⋊Dic3C2×D311yescharacteristic
79C22×C3⋊Dic3C2×D311yescharacteristic
80C22×C3⋊Dic3C2×D311yescharacteristic
81C22×C3⋊Dic3C2×D314yescomplemented
82C2×C3×#67—38no—
83C2×C3×#67—34no—
84C22×C3×Dic3—31no—
85C22×C3×Dic3—31no—
86C22×C3×Dic3—31no—
87C22×C3×Dic3—31no—
88C22×C3×Dic3—31no—
89C22×C3×Dic3—31no—
90C22×C3×Dic3—31no—
91C22×C3×Dic3—34no—
92C22×C3×Dic3—64no—
93C22×C3×Dic3—61no—
94C22×C3×Dic3—61no—
95C2×#624—34no—
96C2×#624—64no—
97C2×D3×Dic3—34no—
98C2×D3×Dic3—64no—
99C3×#217—34no—
100C3×#217—34no—
101C3×#217—64no—
102#3724—34no—
103Dic3⋊Dic3—34no—
104#3722—34no—
105(C2×D3)⋊Dic3—34no—
106Dic32—34no—
107Dic32—34no—
108C22×H(3,𝔽3)—28no—
109C22×H(3,𝔽3)—22no—
110C22×H(3,𝔽3)—22no—
111C22×H(3,𝔽3)C22×C412yes—
112C22×H(3,𝔽3)C22×C414yescomplemented
113C22×H(3,𝔽3)C2×D412yes—
114C22×H(3,𝔽3)#4112yes—
115C22×H(3,𝔽3)C2411yescharacteristic
116C2×(C3⋊D3)⋊C3—28no—
117C2×(C3⋊D3)⋊C3—22no—
118C2×(C3⋊D3)⋊C3—22no—
119C2×(C3⋊D3)⋊C3C22×C414yescomplemented
120#1084—22no—
121#1084—22no—
122(C3⋊Dic3)⋊C3—22no—
123(C3⋊Dic3)⋊C3C22×C414yescomplemented
124(C3⋊Dic3)⋊C3C2×D414yescomplemented
125C22×C3×D4—91no—
126C23×C3×C4—91no—
127C22×#67—31no—
128C22×#67—91no—
129C23×Dic3—31no—
130C23×Dic3—91no—
131C23×Dic3—91no—
132C22×C4×D3—91no—
133C22×C4×D3—91no—
134C2×C3×C22⋊C4—91no—
135C2×#217—31no—
136C2×#217—31no—
137C2×#217—61no—
138C2×#217—91no—
139D4×Dic3—98no—
140C4×#67—98no—
141C2×(C2×D3)⋊C4—91no—
142C2×(C2×D3)⋊C4—91no—
143C2×C4⋊Dic3—91no—
144C2×Dic3⋊C4—91no—
145C2×Dic3⋊C4—91no—
146C2×C4×Dic3—91no—
147C2×C4×Dic3—91no—
148C2×C4×Dic3—91no—
149#67⋊C4—98no—
150C23×C32—22no—
151C23×C32—22no—
152C23×C32—22no—
153C23×C32—22no—
154C23×C32—24no—
155C23×C32—21no—
156C23×C32—22no—
157C23×C32—42no—
158C23×C32—42no—
159C23×C32C4×D314yescomplemented
160C23×C32C2×Dic314yescomplemented
161C23×C32C22×D312yes—
162C23×C32C22×D312yes—
163C23×C32C22×D311yescharacteristic
164C23×C32C22×D311yescharacteristic
165C22×C3⋊D3—22no—
166C22×C3⋊D3—22no—
167C22×C3⋊D3C2×Dic314yescomplemented
168C22×C3⋊D3C22×D312yes—
169C22×C3×D3—34no—
170C22×C3×D3—32no—
171C22×C3×D3—62no—
172C22×C3×D3—62no—
173#638C2×Dic3116yescomplemented
174C2×C3⋊Dic3—22no—
175C2×C3⋊Dic3—28no—
176C2×C3⋊Dic3C2×Dic314yescomplemented
177C2×C3⋊Dic3#6714yescomplemented
178C2×C3⋊Dic3C22×D312yes—
179C2×C3⋊Dic3C22×D312yes—
180C2×C3⋊Dic3C22×D312yes—
181C3×#67—316no—
182C2×C3×Dic3—34no—
183C2×C3×Dic3—34no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—32no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3—32no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3—32no—
190C2×C3×Dic3—32no—
191C2×C3×Dic3—62no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3—68no—
194C2×C3×Dic3—62no—
195C2×C3×Dic3—62no—
196C2×C3×Dic3—62no—
197C2×C3×Dic3—62no—
198C2×C3×Dic3—62no—
199C2×C3×Dic3—122no—
200C2×C3×Dic3—122no—
201#624—68no—
202D3×Dic3—68no—
203C2×C4×D4—271noSylow
204C2×H(3,𝔽3)—24no—
205C2×H(3,𝔽3)C4×D414yescomplemented
206C2×H(3,𝔽3)C23×C411yescharacteristic
207C2×H(3,𝔽3)C22×D411yescharacteristic
208C2×H(3,𝔽3)C2×#4111yescharacteristic
209(C3⋊D3)⋊C3—24no—
210C24×C3—31no—
211C24×C3—31no—
212C24×C3—61no—
213C24×C3—91no—
214C24×C3D3211yescharacteristic
215C23×D3—31no—
216C23×D3—91no—
217C2×C3×D4—94no—
218C2×C3×D4—98no—
219C22×C3×C4—91no—
220C22×C3×C4—91no—
221C22×C3×C4—91no—
222C22×C3×C4—94no—
223C22×C3×C4—91no—
224C22×C3×C4—91no—
225C22×C3×C4—184no—
226C2×#67—38no—
227C2×#67—34no—
228C2×#67—94no—
229C2×#67—98no—
230C22×Dic3—31no—
231C22×Dic3—31no—
232C22×Dic3—31no—
233C22×Dic3—31no—
234C22×Dic3—31no—
235C22×Dic3—31no—
236C22×Dic3—34no—
237C22×Dic3—31no—
238C22×Dic3—61no—
239C22×Dic3—61no—
240C22×Dic3—64no—
241C22×Dic3—91no—
242C22×Dic3—91no—
243C22×Dic3—91no—
244C22×Dic3—94no—
245C22×Dic3—94no—
246C22×Dic3—184no—
247C22×Dic3—184no—
248C2×C4×D3—94no—
249C2×C4×D3—94no—
250C2×C4×D3—184no—
251C2×C4×D3—184no—
252C3×C22⋊C4—94no—
253#217—34no—
254#217—34no—
255#217—64no—
256#217—94no—
257(C2×D3)⋊C4—94no—
258(C2×D3)⋊C4—94no—
259C4⋊Dic3—94no—
260Dic3⋊C4—94no—
261Dic3⋊C4—94no—
262C4×Dic3—94no—
263C4×Dic3—94no—
264C4×Dic3—94no—
265C22×C32—28no—
266C22×C32—28no—
267C22×C32—22no—
268C22×C32—22no—
269C22×C32—22no—
270C22×C32—22no—
271C22×C32—21no—
272C22×C32—22no—
273C22×C32—24no—
274C22×C32—22no—
275C22×C32—22no—
276C22×C32—42no—
277C22×C32—42no—
278C22×C32—48no—
279C22×C32C2×C4×D312yes—
280C22×C32C2×C4×D314yescomplemented
281C22×C32#23512yes—
282C22×C32D3×D412yes—
283C22×C32#23712yes—
284C22×C32C22×Dic312yes—
285C22×C32C22×Dic314yescomplemented
286C22×C32C2×#6712yes—
287C22×C32C23×D311yescharacteristic
288C22×C32C23×D311yescharacteristic
289C2×C3⋊D3—28no—
290C2×C3⋊D3—22no—
291C2×C3⋊D3—22no—
292C2×C3⋊D3C22×Dic314yescomplemented
293C2×C3×D3—34no—
294C2×C3×D3—68no—
295C2×C3×D3—62no—
296C2×C3×D3—62no—
297C3⋊Dic3—22no—
298C3⋊Dic3C22×Dic314yescomplemented
299C3⋊Dic3C2×#6714yescomplemented
300C3×Dic3—34no—
301C3×Dic3—34no—
302C3×Dic3—62no—
303C3×Dic3—62no—
304C3×Dic3—62no—
305C3×Dic3—62no—
306C3×Dic3—62no—
307C3×Dic3—122no—
308C3×Dic3—122no—
309C22×D4—91no—
310C23×C4—91no—
311C23×C4—271no—
312C4×D4—278no—
313C2×C4⋊C4—271no—
314C2×C22⋊C4—91no—
315C2×C22⋊C4—271no—
316C2×C42—271no—
317H(3,𝔽3)C2×C4×D411yescharacteristic, complemented, Sylow
318C23×C3—22no—
319C23×C3—22no—
320C23×C3—34no—
321C23×C3—34no—
322C23×C3—32no—
323C23×C3—32no—
324C23×C3—31no—
325C23×C3—31no—
326C23×C3—62no—
327C23×C3—62no—
328C23×C3—62no—
329C23×C3—62no—
330C23×C3—64no—
331C23×C3—62no—
332C23×C3—61no—
333C23×C3—92no—
334C23×C3—94no—
335C23×C3—122no—
336C23×C3—122no—
337C23×C3—182no—
338C23×C3—182no—
339C23×C3D3×Dic314yes—
340C23×C3C2×D3212yes—
341C23×C3C2×D3211yescharacteristic
342C22×D3—32no—
343C22×D3—34no—
344C22×D3—62no—
345C22×D3—62no—
346C22×D3—94no—
347C22×D3—92no—
348C22×D3—182no—
349C22×D3—182no—
350C3×D4—916no—
351C2×C3×C4—92no—
352C2×C3×C4—92no—
353C2×C3×C4—94no—
354C2×C3×C4—94no—
355C2×C3×C4—92no—
356C2×C3×C4—92no—
357C2×C3×C4—92no—
358C2×C3×C4—188no—
359C2×C3×C4—182no—
360C2×C3×C4—182no—
361C2×C3×C4—182no—
362#67—316no—
363#67—916no—
364C2×Dic3—32no—
365C2×Dic3—32no—
366C2×Dic3—34no—
367C2×Dic3—34no—
368C2×Dic3—32no—
369C2×Dic3—32no—
370C2×Dic3—32no—
371C2×Dic3—32no—
372C2×Dic3—32no—
373C2×Dic3—62no—
374C2×Dic3—62no—
375C2×Dic3—68no—
376C2×Dic3—62no—
377C2×Dic3—62no—
378C2×Dic3—62no—
379C2×Dic3—62no—
380C2×Dic3—62no—
381C2×Dic3—94no—
382C2×Dic3—94no—
383C2×Dic3—92no—
384C2×Dic3—92no—
385C2×Dic3—92no—
386C2×Dic3—122no—
387C2×Dic3—122no—
388C2×Dic3—188no—
389C2×Dic3—182no—
390C2×Dic3—188no—
391C4×D3—188no—
392C4×D3—188no—
393C2×C32—24no—
394C2×C32—24no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32—24no—
399C2×C32—44no—
400C2×C32C4×#6714yescomplemented
401C2×C32D4×Dic314yescomplemented
402C2×C32C22×C4×D311yescharacteristic
403C2×C32C2×#23511yescharacteristic
404C2×C32C2×D3×D411yescharacteristic
405C2×C32C2×#23711yescharacteristic
406C2×C32C23×Dic311yescharacteristic
407C2×C32C22×#6711yescharacteristic
408C3⋊D3—24no—
409C3×D3—64no—
410C24—91no—
411C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
412C2×D4—98no—
413C2×D4—94no—
414C22×C4—91no—
415C22×C4—91no—
416C22×C4—94no—
417C22×C4—91no—
418C22×C4—91no—
419C22×C4—91no—
420C22×C4—184no—
421C22×C4—274no—
422C22×C4—544no—
423C4⋊C4—274no—
424C22⋊C4—94no—
425C22⋊C4—274no—
426C42—274no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—28no—
430C22×C3—31no—
431C22×C3—31no—
432C22×C3—32no—
433C22×C3—32no—
434C22×C3—34no—
435C22×C3—34no—
436C22×C3—32no—
437C22×C3—32no—
438C22×C3—32no—
439C22×C3—32no—
440C22×C3—62no—
441C22×C3—62no—
442C22×C3—62no—
443C22×C3—62no—
444C22×C3—68no—
445C22×C3—68no—
446C22×C3—61no—
447C22×C3—62no—
448C22×C3—64no—
449C22×C3—62no—
450C22×C3—62no—
451C22×C3—94no—
452C22×C3—122no—
453C22×C3—122no—
454C22×C3—128no—
455C22×C3—182no—
456C22×C3—182no—
457C22×C3—188no—
458C22×C3C2×D3×Dic312yes—
459C22×C3C2×D3×Dic314yes—
460C22×C3#380412yes—
461C22×C3D3×#6712yes—
462C22×C3C22×D3211yescharacteristic
463C2×D3—34no—
464C2×D3—62no—
465C2×D3—62no—
466C2×D3—68no—
467C2×D3—94no—
468C2×D3—182no—
469C2×D3—182no—
470C2×D3—188no—
471C3×C4—94no—
472C3×C4—94no—
473C3×C4—182no—
474C3×C4—182no—
475C3×C4—182no—
476Dic3—34no—
477Dic3—34no—
478Dic3—62no—
479Dic3—62no—
480Dic3—62no—
481Dic3—62no—
482Dic3—62no—
483Dic3—94no—
484Dic3—94no—
485Dic3—122no—
486Dic3—122no—
487Dic3—182no—
488C32—21no—
489C32C2×C4×#6711yescharacteristic, complemented
490C32C2×D4×Dic311yescharacteristic, complemented
491C23—22no—
492C23—22no—
493C23—92no—
494C23—94no—
495C23—182no—
496C23—182no—
497C23((C3⋊D3)⋊C3)⋊C414yescomplemented
498C23C2×((C3⋊D3)⋊C3)⋊C212yes—
499C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
500D4—916no—
501C2×C4—92no—
502C2×C4—92no—
503C2×C4—94no—
504C2×C4—94no—
505C2×C4—92no—
506C2×C4—92no—
507C2×C4—92no—
508C2×C4—188no—
509C2×C4—182no—
510C2×C4—182no—
511C2×C4—182no—
512C2×C4—548no—
513C2×C3—24no—
514C2×C3—31no—
515C2×C3—31no—
516C2×C3—31no—
517C2×C3—31no—
518C2×C3—31no—
519C2×C3—31no—
520C2×C3—34no—
521C2×C3—34no—
522C2×C3—64no—
523C2×C3—64no—
524C2×C3—61no—
525C2×C3—61no—
526C2×C3—61no—
527C2×C3—64no—
528C2×C3—124no—
529C2×C3—184no—
530C2×C3Dic3×#6714yes—
531C2×C3C22×D3×Dic311yescharacteristic
532C2×C3C2×#380411yescharacteristic
533C2×C3C2×D3×#6711yescharacteristic
534D3—64no—
535D3—184no—
536C22—22no—
537C22—22no—
538C22—28no—
539C22—94no—
540C22—182no—
541C22—182no—
542C22—188no—
543C22C2×((C3⋊D3)⋊C3)⋊C412yes—
544C22C2×((C3⋊D3)⋊C3)⋊C414yescomplemented
545C22#8867112yes—
546C22#8867612yes—
547C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
548C4—94no—
549C4—94no—
550C4—182no—
551C4—182no—
552C4—182no—
553C3—31no—
554C3—31no—
555C3—61no—
556C3C2×Dic3×#6711yescharacteristic
557C2—24no—
558C2—184no—
559C2#1172131414yesdirect product factor
560C2C22×((C3⋊D3)⋊C3)⋊C411yescharacteristic
561C2C2×#8867111yescharacteristic
562C2C2×#8867611yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC22×C3C22×D32
FittingC24×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC22×#638⋊C3C2
cosocleC22×H(3,𝔽3)C24
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C4×D4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C23×(C3⋊Dic3)⋊C3C2semidirect
3. C22×((C3⋊D3)⋊C3)⋊C4C2semidirect
4. C2×#88552C2semidirect
5. #11721314C2direct (N × H)
6. C2×#88450C2semidirect
7. C2×#88448C2semidirect
8. C2×#88447C2semidirect
15. C2×#638⋊C3C4semidirect
21. C22×(C3⋊Dic3)⋊C3C22semidirect
23. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
24. #88552C22semidirect
25. #88450C22semidirect
26. #88448C22semidirect
27. #88447C22semidirect
29. C23×C3⋊Dic3D3semidirect
47. C23×H(3,𝔽3)C2×C4semidirect
52. C22×(C3⋊D3)⋊C3C2×C4semidirect
58. #638⋊C3C2×C4semidirect
61. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
62. C2×(C3⋊Dic3)⋊C3D4semidirect
75. C2×#638Dic3semidirect
81. C22×C3⋊Dic3C2×D3semidirect
112. C22×H(3,𝔽3)C22×C4semidirect
119. C2×(C3⋊D3)⋊C3C22×C4semidirect
123. (C3⋊Dic3)⋊C3C22×C4semidirect
124. (C3⋊Dic3)⋊C3C2×D4semidirect
159. C23×C32C4×D3semidirect
160. C23×C32C2×Dic3semidirect
167. C22×C3⋊D3C2×Dic3semidirect
173. #638C2×Dic3semidirect
176. C2×C3⋊Dic3C2×Dic3semidirect
177. C2×C3⋊Dic3#67semidirect
205. C2×H(3,𝔽3)C4×D4semidirect
280. C22×C32C2×C4×D3semidirect
285. C22×C32C22×Dic3semidirect
292. C2×C3⋊D3C22×Dic3semidirect
298. C3⋊Dic3C22×Dic3semidirect
299. C3⋊Dic3C2×#67semidirect
317. H(3,𝔽3)C2×C4×D4semidirect
400. C2×C32C4×#67semidirect
401. C2×C32D4×Dic3semidirect
489. C32C2×C4×#67semidirect
490. C32C2×D4×Dic3semidirect
497. C23((C3⋊D3)⋊C3)⋊C4semidirect
544. C22C2×((C3⋊D3)⋊C3)⋊C4semidirect
559. C2#11721314direct (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
C23×(C3⋊Dic3)⋊C32
C22×((C3⋊D3)⋊C3)⋊C42
C2×#885522
#117213142
C2×#884502
C2×#884482
C2×#884472
C2×#67⋊Dic33
C2×#639523