Finite Group Databasebeta

← C2×#11721315 #13152065

Subgroups — C2×#11721315 #13152065

Order 1728.

1727 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#1084C211yesmaximal, characteristic, complemented
2C22×#638⋊C3C211yesmaximal, characteristic
3C22×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
4C2×#3757⋊C3C211yesmaximal, characteristic, complemented
5#11721315C218yesmaximal, direct product factor
6C2×#88450C211yesmaximal, characteristic, complemented
7C2×#88448C211yesmaximal, characteristic, complemented
8C2×#88447C211yesmaximal, characteristic, complemented
9C2×Dic3×#67—31nomaximal
10C2×#63952—31nomaximal
11C24×H(3,𝔽3)C2211yescharacteristic
12C23×(C3⋊D3)⋊C3C2211yescharacteristic
13C22×#1084—24no—
14C22×#1084C2211yescharacteristic
15C22×#1084C2211yescharacteristic
16C22×#1084C2214yescomplemented
17C2×#638⋊C3C418yescomplemented
18C2×#638⋊C3C2214yes—
19C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
22C2×((C3⋊D3)⋊C3)⋊C4—24no—
23C2×((C3⋊D3)⋊C3)⋊C4C2214yescomplemented
24#3757⋊C3C2214yescomplemented
25#88450C2214yescomplemented
26#88448C2214yescomplemented
27#88447C2214yescomplemented
28C22×#638D311yescharacteristic
29C22×C3×#67—31no—
30C23×C3×Dic3—31no—
31C23×C3×Dic3—31no—
32C23×C3×Dic3—61no—
33C22×#624—31no—
34C22×D3×Dic3—31no—
35C2×#3757D311yescharacteristic, complemented
36C2×C3×#217—31no—
37Dic3×#67—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—28no—
56C2×#1084D414yescomplemented
57C2×#1084C2312yes—
58C2×#1084C2312yes—
59#638⋊C3C2×C4116yescomplemented
60C2×(C3⋊Dic3)⋊C3—22no—
61C2×(C3⋊Dic3)⋊C3—22no—
62C2×(C3⋊Dic3)⋊C3C2×C414yescomplemented
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⋊Dic3C2×D311yescharacteristic
78C22×C3⋊Dic3C2×D311yescharacteristic
79C22×C3⋊Dic3C2×D311yescharacteristic
80C2×C3×#67—38no—
81C2×C3×#67—34no—
82C22×C3×Dic3—31no—
83C22×C3×Dic3—31no—
84C22×C3×Dic3—31no—
85C22×C3×Dic3—31no—
86C22×C3×Dic3—31no—
87C22×C3×Dic3—31no—
88C22×C3×Dic3—31no—
89C22×C3×Dic3—34no—
90C22×C3×Dic3—34no—
91C22×C3×Dic3—64no—
92C22×C3×Dic3—64no—
93C22×C3×Dic3—61no—
94C22×C3×Dic3—61no—
95C22×C3×Dic3—64no—
96C22×C3×Dic3—68no—
97C2×#624—34no—
98C2×#624—64no—
99C2×D3×Dic3—34no—
100C2×D3×Dic3—64no—
101#3757C2×D314yescomplemented
102C3×#217—34no—
103#3724—34no—
104Dic3⋊Dic3—34no—
105#3722—34no—
106(C2×D3)⋊Dic3—34no—
107Dic32—34no—
108Dic32—34no—
109C22×H(3,𝔽3)—28no—
110C22×H(3,𝔽3)—22no—
111C22×H(3,𝔽3)—22no—
112C22×H(3,𝔽3)C22×C412yes—
113C22×H(3,𝔽3)C22×C414yescomplemented
114C22×H(3,𝔽3)C2×D412yes—
115C22×H(3,𝔽3)#4112yes—
116C22×H(3,𝔽3)C2411yescharacteristic
117C2×(C3⋊D3)⋊C3—28no—
118C2×(C3⋊D3)⋊C3—22no—
119C2×(C3⋊D3)⋊C3—22no—
120C2×(C3⋊D3)⋊C3C22×C414yescomplemented
121#1084—22no—
122#1084C2×D414yescomplemented
123(C3⋊Dic3)⋊C3—22no—
124(C3⋊Dic3)⋊C3—22no—
125(C3⋊Dic3)⋊C3C22×C414yescomplemented
126C22×C3×D4—91no—
127C23×C3×C4—91no—
128C22×#67—31no—
129C22×#67—91no—
130C23×Dic3—31no—
131C23×Dic3—31no—
132C23×Dic3—61no—
133C23×Dic3—91no—
134C22×C4×D3—91no—
135C22×C4×D3—91no—
136C2×C3×C22⋊C4—91no—
137C2×#217—31no—
138C2×#217—91no—
139C2×#217—91no—
140D4×Dic3—98no—
141C4×#67—98no—
142C2×(C2×D3)⋊C4—91no—
143C2×(C2×D3)⋊C4—91no—
144C2×C4⋊Dic3—91no—
145C2×Dic3⋊C4—91no—
146C2×Dic3⋊C4—91no—
147C2×C4×Dic3—91no—
148C2×C4×Dic3—91no—
149C2×C4×Dic3—91no—
150#67⋊C4—98no—
151C23×C32—24no—
152C23×C32—24no—
153C23×C32—22no—
154C23×C32—22no—
155C23×C32—22no—
156C23×C32—22no—
157C23×C32—24no—
158C23×C32—22no—
159C23×C32—21no—
160C23×C32C4×D314yescomplemented
161C23×C32C2×Dic314yescomplemented
162C23×C32C22×D312yes—
163C23×C32C22×D312yes—
164C23×C32C22×D311yescharacteristic
165C23×C32C22×D311yescharacteristic
166C22×C3⋊D3—22no—
167C22×C3⋊D3—22no—
168C22×C3⋊D3C2×Dic314yescomplemented
169C22×C3⋊D3C22×D312yes—
170C22×C3×D3—34no—
171C22×C3×D3—32no—
172C22×C3×D3—62no—
173C22×C3×D3—62no—
174#638C2×Dic3116yescomplemented
175C2×C3⋊Dic3—22no—
176C2×C3⋊Dic3—22no—
177C2×C3⋊Dic3C2×Dic314yescomplemented
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—34no—
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—32no—
192C2×C3×Dic3—64no—
193C2×C3×Dic3—62no—
194C2×C3×Dic3—62no—
195C2×C3×Dic3—68no—
196C2×C3×Dic3—68no—
197C2×C3×Dic3—616no—
198C2×C3×Dic3—64no—
199C2×C3×Dic3—62no—
200C2×C3×Dic3—62no—
201C2×C3×Dic3—62no—
202C2×C3×Dic3—62no—
203#624—68no—
204D3×Dic3—68no—
205C2×C4×D4—271noSylow
206C2×H(3,𝔽3)—24no—
207C2×H(3,𝔽3)C4×D414yescomplemented
208C2×H(3,𝔽3)C23×C411yescharacteristic
209C2×H(3,𝔽3)C22×D411yescharacteristic
210C2×H(3,𝔽3)C2×#4111yescharacteristic
211(C3⋊D3)⋊C3—24no—
212C24×C3—31no—
213C24×C3—31no—
214C24×C3—61no—
215C24×C3—91no—
216C24×C3D3211yescharacteristic
217C23×D3—31no—
218C23×D3—91no—
219C2×C3×D4—94no—
220C2×C3×D4—98no—
221C22×C3×C4—91no—
222C22×C3×C4—91no—
223C22×C3×C4—91no—
224C22×C3×C4—91no—
225C22×C3×C4—91no—
226C22×C3×C4—94no—
227C22×C3×C4—184no—
228C2×#67—38no—
229C2×#67—34no—
230C2×#67—94no—
231C2×#67—98no—
232C22×Dic3—31no—
233C22×Dic3—31no—
234C22×Dic3—31no—
235C22×Dic3—34no—
236C22×Dic3—31no—
237C22×Dic3—31no—
238C22×Dic3—34no—
239C22×Dic3—31no—
240C22×Dic3—31no—
241C22×Dic3—61no—
242C22×Dic3—64no—
243C22×Dic3—61no—
244C22×Dic3—64no—
245C22×Dic3—64no—
246C22×Dic3—68no—
247C22×Dic3—91no—
248C22×Dic3—91no—
249C22×Dic3—91no—
250C22×Dic3—94no—
251C22×Dic3—184no—
252C2×C4×D3—94no—
253C2×C4×D3—94no—
254C2×C4×D3—184no—
255C2×C4×D3—184no—
256C3×C22⋊C4—94no—
257#217—34no—
258#217—94no—
259#217—94no—
260(C2×D3)⋊C4—94no—
261(C2×D3)⋊C4—94no—
262C4⋊Dic3—94no—
263Dic3⋊C4—94no—
264Dic3⋊C4—94no—
265C4×Dic3—94no—
266C4×Dic3—94no—
267C4×Dic3—94no—
268C22×C32—24no—
269C22×C32—24no—
270C22×C32—216no—
271C22×C32—28no—
272C22×C32—28no—
273C22×C32—22no—
274C22×C32—22no—
275C22×C32—22no—
276C22×C32—22no—
277C22×C32—21no—
278C22×C32—22no—
279C22×C32—24no—
280C22×C32—22no—
281C22×C32—22no—
282C22×C32C2×C4×D312yes—
283C22×C32C2×C4×D314yescomplemented
284C22×C32D3×D412yes—
285C22×C32D3×D412yes—
286C22×C32#23712yes—
287C22×C32#23712yes—
288C22×C32C22×Dic312yes—
289C22×C32C22×Dic314yescomplemented
290C22×C32C23×D311yescharacteristic
291C22×C32C23×D311yescharacteristic
292C2×C3⋊D3—28no—
293C2×C3⋊D3—22no—
294C2×C3⋊D3—22no—
295C2×C3⋊D3C22×Dic314yescomplemented
296C2×C3×D3—34no—
297C2×C3×D3—68no—
298C2×C3×D3—62no—
299C2×C3×D3—62no—
300C3⋊Dic3—22no—
301C3⋊Dic3—22no—
302C3⋊Dic3C22×Dic314yescomplemented
303C3×Dic3—34no—
304C3×Dic3—34no—
305C3×Dic3—34no—
306C3×Dic3—64no—
307C3×Dic3—64no—
308C3×Dic3—62no—
309C3×Dic3—62no—
310C3×Dic3—62no—
311C3×Dic3—62no—
312C22×D4—91no—
313C23×C4—91no—
314C23×C4—271no—
315C4×D4—278no—
316C2×C4⋊C4—271no—
317C2×C22⋊C4—91no—
318C2×C22⋊C4—271no—
319C2×C42—271no—
320H(3,𝔽3)C2×C4×D411yescharacteristic, complemented, Sylow
321C23×C3—22no—
322C23×C3—22no—
323C23×C3—34no—
324C23×C3—34no—
325C23×C3—32no—
326C23×C3—32no—
327C23×C3—31no—
328C23×C3—31no—
329C23×C3—62no—
330C23×C3—62no—
331C23×C3—62no—
332C23×C3—62no—
333C23×C3—64no—
334C23×C3—64no—
335C23×C3—64no—
336C23×C3—62no—
337C23×C3—61no—
338C23×C3—92no—
339C23×C3—94no—
340C23×C3—182no—
341C23×C3—182no—
342C23×C3D3×Dic314yes—
343C23×C3C2×D3212yes—
344C23×C3C2×D3211yescharacteristic
345C22×D3—32no—
346C22×D3—34no—
347C22×D3—62no—
348C22×D3—62no—
349C22×D3—94no—
350C22×D3—92no—
351C22×D3—182no—
352C22×D3—182no—
353C3×D4—916no—
354C2×C3×C4—92no—
355C2×C3×C4—94no—
356C2×C3×C4—92no—
357C2×C3×C4—94no—
358C2×C3×C4—92no—
359C2×C3×C4—92no—
360C2×C3×C4—92no—
361C2×C3×C4—188no—
362C2×C3×C4—182no—
363C2×C3×C4—182no—
364C2×C3×C4—182no—
365#67—316no—
366#67—916no—
367C2×Dic3—32no—
368C2×Dic3—32no—
369C2×Dic3—34no—
370C2×Dic3—34no—
371C2×Dic3—32no—
372C2×Dic3—34no—
373C2×Dic3—32no—
374C2×Dic3—32no—
375C2×Dic3—32no—
376C2×Dic3—32no—
377C2×Dic3—62no—
378C2×Dic3—64no—
379C2×Dic3—62no—
380C2×Dic3—68no—
381C2×Dic3—68no—
382C2×Dic3—62no—
383C2×Dic3—62no—
384C2×Dic3—616no—
385C2×Dic3—64no—
386C2×Dic3—62no—
387C2×Dic3—62no—
388C2×Dic3—94no—
389C2×Dic3—92no—
390C2×Dic3—92no—
391C2×Dic3—92no—
392C2×Dic3—182no—
393C2×Dic3—182no—
394C2×Dic3—188no—
395C4×D3—188no—
396C4×D3—188no—
397C2×C32—28no—
398C2×C32—24no—
399C2×C32—24no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—24no—
404C2×C32#67⋊C414yescomplemented
405C2×C32D4×Dic314yescomplemented
406C2×C32C22×C4×D311yescharacteristic
407C2×C32C2×D3×D411yescharacteristic
408C2×C32C2×D3×D411yescharacteristic
409C2×C32C2×#23711yescharacteristic
410C2×C32C2×#23711yescharacteristic
411C2×C32C23×Dic311yescharacteristic
412C3⋊D3—24no—
413C3×D3—64no—
414C24—91no—
415C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
416C2×D4—98no—
417C2×D4—94no—
418C22×C4—91no—
419C22×C4—91no—
420C22×C4—94no—
421C22×C4—91no—
422C22×C4—91no—
423C22×C4—91no—
424C22×C4—184no—
425C22×C4—274no—
426C22×C4—544no—
427C4⋊C4—274no—
428C22⋊C4—94no—
429C22⋊C4—274no—
430C42—274no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—28no—
434C22×C3—31no—
435C22×C3—31no—
436C22×C3—32no—
437C22×C3—32no—
438C22×C3—34no—
439C22×C3—34no—
440C22×C3—32no—
441C22×C3—32no—
442C22×C3—32no—
443C22×C3—32no—
444C22×C3—62no—
445C22×C3—62no—
446C22×C3—62no—
447C22×C3—62no—
448C22×C3—68no—
449C22×C3—68no—
450C22×C3—64no—
451C22×C3—64no—
452C22×C3—616no—
453C22×C3—61no—
454C22×C3—62no—
455C22×C3—64no—
456C22×C3—62no—
457C22×C3—62no—
458C22×C3—94no—
459C22×C3—182no—
460C22×C3—182no—
461C22×C3—188no—
462C22×C3C2×D3×Dic312yes—
463C22×C3C2×D3×Dic314yes—
464C22×C3#380512yes—
465C22×C3#381112yes—
466C22×C3C22×D3211yescharacteristic
467C2×D3—34no—
468C2×D3—62no—
469C2×D3—62no—
470C2×D3—68no—
471C2×D3—94no—
472C2×D3—182no—
473C2×D3—182no—
474C2×D3—188no—
475C3×C4—94no—
476C3×C4—94no—
477C3×C4—182no—
478C3×C4—182no—
479C3×C4—182no—
480Dic3—34no—
481Dic3—34no—
482Dic3—34no—
483Dic3—64no—
484Dic3—62no—
485Dic3—62no—
486Dic3—64no—
487Dic3—62no—
488Dic3—62no—
489Dic3—94no—
490Dic3—182no—
491Dic3—182no—
492C32—21no—
493C32C2×#67⋊C411yescharacteristic, complemented
494C32C2×D4×Dic311yescharacteristic, complemented
495C23—22no—
496C23—22no—
497C23—92no—
498C23—94no—
499C23—182no—
500C23—182no—
501C23((C3⋊D3)⋊C3)⋊C414yescomplemented
502C23C2×((C3⋊D3)⋊C3)⋊C212yes—
503C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
504D4—916no—
505C2×C4—92no—
506C2×C4—92no—
507C2×C4—94no—
508C2×C4—94no—
509C2×C4—92no—
510C2×C4—92no—
511C2×C4—92no—
512C2×C4—188no—
513C2×C4—182no—
514C2×C4—182no—
515C2×C4—182no—
516C2×C4—548no—
517C2×C3—24no—
518C2×C3—31no—
519C2×C3—31no—
520C2×C3—31no—
521C2×C3—31no—
522C2×C3—31no—
523C2×C3—31no—
524C2×C3—34no—
525C2×C3—34no—
526C2×C3—64no—
527C2×C3—64no—
528C2×C3—68no—
529C2×C3—61no—
530C2×C3—61no—
531C2×C3—61no—
532C2×C3—64no—
533C2×C3—184no—
534C2×C3#67⋊Dic314yes—
535C2×C3C22×D3×Dic311yescharacteristic
536C2×C3C2×#380511yescharacteristic
537C2×C3C2×#381111yescharacteristic
538D3—64no—
539D3—184no—
540C22—22no—
541C22—22no—
542C22—28no—
543C22—94no—
544C22—182no—
545C22—182no—
546C22—188no—
547C22C2×((C3⋊D3)⋊C3)⋊C412yes—
548C22C2×((C3⋊D3)⋊C3)⋊C414yescomplemented
549C22#8867212yes—
550C22#8867712yes—
551C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
552C4—94no—
553C4—94no—
554C4—182no—
555C4—182no—
556C4—182no—
557C3—31no—
558C3—31no—
559C3—61no—
560C3C2×#67⋊Dic311yescharacteristic
561C2—24no—
562C2—184no—
563C2#1172131514yesdirect product factor
564C2C22×((C3⋊D3)⋊C3)⋊C411yescharacteristic
565C2C2×#8867211yescharacteristic
566C2C2×#8867711yescharacteristic
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
1. C23×#1084C2semidirect
3. C22×((C3⋊D3)⋊C3)⋊C4C2semidirect
4. C2×#3757⋊C3C2semidirect
5. #11721315C2direct (N × H)
6. C2×#88450C2semidirect
7. C2×#88448C2semidirect
8. C2×#88447C2semidirect
16. C22×#1084C22semidirect
17. C2×#638⋊C3C4semidirect
23. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
24. #3757⋊C3C22semidirect
25. #88450C22semidirect
26. #88448C22semidirect
27. #88447C22semidirect
35. C2×#3757D3semidirect
47. C23×H(3,𝔽3)C2×C4semidirect
52. C22×(C3⋊D3)⋊C3C2×C4semidirect
56. C2×#1084D4semidirect
59. #638⋊C3C2×C4semidirect
62. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
75. C2×#638Dic3semidirect
101. #3757C2×D3semidirect
113. C22×H(3,𝔽3)C22×C4semidirect
120. C2×(C3⋊D3)⋊C3C22×C4semidirect
122. #1084C2×D4semidirect
125. (C3⋊Dic3)⋊C3C22×C4semidirect
160. C23×C32C4×D3semidirect
161. C23×C32C2×Dic3semidirect
168. C22×C3⋊D3C2×Dic3semidirect
174. #638C2×Dic3semidirect
177. C2×C3⋊Dic3C2×Dic3semidirect
207. C2×H(3,𝔽3)C4×D4semidirect
283. C22×C32C2×C4×D3semidirect
289. C22×C32C22×Dic3semidirect
295. C2×C3⋊D3C22×Dic3semidirect
302. C3⋊Dic3C22×Dic3semidirect
320. H(3,𝔽3)C2×C4×D4semidirect
404. C2×C32#67⋊C4semidirect
405. C2×C32D4×Dic3semidirect
493. C32C2×#67⋊C4semidirect
494. C32C2×D4×Dic3semidirect
501. C23((C3⋊D3)⋊C3)⋊C4semidirect
548. C22C2×((C3⋊D3)⋊C3)⋊C4semidirect
563. C2#11721315direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#10842
C22×#638⋊C32
C22×((C3⋊D3)⋊C3)⋊C42
C2×#3757⋊C32
#117213152
C2×#884502
C2×#884482
C2×#884472
C2×Dic3×#673
C2×#639523