Finite Group Databasebeta

← C2×#11721311 #13152072

Subgroups — C2×#11721311 #13152072

Order 1728.

1329 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#638⋊C3C211yesmaximal, characteristic
2C22×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
3C2×#88552C211yesmaximal, characteristic, complemented
4C2×#3757⋊C3C211yesmaximal, characteristic, complemented
5#11721311C218yesmaximal, direct product factor
6C2×#88450C211yesmaximal, characteristic, complemented
7C2×#88449C211yesmaximal, characteristic, complemented
8C2×#88448C211yesmaximal, characteristic, complemented
9C2×#63975—31nomaximal
10C2×#63969—31nomaximal
11C24×H(3,𝔽3)C2211yescharacteristic
12C23×(C3⋊D3)⋊C3C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C2×#638⋊C3—24no—
16C2×#638⋊C3C2214yes—
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×((C3⋊D3)⋊C3)⋊C4—24no—
21C2×((C3⋊D3)⋊C3)⋊C4C2214yescomplemented
22#88552C2214yescomplemented
23#3757⋊C3C2214yescomplemented
24#88450C2214yescomplemented
25#88449C2214yescomplemented
26#88448C2214yescomplemented
27C22×#638D311yescharacteristic
28C22×C3×#67—31no—
29C22×#624—31no—
30C22×D3×Dic3—31no—
31C2×#3757D311yescharacteristic, complemented
32C2×C3×#217—31no—
33C2×C3×#217—31no—
34C2×C3×#217—31no—
35C2×C3×#217—61no—
36#63975—38no—
37#63969—38no—
38C2×#3724—31no—
39C2×Dic3⋊Dic3—31no—
40C2×Dic3⋊Dic3—31no—
41C2×Dic3⋊Dic3—31no—
42C2×#3722—31no—
43C2×(C2×D3)⋊Dic3—31no—
44C23×H(3,𝔽3)—22no—
45C23×H(3,𝔽3)—22no—
46C23×H(3,𝔽3)—22no—
47C23×H(3,𝔽3)C2312yes—
48C23×H(3,𝔽3)C2311yescharacteristic
49C22×(C3⋊D3)⋊C3—22no—
50C22×(C3⋊D3)⋊C3—22no—
51C22×(C3⋊D3)⋊C3D414yes—
52C22×(C3⋊D3)⋊C3C2312yes—
53C2×#1084—22no—
54C2×#1084—22no—
55C2×#1084C2312yes—
56C2×#1084C2312yes—
57#638⋊C3—28no—
58C2×(C3⋊Dic3)⋊C3—22no—
59C2×(C3⋊Dic3)⋊C3—22no—
60C2×(C3⋊Dic3)⋊C3—22no—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64((C3⋊D3)⋊C3)⋊C4—28no—
65C2×#935—91no—
66C2×#883—91no—
67C2×#883—91no—
68C24×C32—21no—
69C24×C32C2×D311yescharacteristic
70C24×C32C2×D311yescharacteristic
71C23×C3⋊D3C2×D311yescharacteristic
72C23×C3×D3—31no—
73C2×#638—24no—
74C2×#638C2×D314yes—
75C22×C3⋊Dic3C2×D311yescharacteristic
76C22×C3⋊Dic3C2×D311yescharacteristic
77C22×C3⋊Dic3C2×D311yescharacteristic
78C2×C3×#67—34no—
79C2×C3×#67—64no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3—31no—
83C22×C3×Dic3—31no—
84C22×C3×Dic3—31no—
85C22×C3×Dic3—31no—
86C22×C3×Dic3—31no—
87C22×C3×Dic3—61no—
88C22×C3×Dic3—61no—
89C2×#624—34no—
90C2×#624—64no—
91C2×D3×Dic3—34no—
92C2×D3×Dic3—64no—
93#3757C2×D314yescomplemented
94C3×#217—34no—
95C3×#217—34no—
96C3×#217—34no—
97C3×#217—64no—
98#3724—34no—
99Dic3⋊Dic3—34no—
100Dic3⋊Dic3—34no—
101Dic3⋊Dic3—34no—
102#3722—34no—
103(C2×D3)⋊Dic3—34no—
104C22×H(3,𝔽3)—22no—
105C22×H(3,𝔽3)—22no—
106C22×H(3,𝔽3)—22no—
107C22×H(3,𝔽3)—44no—
108C22×H(3,𝔽3)C2×D412yes—
109C22×H(3,𝔽3)#4112yes—
110C22×H(3,𝔽3)#4112yes—
111C22×H(3,𝔽3)C2411yescharacteristic
112C2×(C3⋊D3)⋊C3—28no—
113C2×(C3⋊D3)⋊C3—22no—
114C2×(C3⋊D3)⋊C3—22no—
115C2×(C3⋊D3)⋊C3C2×D414yes—
116#1084—22no—
117#1084—22no—
118(C3⋊Dic3)⋊C3—22no—
119(C3⋊Dic3)⋊C3—22no—
120(C3⋊Dic3)⋊C3—22no—
121C22×C3×D4—91no—
122C22×#67—31no—
123C22×#67—91no—
124C23×Dic3—91no—
125C22×C4×D3—91no—
126C22×C4×D3—91no—
127C2×C3×C22⋊C4—91no—
128C2×C3×C22⋊C4—91no—
129C2×#217—31no—
130C2×#217—31no—
131C2×#217—31no—
132C2×#217—61no—
133C2×#217—91no—
134C2×#217—91no—
135#935—98no—
136C2×(C2×D3)⋊C4—91no—
137C2×(C2×D3)⋊C4—91no—
138C2×C4⋊Dic3—91no—
139C2×C4⋊Dic3—91no—
140C2×Dic3⋊C4—91no—
141C2×Dic3⋊C4—91no—
142C2×Dic3⋊C4—91no—
143C2×Dic3⋊C4—91no—
144#883—98no—
145#883—98no—
146C23×C32—24no—
147C23×C32—22no—
148C23×C32—22no—
149C23×C32—22no—
150C23×C32—22no—
151C23×C32—22no—
152C23×C32—22no—
153C23×C32—21no—
154C23×C32—22no—
155C23×C32—42no—
156C23×C32—42no—
157C23×C32C22×D312yes—
158C23×C32C22×D312yes—
159C23×C32C22×D311yescharacteristic
160C23×C32C22×D311yescharacteristic
161C22×C3⋊D3—22no—
162C22×C3⋊D3—22no—
163C22×C3⋊D3#6714yes—
164C22×C3⋊D3C22×D312yes—
165C22×C3×D3—34no—
166C22×C3×D3—32no—
167C22×C3×D3—62no—
168C22×C3×D3—62no—
169#638—28no—
170C2×C3⋊Dic3—22no—
171C2×C3⋊Dic3—22no—
172C2×C3⋊Dic3—22no—
173C2×C3⋊Dic3C22×D312yes—
174C2×C3⋊Dic3C22×D312yes—
175C2×C3⋊Dic3C22×D312yes—
176C3×#67—68no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—62no—
185C2×C3×Dic3—62no—
186C2×C3×Dic3—62no—
187C2×C3×Dic3—62no—
188C2×C3×Dic3—62no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—62no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3—122no—
194C2×C3×Dic3—122no—
195#624—68no—
196D3×Dic3—68no—
197C2×#123—271noSylow
198C2×H(3,𝔽3)—42no—
199C2×H(3,𝔽3)#12314yescomplemented
200C2×H(3,𝔽3)C22×D411yescharacteristic
201C2×H(3,𝔽3)C2×#4111yescharacteristic
202C2×H(3,𝔽3)C2×#4111yescharacteristic
203(C3⋊D3)⋊C3—24no—
204C24×C3—31no—
205C24×C3—31no—
206C24×C3—61no—
207C24×C3—91no—
208C24×C3D3211yescharacteristic
209C23×D3—31no—
210C23×D3—91no—
211C2×C3×D4—94no—
212C2×C3×D4—184no—
213C22×C3×C4—91no—
214C22×C3×C4—91no—
215C22×C3×C4—91no—
216C22×C3×C4—91no—
217C22×C3×C4—91no—
218C2×#67—34no—
219C2×#67—64no—
220C2×#67—94no—
221C2×#67—184no—
222C22×Dic3—31no—
223C22×Dic3—31no—
224C22×Dic3—31no—
225C22×Dic3—31no—
226C22×Dic3—31no—
227C22×Dic3—31no—
228C22×Dic3—31no—
229C22×Dic3—61no—
230C22×Dic3—61no—
231C22×Dic3—91no—
232C22×Dic3—91no—
233C22×Dic3—91no—
234C22×Dic3—94no—
235C22×Dic3—184no—
236C2×C4×D3—94no—
237C2×C4×D3—94no—
238C2×C4×D3—184no—
239C2×C4×D3—184no—
240C3×C22⋊C4—94no—
241C3×C22⋊C4—94no—
242#217—34no—
243#217—34no—
244#217—34no—
245#217—64no—
246#217—94no—
247#217—94no—
248(C2×D3)⋊C4—94no—
249(C2×D3)⋊C4—94no—
250C4⋊Dic3—94no—
251C4⋊Dic3—94no—
252Dic3⋊C4—94no—
253Dic3⋊C4—94no—
254Dic3⋊C4—94no—
255Dic3⋊C4—94no—
256C22×C32—24no—
257C22×C32—22no—
258C22×C32—22no—
259C22×C32—22no—
260C22×C32—22no—
261C22×C32—22no—
262C22×C32—22no—
263C22×C32—21no—
264C22×C32—22no—
265C22×C32—22no—
266C22×C32—22no—
267C22×C32—42no—
268C22×C32—42no—
269C22×C32—48no—
270C22×C32—44no—
271C22×C32—44no—
272C22×C32#23512yes—
273C22×C32D3×D412yes—
274C22×C32#23712yes—
275C22×C32#23712yes—
276C22×C32#23712yes—
277C22×C32C2×#6712yes—
278C22×C32C23×D311yescharacteristic
279C22×C32C23×D311yescharacteristic
280C2×C3⋊D3—28no—
281C2×C3⋊D3—22no—
282C2×C3⋊D3—22no—
283C2×C3⋊D3C2×#6714yes—
284C2×C3×D3—34no—
285C2×C3×D3—68no—
286C2×C3×D3—62no—
287C2×C3×D3—62no—
288C3⋊Dic3—22no—
289C3⋊Dic3—22no—
290C3⋊Dic3—22no—
291C3×Dic3—62no—
292C3×Dic3—62no—
293C3×Dic3—62no—
294C3×Dic3—62no—
295C3×Dic3—62no—
296C3×Dic3—62no—
297C3×Dic3—62no—
298C3×Dic3—122no—
299C3×Dic3—122no—
300C22×D4—91no—
301C23×C4—271no—
302#123—278no—
303C2×C4⋊C4—271no—
304C2×C4⋊C4—271no—
305C2×C22⋊C4—91no—
306C2×C22⋊C4—91no—
307C2×C22⋊C4—271no—
308H(3,𝔽3)C2×#12311yescharacteristic, complemented, Sylow
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—32no—
313C23×C3—32no—
314C23×C3—31no—
315C23×C3—31no—
316C23×C3—62no—
317C23×C3—62no—
318C23×C3—62no—
319C23×C3—62no—
320C23×C3—62no—
321C23×C3—62no—
322C23×C3—64no—
323C23×C3—62no—
324C23×C3—61no—
325C23×C3—92no—
326C23×C3—94no—
327C23×C3—122no—
328C23×C3—122no—
329C23×C3—182no—
330C23×C3—182no—
331C23×C3C2×D3212yes—
332C23×C3C2×D3211yescharacteristic
333C22×D3—32no—
334C22×D3—34no—
335C22×D3—62no—
336C22×D3—62no—
337C22×D3—94no—
338C22×D3—92no—
339C22×D3—182no—
340C22×D3—182no—
341C3×D4—188no—
342C2×C3×C4—92no—
343C2×C3×C4—92no—
344C2×C3×C4—92no—
345C2×C3×C4—92no—
346C2×C3×C4—92no—
347C2×C3×C4—182no—
348C2×C3×C4—182no—
349C2×C3×C4—182no—
350C2×C3×C4—182no—
351C2×C3×C4—182no—
352#67—68no—
353#67—188no—
354C2×Dic3—32no—
355C2×Dic3—32no—
356C2×Dic3—32no—
357C2×Dic3—32no—
358C2×Dic3—32no—
359C2×Dic3—32no—
360C2×Dic3—32no—
361C2×Dic3—62no—
362C2×Dic3—62no—
363C2×Dic3—62no—
364C2×Dic3—62no—
365C2×Dic3—62no—
366C2×Dic3—62no—
367C2×Dic3—62no—
368C2×Dic3—62no—
369C2×Dic3—62no—
370C2×Dic3—92no—
371C2×Dic3—92no—
372C2×Dic3—92no—
373C2×Dic3—122no—
374C2×Dic3—122no—
375C2×Dic3—182no—
376C2×Dic3—182no—
377C2×Dic3—182no—
378C2×Dic3—188no—
379C4×D3—188no—
380C4×D3—188no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—24no—
385C2×C32—44no—
386C2×C32—42no—
387C2×C32—42no—
388C2×C32#88314yescomplemented
389C2×C32#93514yescomplemented
390C2×C32C2×#23511yescharacteristic
391C2×C32C2×D3×D411yescharacteristic
392C2×C32C2×#23711yescharacteristic
393C2×C32C2×#23711yescharacteristic
394C2×C32C2×#23711yescharacteristic
395C2×C32C22×#6711yescharacteristic
396C3⋊D3—24no—
397C3×D3—64no—
398C24—91no—
399C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
400C2×D4—94no—
401C2×D4—184no—
402C22×C4—91no—
403C22×C4—91no—
404C22×C4—91no—
405C22×C4—91no—
406C22×C4—91no—
407C22×C4—274no—
408C22×C4—544no—
409C4⋊C4—274no—
410C4⋊C4—274no—
411C22⋊C4—94no—
412C22⋊C4—94no—
413C22⋊C4—274no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—31no—
418C22×C3—31no—
419C22×C3—32no—
420C22×C3—32no—
421C22×C3—32no—
422C22×C3—32no—
423C22×C3—32no—
424C22×C3—32no—
425C22×C3—44no—
426C22×C3—62no—
427C22×C3—62no—
428C22×C3—62no—
429C22×C3—62no—
430C22×C3—62no—
431C22×C3—62no—
432C22×C3—64no—
433C22×C3—61no—
434C22×C3—62no—
435C22×C3—62no—
436C22×C3—62no—
437C22×C3—94no—
438C22×C3—122no—
439C22×C3—122no—
440C22×C3—124no—
441C22×C3—124no—
442C22×C3—128no—
443C22×C3—182no—
444C22×C3—182no—
445C22×C3—188no—
446C22×C3#380412yes—
447C22×C3#380512yes—
448C22×C3D3×#6712yes—
449C22×C3C22×D3211yescharacteristic
450C2×D3—34no—
451C2×D3—62no—
452C2×D3—62no—
453C2×D3—68no—
454C2×D3—94no—
455C2×D3—182no—
456C2×D3—182no—
457C2×D3—188no—
458C3×C4—182no—
459C3×C4—182no—
460C3×C4—182no—
461C3×C4—182no—
462C3×C4—182no—
463Dic3—62no—
464Dic3—62no—
465Dic3—62no—
466Dic3—62no—
467Dic3—62no—
468Dic3—62no—
469Dic3—62no—
470Dic3—122no—
471Dic3—122no—
472Dic3—182no—
473Dic3—182no—
474Dic3—182no—
475C32—21no—
476C32C2×#88311yescharacteristic, complemented
477C32C2×#93511yescharacteristic, complemented
478C23—22no—
479C23—22no—
480C23—22no—
481C23—92no—
482C23—94no—
483C23—182no—
484C23—182no—
485C23C2×((C3⋊D3)⋊C3)⋊C212yes—
486C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
487D4—188no—
488C2×C4—92no—
489C2×C4—92no—
490C2×C4—92no—
491C2×C4—92no—
492C2×C4—92no—
493C2×C4—182no—
494C2×C4—182no—
495C2×C4—182no—
496C2×C4—182no—
497C2×C4—182no—
498C2×C4—548no—
499C2×C3—31no—
500C2×C3—31no—
501C2×C3—31no—
502C2×C3—31no—
503C2×C3—31no—
504C2×C3—31no—
505C2×C3—34no—
506C2×C3—34no—
507C2×C3—42no—
508C2×C3—61no—
509C2×C3—61no—
510C2×C3—61no—
511C2×C3—64no—
512C2×C3—122no—
513C2×C3—122no—
514C2×C3—124no—
515C2×C3—184no—
516C2×C3#6396914yes—
517C2×C3C2×#380411yescharacteristic
518C2×C3C2×#380511yescharacteristic
519C2×C3C2×D3×#6711yescharacteristic
520D3—64no—
521D3—184no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—44no—
526C22—94no—
527C22—182no—
528C22—182no—
529C22—188no—
530C22#8867112yes—
531C22#8867212yes—
532C22#8867612yes—
533C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
534C4—182no—
535C4—182no—
536C4—182no—
537C4—182no—
538C4—182no—
539C3—31no—
540C3—31no—
541C3—61no—
542C3C2×#6396911yescharacteristic
543C2—42no—
544C2—184no—
545C2#1172131114yesdirect product factor
546C2C2×#8867111yescharacteristic
547C2C2×#8867211yescharacteristic
548C2C2×#8867611yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC24×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC22×#638⋊C3C2
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#123
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C22×((C3⋊D3)⋊C3)⋊C4C2semidirect
3. C2×#88552C2semidirect
4. C2×#3757⋊C3C2semidirect
5. #11721311C2direct (N × H)
6. C2×#88450C2semidirect
7. C2×#88449C2semidirect
8. C2×#88448C2semidirect
21. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
22. #88552C22semidirect
23. #3757⋊C3C22semidirect
24. #88450C22semidirect
25. #88449C22semidirect
26. #88448C22semidirect
31. C2×#3757D3semidirect
93. #3757C2×D3semidirect
199. C2×H(3,𝔽3)#123semidirect
308. H(3,𝔽3)C2×#123semidirect
388. C2×C32#883semidirect
389. C2×C32#935semidirect
476. C32C2×#883semidirect
477. C32C2×#935semidirect
545. C2#11721311direct (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
C22×((C3⋊D3)⋊C3)⋊C42
C2×#885522
C2×#3757⋊C32
#117213112
C2×#884502
C2×#884492
C2×#884482
C2×#639753
C2×#639693