Finite Group Databasebeta

← C2×#11721127 #13150833

Subgroups — C2×#11721127 #13150833

Order 1728.

1623 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
2C22×#6200C211yesmaximal, characteristic, complemented
3C22×#6200C211yesmaximal, characteristic, complemented
4C22×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
5C2×#88542C211yesmaximal, characteristic, complemented
6C2×#3750⋊C3C211yesmaximal, characteristic, complemented
7C2×#88450C211yesmaximal, characteristic, complemented
8#11721127C218yesmaximal, direct product factor
9C2×#63910—31nomaximal
10C2×#63885—31nomaximal
11C23×#272C2211yescharacteristic
12C23×(C3⋊D3)⋊C3C2211yescharacteristic
13C23×(C3⋊D3)⋊C3C2211yescharacteristic
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C22×#1084C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×C4⋊((C3⋊D3)⋊C3)—24no—
19C2×C4⋊((C3⋊D3)⋊C3)C2214yescomplemented
20C2×#6200—24no—
21C2×#6200—24no—
22C2×#6200C2214yescomplemented
23C2×#6200C2214yescomplemented
24C2×((C3⋊D3)⋊C3)⋊C4—24no—
25C2×((C3⋊D3)⋊C3)⋊C4C2214yescomplemented
26#88542C2214yescomplemented
27#3750⋊C3C2214yescomplemented
28#88450C2214yescomplemented
29C22×C4⋊(C3⋊D3)D311yescharacteristic, complemented
30C22×C3×D12—31no—
31C22×#626—31no—
32C22×#626—31no—
33C22×#625—31no—
34C22×#625—31no—
35C22×#624—31no—
36C22×D3×Dic3—31no—
37C2×#3750D311yescharacteristic, complemented
38C2×C3×(C2×D3)⋊C4—31no—
39C2×C3×(C2×D3)⋊C4—31no—
40C2×C3×(C2×D3)⋊C4—61no—
41C2×C3×Dic3⋊C4—31no—
42C2×#3722—31no—
43C2×(C2×D3)⋊Dic3—31no—
44#63910—38no—
45#63885—38no—
46C23×H(3,𝔽3)C2311yescharacteristic
47C22×#272—22no—
48C22×#272—22no—
49C22×#272—22no—
50C22×#272C2312yes—
51C22×(C3⋊D3)⋊C3—22no—
52C22×(C3⋊D3)⋊C3—22no—
53C22×(C3⋊D3)⋊C3—22no—
54C22×(C3⋊D3)⋊C3—22no—
55C22×(C3⋊D3)⋊C3—22no—
56C22×(C3⋊D3)⋊C3D414yescomplemented
57C22×(C3⋊D3)⋊C3C2312yes—
58C22×(C3⋊D3)⋊C3C2312yes—
59C2×C4×H(3,𝔽3)—22no—
60C2×C4×H(3,𝔽3)C2312yes—
61C2×#1084—22no—
62C2×#1084C2312yes—
63C2×(C3⋊Dic3)⋊C3—22no—
64C2×(C3⋊Dic3)⋊C3D414yescomplemented
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C2×(C3⋊Dic3)⋊C3C2312yes—
67C4⋊((C3⋊D3)⋊C3)—28no—
68#6200—28no—
69#6200—28no—
70((C3⋊D3)⋊C3)⋊C4—28no—
71C2×#939—91no—
72C2×#895—91no—
73C2×#884—91no—
74C23×C3⋊D3C2×D311yescharacteristic
75C23×C3⋊D3C2×D311yescharacteristic
76C23×C3×D3—31no—
77C23×C3×D3—31no—
78C23×C3×D3—31no—
79C23×C3×D3—31no—
80C23×C3×D3—61no—
81C22×C32×C4—21no—
82C22×C32×C4C2×D311yescharacteristic
83C22×C32×C4C2×D311yescharacteristic
84C22×C3⋊Dic3C2×D311yescharacteristic
85C22×C3⋊Dic3C2×D311yescharacteristic
86C2×C4⋊(C3⋊D3)—24no—
87C2×C4⋊(C3⋊D3)C2×D314yescomplemented
88C22×C3×Dic3—31no—
89C22×C3×Dic3—31no—
90C22×C3×Dic3—31no—
91C22×C3×Dic3—31no—
92C22×C3×Dic3—61no—
93C2×C3×D12—34no—
94C2×C3×D12—64no—
95C2×#626—34no—
96C2×#626—34no—
97C2×#626—64no—
98C2×#626—64no—
99C2×#625—34no—
100C2×#625—34no—
101C2×#625—64no—
102C2×#625—64no—
103C2×#624—34no—
104C2×#624—64no—
105C2×D3×Dic3—34no—
106C2×D3×Dic3—64no—
107#3750C2×D314yescomplemented
108C3×(C2×D3)⋊C4—34no—
109C3×(C2×D3)⋊C4—34no—
110C3×(C2×D3)⋊C4—64no—
111C3×Dic3⋊C4—34no—
112#3722—34no—
113(C2×D3)⋊Dic3—34no—
114C22×H(3,𝔽3)C2×D412yes—
115C22×H(3,𝔽3)C2×D412yes—
116C22×H(3,𝔽3)#4112yes—
117C22×H(3,𝔽3)C2411yescharacteristic
118C2×#272—22no—
119C2×#272—22no—
120C2×#272—22no—
121C2×#272—44no—
122C2×(C3⋊D3)⋊C3—28no—
123C2×(C3⋊D3)⋊C3—22no—
124C2×(C3⋊D3)⋊C3—22no—
125C2×(C3⋊D3)⋊C3—22no—
126C2×(C3⋊D3)⋊C3—22no—
127C2×(C3⋊D3)⋊C3—22no—
128C2×(C3⋊D3)⋊C3—44no—
129C2×(C3⋊D3)⋊C3C2×D414yescomplemented
130C4×H(3,𝔽3)—22no—
131#1084—22no—
132(C3⋊Dic3)⋊C3—22no—
133(C3⋊Dic3)⋊C3C2×D414yescomplemented
134C22×C3×D4—91no—
135C22×#67—91no—
136C22×#67—91no—
137C22×#67—91no—
138C22×#67—91no—
139C23×Dic3—91no—
140C22×D12—31no—
141C22×D12—91no—
142C22×D12—91no—
143C22×D12—91no—
144C22×C4×D3—91no—
145C22×C4×D3—91no—
146C2×C3×C4⋊C4—91no—
147C2×C3×C22⋊C4—91no—
148C2×#217—91no—
149#939—98no—
150C2×(C2×D3)⋊C4—31no—
151C2×(C2×D3)⋊C4—31no—
152C2×(C2×D3)⋊C4—61no—
153C2×(C2×D3)⋊C4—91no—
154C2×(C2×D3)⋊C4—91no—
155C2×Dic3⋊C4—31no—
156C2×Dic3⋊C4—91no—
157#895—98no—
158#884—98no—
159C23×C32—21no—
160C23×C32C22×D311yescharacteristic
161C23×C32C22×D311yescharacteristic
162C22×C3⋊D3—22no—
163C22×C3⋊D3—22no—
164C22×C3⋊D3—22no—
165C22×C3⋊D3—22no—
166C22×C3⋊D3—22no—
167C22×C3⋊D3#6714yescomplemented
168C22×C3⋊D3C22×D312yes—
169C22×C3⋊D3C22×D312yes—
170C22×C3×D3—34no—
171C22×C3×D3—32no—
172C22×C3×D3—32no—
173C22×C3×D3—32no—
174C22×C3×D3—32no—
175C22×C3×D3—62no—
176C22×C3×D3—62no—
177C22×C3×D3—62no—
178C22×C3×D3—64no—
179C22×C3×D3—62no—
180C22×C3×D3—62no—
181C22×C3×D3—62no—
182C22×C3×D3—62no—
183C22×C3×D3—62no—
184C22×C3×D3—62no—
185C22×C3×D3—62no—
186C22×C3×D3—62no—
187C22×C3×D3—62no—
188C22×C3×D3—122no—
189C22×C3×D3—122no—
190C2×C32×C4—22no—
191C2×C32×C4—22no—
192C2×C32×C4—22no—
193C2×C32×C4—42no—
194C2×C32×C4C22×D312yes—
195C2×C32×C4C22×D312yes—
196C2×C3⋊Dic3—22no—
197C2×C3⋊Dic3D1214yescomplemented
198C2×C3⋊Dic3C22×D312yes—
199C2×C3⋊Dic3C22×D312yes—
200C4⋊(C3⋊D3)—28no—
201C2×C3×Dic3—34no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—62no—
207C2×C3×Dic3—62no—
208C2×C3×Dic3—62no—
209C2×C3×Dic3—62no—
210C2×C3×Dic3—122no—
211C3×D12—68no—
212#626—68no—
213#626—68no—
214#625—68no—
215#625—68no—
216#624—68no—
217D3×Dic3—68no—
218C2×#121—271noSylow
219C2×H(3,𝔽3)#12114yescomplemented
220C2×H(3,𝔽3)C22×D411yescharacteristic
221C2×H(3,𝔽3)C22×D411yescharacteristic
222C2×H(3,𝔽3)C2×#4111yescharacteristic
223#272—42no—
224(C3⋊D3)⋊C3—24no—
225(C3⋊D3)⋊C3—42no—
226C24×C3—91no—
227C24×C3—91no—
228C24×C3—91no—
229C23×D3—31no—
230C23×D3—31no—
231C23×D3—31no—
232C23×D3—31no—
233C23×D3—61no—
234C23×D3—91no—
235C23×D3—91no—
236C2×C3×D4—94no—
237C2×C3×D4—184no—
238C22×C3×C4—31no—
239C22×C3×C4—31no—
240C22×C3×C4—61no—
241C22×C3×C4—91no—
242C22×C3×C4—91no—
243C22×C3×C4—91no—
244C22×C3×C4D3211yescharacteristic
245C2×#67—94no—
246C2×#67—94no—
247C2×#67—94no—
248C2×#67—94no—
249C2×#67—184no—
250C2×#67—184no—
251C2×#67—184no—
252C2×#67—184no—
253C22×Dic3—31no—
254C22×Dic3—31no—
255C22×Dic3—31no—
256C22×Dic3—31no—
257C22×Dic3—61no—
258C22×Dic3—91no—
259C22×Dic3—91no—
260C22×Dic3—94no—
261C22×Dic3—184no—
262C2×D12—34no—
263C2×D12—64no—
264C2×D12—94no—
265C2×D12—94no—
266C2×D12—94no—
267C2×D12—184no—
268C2×D12—184no—
269C2×D12—184no—
270C2×C4×D3—94no—
271C2×C4×D3—94no—
272C2×C4×D3—184no—
273C2×C4×D3—184no—
274C3×C4⋊C4—94no—
275C3×C22⋊C4—94no—
276#217—94no—
277(C2×D3)⋊C4—34no—
278(C2×D3)⋊C4—34no—
279(C2×D3)⋊C4—64no—
280(C2×D3)⋊C4—94no—
281(C2×D3)⋊C4—94no—
282Dic3⋊C4—34no—
283Dic3⋊C4—94no—
284C22×C32—21no—
285C22×C32—22no—
286C22×C32—22no—
287C22×C32—22no—
288C22×C32C2×D1212yes—
289C22×C32D3×D412yes—
290C22×C32D3×D412yes—
291C22×C32#23712yes—
292C22×C32#23912yes—
293C22×C32C2×#6712yes—
294C22×C32C23×D311yescharacteristic
295C22×C32C23×D311yescharacteristic
296C2×C3⋊D3—28no—
297C2×C3⋊D3—22no—
298C2×C3⋊D3—22no—
299C2×C3⋊D3—22no—
300C2×C3⋊D3—22no—
301C2×C3⋊D3—22no—
302C2×C3⋊D3—44no—
303C2×C3⋊D3C2×#6714yescomplemented
304C2×C3×D3—34no—
305C2×C3×D3—68no—
306C2×C3×D3—62no—
307C2×C3×D3—62no—
308C2×C3×D3—64no—
309C2×C3×D3—62no—
310C2×C3×D3—62no—
311C2×C3×D3—62no—
312C2×C3×D3—62no—
313C2×C3×D3—62no—
314C2×C3×D3—62no—
315C2×C3×D3—62no—
316C2×C3×D3—62no—
317C2×C3×D3—62no—
318C2×C3×D3—122no—
319C2×C3×D3—122no—
320C2×C3×D3—128no—
321C2×C3×D3—124no—
322C2×C3×D3—124no—
323C2×C3×D3—124no—
324C32×C4—22no—
325C32×C4—22no—
326C32×C4—42no—
327C3⋊Dic3—22no—
328C3⋊Dic3C2×D1214yescomplemented
329C3×Dic3—34no—
330C3×Dic3—62no—
331C3×Dic3—62no—
332C3×Dic3—62no—
333C3×Dic3—122no—
334C22×D4—91no—
335C22×D4—271no—
336C22×D4—271no—
337C23×C4—271no—
338#121—278no—
339C2×C4⋊C4—91no—
340C2×C22⋊C4—91no—
341C2×C22⋊C4—271no—
342H(3,𝔽3)C2×#12111yescharacteristic, complemented, Sylow
343C23×C3—31no—
344C23×C3—31no—
345C23×C3—61no—
346C23×C3—92no—
347C23×C3—94no—
348C23×C3—92no—
349C23×C3—92no—
350C23×C3—182no—
351C23×C3—182no—
352C23×C3—182no—
353C23×C3—182no—
354C23×C3—182no—
355C23×C3—182no—
356C23×C3—182no—
357C23×C3—182no—
358C23×C3C2×D3211yescharacteristic
359C22×D3—32no—
360C22×D3—34no—
361C22×D3—32no—
362C22×D3—32no—
363C22×D3—32no—
364C22×D3—62no—
365C22×D3—62no—
366C22×D3—62no—
367C22×D3—62no—
368C22×D3—62no—
369C22×D3—62no—
370C22×D3—62no—
371C22×D3—62no—
372C22×D3—62no—
373C22×D3—64no—
374C22×D3—62no—
375C22×D3—62no—
376C22×D3—62no—
377C22×D3—94no—
378C22×D3—92no—
379C22×D3—92no—
380C22×D3—122no—
381C22×D3—122no—
382C22×D3—182no—
383C22×D3—182no—
384C22×D3—182no—
385C22×D3—182no—
386C22×D3—182no—
387C3×D4—188no—
388C2×C3×C4—22no—
389C2×C3×C4—32no—
390C2×C3×C4—32no—
391C2×C3×C4—62no—
392C2×C3×C4—62no—
393C2×C3×C4—62no—
394C2×C3×C4—92no—
395C2×C3×C4—94no—
396C2×C3×C4—92no—
397C2×C3×C4—92no—
398C2×C3×C4—122no—
399C2×C3×C4—182no—
400C2×C3×C4—182no—
401C2×C3×C4C2×D3212yes—
402#67—188no—
403#67—188no—
404#67—188no—
405#67—188no—
406C2×Dic3—32no—
407C2×Dic3—34no—
408C2×Dic3—32no—
409C2×Dic3—32no—
410C2×Dic3—32no—
411C2×Dic3—62no—
412C2×Dic3—62no—
413C2×Dic3—62no—
414C2×Dic3—62no—
415C2×Dic3—94no—
416C2×Dic3—92no—
417C2×Dic3—92no—
418C2×Dic3—122no—
419C2×Dic3—182no—
420C2×Dic3—188no—
421D12—68no—
422D12—188no—
423D12—188no—
424D12—188no—
425C4×D3—188no—
426C4×D3—188no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—24no—
431C2×C32#89514yescomplemented
432C2×C32#93914yescomplemented
433C2×C32C22×D1211yescharacteristic
434C2×C32C2×D3×D411yescharacteristic
435C2×C32C2×D3×D411yescharacteristic
436C2×C32C2×#23711yescharacteristic
437C2×C32C2×#23911yescharacteristic
438C2×C32C22×#6711yescharacteristic
439C3⋊D3—24no—
440C3⋊D3—42no—
441C3×D3—64no—
442C3×D3—124no—
443C3×D3—122no—
444C3×D3—122no—
445C3×D3—122no—
446C24—91no—
447C24—91no—
448C24—91no—
449C2×D4—94no—
450C2×D4—184no—
451C2×D4—274no—
452C2×D4—274no—
453C2×D4—544no—
454C2×D4—544no—
455C22×C4—91no—
456C22×C4—91no—
457C22×C4—91no—
458C22×C4—274no—
459C22×C4—544no—
460C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
461C4⋊C4—94no—
462C22⋊C4—94no—
463C22⋊C4—274no—
464C22×C3—31no—
465C22×C3—31no—
466C22×C3—32no—
467C22×C3—32no—
468C22×C3—32no—
469C22×C3—32no—
470C22×C3—32no—
471C22×C3—32no—
472C22×C3—61no—
473C22×C3—62no—
474C22×C3—62no—
475C22×C3—62no—
476C22×C3—94no—
477C22×C3—182no—
478C22×C3—188no—
479C22×C3—182no—
480C22×C3—182no—
481C22×C3—182no—
482C22×C3—182no—
483C22×C3—182no—
484C22×C3—182no—
485C22×C3—182no—
486C22×C3—364no—
487C22×C3—364no—
488C22×C3#379612yes—
489C22×C3D3×D1212yes—
490C22×C3D3×#6712yes—
491C22×C3C22×D3211yescharacteristic
492C2×D3—34no—
493C2×D3—62no—
494C2×D3—62no—
495C2×D3—68no—
496C2×D3—62no—
497C2×D3—62no—
498C2×D3—62no—
499C2×D3—62no—
500C2×D3—62no—
501C2×D3—62no—
502C2×D3—64no—
503C2×D3—62no—
504C2×D3—62no—
505C2×D3—62no—
506C2×D3—94no—
507C2×D3—122no—
508C2×D3—122no—
509C2×D3—124no—
510C2×D3—124no—
511C2×D3—128no—
512C2×D3—124no—
513C2×D3—182no—
514C2×D3—188no—
515C2×D3—182no—
516C2×D3—182no—
517C2×D3—182no—
518C2×D3—182no—
519C2×D3—364no—
520C3×C4—22no—
521C3×C4—62no—
522C3×C4—62no—
523C3×C4—94no—
524C3×C4—122no—
525C3×C4—182no—
526C3×C4—182no—
527Dic3—34no—
528Dic3—62no—
529Dic3—62no—
530Dic3—62no—
531Dic3—94no—
532Dic3—122no—
533Dic3—182no—
534C32—21no—
535C32C2×#89511yescharacteristic, complemented
536C32C2×#93911yescharacteristic, complemented
537C23—92no—
538C23—94no—
539C23—92no—
540C23—92no—
541C23—182no—
542C23—182no—
543C23—182no—
544C23—182no—
545C23—182no—
546C23—182no—
547C23—182no—
548C23—182no—
549C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
550D4—188no—
551D4—548no—
552D4—548no—
553C2×C4—22no—
554C2×C4—92no—
555C2×C4—94no—
556C2×C4—92no—
557C2×C4—92no—
558C2×C4—182no—
559C2×C4—182no—
560C2×C4—548no—
561C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
562C2×C3—31no—
563C2×C3—31no—
564C2×C3—34no—
565C2×C3—34no—
566C2×C3—31no—
567C2×C3—31no—
568C2×C3—31no—
569C2×C3—31no—
570C2×C3—61no—
571C2×C3—64no—
572C2×C3—61no—
573C2×C3—61no—
574C2×C3—184no—
575C2×C3—362no—
576C2×C3—362no—
577C2×C3#6388614yes—
578C2×C3C2×#379611yescharacteristic
579C2×C3C2×D3×D1211yescharacteristic
580C2×C3C2×D3×#6711yescharacteristic
581D3—64no—
582D3—122no—
583D3—122no—
584D3—124no—
585D3—122no—
586D3—184no—
587D3—362no—
588C22—94no—
589C22—188no—
590C22—182no—
591C22—182no—
592C22—182no—
593C22—182no—
594C22—182no—
595C22—182no—
596C22—182no—
597C22—182no—
598C22—364no—
599C22—364no—
600C22#8864912yes—
601C22#8865412yes—
602C22#8867612yes—
603C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
604C4—22no—
605C4—94no—
606C4—182no—
607C4—182no—
608C3—31no—
609C3—31no—
610C3—61no—
611C3C2×#6388611yescharacteristic
612C2—184no—
613C2—362no—
614C2—362no—
615C2#1172112714yesdirect product factor
616C2C2×#8864911yescharacteristic
617C2C2×#8865411yescharacteristic
618C2C2×#8867611yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#121
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊((C3⋊D3)⋊C3)C2semidirect
2. C22×#6200C2semidirect
3. C22×#6200C2semidirect
4. C22×((C3⋊D3)⋊C3)⋊C4C2semidirect
5. C2×#88542C2semidirect
6. C2×#3750⋊C3C2semidirect
7. C2×#88450C2semidirect
8. #11721127C2direct (N × H)
19. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
22. C2×#6200C22semidirect
23. C2×#6200C22semidirect
25. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
26. #88542C22semidirect
27. #3750⋊C3C22semidirect
28. #88450C22semidirect
29. C22×C4⋊(C3⋊D3)D3semidirect
37. C2×#3750D3semidirect
56. C22×(C3⋊D3)⋊C3D4semidirect
64. C2×(C3⋊Dic3)⋊C3D4semidirect
87. C2×C4⋊(C3⋊D3)C2×D3semidirect
107. #3750C2×D3semidirect
129. C2×(C3⋊D3)⋊C3C2×D4semidirect
133. (C3⋊Dic3)⋊C3C2×D4semidirect
167. C22×C3⋊D3#67semidirect
197. C2×C3⋊Dic3D12semidirect
219. C2×H(3,𝔽3)#121semidirect
303. C2×C3⋊D3C2×#67semidirect
328. C3⋊Dic3C2×D12semidirect
342. H(3,𝔽3)C2×#121semidirect
431. C2×C32#895semidirect
432. C2×C32#939semidirect
535. C32C2×#895semidirect
536. C32C2×#939semidirect
615. C2#11721127direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊((C3⋊D3)⋊C3)2
C22×#62002
C22×((C3⋊D3)⋊C3)⋊C42
C2×#885422
C2×#3750⋊C32
C2×#884502
#117211272
C2×#639103
C2×#638853