Finite Group Databasebeta

← C4×#88649 #13147775

Subgroups — C4×#88649 #13147775

Order 1728.

1193 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88649C211yesmaximal, characteristic
2C42×#272C211yesmaximal, characteristic, complemented
3C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
4C4×#634⋊C3C211yesmaximal, characteristic, complemented
5C4×#6200C212yesmaximal, complemented
6#11721136C211yesmaximal, characteristic, complemented
7#11721124C212yesmaximal, complemented
8#11721121C212yesmaximal, complemented
9C4×((C3⋊D3)⋊C3)⋊C4C212yesmaximal, complemented
10#11721089C211yesmaximal, characteristic, complemented
11#11721084C211yesmaximal, characteristic, complemented
12C4×#3799—31nomaximal
13C4×#3795—31nomaximal
14C2×C4×#272C2211yescharacteristic
15C2×C4×#272C2211yescharacteristic
16C2×C4×#272C2211yescharacteristic
17C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2212yes—
19C2×#634⋊C3C2211yescharacteristic
20C2×#6200C2212yes—
21C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
22C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
23#88649C418yesdirect product factor
24C42×H(3,𝔽3)C2211yescharacteristic
25C4×#1084C2211yescharacteristic
26C4×#1084C2211yescharacteristic
27C4×#1084C2211yescharacteristic
28#3752⋊C3C2212yes—
29C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
30C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
31#3750⋊C3C2212yes—
32C4×(C3⋊Dic3)⋊C3C2212yes—
33#88451C2212yes—
34#88450C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37#88447C2212yes—
38C2×#3799—31no—
39C2×#3795—31no—
40C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
41C4×#634D311yescharacteristic, complemented
42C3×C4×D12—31no—
43C3×C42×D3—31no—
44C3×C42×D3—31no—
45C3×C42×D3—61no—
46C3×C4×Dic6—31no—
47#63904—32no—
48C4×#626—32no—
49C4×#625—32no—
50#63894—31no—
51Dic3×D12—31no—
52C4×#624—32no—
53#63879—32no—
54#63877—32no—
55C4×D3×Dic3—32no—
56#63844—31no—
57Dic3×Dic6—31no—
58#63841—31no—
59#63841—31no—
60#63836—32no—
61C22×#272C2311yescharacteristic
62C22×(C3⋊D3)⋊C3C2312yes—
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×#1084C2311yescharacteristic
67C2×#1084C2311yescharacteristic
68C2×#1084C2311yescharacteristic
69C4×#272—22no—
70C4×#272—22no—
71C4×#272C2×C414yescomplemented
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C2×(C3⋊Dic3)⋊C3C2312yes—
74C2×(C3⋊Dic3)⋊C3C2312yes—
75C4⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
76C4×(C3⋊D3)⋊C3—24no—
77#634⋊C3C2×C414yescomplemented
78#6200C2×C418yescomplemented
79((C3⋊D3)⋊C3)⋊C4—24no—
80((C3⋊D3)⋊C3)⋊C4C2×C418yescomplemented
81C4×#239—91no—
82C4×#237—91no—
83C4×#235—91no—
84C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
85C2×C4×C3⋊D3C2×D312yes—
86C2×#634C2×D311yescharacteristic
87C2×C3×D12—31no—
88C2×C3×C4×D3—31no—
89C2×C3×C4×D3—31no—
90C2×C3×C4×D3—32no—
91C2×C3×C4×D3—31no—
92C2×C3×C4×D3—31no—
93C2×C3×C4×D3—31no—
94C2×C3×C4×D3—31no—
95C2×C3×C4×D3—61no—
96C2×C3×C4×D3—61no—
97C2×C3×C4×D3—61no—
98C2×C3×Dic6—31no—
99C2×#626—32no—
100C2×#625—32no—
101C2×#624—32no—
102C2×#624—32no—
103C2×D3×Dic3—32no—
104C2×D3×Dic3—32no—
105#3799—38no—
106#3795—38no—
107C32×C42—21no—
108C32×C42C2×D311yescharacteristic
109C32×C42C2×D311yescharacteristic
110#3752C2×D312yes—
111C4⋊(C3⋊Dic3)C2×D311yescharacteristic
112C4⋊(C3⋊Dic3)C2×D311yescharacteristic
113#3750C2×D312yes—
114C4×C3⋊Dic3C2×D312yes—
115C3×(C2×D3)⋊C4—32no—
116C3×C4⋊Dic3—31no—
117C3×C4⋊Dic3—31no—
118C3×Dic3⋊C4—32no—
119C3×C4×Dic3—31no—
120C3×C4×Dic3—31no—
121C3×C4×Dic3—31no—
122C3×C4×Dic3—31no—
123C3×C4×Dic3—31no—
124C3×C4×Dic3—31no—
125C3×C4×Dic3—32no—
126C3×C4×Dic3—61no—
127C3×C4×Dic3—61no—
128C3×C4×Dic3—61no—
129#3724—32no—
130Dic3⋊Dic3—32no—
131#3722—32no—
132(C2×D3)⋊Dic3—32no—
133(C2×D3)⋊Dic3—32no—
134(C2×D3)⋊Dic3—32no—
135Dic32—32no—
136Dic32—32no—
137Dic32—32no—
138Dic32—32no—
139C22×H(3,𝔽3)C2411yescharacteristic
140C2×#272—21no—
141C2×#272—21no—
142C2×#272C22×C412yes—
143C2×(C3⋊D3)⋊C3—22no—
144C2×(C3⋊D3)⋊C3—22no—
145C2×(C3⋊D3)⋊C3C22×C414yes—
146C4×H(3,𝔽3)—21no—
147C4×H(3,𝔽3)C22×C412yes—
148C4×H(3,𝔽3)#4112yescomplemented
149#1084—21no—
150#1084C22×C412yes—
151#1084#4112yescomplemented
152(C3⋊Dic3)⋊C3—22no—
153(C3⋊Dic3)⋊C3—22no—
154(C3⋊Dic3)⋊C3C22×C414yes—
155C2×#239—91no—
156C2×#237—91no—
157C2×#235—91no—
158C3×C4×Q8—91no—
159C3×C4×D4—91no—
160C2×C3×C42—91no—
161Q8×Dic3—91no—
162D4×Dic3—91no—
163C4×#67—92no—
164C4×#67—92no—
165C4×#67—92no—
166#926—91no—
167C2×C4×Dic3—92no—
168#893—91no—
169#893—92no—
170#892—92no—
171#892—91no—
172#892—91no—
173#887—91no—
174#67⋊C4—92no—
175#877—92no—
176C4×D12—31no—
177C4×D12—91no—
178C4×D12—91no—
179C4×D12—92no—
180#872—92no—
181C42×D3—31no—
182C42×D3—31no—
183C42×D3—61no—
184C42×D3—92no—
185C42×D3—92no—
186C4×Dic6—31no—
187C4×Dic6—91no—
188C4×Dic6—91no—
189C22×C3⋊D3C22×D312yes—
190C22×C3×D3—32no—
191C22×C3×D3—31no—
192C22×C3×D3—31no—
193C22×C3×D3—61no—
194C2×C32×C4—21no—
195C2×C32×C4—21no—
196C2×C32×C4—21no—
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200C2×C32×C4C22×D311yescharacteristic
201C2×C32×C4C22×D311yescharacteristic
202C2×C32×C4C22×D311yescharacteristic
203C2×C3⋊Dic3C22×D312yes—
204C2×C3⋊Dic3C22×D312yes—
205C2×C3⋊Dic3C22×D312yes—
206C4⋊(C3⋊D3)C4×D314yescomplemented
207C4×C3⋊D3—24no—
208#634C4×D314yescomplemented
209C2×C3×Dic3—32no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—31no—
213C2×C3×Dic3—31no—
214C2×C3×Dic3—32no—
215C2×C3×Dic3—32no—
216C2×C3×Dic3—31no—
217C2×C3×Dic3—31no—
218C2×C3×Dic3—61no—
219C2×C3×Dic3—61no—
220C2×C3×Dic3—61no—
221C3×D12—34no—
222C3×C4×D3—34no—
223C3×C4×D3—34no—
224C3×C4×D3—64no—
225C3×C4×D3—62no—
226C3×C4×D3—62no—
227C3×C4×D3—64no—
228C3×C4×D3—64no—
229C3×C4×D3—62no—
230C3×C4×D3—64no—
231C3×C4×D3—62no—
232C3×Dic6—34no—
233#626—38no—
234#625—38no—
235#624—38no—
236#624—64no—
237D3×Dic3—38no—
238D3×Dic3—64no—
239C4×#41—271noSylow
240C2×H(3,𝔽3)C23×C411yescharacteristic
241C2×H(3,𝔽3)C2×#4111yescharacteristic
242C2×H(3,𝔽3)C2×#4111yescharacteristic
243#272—22no—
244(C3⋊D3)⋊C3—24no—
245C2×C3×Q8—91no—
246C2×C3×D4—91no—
247C22×C3×C4—92no—
248C22×C3×C4—91no—
249C22×C3×C4—91no—
250C22×C3×C4—91no—
251C2×#67—92no—
252C2×#67—92no—
253C22×Dic3—92no—
254C22×Dic3—92no—
255#239—98no—
256#237—98no—
257#235—98no—
258C2×D12—31no—
259C2×D12—91no—
260C2×D12—92no—
261C2×C4×D3—31no—
262C2×C4×D3—31no—
263C2×C4×D3—31no—
264C2×C4×D3—31no—
265C2×C4×D3—31no—
266C2×C4×D3—31no—
267C2×C4×D3—32no—
268C2×C4×D3—61no—
269C2×C4×D3—61no—
270C2×C4×D3—61no—
271C2×C4×D3—92no—
272C2×C4×D3—92no—
273C2×C4×D3—92no—
274C2×C4×D3—92no—
275C2×C4×D3—92no—
276C2×Dic6—31no—
277C2×Dic6—91no—
278C3×C4⋊C4—91no—
279C3×C4⋊C4—92no—
280C3×C4⋊C4—91no—
281C3×C22⋊C4—92no—
282C3×C42—31no—
283C3×C42—31no—
284C3×C42—61no—
285C3×C42—92no—
286C3×C42—91no—
287C3×C42—91no—
288C3×C42—91no—
289C3×C42D3211yescharacteristic
290#217—92no—
291#217—92no—
292(C2×D3)⋊C4—32no—
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4—92no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—92no—
298C4⋊Dic3—31no—
299C4⋊Dic3—31no—
300C4⋊Dic3—91no—
301C4⋊Dic3—91no—
302C4⋊Dic3—92no—
303Dic3⋊C4—32no—
304Dic3⋊C4—92no—
305Dic3⋊C4—92no—
306Dic3⋊C4—92no—
307C4×Dic3—31no—
308C4×Dic3—31no—
309C4×Dic3—31no—
310C4×Dic3—31no—
311C4×Dic3—31no—
312C4×Dic3—31no—
313C4×Dic3—32no—
314C4×Dic3—61no—
315C4×Dic3—61no—
316C4×Dic3—61no—
317C4×Dic3—92no—
318C4×Dic3—92no—
319C4×Dic3—92no—
320C4×Dic3—92no—
321C4×Dic3—92no—
322C4×Dic3—92no—
323C4×Dic3—92no—
324C22×C32—21no—
325C22×C32C23×D311yescharacteristic
326C22×C32C23×D311yescharacteristic
327C2×C3⋊D3—22no—
328C2×C3⋊D3—22no—
329C2×C3⋊D3C2×C4×D314yes—
330C2×C3×D3—34no—
331C2×C3×D3—32no—
332C2×C3×D3—32no—
333C2×C3×D3—62no—
334C2×C3×D3—62no—
335C2×C3×D3—62no—
336C2×C3×D3—61no—
337C2×C3×D3—62no—
338C2×C3×D3—61no—
339C2×C3×D3—62no—
340C2×C3×D3—61no—
341C2×C3×D3—61no—
342C32×C4—22no—
343C32×C4—21no—
344C32×C4—21no—
345C32×C4—22no—
346C32×C4—22no—
347C32×C4C2×C4×D312yes—
348C32×C4C2×C4×D312yes—
349C32×C4#23712yescomplemented
350C32×C4#23912yescomplemented
351C3⋊Dic3—22no—
352C3⋊Dic3—22no—
353C3⋊Dic3C2×C4×D314yes—
354C3×Dic3—34no—
355C3×Dic3—32no—
356C3×Dic3—32no—
357C3×Dic3—32no—
358C3×Dic3—32no—
359C3×Dic3—62no—
360C3×Dic3—62no—
361C3×Dic3—62no—
362C3×Dic3—61no—
363C3×Dic3—62no—
364C3×Dic3—61no—
365C3×Dic3—62no—
366C2×#41—271no—
367C4×Q8—91no—
368C4×Q8—271no—
369C4×D4—91no—
370C4×D4—271no—
371C4×D4—272no—
372C4×D4—272no—
373#117—272no—
374#117—271no—
375C2×C42—91no—
376C2×C42—272no—
377H(3,𝔽3)C4×#4111yescharacteristic, complemented, Sylow
378C23×C3—92no—
379C23×C3—91no—
380C22×D3—31no—
381C22×D3—31no—
382C22×D3—32no—
383C22×D3—61no—
384C22×D3—92no—
385C3×Q8—94no—
386C3×D4—94no—
387C2×C3×C4—31no—
388C2×C3×C4—31no—
389C2×C3×C4—31no—
390C2×C3×C4—31no—
391C2×C3×C4—31no—
392C2×C3×C4—31no—
393C2×C3×C4—61no—
394C2×C3×C4—61no—
395C2×C3×C4—61no—
396C2×C3×C4—92no—
397C2×C3×C4—91no—
398C2×C3×C4—91no—
399C2×C3×C4—94no—
400C2×C3×C4—92no—
401C2×C3×C4—92no—
402C2×C3×C4—91no—
403C2×C3×C4—182no—
404C2×C3×C4—184no—
405C2×C3×C4—182no—
406C2×C3×C4C2×D3211yescharacteristic
407C2×C3×C4C2×D3211yescharacteristic
408C2×C3×C4C2×D3211yescharacteristic
409#67—98no—
410#67—98no—
411C2×Dic3—31no—
412C2×Dic3—31no—
413C2×Dic3—31no—
414C2×Dic3—31no—
415C2×Dic3—32no—
416C2×Dic3—31no—
417C2×Dic3—31no—
418C2×Dic3—32no—
419C2×Dic3—32no—
420C2×Dic3—61no—
421C2×Dic3—61no—
422C2×Dic3—61no—
423C2×Dic3—92no—
424C2×Dic3—92no—
425C2×Dic3—92no—
426C2×Dic3—98no—
427C2×Dic3—184no—
428D12—34no—
429D12—94no—
430D12—98no—
431C4×D3—34no—
432C4×D3—34no—
433C4×D3—64no—
434C4×D3—62no—
435C4×D3—62no—
436C4×D3—62no—
437C4×D3—62no—
438C4×D3—64no—
439C4×D3—64no—
440C4×D3—64no—
441C4×D3—98no—
442C4×D3—98no—
443C4×D3—184no—
444C4×D3—184no—
445C4×D3—184no—
446Dic6—34no—
447Dic6—94no—
448C2×C32—21no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32C22×C4×D311yescharacteristic
452C2×C32C22×C4×D311yescharacteristic
453C2×C32C2×#23711yescharacteristic
454C2×C32C2×#23911yescharacteristic
455C2×C32D3×#4111yescharacteristic
456C2×C32D3×#4111yescharacteristic
457C3⋊D3—24no—
458C3×D3—64no—
459C3×D3—62no—
460C3×D3—64no—
461C3×D3—62no—
462#41—278no—
463C2×Q8—91no—
464C2×D4—91no—
465C2×D4—272no—
466C22×C4—91no—
467C22×C4—91no—
468C22×C4—91no—
469C22×C4—92no—
470C22×C4—272no—
471C22×C4—272no—
472C4⋊C4—91no—
473C4⋊C4—92no—
474C4⋊C4—91no—
475C4⋊C4—272no—
476C22⋊C4—92no—
477C22⋊C4—272no—
478C22⋊C4—272no—
479C42—91no—
480C42—91no—
481C42—91no—
482C42—92no—
483C42—272no—
484C42—272no—
485C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
486C22×C3—31no—
487C22×C3—31no—
488C22×C3—61no—
489C22×C3—94no—
490C22×C3—92no—
491C22×C3—182no—
492C22×C3—182no—
493C22×C3—181no—
494C22×C3—181no—
495C22×C3C22×D3211yescharacteristic
496C2×D3—32no—
497C2×D3—32no—
498C2×D3—34no—
499C2×D3—62no—
500C2×D3—61no—
501C2×D3—61no—
502C2×D3—61no—
503C2×D3—61no—
504C2×D3—62no—
505C2×D3—62no—
506C2×D3—62no—
507C2×D3—62no—
508C2×D3—94no—
509C2×D3—182no—
510C2×D3—182no—
511C3×C4—21no—
512C3×C4—32no—
513C3×C4—32no—
514C3×C4—32no—
515C3×C4—32no—
516C3×C4—61no—
517C3×C4—61no—
518C3×C4—62no—
519C3×C4—62no—
520C3×C4—62no—
521C3×C4—94no—
522C3×C4—92no—
523C3×C4—92no—
524C3×C4—182no—
525C3×C4—182no—
526C3×C4—181no—
527C3×C4#379612yes—
528C3×C4C4×D3212yes—
529Dic3—32no—
530Dic3—32no—
531Dic3—32no—
532Dic3—32no—
533Dic3—34no—
534Dic3—62no—
535Dic3—62no—
536Dic3—61no—
537Dic3—61no—
538Dic3—62no—
539Dic3—62no—
540Dic3—62no—
541Dic3—94no—
542Dic3—182no—
543Dic3—182no—
544C32—21no—
545C32C4×#23711yescharacteristic, complemented
546C32C4×#23911yescharacteristic, complemented
547C23—92no—
548C23—91no—
549Q8—94no—
550D4—94no—
551D4—278no—
552C2×C4—91no—
553C2×C4—91no—
554C2×C4—92no—
555C2×C4—94no—
556C2×C4—91no—
557C2×C4—92no—
558C2×C4—92no—
559C2×C4—182no—
560C2×C4—182no—
561C2×C4—184no—
562C2×C4—278no—
563C2×C4—544no—
564C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
565C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
566C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
567C2×C3—31no—
568C2×C3—31no—
569C2×C3—31no—
570C2×C3—31no—
571C2×C3—31no—
572C2×C3—31no—
573C2×C3—61no—
574C2×C3—61no—
575C2×C3—61no—
576C2×C3—184no—
577C2×C3—182no—
578C2×C3C2×#379611yescharacteristic
579C2×C3C2×C4×D3211yescharacteristic
580C2×C3#6430611yescharacteristic
581D3—62no—
582D3—62no—
583D3—64no—
584D3—64no—
585D3—184no—
586C22—92no—
587C22—94no—
588C22—181no—
589C22—182no—
590C22—181no—
591C22—182no—
592C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
593C4—21no—
594C4—92no—
595C4—94no—
596C4—92no—
597C4—181no—
598C4—182no—
599C4—182no—
600C4#8864912yesdirect product factor
601C4C4×((C3⋊D3)⋊C3)⋊C212yes—
602C3—31no—
603C3—31no—
604C3—61no—
605C3C4×#379611yescharacteristic
606C2—182no—
607C2—184no—
608C2C2×#8864911yescharacteristic
609C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
610C2#1172249111yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC2×#88649C2
cosocleC22×H(3,𝔽3)C24
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×#41
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C42×#272C2semidirect
3. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. C4×#634⋊C3C2semidirect
5. C4×#6200C2semidirect
6. #11721136C2semidirect
7. #11721124C2semidirect
8. #11721121C2semidirect
9. C4×((C3⋊D3)⋊C3)⋊C4C2semidirect
10. #11721089C2semidirect
11. #11721084C2semidirect
23. #88649C4direct (N × H)
40. C4×C4⋊(C3⋊D3)D3semidirect
41. C4×#634D3semidirect
71. C4×#272C2×C4semidirect
75. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
77. #634⋊C3C2×C4semidirect
78. #6200C2×C4semidirect
80. ((C3⋊D3)⋊C3)⋊C4C2×C4semidirect
148. C4×H(3,𝔽3)#41semidirect
151. #1084#41semidirect
206. C4⋊(C3⋊D3)C4×D3semidirect
208. #634C4×D3semidirect
349. C32×C4#237semidirect
350. C32×C4#239semidirect
377. H(3,𝔽3)C4×#41semidirect
545. C32C4×#237semidirect
546. C32C4×#239semidirect
600. C4#88649direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#886492
C42×#2722
C4×C4⋊((C3⋊D3)⋊C3)2
C4×#634⋊C32
C4×#62002
#117211362
#117211242
#117211212
C4×((C3⋊D3)⋊C3)⋊C42
#117210892
#117210842
C4×#37993
C4×#37953