Finite Group Databasebeta

← C4×#88652 #13147873

Subgroups — C4×#88652 #13147873

Order 1728.

1179 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88652C211yesmaximal, characteristic
2C4×C4⋊#272C211yesmaximal, characteristic, complemented
3C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4C4×#634⋊C3C211yesmaximal, characteristic, complemented
5#11721154C212yesmaximal, complemented
6C4×#6200C212yesmaximal, complemented
7#11721149C211yesmaximal, characteristic, complemented
8C4×#6199C212yesmaximal, complemented
9#11721082C211yesmaximal, characteristic, complemented
10#11721081C211yesmaximal, characteristic, complemented
11#11721076C212yesmaximal, complemented
12C4×#3796—31nomaximal
13C4×#3795—31nomaximal
14C2×C4⋊#272C2211yescharacteristic
15C2×C4×#272C2212yes—
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
19C2×#634⋊C3C2211yescharacteristic
20C2×#6200C2212yes—
21C2×#6199C2212yes—
22C2×#6199C2212yes—
23#88652C418yesdirect product factor
24C42×H(3,𝔽3)C2211yescharacteristic
25#88542C2212yes—
26C4⋊#1084C2211yescharacteristic
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28#3750⋊C3C2212yes—
29C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
30C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
31C4×(C3⋊Dic3)⋊C3C2212yes—
32C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
33#88451C2212yes—
34#88450C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37#88447C2212yes—
38C2×#3796—31no—
39C2×#3795—31no—
40C42×C3⋊D3D311yescharacteristic, complemented
41C4×#634D311yescharacteristic, complemented
42C3×C4×D12—31no—
43C3×C4×D12—31no—
44C3×C4×D12—61no—
45C3×C42×D3—31no—
46C3×C4×Dic6—31no—
47C4×#626—32no—
48C4×#625—32no—
49#63898—31no—
50Dic3×D12—31no—
51#63881—32no—
52#63879—32no—
53#63878—32no—
54#63877—32no—
55C4×D3×Dic3—32no—
56C4×D3×Dic3—32no—
57#63849—31no—
58#63844—31no—
59#63843—31no—
60#63841—31no—
61C22×#272C2312yes—
62C22×(C3⋊D3)⋊C3C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×#1084C2312yes—
67C4⋊#272C2×C414yescomplemented
68C4×#272—24no—
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
74C4×(C3⋊D3)⋊C3—22no—
75C4×(C3⋊D3)⋊C3—22no—
76C4×(C3⋊D3)⋊C3C2×C414yescomplemented
77#634⋊C3C2×C414yescomplemented
78#6200C2×C418yescomplemented
79#6199—24no—
80#6199C2×C418yescomplemented
81C4×#239—91no—
82C4×#237—91no—
83C4×#235—91no—
84C2×C4×C3⋊D3C2×D311yescharacteristic
85C2×C4×C3⋊D3C2×D311yescharacteristic
86C2×C4×C3⋊D3C2×D311yescharacteristic
87C2×#634C2×D311yescharacteristic
88C2×C3×D12—31no—
89C2×C3×D12—31no—
90C2×C3×D12—61no—
91C2×C3×C4×D3—31no—
92C2×C3×C4×D3—31no—
93C2×C3×C4×D3—31no—
94C2×C3×C4×D3—32no—
95C2×C3×C4×D3—32no—
96C2×C3×C4×D3—62no—
97C2×C3×Dic6—31no—
98C2×#626—32no—
99C2×#625—32no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—32no—
102C2×D3×Dic3—32no—
103C2×D3×Dic3—32no—
104#3796—38no—
105#3795—38no—
106C32×C42—21no—
107C32×C42C2×D311yescharacteristic
108C32×C42C2×D311yescharacteristic
109C4⋊(C3⋊Dic3)C2×D311yescharacteristic
110#3750C2×D312yes—
111C4×C3⋊Dic3C2×D312yes—
112C4×C3⋊Dic3C2×D311yescharacteristic
113C4×C3⋊Dic3C2×D311yescharacteristic
114C4×C3⋊Dic3C2×D311yescharacteristic
115C3×(C2×D3)⋊C4—32no—
116C3×(C2×D3)⋊C4—32no—
117C3×(C2×D3)⋊C4—62no—
118C3×C4⋊Dic3—31no—
119C3×C4⋊Dic3—31no—
120C3×C4⋊Dic3—31no—
121C3×C4⋊Dic3—61no—
122C3×Dic3⋊C4—32no—
123C3×C4×Dic3—31no—
124C3×C4×Dic3—31no—
125C3×C4×Dic3—32no—
126C3×C4×Dic3—31no—
127#3724—32no—
128Dic3⋊Dic3—32no—
129#3722—32no—
130(C2×D3)⋊Dic3—32no—
131(C2×D3)⋊Dic3—32no—
132(C2×D3)⋊Dic3—32no—
133Dic32—32no—
134Dic32—32no—
135Dic32—32no—
136Dic32—32no—
137C22×H(3,𝔽3)C2411yescharacteristic
138C2×#272—22no—
139C2×#272—22no—
140C2×#272C22×C414yes—
141C2×(C3⋊D3)⋊C3—21no—
142C2×(C3⋊D3)⋊C3—21no—
143C2×(C3⋊D3)⋊C3C22×C412yes—
144C4×H(3,𝔽3)—21no—
145C4×H(3,𝔽3)C22×C412yes—
146C4×H(3,𝔽3)#4112yescomplemented
147#1084—22no—
148(C3⋊Dic3)⋊C3—22no—
149(C3⋊Dic3)⋊C3—21no—
150(C3⋊Dic3)⋊C3C22×C412yes—
151(C3⋊Dic3)⋊C3C22×C414yes—
152(C3⋊Dic3)⋊C3#4112yescomplemented
153C2×#239—91no—
154C2×#237—91no—
155C2×#235—91no—
156C3×C4×Q8—91no—
157C3×C4×D4—91no—
158C2×C3×C42—91no—
159Q8×Dic3—91no—
160D4×Dic3—91no—
161C4×#67—92no—
162C4×#67—92no—
163C4×#67—92no—
164#926—91no—
165C2×C4×Dic3—92no—
166#893—92no—
167#893—91no—
168#892—91no—
169#892—91no—
170#892—92no—
171#887—91no—
172#67⋊C4—92no—
173#877—92no—
174C4×D12—31no—
175C4×D12—31no—
176C4×D12—61no—
177C4×D12—92no—
178C4×D12—91no—
179#872—92no—
180C42×D3—31no—
181C42×D3—91no—
182C42×D3—92no—
183C42×D3—92no—
184C4×Dic6—31no—
185C4×Dic6—91no—
186C4×Dic6—91no—
187C22×C3⋊D3C22×D311yescharacteristic
188C22×C3×D3—31no—
189C22×C3×D3—32no—
190C22×C3×D3—32no—
191C22×C3×D3—62no—
192C2×C32×C4—21no—
193C2×C32×C4—21no—
194C2×C32×C4—21no—
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200C2×C32×C4C22×D311yescharacteristic
201C2×C3⋊Dic3C22×D312yes—
202C2×C3⋊Dic3C22×D311yescharacteristic
203C2×C3⋊Dic3C22×D311yescharacteristic
204C2×C3⋊Dic3C22×D312yes—
205C2×C3⋊Dic3C22×D311yescharacteristic
206C4×C3⋊D3—22no—
207C4×C3⋊D3—22no—
208C4×C3⋊D3C4×D314yescomplemented
209#634C4×D314yescomplemented
210C2×C3×Dic3—31no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—31no—
215C2×C3×Dic3—32no—
216C2×C3×Dic3—32no—
217C2×C3×Dic3—62no—
218C3×D12—34no—
219C3×D12—34no—
220C3×D12—64no—
221C3×C4×D3—34no—
222C3×C4×D3—62no—
223C3×C4×D3—64no—
224C3×C4×D3—62no—
225C3×C4×D3—64no—
226C3×C4×D3—124no—
227C3×Dic6—34no—
228#626—38no—
229#625—38no—
230D3×Dic3—38no—
231D3×Dic3—38no—
232D3×Dic3—64no—
233D3×Dic3—64no—
234C4×#41—271noSylow
235C2×H(3,𝔽3)C23×C411yescharacteristic
236C2×H(3,𝔽3)C2×#4111yescharacteristic
237C2×H(3,𝔽3)C2×#4111yescharacteristic
238#272—24no—
239(C3⋊D3)⋊C3—22no—
240C2×C3×Q8—91no—
241C2×C3×D4—91no—
242C22×C3×C4—91no—
243C22×C3×C4—91no—
244C22×C3×C4—91no—
245C22×C3×C4—92no—
246C2×#67—92no—
247C2×#67—92no—
248C22×Dic3—92no—
249C22×Dic3—92no—
250#239—98no—
251#237—98no—
252#235—98no—
253C2×D12—31no—
254C2×D12—31no—
255C2×D12—61no—
256C2×D12—92no—
257C2×C4×D3—31no—
258C2×C4×D3—32no—
259C2×C4×D3—32no—
260C2×C4×D3—31no—
261C2×C4×D3—31no—
262C2×C4×D3—62no—
263C2×C4×D3—91no—
264C2×C4×D3—91no—
265C2×C4×D3—91no—
266C2×C4×D3—92no—
267C2×C4×D3—92no—
268C2×C4×D3—92no—
269C2×C4×D3—92no—
270C2×Dic6—31no—
271C2×Dic6—91no—
272C3×C4⋊C4—92no—
273C3×C4⋊C4—91no—
274C3×C4⋊C4—91no—
275C3×C22⋊C4—92no—
276C3×C42—31no—
277C3×C42—31no—
278C3×C42—61no—
279C3×C42—91no—
280C3×C42—91no—
281C3×C42—92no—
282C3×C42—91no—
283C3×C42D3211yescharacteristic
284#217—92no—
285#217—92no—
286(C2×D3)⋊C4—32no—
287(C2×D3)⋊C4—32no—
288(C2×D3)⋊C4—62no—
289(C2×D3)⋊C4—92no—
290(C2×D3)⋊C4—92no—
291(C2×D3)⋊C4—92no—
292(C2×D3)⋊C4—92no—
293C4⋊Dic3—31no—
294C4⋊Dic3—31no—
295C4⋊Dic3—31no—
296C4⋊Dic3—61no—
297C4⋊Dic3—91no—
298C4⋊Dic3—92no—
299Dic3⋊C4—32no—
300Dic3⋊C4—92no—
301Dic3⋊C4—92no—
302Dic3⋊C4—92no—
303C4×Dic3—32no—
304C4×Dic3—31no—
305C4×Dic3—31no—
306C4×Dic3—31no—
307C4×Dic3—92no—
308C4×Dic3—91no—
309C4×Dic3—91no—
310C4×Dic3—91no—
311C4×Dic3—92no—
312C4×Dic3—92no—
313C4×Dic3—92no—
314C4×Dic3—92no—
315C4×Dic3—92no—
316C4×Dic3—92no—
317C22×C32—21no—
318C22×C32C23×D311yescharacteristic
319C22×C32C23×D311yescharacteristic
320C2×C3⋊D3—21no—
321C2×C3⋊D3—21no—
322C2×C3⋊D3C2×C4×D312yes—
323C2×C3×D3—32no—
324C2×C3×D3—34no—
325C2×C3×D3—34no—
326C2×C3×D3—64no—
327C2×C3×D3—62no—
328C2×C3×D3—62no—
329C2×C3×D3—61no—
330C2×C3×D3—62no—
331C2×C3×D3—61no—
332C2×C3×D3—62no—
333C2×C3×D3—122no—
334C2×C3×D3—122no—
335C32×C4—21no—
336C32×C4—21no—
337C32×C4—22no—
338C32×C4—22no—
339C32×C4—41no—
340C32×C4C2×C4×D312yes—
341C32×C4C2×C4×D312yes—
342C32×C4#23512yescomplemented
343C32×C4#23912yescomplemented
344C3⋊Dic3—21no—
345C3⋊Dic3—22no—
346C3⋊Dic3C2×C4×D312yes—
347C3⋊Dic3C2×C4×D314yes—
348C3⋊Dic3#23512yescomplemented
349C3×Dic3—32no—
350C3×Dic3—34no—
351C3×Dic3—32no—
352C3×Dic3—62no—
353C3×Dic3—62no—
354C3×Dic3—61no—
355C3×Dic3—62no—
356C3×Dic3—122no—
357C2×#41—271no—
358C4×Q8—91no—
359C4×Q8—271no—
360C4×D4—91no—
361C4×D4—271no—
362C4×D4—272no—
363C4×D4—272no—
364#117—272no—
365#117—271no—
366C2×C42—91no—
367C2×C42—272no—
368H(3,𝔽3)C4×#4111yescharacteristic, complemented, Sylow
369C23×C3—91no—
370C23×C3—92no—
371C22×D3—32no—
372C22×D3—32no—
373C22×D3—31no—
374C22×D3—62no—
375C22×D3—91no—
376C3×Q8—94no—
377C3×D4—94no—
378C2×C3×C4—31no—
379C2×C3×C4—31no—
380C2×C3×C4—31no—
381C2×C3×C4—31no—
382C2×C3×C4—31no—
383C2×C3×C4—31no—
384C2×C3×C4—61no—
385C2×C3×C4—61no—
386C2×C3×C4—61no—
387C2×C3×C4—94no—
388C2×C3×C4—91no—
389C2×C3×C4—91no—
390C2×C3×C4—92no—
391C2×C3×C4—92no—
392C2×C3×C4—91no—
393C2×C3×C4—92no—
394C2×C3×C4—182no—
395C2×C3×C4—182no—
396C2×C3×C4—184no—
397C2×C3×C4C2×D3211yescharacteristic
398C2×C3×C4C2×D3211yescharacteristic
399C2×C3×C4C2×D3211yescharacteristic
400#67—98no—
401#67—98no—
402C2×Dic3—31no—
403C2×Dic3—31no—
404C2×Dic3—32no—
405C2×Dic3—32no—
406C2×Dic3—32no—
407C2×Dic3—31no—
408C2×Dic3—32no—
409C2×Dic3—62no—
410C2×Dic3—91no—
411C2×Dic3—91no—
412C2×Dic3—92no—
413C2×Dic3—92no—
414C2×Dic3—91no—
415C2×Dic3—98no—
416C2×Dic3—184no—
417D12—34no—
418D12—34no—
419D12—64no—
420D12—98no—
421C4×D3—34no—
422C4×D3—62no—
423C4×D3—64no—
424C4×D3—64no—
425C4×D3—62no—
426C4×D3—94no—
427C4×D3—98no—
428C4×D3—98no—
429C4×D3—124no—
430C4×D3—182no—
431C4×D3—182no—
432C4×D3—184no—
433C4×D3—184no—
434Dic6—34no—
435Dic6—94no—
436C2×C32—21no—
437C2×C32—21no—
438C2×C32—21no—
439C2×C32C22×C4×D311yescharacteristic
440C2×C32C22×C4×D311yescharacteristic
441C2×C32C2×#23511yescharacteristic
442C2×C32C2×#23511yescharacteristic
443C2×C32C2×#23911yescharacteristic
444C2×C32D3×#4111yescharacteristic
445C3⋊D3—22no—
446C3×D3—64no—
447C3×D3—62no—
448C3×D3—64no—
449C3×D3—124no—
450#41—278no—
451C2×Q8—91no—
452C2×D4—91no—
453C2×D4—272no—
454C22×C4—91no—
455C22×C4—91no—
456C22×C4—91no—
457C22×C4—92no—
458C22×C4—272no—
459C22×C4—272no—
460C4⋊C4—91no—
461C4⋊C4—92no—
462C4⋊C4—91no—
463C4⋊C4—272no—
464C22⋊C4—92no—
465C22⋊C4—272no—
466C22⋊C4—272no—
467C42—91no—
468C42—91no—
469C42—91no—
470C42—92no—
471C42—272no—
472C42—272no—
473C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
474C22×C3—31no—
475C22×C3—31no—
476C22×C3—61no—
477C22×C3—92no—
478C22×C3—94no—
479C22×C3—181no—
480C22×C3—181no—
481C22×C3—182no—
482C22×C3—182no—
483C22×C3C22×D3211yescharacteristic
484C2×D3—34no—
485C2×D3—34no—
486C2×D3—32no—
487C2×D3—64no—
488C2×D3—62no—
489C2×D3—62no—
490C2×D3—62no—
491C2×D3—62no—
492C2×D3—61no—
493C2×D3—61no—
494C2×D3—92no—
495C2×D3—122no—
496C2×D3—122no—
497C2×D3—181no—
498C2×D3—181no—
499C3×C4—21no—
500C3×C4—32no—
501C3×C4—32no—
502C3×C4—32no—
503C3×C4—32no—
504C3×C4—61no—
505C3×C4—61no—
506C3×C4—62no—
507C3×C4—62no—
508C3×C4—92no—
509C3×C4—94no—
510C3×C4—92no—
511C3×C4—121no—
512C3×C4—182no—
513C3×C4—181no—
514C3×C4—182no—
515C3×C4#379912yes—
516C3×C4C4×D3212yes—
517Dic3—34no—
518Dic3—32no—
519Dic3—32no—
520Dic3—62no—
521Dic3—62no—
522Dic3—61no—
523Dic3—62no—
524Dic3—94no—
525Dic3—92no—
526Dic3—92no—
527Dic3—122no—
528Dic3—181no—
529Dic3—182no—
530C32—21no—
531C32C4×#23511yescharacteristic, complemented
532C32C4×#23911yescharacteristic, complemented
533C23—92no—
534C23—91no—
535Q8—94no—
536D4—94no—
537D4—278no—
538C2×C4—91no—
539C2×C4—91no—
540C2×C4—92no—
541C2×C4—94no—
542C2×C4—91no—
543C2×C4—92no—
544C2×C4—92no—
545C2×C4—182no—
546C2×C4—184no—
547C2×C4—182no—
548C2×C4—278no—
549C2×C4—544no—
550C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
551C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
552C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
553C2×C3—31no—
554C2×C3—31no—
555C2×C3—31no—
556C2×C3—31no—
557C2×C3—31no—
558C2×C3—31no—
559C2×C3—61no—
560C2×C3—61no—
561C2×C3—61no—
562C2×C3—182no—
563C2×C3—184no—
564C2×C3C2×#379911yescharacteristic
565C2×C3C2×C4×D3211yescharacteristic
566C2×C3D3×#23511yescharacteristic
567D3—64no—
568D3—64no—
569D3—62no—
570D3—124no—
571D3—182no—
572C22—92no—
573C22—94no—
574C22—182no—
575C22—181no—
576C22—181no—
577C22—182no—
578C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
579C4—21no—
580C4—92no—
581C4—94no—
582C4—92no—
583C4—182no—
584C4—181no—
585C4—182no—
586C4#8865212yesdirect product factor
587C4C4×((C3⋊D3)⋊C3)⋊C212yes—
588C3—31no—
589C3—31no—
590C3—61no—
591C3C4×#379911yescharacteristic
592C2—184no—
593C2—182no—
594C2C2×#8865211yescharacteristic
595C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
596C2#1172249011yescharacteristic
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×#88652C2
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. C4×C4⋊#272C2semidirect
3. C42×(C3⋊D3)⋊C3C2semidirect
4. C4×#634⋊C3C2semidirect
5. #11721154C2semidirect
6. C4×#6200C2semidirect
7. #11721149C2semidirect
8. C4×#6199C2semidirect
9. #11721082C2semidirect
10. #11721081C2semidirect
11. #11721076C2semidirect
23. #88652C4direct (N × H)
40. C42×C3⋊D3D3semidirect
41. C4×#634D3semidirect
67. C4⋊#272C2×C4semidirect
76. C4×(C3⋊D3)⋊C3C2×C4semidirect
77. #634⋊C3C2×C4semidirect
78. #6200C2×C4semidirect
80. #6199C2×C4semidirect
146. C4×H(3,𝔽3)#41semidirect
152. (C3⋊Dic3)⋊C3#41semidirect
208. C4×C3⋊D3C4×D3semidirect
209. #634C4×D3semidirect
342. C32×C4#235semidirect
343. C32×C4#239semidirect
348. C3⋊Dic3#235semidirect
368. H(3,𝔽3)C4×#41semidirect
531. C32C4×#235semidirect
532. C32C4×#239semidirect
586. C4#88652direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#886522
C4×C4⋊#2722
C42×(C3⋊D3)⋊C32
C4×#634⋊C32
#117211542
C4×#62002
#117211492
C4×#61992
#117210822
#117210812
#117210762
C4×#37963
C4×#37953