Finite Group Databasebeta

← C2×C4×#6200 #13150828

Subgroups — C2×C4×#6200 #13150828

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
1C22×C4×#272C211yesmaximal, characteristic, complemented
2C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3C22×#6200C211yesmaximal, characteristic
4C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
5C2×#88451C211yesmaximal, characteristic, complemented
6C2×#88450C211yesmaximal, characteristic, complemented
7C2×#88448C211yesmaximal, characteristic, complemented
8C4×#6200C218yesmaximal, direct product factor
9C2×C4×#626—31nomaximal
10C2×C4×#625—31nomaximal
11C23×#272C2211yescharacteristic
12C23×(C3⋊D3)⋊C3C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C2×C4×#272—24no—
16C2×C4×#272C2214yescomplemented
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×C4×(C3⋊D3)⋊C3—24no—
21C2×C4×(C3⋊D3)⋊C3C2214yescomplemented
22C2×#6200C418yesdirect product factor
23C2×#6200C2214yes—
24C4×(C3⋊Dic3)⋊C3C2214yescomplemented
25#88451C2214yescomplemented
26#88450C2214yescomplemented
27#88448C2214yescomplemented
28C22×C4×C3⋊D3D311yescharacteristic, complemented
29C22×C3×C4×D3—31no—
30C22×C3×C4×D3—31no—
31C22×C3×C4×D3—31no—
32C22×C3×C4×D3—61no—
33C22×#626—31no—
34C22×#625—31no—
35C2×C4×C3⋊Dic3D311yescharacteristic, complemented
36C2×C3×C4×Dic3—31no—
37C2×#3724—31no—
38C2×Dic3⋊Dic3—31no—
39C2×#3722—31no—
40C2×(C2×D3)⋊Dic3—31no—
41C2×(C2×D3)⋊Dic3—31no—
42C2×(C2×D3)⋊Dic3—31no—
43C4×#626—38no—
44C4×#625—38no—
45C23×H(3,𝔽3)C2311yescharacteristic
46C22×#272—22no—
47C22×#272—22no—
48C22×#272C2×C414yescomplemented
49C22×#272C2312yes—
50C22×(C3⋊D3)⋊C3—22no—
51C22×(C3⋊D3)⋊C3—22no—
52C22×(C3⋊D3)⋊C3C2×C414yescomplemented
53C22×(C3⋊D3)⋊C3C2312yes—
54C2×C4×H(3,𝔽3)D414yescomplemented
55C2×C4×H(3,𝔽3)C2312yes—
56C2×#1084—22no—
57C2×#1084C2312yes—
58C4×#272—28no—
59C2×(C3⋊Dic3)⋊C3—22no—
60C2×(C3⋊Dic3)⋊C3—22no—
61C2×(C3⋊Dic3)⋊C3C2×C414yescomplemented
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C4×(C3⋊D3)⋊C3—28no—
66#6200C2×C4116yesdirect product factor
67C2×C4×#67—91no—
68C2×C4×#67—91no—
69C2×C4×D12—91no—
70C23×C3⋊D3C2×D311yescharacteristic
71C23×C3×D3—31no—
72C23×C3×D3—31no—
73C23×C3×D3—31no—
74C23×C3×D3—61no—
75C22×C32×C4—21no—
76C22×C32×C4C2×D311yescharacteristic
77C22×C32×C4C2×D311yescharacteristic
78C22×C3⋊Dic3C2×D311yescharacteristic
79C22×C3⋊Dic3C2×D311yescharacteristic
80C22×C3⋊Dic3C2×D311yescharacteristic
81C2×C4×C3⋊D3—24no—
82C2×C4×C3⋊D3C2×D314yescomplemented
83C22×C3×Dic3—31no—
84C22×C3×Dic3—31no—
85C22×C3×Dic3—31no—
86C22×C3×Dic3—31no—
87C22×C3×Dic3—31no—
88C22×C3×Dic3—61no—
89C2×C3×C4×D3—34no—
90C2×C3×C4×D3—34no—
91C2×C3×C4×D3—34no—
92C2×C3×C4×D3—64no—
93C2×C3×C4×D3—68no—
94C2×C3×C4×D3—64no—
95C2×C3×C4×D3—64no—
96C2×C3×C4×D3—64no—
97C2×#626—38no—
98C2×#626—34no—
99C2×#625—38no—
100C2×#625—34no—
101C4×C3⋊Dic3C2×D314yescomplemented
102C3×C4×Dic3—34no—
103#3724—34no—
104Dic3⋊Dic3—34no—
105#3722—34no—
106(C2×D3)⋊Dic3—34no—
107(C2×D3)⋊Dic3—34no—
108(C2×D3)⋊Dic3—34no—
109C22×H(3,𝔽3)C22×C412yes—
110C22×H(3,𝔽3)C2×D412yes—
111C22×H(3,𝔽3)#4112yes—
112C22×H(3,𝔽3)C2411yescharacteristic
113C2×#272—28no—
114C2×#272—22no—
115C2×#272—22no—
116C2×#272C22×C414yescomplemented
117C2×(C3⋊D3)⋊C3—28no—
118C2×(C3⋊D3)⋊C3—22no—
119C2×(C3⋊D3)⋊C3—22no—
120C2×(C3⋊D3)⋊C3C22×C414yescomplemented
121C4×H(3,𝔽3)C2×D414yescomplemented
122#1084—22no—
123(C3⋊Dic3)⋊C3—22no—
124(C3⋊Dic3)⋊C3—22no—
125(C3⋊Dic3)⋊C3C22×C414yescomplemented
126C23×C3×C4—91no—
127C23×C3×C4—91no—
128C22×#67—91no—
129C22×#67—91no—
130C22×D12—91no—
131C22×C4×D3—31no—
132C22×C4×D3—31no—
133C22×C4×D3—31no—
134C22×C4×D3—61no—
135C22×C4×D3—91no—
136C2×C3×C42—91no—
137C2×#217—91no—
138C2×#217—91no—
139C4×#67—98no—
140C4×#67—98no—
141C2×(C2×D3)⋊C4—91no—
142C2×(C2×D3)⋊C4—91no—
143C2×(C2×D3)⋊C4—91no—
144C2×(C2×D3)⋊C4—91no—
145C2×C4⋊Dic3—91no—
146C2×Dic3⋊C4—91no—
147C2×Dic3⋊C4—91no—
148C2×C4×Dic3—31no—
149C2×C4×Dic3—91no—
150C4×D12—98no—
151C23×C32—21no—
152C23×C32C22×D311yescharacteristic
153C23×C32C22×D311yescharacteristic
154C22×C3⋊D3—22no—
155C22×C3⋊D3—22no—
156C22×C3⋊D3C4×D314yescomplemented
157C22×C3⋊D3C22×D312yes—
158C22×C3×D3—34no—
159C22×C3×D3—34no—
160C22×C3×D3—34no—
161C22×C3×D3—32no—
162C22×C3×D3—32no—
163C22×C3×D3—32no—
164C22×C3×D3—64no—
165C22×C3×D3—62no—
166C22×C3×D3—64no—
167C22×C3×D3—64no—
168C22×C3×D3—62no—
169C22×C3×D3—62no—
170C22×C3×D3—62no—
171C22×C3×D3—62no—
172C22×C3×D3—62no—
173C22×C3×D3—62no—
174C2×C32×C4—24no—
175C2×C32×C4—22no—
176C2×C32×C4D1214yescomplemented
177C2×C32×C4#6714yescomplemented
178C2×C32×C4C22×D312yes—
179C2×C32×C4C22×D312yes—
180C2×C3⋊Dic3—22no—
181C2×C3⋊Dic3—22no—
182C2×C3⋊Dic3C4×D314yescomplemented
183C2×C3⋊Dic3C22×D312yes—
184C2×C3⋊Dic3C22×D312yes—
185C2×C3⋊Dic3C22×D312yes—
186C4×C3⋊D3—28no—
187C2×C3×Dic3—34no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3—32no—
190C2×C3×Dic3—32no—
191C2×C3×Dic3—32no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3—62no—
194C2×C3×Dic3—64no—
195C2×C3×Dic3—62no—
196C2×C3×Dic3—62no—
197C2×C3×Dic3—62no—
198C2×C3×Dic3—62no—
199C3×C4×D3—616no—
200C3×C4×D3—68no—
201C3×C4×D3—68no—
202C3×C4×D3—68no—
203#626—316no—
204#625—316no—
205C2×C4×D4—271noSylow
206C2×H(3,𝔽3)C4×D414yescomplemented
207C2×H(3,𝔽3)C23×C411yescharacteristic
208C2×H(3,𝔽3)C22×D411yescharacteristic
209C2×H(3,𝔽3)C2×#4111yescharacteristic
210#272—24no—
211(C3⋊D3)⋊C3—24no—
212C24×C3—91no—
213C24×C3—91no—
214C23×D3—31no—
215C23×D3—31no—
216C23×D3—31no—
217C23×D3—61no—
218C23×D3—91no—
219C22×C3×C4—31no—
220C22×C3×C4—31no—
221C22×C3×C4—61no—
222C22×C3×C4—94no—
223C22×C3×C4—91no—
224C22×C3×C4—91no—
225C22×C3×C4—91no—
226C22×C3×C4—91no—
227C22×C3×C4—94no—
228C22×C3×C4—184no—
229C22×C3×C4—184no—
230C22×C3×C4D3211yescharacteristic
231C2×#67—94no—
232C2×#67—98no—
233C2×#67—94no—
234C2×#67—98no—
235C22×Dic3—31no—
236C22×Dic3—31no—
237C22×Dic3—31no—
238C22×Dic3—31no—
239C22×Dic3—31no—
240C22×Dic3—61no—
241C22×Dic3—91no—
242C22×Dic3—91no—
243C22×Dic3—91no—
244C2×D12—94no—
245C2×D12—98no—
246C2×C4×D3—34no—
247C2×C4×D3—34no—
248C2×C4×D3—34no—
249C2×C4×D3—64no—
250C2×C4×D3—64no—
251C2×C4×D3—64no—
252C2×C4×D3—68no—
253C2×C4×D3—64no—
254C2×C4×D3—94no—
255C2×C4×D3—184no—
256C3×C42—94no—
257#217—94no—
258#217—94no—
259(C2×D3)⋊C4—94no—
260(C2×D3)⋊C4—94no—
261(C2×D3)⋊C4—94no—
262(C2×D3)⋊C4—94no—
263C4⋊Dic3—94no—
264Dic3⋊C4—94no—
265Dic3⋊C4—94no—
266C4×Dic3—34no—
267C4×Dic3—94no—
268C22×C32—21no—
269C22×C32—22no—
270C22×C32—22no—
271C22×C32—22no—
272C22×C32C2×C4×D312yes—
273C22×C32C2×C4×D312yes—
274C22×C32C2×D1212yes—
275C22×C32#23512yes—
276C22×C32#23512yes—
277C22×C32C2×#6712yes—
278C22×C32C23×D311yescharacteristic
279C22×C32C23×D311yescharacteristic
280C2×C3⋊D3—28no—
281C2×C3⋊D3—22no—
282C2×C3⋊D3—22no—
283C2×C3⋊D3C2×C4×D314yescomplemented
284C2×C3×D3—34no—
285C2×C3×D3—34no—
286C2×C3×D3—34no—
287C2×C3×D3—64no—
288C2×C3×D3—64no—
289C2×C3×D3—64no—
290C2×C3×D3—616no—
291C2×C3×D3—68no—
292C2×C3×D3—68no—
293C2×C3×D3—68no—
294C2×C3×D3—62no—
295C2×C3×D3—62no—
296C2×C3×D3—62no—
297C2×C3×D3—62no—
298C2×C3×D3—62no—
299C2×C3×D3—62no—
300C32×C4—24no—
301C32×C4C2×D1214yescomplemented
302C32×C4C2×#6714yescomplemented
303C3⋊Dic3—22no—
304C3⋊Dic3—22no—
305C3⋊Dic3C2×C4×D314yescomplemented
306C3×Dic3—34no—
307C3×Dic3—64no—
308C3×Dic3—62no—
309C3×Dic3—62no—
310C3×Dic3—62no—
311C3×Dic3—62no—
312C22×D4—271no—
313C23×C4—91no—
314C23×C4—91no—
315C4×D4—278no—
316C2×C4⋊C4—271no—
317C2×C22⋊C4—271no—
318C2×C22⋊C4—271no—
319C2×C42—91no—
320H(3,𝔽3)C2×C4×D411yescharacteristic, complemented, Sylow
321C23×C3—31no—
322C23×C3—31no—
323C23×C3—61no—
324C23×C3—92no—
325C23×C3—94no—
326C23×C3—92no—
327C23×C3—94no—
328C23×C3—182no—
329C23×C3—182no—
330C23×C3—182no—
331C23×C3—182no—
332C23×C3C2×D3211yescharacteristic
333C22×D3—32no—
334C22×D3—32no—
335C22×D3—34no—
336C22×D3—34no—
337C22×D3—32no—
338C22×D3—34no—
339C22×D3—62no—
340C22×D3—64no—
341C22×D3—62no—
342C22×D3—62no—
343C22×D3—62no—
344C22×D3—62no—
345C22×D3—64no—
346C22×D3—64no—
347C22×D3—62no—
348C22×D3—62no—
349C22×D3—94no—
350C22×D3—92no—
351C22×D3—182no—
352C22×D3—182no—
353C2×C3×C4—34no—
354C2×C3×C4—34no—
355C2×C3×C4—32no—
356C2×C3×C4—32no—
357C2×C3×C4—64no—
358C2×C3×C4—62no—
359C2×C3×C4—92no—
360C2×C3×C4—94no—
361C2×C3×C4—92no—
362C2×C3×C4—92no—
363C2×C3×C4—92no—
364C2×C3×C4—188no—
365C2×C3×C4—182no—
366C2×C3×C4—182no—
367C2×C3×C4—182no—
368C2×C3×C4—188no—
369C2×C3×C4#62614yes—
370C2×C3×C4C2×D3212yes—
371#67—916no—
372#67—916no—
373C2×Dic3—32no—
374C2×Dic3—34no—
375C2×Dic3—32no—
376C2×Dic3—32no—
377C2×Dic3—32no—
378C2×Dic3—32no—
379C2×Dic3—62no—
380C2×Dic3—62no—
381C2×Dic3—62no—
382C2×Dic3—64no—
383C2×Dic3—62no—
384C2×Dic3—62no—
385C2×Dic3—94no—
386C2×Dic3—92no—
387C2×Dic3—92no—
388C2×Dic3—92no—
389C2×Dic3—182no—
390C2×Dic3—182no—
391D12—916no—
392C4×D3—68no—
393C4×D3—68no—
394C4×D3—616no—
395C4×D3—68no—
396C4×D3—188no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—21no—
400C2×C32—24no—
401C2×C32C4×D1214yescomplemented
402C2×C32C4×#6714yescomplemented
403C2×C32C22×C4×D311yescharacteristic
404C2×C32C22×C4×D311yescharacteristic
405C2×C32C22×D1211yescharacteristic
406C2×C32C2×#23511yescharacteristic
407C2×C32C2×#23511yescharacteristic
408C2×C32C22×#6711yescharacteristic
409C3⋊D3—24no—
410C3×D3—68no—
411C3×D3—64no—
412C3×D3—64no—
413C3×D3—64no—
414C24—91no—
415C24—91no—
416C2×D4—274no—
417C2×D4—278no—
418C22×C4—91no—
419C22×C4—91no—
420C22×C4—91no—
421C22×C4—91no—
422C22×C4—94no—
423C22×C4—94no—
424C22×C4—184no—
425C22×C4—184no—
426C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
427C4⋊C4—274no—
428C22⋊C4—274no—
429C22⋊C4—274no—
430C42—94no—
431C22×C3—31no—
432C22×C3—31no—
433C22×C3—32no—
434C22×C3—32no—
435C22×C3—32no—
436C22×C3—32no—
437C22×C3—32no—
438C22×C3—32no—
439C22×C3—61no—
440C22×C3—62no—
441C22×C3—62no—
442C22×C3—62no—
443C22×C3—94no—
444C22×C3—94no—
445C22×C3—182no—
446C22×C3—188no—
447C22×C3—182no—
448C22×C3—182no—
449C22×C3—182no—
450C22×C3—188no—
451C22×C3#379812yes—
452C22×C3C4×D3212yes—
453C22×C3C2×#62612yes—
454C22×C3C22×D3211yescharacteristic
455C2×D3—34no—
456C2×D3—34no—
457C2×D3—34no—
458C2×D3—64no—
459C2×D3—62no—
460C2×D3—62no—
461C2×D3—68no—
462C2×D3—68no—
463C2×D3—62no—
464C2×D3—62no—
465C2×D3—64no—
466C2×D3—616no—
467C2×D3—64no—
468C2×D3—62no—
469C2×D3—62no—
470C2×D3—68no—
471C2×D3—94no—
472C2×D3—182no—
473C2×D3—188no—
474C2×D3—182no—
475C3×C4—34no—
476C3×C4—34no—
477C3×C4—64no—
478C3×C4—94no—
479C3×C4—182no—
480C3×C4—182no—
481C3×C4—182no—
482C3×C4C2×#62614yes—
483Dic3—34no—
484Dic3—62no—
485Dic3—62no—
486Dic3—64no—
487Dic3—62no—
488Dic3—62no—
489Dic3—94no—
490Dic3—182no—
491Dic3—182no—
492C32—21no—
493C32C2×C4×D1211yescharacteristic, complemented
494C32C2×C4×#6711yescharacteristic, complemented
495C23—92no—
496C23—92no—
497C23—94no—
498C23—94no—
499C23—182no—
500C23—182no—
501C23—182no—
502C23—182no—
503C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
504D4—2716no—
505C2×C4—92no—
506C2×C4—94no—
507C2×C4—92no—
508C2×C4—92no—
509C2×C4—92no—
510C2×C4—182no—
511C2×C4—182no—
512C2×C4—182no—
513C2×C4—188no—
514C2×C4—188no—
515C2×C4#620014yesdirect product factor
516C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
517C2×C3—31no—
518C2×C3—31no—
519C2×C3—34no—
520C2×C3—34no—
521C2×C3—31no—
522C2×C3—31no—
523C2×C3—31no—
524C2×C3—31no—
525C2×C3—61no—
526C2×C3—64no—
527C2×C3—61no—
528C2×C3—61no—
529C2×C3—184no—
530C2×C3—184no—
531C2×C3C4×#62614yes—
532C2×C3C2×#379811yescharacteristic
533C2×C3C2×C4×D3211yescharacteristic
534C2×C3C22×#62611yescharacteristic
535D3—64no—
536D3—64no—
537D3—68no—
538D3—64no—
539D3—184no—
540C22—94no—
541C22—94no—
542C22—182no—
543C22—182no—
544C22—188no—
545C22—188no—
546C22—182no—
547C22—182no—
548C22#8865012yes—
549C22C4×((C3⋊D3)⋊C3)⋊C212yes—
550C22C2×#620012yes—
551C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
552C4—94no—
553C4—182no—
554C4—182no—
555C4—182no—
556C4C2×#620014yesdirect product factor
557C3—31no—
558C3—31no—
559C3—61no—
560C3C2×C4×#62611yescharacteristic
561C2—184no—
562C2—184no—
563C2C4×#620014yesdirect product factor
564C2C2×#8865011yescharacteristic
565C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
566C2C22×#620011yescharacteristic
subgroupG/H
center, hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC22×#6200C2
cosocleC22×H(3,𝔽3)C24
hypocenterH(3,𝔽3)C2×C4×D4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×#272C2semidirect
2. C22×C4×(C3⋊D3)⋊C3C2semidirect
4. C2×C4×(C3⋊Dic3)⋊C3C2semidirect
5. C2×#88451C2semidirect
6. C2×#88450C2semidirect
7. C2×#88448C2semidirect
8. C4×#6200C2direct (N × H)
16. C2×C4×#272C22semidirect
21. C2×C4×(C3⋊D3)⋊C3C22semidirect
22. C2×#6200C4direct (N × H)
24. C4×(C3⋊Dic3)⋊C3C22semidirect
25. #88451C22semidirect
26. #88450C22semidirect
27. #88448C22semidirect
28. C22×C4×C3⋊D3D3semidirect
35. C2×C4×C3⋊Dic3D3semidirect
48. C22×#272C2×C4semidirect
52. C22×(C3⋊D3)⋊C3C2×C4semidirect
54. C2×C4×H(3,𝔽3)D4semidirect
61. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
66. #6200C2×C4direct (N × H)
82. C2×C4×C3⋊D3C2×D3semidirect
101. C4×C3⋊Dic3C2×D3semidirect
116. C2×#272C22×C4semidirect
120. C2×(C3⋊D3)⋊C3C22×C4semidirect
121. C4×H(3,𝔽3)C2×D4semidirect
125. (C3⋊Dic3)⋊C3C22×C4semidirect
156. C22×C3⋊D3C4×D3semidirect
176. C2×C32×C4D12semidirect
177. C2×C32×C4#67semidirect
182. C2×C3⋊Dic3C4×D3semidirect
206. C2×H(3,𝔽3)C4×D4semidirect
283. C2×C3⋊D3C2×C4×D3semidirect
301. C32×C4C2×D12semidirect
302. C32×C4C2×#67semidirect
305. C3⋊Dic3C2×C4×D3semidirect
320. H(3,𝔽3)C2×C4×D4semidirect
401. C2×C32C4×D12semidirect
402. C2×C32C4×#67semidirect
493. C32C2×C4×D12semidirect
494. C32C2×C4×#67semidirect
515. C2×C4#6200direct (N × H)
556. C4C2×#6200direct (N × H)
563. C2C4×#6200direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C22×C4×(C3⋊D3)⋊C32
C22×#62002
C2×C4×(C3⋊Dic3)⋊C32
C2×#884512
C2×#884502
C2×#884482
C4×#62002
C2×C4×#6263
C2×C4×#6253