Finite Group Databasebeta

← C2×C4×#66977 #14912223

Subgroups — C2×C4×#66977 #14912223

Order 2816.

3854 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66977C211yesmaximal, characteristic
2C22×C4×Dic22C211yesmaximal, characteristic, complemented
3C4×#66977C2116yesmaximal, direct product factor
4C2×#10622542C211yesmaximal, characteristic, complemented
5C2×#10622541C211yesmaximal, characteristic, complemented
6C2×#10622546C212yesmaximal, complemented
7C2×#10622823C212yesmaximal
8C2×#10622825C211yesmaximal, characteristic
9C2×C4×Dic11⋊C4C212yesmaximal, complemented
10C2×C4×C4⋊Dic11C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊Dic11C212yesmaximal
12C23×C42×C11C211yesmaximal, characteristic
13C24×C4×C11C2211yescharacteristic
14C24×C4×C11C2211yescharacteristic
15C24×C4×C11C2211yescharacteristic
16C23×Dic22C2211yescharacteristic
17C22×C42×C11—28no—
18C22×C42×C11C2218yes—
19C22×C42×C11C2211yescharacteristic
20C22×C42×C11C2211yescharacteristic
21C22×C42×C11C2211yescharacteristic
22C22×C42×C11C2211yescharacteristic
23C22×C22⋊Dic11C2212yes—
24C22×C22⋊Dic11C2212yes—
25C22×C4⋊Dic11C2211yescharacteristic
26C22×C4⋊Dic11C2211yescharacteristic
27C22×C4⋊Dic11C2211yescharacteristic
28C2×#66977C4116yesdirect product factor
29C2×#66977C2218yes—
30C22×Dic11⋊C4C2212yes—
31C22×Dic11⋊C4C2212yes—
32C22×Dic11⋊C4C2212yes—
33C22×C4×Dic11C2212yes—
34C22×C4×Dic11C2212yes—
35C2×C4×Dic22—28no—
36C2×C4×Dic22C2218yescomplemented
37#10622825C2218yes—
38#10622823C22116yes—
39C4×C22⋊Dic11C22116yes—
40C2×C11⋊#95C2212yes—
41C2×C11⋊#95C2212yes—
42C2×C11⋊#95C2212yes—
43C2×C11⋊#95C2212yes—
44C2×C11⋊#95C2212yes—
45C4×C4⋊Dic11C2218yescomplemented
46#10622546C22116yescomplemented
47C4×Dic11⋊C4C22116yescomplemented
48#10622542C2218yescomplemented
49#10622541C2218yescomplemented
50C25×C11C2311yescharacteristic
51C23×C4×C11—24no—
52C23×C4×C11—24no—
53C23×C4×C11—24no—
54C23×C4×C11—24no—
55C23×C4×C11—24no—
56C23×C4×C11—24no—
57C23×C4×C11—24no—
58C23×C4×C11C2×C418yes—
59C23×C4×C11Q818yescomplemented
60C23×C4×C11C2314yes—
61C23×C4×C11C2311yescharacteristic
62C23×C4×C11C2314yes—
63C23×C4×C11C2311yescharacteristic
64C23×C4×C11C2311yescharacteristic
65C23×C4×C11C2314yes—
66C23×C4×C11C2311yescharacteristic
67C23×C4×C11C2311yescharacteristic
68C23×C4×C11C2311yescharacteristic
69C23×Dic11C2312yes—
70C23×Dic11C2312yes—
71C23×Dic11C2312yes—
72C23×Dic11C2312yes—
73C22×Dic22—24no—
74C22×Dic22—24no—
75C22×Dic22C2×C418yescomplemented
76C22×Dic22C2314yes—
77C2×C42×C11—24no—
78C2×C42×C11—24no—
79C2×C42×C11—24no—
80C2×C42×C11—232no—
81C2×C42×C11D418yes—
82C2×C42×C11C2314yes—
83C2×C42×C11C2314yes—
84C2×C42×C11C2314yes—
85C2×C42×C11C2314yes—
86C2×C22⋊Dic11—28no—
87C2×C22⋊Dic11C2×C4116yes—
88C2×C22⋊Dic11C2318yes—
89C2×C22⋊Dic11C2318yes—
90C2×C4⋊Dic11—24no—
91C2×C4⋊Dic11—24no—
92C2×C4⋊Dic11C2×C418yescomplemented
93C2×C4⋊Dic11C2314yes—
94C2×C4⋊Dic11C2314yes—
95C2×C4⋊Dic11C2314yes—
96#66977C2×C4164yesdirect product factor
97C2×Dic11⋊C4—28no—
98C2×Dic11⋊C4—28no—
99C2×Dic11⋊C4C2×C4116yescomplemented
100C2×Dic11⋊C4C2318yes—
101C2×Dic11⋊C4C2318yes—
102C2×Dic11⋊C4C2318yes—
103C2×C4×Dic11—28no—
104C2×C4×Dic11—28no—
105C2×C4×Dic11C2318yes—
106C2×C4×Dic11C2318yes—
107C4×Dic22—232no—
108C11⋊#95C2318yes—
109C11⋊#95C2318yes—
110C11⋊#95C2318yes—
111C11⋊#95C2318yes—
112C11⋊#95C2318yes—
113C2×C4×C22⋊Q8—111nomaximal, Sylow
114C24×C11—22no—
115C24×C11—22no—
116C24×C11—22no—
117C24×C11—22no—
118C24×C11C22×C414yes—
119C24×C11C2×Q814yes—
120C24×C11#4114yes—
121C24×C11C2412yes—
122C24×C11C2411yescharacteristic
123C22×C4×C11—232no—
124C22×C4×C11—22no—
125C22×C4×C11—22no—
126C22×C4×C11—22no—
127C22×C4×C11—28no—
128C22×C4×C11—22no—
129C22×C4×C11—22no—
130C22×C4×C11—22no—
131C22×C4×C11—22no—
132C22×C4×C11—22no—
133C22×C4×C11—232no—
134C22×C4×C11—232no—
135C22×C4×C11—28no—
136C22×C4×C11—28no—
137C22×C4×C11—28no—
138C22×C4×C11—28no—
139C22×C4×C11—28no—
140C22×C4×C11—28no—
141C22×C4×C11C22×C414yes—
142C22×C4×C11C22×C4116yes—
143C22×C4×C11C22×C414yes—
144C22×C4×C11C2×D414yes—
145C22×C4×C11C2×D414yes—
146C22×C4×C11C2×D414yes—
147C22×C4×C11C2×Q814yes—
148C22×C4×C11C2×Q814yes—
149C22×C4×C11C2×Q8116yescomplemented
150C22×C4×C11#4114yes—
151C22×C4×C11#4114yes—
152C22×C4×C11#4114yes—
153C22×C4×C11C2412yes—
154C22×C4×C11C2412yes—
155C22×C4×C11C2412yes—
156C22×C4×C11C2412yes—
157C22×C4×C11C2412yes—
158C22×C4×C11C2412yes—
159C22×Dic11—24no—
160C22×Dic11—24no—
161C22×Dic11—24no—
162C22×Dic11—24no—
163C22×Dic11—24no—
164C22×Dic11—24no—
165C22×Dic11—24no—
166C22×Dic11—24no—
167C22×Dic11—24no—
168C22×Dic11—24no—
169C22×Dic11C22×C418yes—
170C22×Dic11C22×C418yes—
171C22×Dic11C2414yes—
172C22×Dic11C2414yes—
173C22×Dic11C2414yes—
174C22×Dic11C2414yes—
175C2×Dic22—232no—
176C2×Dic22—28no—
177C2×Dic22—28no—
178C2×Dic22C22×C4116yescomplemented
179C42×C11—28no—
180C42×C11—28no—
181C42×C11—28no—
182C42×C11C2×D4116yes—
183C22⋊Dic11—216no—
184C22⋊Dic11C22×C4132yes—
185C4⋊Dic11—28no—
186C4⋊Dic11—28no—
187C4⋊Dic11C22×C4116yescomplemented
188Dic11⋊C4—216no—
189Dic11⋊C4—216no—
190Dic11⋊C4C22×C4132yescomplemented
191C4×Dic11—216no—
192C4×Dic11—216no—
193C22×C22⋊Q8—111no—
194C22×C4×Q8—111no—
195C23×C42D1111yescharacteristic
196C4×C22⋊Q8—1116no—
197C2×#391—111no—
198C2×#389—111no—
199C2×#387—112no—
200C2×#385—112no—
201C2×#380—111no—
202C2×C4×C4⋊C4—111no—
203C2×C4×C4⋊C4—112no—
204C2×C4×C22⋊C4—112no—
205C23×C11—28no—
206C23×C11—28no—
207C23×C11—28no—
208C23×C11—28no—
209C23×C11—28no—
210C23×C11—28no—
211C23×C11—22no—
212C23×C11—22no—
213C23×C11—22no—
214C23×C11—22no—
215C23×C11C4×Q8116yescomplemented
216C23×C11C23×C414yes—
217C23×C11C23×C412yes—
218C23×C11C22×D412yes—
219C23×C11C22×Q814yes—
220C23×C11C22×Q812yes—
221C23×C11C2×#4114yes—
222C23×C11C2×#4112yes—
223C23×C11C2×#4112yes—
224C23×C11C2×#4112yes—
225C23×C11C2×#4112yes—
226C23×C11C2511yescharacteristic
227C2×C4×C11—232no—
228C2×C4×C11—28no—
229C2×C4×C11—28no—
230C2×C4×C11—28no—
231C2×C4×C11—28no—
232C2×C4×C11—22no—
233C2×C4×C11—22no—
234C2×C4×C11—22no—
235C2×C4×C11—22no—
236C2×C4×C11—22no—
237C2×C4×C11—22no—
238C2×C4×C11—22no—
239C2×C4×C11—22no—
240C2×C4×C11—232no—
241C2×C4×C11—232no—
242C2×C4×C11C4×D4116yes—
243C2×C4×C11C22⋊Q8116yescomplemented
244C2×C4×C11C23×C414yes—
245C2×C4×C11C23×C414yes—
246C2×C4×C11C22×D414yes—
247C2×C4×C11C22×D414yes—
248C2×C4×C11C22×D414yes—
249C2×C4×C11C22×Q814yes—
250C2×C4×C11C22×Q814yes—
251C2×C4×C11C2×#4114yes—
252C2×C4×C11C2×#4114yes—
253C2×C4×C11C2×#4114yes—
254C2×Dic11—216no—
255C2×Dic11—216no—
256C2×Dic11—24no—
257C2×Dic11—24no—
258C2×Dic11—24no—
259C2×Dic11—24no—
260C2×Dic11—24no—
261C2×Dic11—24no—
262C2×Dic11—24no—
263C2×Dic11—24no—
264C2×Dic11—24no—
265C2×Dic11—24no—
266C2×Dic11—48no—
267C2×Dic11—48no—
268C2×Dic11C23×C418yes—
269C2×Dic11C23×C418yes—
270Dic22—232no—
271C23×Q8—111no—
272C24×C4C2×D1111yescharacteristic
273C24×C4C2×D1111yescharacteristic
274C24×C4C2×D1111yescharacteristic
275C2×C22⋊Q8—118no—
276C2×C22⋊Q8—1116no—
277C2×C4×Q8—118no—
278C2×C4×Q8—228no—
279C22×C4⋊C4—112no—
280C22×C4⋊C4—111no—
281C22×C4⋊C4—112no—
282C22×C4⋊C4—112no—
283C22×C4⋊C4—111no—
284C22×C4⋊C4—111no—
285C22×C22⋊C4—112no—
286C22×C22⋊C4—112no—
287C22×C42—28no—
288C22×C42—112no—
289C22×C42—112no—
290C22×C42C2×D1111yescharacteristic
291C22×C42C2×D1111yescharacteristic
292C22×C42C2×D1118yes—
293C22×C42C2×D1111yescharacteristic
294C22×C42C2×D1111yescharacteristic
295#391—118no—
296#389—118no—
297#387—1116no—
298#385—1116no—
299#380—118no—
300C4×C4⋊C4—1116no—
301C4×C4⋊C4—118no—
302C4×C22⋊C4—1116no—
303C2×#95—112no—
304C2×#95—112no—
305C2×#95—112no—
306C2×#95—112no—
307C2×#95—112no—
308C22×C11—24no—
309C22×C11—24no—
310C22×C11—24no—
311C22×C11—24no—
312C22×C11—24no—
313C22×C11—24no—
314C22×C11—232no—
315C22×C11C2×C4×D414yes—
316C22×C11C2×C4×Q814yes—
317C22×C11C2×C4×Q818yescomplemented
318C22×C11C4×#4114yes—
319C22×C11C2×C22⋊Q814yes—
320C22×C11#52514yes—
321C22×C11#52914yes—
322C22×C11#53314yes—
323C22×C11C24×C411yescharacteristic
324C22×C11C23×D411yescharacteristic
325C22×C11C23×Q811yescharacteristic
326C22×C11C22×#4111yescharacteristic
327C22×C11C22×#4111yescharacteristic
328C22×C11C22×#4111yescharacteristic
329C22×C11C22×#4111yescharacteristic
330C4×C11—24no—
331C4×C11—24no—
332C4×C11—24no—
333C4×C11—24no—
334C4×C11C2×C4×D418yes—
335C4×C11C2×C22⋊Q818yescomplemented
336Dic11—28no—
337Dic11—28no—
338Dic11—44no—
339Dic11—44no—
340C25C22×D1111yescharacteristic
341C22×Q8—118no—
342C22×Q8—114no—
343C22×Q8—224no—
344C22×Q8—224no—
345C23×C4—24no—
346C23×C4—24no—
347C23×C4—24no—
348C23×C4—24no—
349C23×C4—24no—
350C23×C4—24no—
351C23×C4—24no—
352C23×C4—112no—
353C23×C4—112no—
354C23×C4—112no—
355C23×C4—112no—
356C23×C4Dic2218yescomplemented
357C23×C4C4×D1118yes—
358C23×C4C22×D1114yes—
359C23×C4C22×D1111yescharacteristic
360C23×C4C22×D1114yes—
361C23×C4C22×D1111yescharacteristic
362C23×C4C22×D1114yes—
363C23×C4C22×D1111yescharacteristic
364C23×C4C22×D1111yescharacteristic
365C23×C4C22×D1111yescharacteristic
366C23×C4C22×D1111yescharacteristic
367C22⋊Q8—1164no—
368C4×Q8—2232no—
369C2×C4⋊C4—1116no—
370C2×C4⋊C4—118no—
371C2×C4⋊C4—118no—
372C2×C4⋊C4—114no—
373C2×C4⋊C4—114no—
374C2×C4⋊C4—118no—
375C2×C4⋊C4—114no—
376C2×C4⋊C4—118no—
377C2×C4⋊C4—228no—
378C2×C4⋊C4—224no—
379C2×C4⋊C4—228no—
380C2×C4⋊C4—224no—
381C2×C22⋊C4—1116no—
382C2×C22⋊C4—118no—
383C2×C22⋊C4—118no—
384C2×C22⋊C4—228no—
385C2×C42—24no—
386C2×C42—24no—
387C2×C42—24no—
388C2×C42—232no—
389C2×C42—118no—
390C2×C42—118no—
391C2×C42—228no—
392C2×C42—228no—
393C2×C42C22⋊D1118yes—
394C2×C42C22×D1114yes—
395C2×C42C22×D1114yes—
396C2×C42C22×D1114yes—
397C2×C42C22×D1114yes—
398#95—118no—
399#95—118no—
400#95—118no—
401#95—118no—
402#95—118no—
403C2×C11—28no—
404C2×C11C4×C22⋊Q818yescomplemented
405C2×C11C22×C4×D411yescharacteristic
406C2×C11C22×C4×Q811yescharacteristic
407C2×C11C2×C4×#4111yescharacteristic
408C2×C11C22×C22⋊Q811yescharacteristic
409C2×C11C2×#52511yescharacteristic
410C2×C11C2×#52911yescharacteristic
411C2×C11C2×#53311yescharacteristic
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24C2×Dic2214yes—
417C24C2×C4×D1114yes—
418C24#433914yes—
419C24C23×D1112yes—
420C24C23×D1111yescharacteristic
421C2×Q8—1116no—
422C2×Q8—2232no—
423C2×Q8—228no—
424C2×Q8—228no—
425C22×C4—232no—
426C22×C4—28no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—232no—
436C22×C4—232no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—28no—
443C22×C4—118no—
444C22×C4—118no—
445C22×C4—114no—
446C22×C4—114no—
447C22×C4—114no—
448C22×C4—114no—
449C22×C4—224no—
450C22×C4—224no—
451C22×C4—224no—
452C22×C4—224no—
453C22×C4—224no—
454C22×C4—224no—
455C22×C4—224no—
456C22×C4—224no—
457C22×C4—224no—
458C22×C4—224no—
459C22×C4C2×Dic2214yes—
460C22×C4C2×Dic2214yes—
461C22×C4C2×Dic22116yescomplemented
462C22×C4C2×C4×D1114yes—
463C22×C4C2×C4×D11116yes—
464C22×C4C2×C4×D1114yes—
465C22×C4#433914yes—
466C22×C4#433914yes—
467C22×C4#433914yes—
468C22×C4C2×C22⋊D1114yes—
469C22×C4C2×C22⋊D1114yes—
470C22×C4C2×C22⋊D1114yes—
471C22×C4C23×D1112yes—
472C22×C4C23×D1112yes—
473C22×C4C23×D1112yes—
474C22×C4C23×D1112yes—
475C22×C4C23×D1112yes—
476C22×C4C23×D1112yes—
477C4⋊C4—1132no—
478C4⋊C4—1116no—
479C4⋊C4—2216no—
480C4⋊C4—228no—
481C4⋊C4—2216no—
482C4⋊C4—228no—
483C22⋊C4—1132no—
484C22⋊C4—2216no—
485C42—28no—
486C42—28no—
487C42—28no—
488C42—2216no—
489C42—2216no—
490C42C2×C22⋊D11116yes—
491C11C2×C4×C22⋊Q811yescharacteristic, complemented, Sylow
492C23—28no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23C4×Dic22116yescomplemented
503C23C22×Dic2214yes—
504C23C22×Dic2212yes—
505C23C22×C4×D1114yes—
506C23C22×C4×D1112yes—
507C23C2×#433914yes—
508C23C2×#433912yes—
509C23C2×#433912yes—
510C23C2×#433912yes—
511C23C2×#433912yes—
512C23C22×C22⋊D1112yes—
513C23C24×D1111yescharacteristic
514Q8—2232no—
515C2×C4—232no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—232no—
529C2×C4—232no—
530C2×C4—118no—
531C2×C4—118no—
532C2×C4—2216no—
533C2×C4—2216no—
534C2×C4—224no—
535C2×C4—224no—
536C2×C4—224no—
537C2×C4—224no—
538C2×C4—224no—
539C2×C4—224no—
540C2×C4—224no—
541C2×C4—224no—
542C2×C4—224no—
543C2×C4—224no—
544C2×C4—448no—
545C2×C4—448no—
546C2×C4#66977116yesdirect product factor
547C2×C4C4×C22⋊D11116yes—
548C2×C4C22×Dic2214yes—
549C2×C4C22×Dic2214yes—
550C2×C4C22×C4×D1114yes—
551C2×C4C22×C4×D1114yes—
552C2×C4C2×#433914yes—
553C2×C4C2×#433914yes—
554C2×C4C2×#433914yes—
555C2×C4C22×C22⋊D1114yes—
556C2×C4C22×C22⋊D1114yes—
557C2×C4C22×C22⋊D1114yes—
558C22—24no—
559C22—24no—
560C22—232no—
561C22—24no—
562C22—24no—
563C22—24no—
564C22—24no—
565C22C2×C4×Dic2214yes—
566C22C2×C4×Dic2218yescomplemented
567C22#1062300114yes—
568C22C4×#433914yes—
569C22#1062301014yes—
570C22C2×#6697714yes—
571C22C2×C4×C22⋊D1114yes—
572C22#1062332214yes—
573C22C23×Dic2211yescharacteristic
574C22C23×C4×D1111yescharacteristic
575C22C22×#433911yescharacteristic
576C22C22×#433911yescharacteristic
577C22C22×#433911yescharacteristic
578C22C22×#433911yescharacteristic
579C22C23×C22⋊D1111yescharacteristic
580C4—24no—
581C4—24no—
582C4—24no—
583C4—24no—
584C4—228no—
585C4—228no—
586C4—444no—
587C4—444no—
588C4C2×#6697718yesdirect product factor
589C4C2×C4×C22⋊D1118yes—
590C2—28no—
591C2C4×#6697718yesdirect product factor
592C2C22×C4×C22⋊D1111yescharacteristic
593C2C22×C4×Dic2211yescharacteristic
594C2C2×C4×#433911yescharacteristic
595C2C22×#6697711yescharacteristic
596C2C2×#1062332211yescharacteristic
597C2C2×#1062301011yescharacteristic
598C2C2×#1062300111yescharacteristic
subgroupG/H
centerC23×C4C22×D11
commutatorC22×C11C24×C4
FrattiniC23C24×D11
FittingC23×C42×C11C2
socleC24×C11C24
prime radicalC25×C11C23
cosocleC23×C11C25
hypercenterC23×C42D11
hypocenterC11C2×C4×C22⋊Q8
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C23×C4 ◁ C23×C42

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic22C2semidirect
3. C4×#66977C2direct (N × H)
4. C2×#10622542C2semidirect
5. C2×#10622541C2semidirect
6. C2×#10622546C2semidirect
9. C2×C4×Dic11⋊C4C2semidirect
10. C2×C4×C4⋊Dic11C2semidirect
28. C2×#66977C4direct (N × H)
36. C2×C4×Dic22C22semidirect
45. C4×C4⋊Dic11C22semidirect
46. #10622546C22semidirect
47. C4×Dic11⋊C4C22semidirect
48. #10622542C22semidirect
49. #10622541C22semidirect
59. C23×C4×C11Q8semidirect
75. C22×Dic22C2×C4semidirect
92. C2×C4⋊Dic11C2×C4semidirect
96. #66977C2×C4direct (N × H)
99. C2×Dic11⋊C4C2×C4semidirect
149. C22×C4×C11C2×Q8semidirect
178. C2×Dic22C22×C4semidirect
187. C4⋊Dic11C22×C4semidirect
190. Dic11⋊C4C22×C4semidirect
215. C23×C11C4×Q8semidirect
243. C2×C4×C11C22⋊Q8semidirect
317. C22×C11C2×C4×Q8semidirect
335. C4×C11C2×C22⋊Q8semidirect
356. C23×C4Dic22semidirect
404. C2×C11C4×C22⋊Q8semidirect
461. C22×C4C2×Dic22semidirect
491. C11C2×C4×C22⋊Q8semidirect
502. C23C4×Dic22semidirect
546. C2×C4#66977direct (N × H)
566. C22C2×C4×Dic22semidirect
588. C4C2×#66977direct (N × H)
591. C2C4×#66977direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#669772
C22×C4×Dic222
C4×#669772
C2×#106225422
C2×#106225412
C2×#106225462
C2×#106228232
C2×#106228252
C2×C4×Dic11⋊C42
C2×C4×C4⋊Dic112
C2×C4×C22⋊Dic112
C23×C42×C112
C2×C4×C22⋊Q811