Finite Group Databasebeta

← C2×#13071625 #14913294

Subgroups — C2×#13071625 #14913294

Order 2816.

3614 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic11C211yesmaximal, characteristic, complemented
2C23×C4⋊Dic11C211yesmaximal, characteristic
3#13071625C2116yesmaximal, direct product factor
4C2×#10622843C211yesmaximal, characteristic, complemented
5C2×#10622598C211yesmaximal, characteristic, complemented
6C2×#10622598C212yesmaximal, complemented
7C2×#10622594C212yesmaximal, complemented
8C2×#10622563C212yesmaximal
9C2×C4⋊C4×Dic11C211yesmaximal, characteristic, complemented
10C2×C22⋊C4×Dic11C212yesmaximal
11C2×C4×C22⋊Dic11C211yesmaximal, characteristic
12C22×C11×C22⋊Q8C211yesmaximal, characteristic
13C23×Q8×C11C2211yescharacteristic
14C24×C4×C11C2211yescharacteristic
15C24×Dic11C2212yes—
16C2×C11×C22⋊Q8C4116yescomplemented
17C2×C11×C22⋊Q8C2218yes—
18C22×C11×C4⋊C4C2211yescharacteristic
19C22×C11×C4⋊C4C2212yes—
20C22×C11×C22⋊C4C2212yes—
21C22×C22⋊Dic11C2211yescharacteristic
22C22×C22⋊Dic11C2211yescharacteristic
23C2×Q8×Dic11—28no—
24C2×Q8×Dic11C2218yescomplemented
25C22×C4⋊Dic11—28no—
26C22×C4⋊Dic11C2218yes—
27C22×C4⋊Dic11C2212yes—
28C22×C4⋊Dic11C2211yescharacteristic
29C22×C4⋊Dic11C2212yes—
30C22×C4⋊Dic11C2212yes—
31C22×C4⋊Dic11C2211yescharacteristic
32C22×C4×Dic11C2211yescharacteristic
33C22×C4×Dic11C2211yescharacteristic
34C22×C4×Dic11C2212yes—
35C22×C4×Dic11C2212yes—
36#10622843C2218yescomplemented
37C4×C22⋊Dic11C2218yes—
38C2×C11⋊#95C2211yescharacteristic
39C2×C11⋊#95C2211yescharacteristic
40C2×C11⋊#95C2212yes—
41C2×C11⋊#95C2212yes—
42C2×C11⋊#95C2212yes—
43C2×C11⋊#95C2212yes—
44#10622598C22116yescomplemented
45#10622598C2218yescomplemented
46#10622594C22116yescomplemented
47C4⋊C4×Dic11C2218yescomplemented
48#10622563C22116yes—
49C22⋊C4×Dic11C22116yes—
50C25×C11C2311yescharacteristic
51C22×Q8×C11—24no—
52C22×Q8×C11—24no—
53C22×Q8×C11C2×C418yescomplemented
54C22×Q8×C11C2314yes—
55C23×C4×C11—24no—
56C23×C4×C11—24no—
57C23×C4×C11C2×C418yes—
58C23×C4×C11C2312yes—
59C23×C4×C11C2312yes—
60C23×C4×C11C2311yescharacteristic
61C23×C4×C11C2314yes—
62C23×C4×C11C2311yescharacteristic
63C23×Dic11—28no—
64C23×Dic11—28no—
65C23×Dic11—44no—
66C23×Dic11C2311yescharacteristic
67C23×Dic11C2311yescharacteristic
68C23×Dic11C2312yes—
69C23×Dic11C2318yes—
70C23×Dic11C2312yes—
71C23×Dic11C2311yescharacteristic
72C23×Dic11C2311yescharacteristic
73C11×C22⋊Q8C2×C4164yescomplemented
74C2×C11×C4⋊C4C2×C418yescomplemented
75C2×C11×C4⋊C4C2×C4116yescomplemented
76C2×C11×C4⋊C4C2314yes—
77C2×C11×C4⋊C4C2318yes—
78C2×C11×C22⋊C4C2×C4116yes—
79C2×C11×C22⋊C4C2318yes—
80C2×C22⋊Dic11—24no—
81C2×C22⋊Dic11Q818yescomplemented
82C2×C22⋊Dic11C2314yes—
83C2×C22⋊Dic11C2314yes—
84Q8×Dic11—232no—
85C2×C4⋊Dic11—24no—
86C2×C4⋊Dic11—28no—
87C2×C4⋊Dic11—28no—
88C2×C4⋊Dic11—232no—
89C2×C4⋊Dic11—44no—
90C2×C4⋊Dic11D418yes—
91C2×C4⋊Dic11C2314yes—
92C2×C4⋊Dic11C2318yes—
93C2×C4⋊Dic11C2318yes—
94C2×C4⋊Dic11C2318yes—
95C2×C4⋊Dic11C2314yes—
96C2×C4×Dic11—24no—
97C2×C4×Dic11—24no—
98C2×C4×Dic11—28no—
99C2×C4×Dic11—44no—
100C2×C4×Dic11C2318yes—
101C2×C4×Dic11C2314yes—
102C2×C4×Dic11C2314yes—
103C2×C4×Dic11C2318yes—
104C11⋊#95C2318yes—
105C11⋊#95C2318yes—
106C11⋊#95C2318yes—
107C11⋊#95C2318yes—
108C11⋊#95C2314yes—
109C11⋊#95C2314yes—
110C2×#2360—111nomaximal, Sylow
111C24×C11—22no—
112C24×C11—22no—
113C24×C11—22no—
114C24×C11—22no—
115C24×C11C22×C414yes—
116C24×C11C2×Q814yes—
117C24×C11#4114yes—
118C24×C11C2412yes—
119C24×C11C2411yescharacteristic
120C2×Q8×C11—232no—
121C2×Q8×C11—28no—
122C2×Q8×C11—28no—
123C2×Q8×C11C22×C4116yescomplemented
124C22×C4×C11—24no—
125C22×C4×C11—22no—
126C22×C4×C11—22no—
127C22×C4×C11—24no—
128C22×C4×C11—24no—
129C22×C4×C11—24no—
130C22×C4×C11—232no—
131C22×C4×C11—28no—
132C22×C4×C11—28no—
133C22×C4×C11C22×C418yes—
134C22×C4×C11C22×C418yes—
135C22×C4×C11C22×C414yes—
136C22×C4×C11C22×C4116yes—
137C22×C4×C11C22×C414yes—
138C22×C4×C11C2×D414yes—
139C22×C4×C11#4114yes—
140C22×C4×C11C2414yes—
141C22×C4×C11C2414yes—
142C22×C4×C11C2412yes—
143C22×C4×C11C2412yes—
144C22×Dic11—216no—
145C22×Dic11—22no—
146C22×Dic11—22no—
147C22×Dic11—24no—
148C22×Dic11—22no—
149C22×Dic11—24no—
150C22×Dic11—22no—
151C22×Dic11—22no—
152C22×Dic11—22no—
153C22×Dic11—216no—
154C22×Dic11—24no—
155C22×Dic11—22no—
156C22×Dic11—24no—
157C22×Dic11—22no—
158C22×Dic11—24no—
159C22×Dic11—22no—
160C22×Dic11—22no—
161C22×Dic11—48no—
162C22×Dic11—432no—
163C22×Dic11C2×D418yes—
164C22×Dic11C2×Q814yes—
165C22×Dic11C2×Q814yes—
166C22×Dic11C2412yes—
167C22×Dic11C2414yes—
168C22×Dic11C2412yes—
169C22×Dic11C2414yes—
170C22×Dic11C2412yes—
171C22×Dic11C2412yes—
172C11×C4⋊C4C22×C4116yescomplemented
173C11×C4⋊C4C22×C4132yescomplemented
174C11×C22⋊C4C22×C4132yes—
175C22⋊Dic11—28no—
176C22⋊Dic11C2×Q8116yescomplemented
177C4⋊Dic11—216no—
178C4⋊Dic11—216no—
179C4⋊Dic11—28no—
180C4⋊Dic11—48no—
181C4⋊Dic11C2×D4116yes—
182C4×Dic11—216no—
183C4×Dic11—28no—
184C4×Dic11—28no—
185C4×Dic11—48no—
186C22×C22⋊Q8D1111yescharacteristic
187C22×C4×Q8—111no—
188C23×C4⋊C4—111no—
189#2360—1116no—
190C2×#391—111no—
191C2×#389—112no—
192C2×#389—111no—
193C2×#387—112no—
194C2×#385—112no—
195C2×C4×C4⋊C4—111no—
196C2×C4×C22⋊C4—111no—
197C2×C4×C22⋊C4—112no—
198C23×C11—28no—
199C23×C11—28no—
200C23×C11—28no—
201C23×C11—28no—
202C23×C11—28no—
203C23×C11—28no—
204C23×C11—22no—
205C23×C11—22no—
206C23×C11—22no—
207C23×C11—22no—
208C23×C11C4×Q8116yescomplemented
209C23×C11C23×C414yes—
210C23×C11C23×C412yes—
211C23×C11C22×D412yes—
212C23×C11C22×Q814yes—
213C23×C11C22×Q812yes—
214C23×C11C2×#4112yes—
215C23×C11C2×#4114yes—
216C23×C11C2×#4112yes—
217C23×C11H(5,𝔽2)12yes—
218C23×C112−1+412yes—
219C23×C11C2511yescharacteristic
220Q8×C11—232no—
221C2×C4×C11—28no—
222C2×C4×C11—216no—
223C2×C4×C11—216no—
224C2×C4×C11—24no—
225C2×C4×C11—22no—
226C2×C4×C11—22no—
227C2×C4×C11—24no—
228C2×C4×C11—24no—
229C2×C4×C11—24no—
230C2×C4×C11—232no—
231C2×C4×C11C4×D4116yes—
232C2×C4×C11C23×C418yes—
233C2×C4×C11C23×C418yes—
234C2×C4×C11C23×C414yes—
235C2×C4×C11C23×C414yes—
236C2×C4×C11C22×D414yes—
237C2×C4×C11C2×#4114yes—
238C2×Dic11—216no—
239C2×Dic11—28no—
240C2×Dic11—28no—
241C2×Dic11—22no—
242C2×Dic11—22no—
243C2×Dic11—24no—
244C2×Dic11—22no—
245C2×Dic11—24no—
246C2×Dic11—22no—
247C2×Dic11—22no—
248C2×Dic11—24no—
249C2×Dic11—22no—
250C2×Dic11—22no—
251C2×Dic11—22no—
252C2×Dic11—24no—
253C2×Dic11—22no—
254C2×Dic11—24no—
255C2×Dic11—22no—
256C2×Dic11—432no—
257C2×Dic11—44no—
258C2×Dic11—48no—
259C2×Dic11—44no—
260C2×Dic11C22×D418yes—
261C2×Dic11C22×Q814yes—
262C2×Dic11C22×Q814yes—
263C23×Q8C2×D1111yescharacteristic
264C24×C4—112no—
265C24×C4C2×D1111yescharacteristic
266C2×C22⋊Q8Dic11116yescomplemented
267C2×C22⋊Q8C2×D1118yes—
268C2×C4×Q8—118no—
269C2×C4×Q8—228no—
270C22×C4⋊C4—111no—
271C22×C4⋊C4—118no—
272C22×C4⋊C4—112no—
273C22×C4⋊C4—111no—
274C22×C4⋊C4—112no—
275C22×C4⋊C4—112no—
276C22×C4⋊C4—228no—
277C22×C4⋊C4C2×D1111yescharacteristic
278C22×C4⋊C4C2×D1112yes—
279C22×C22⋊C4—111no—
280C22×C22⋊C4—111no—
281C22×C22⋊C4C2×D1112yes—
282C22×C42—112no—
283C22×C42—111no—
284C22×C42—111no—
285C22×C42—112no—
286#391—118no—
287#389—118no—
288#389—1116no—
289#387—1116no—
290#385—1116no—
291C4×C4⋊C4—118no—
292C4×C22⋊C4—118no—
293C4×C22⋊C4—1116no—
294C2×#95—111no—
295C2×#95—112no—
296C2×#95—111no—
297C2×#95—112no—
298C2×#95—112no—
299C2×#95—112no—
300C22×C11—232no—
301C22×C11—24no—
302C22×C11—24no—
303C22×C11—24no—
304C22×C11—24no—
305C22×C11—24no—
306C22×C11—24no—
307C22×C11C2×C4×D414yes—
308C22×C11C2×C4×Q814yes—
309C22×C11C2×C4×Q818yescomplemented
310C22×C11#52014yes—
311C22×C11#54614yes—
312C22×C11D4×Q814yes—
313C22×C11#55214yes—
314C22×C11#55414yes—
315C22×C11C24×C411yescharacteristic
316C22×C11C23×D411yescharacteristic
317C22×C11C23×Q811yescharacteristic
318C22×C11C22×#4111yescharacteristic
319C22×C11C22×#4111yescharacteristic
320C22×C11C2×H(5,𝔽2)11yescharacteristic
321C22×C11C2×2−1+411yescharacteristic
322C4×C11—24no—
323C4×C11—28no—
324C4×C11—28no—
325C4×C11C2×C4×D418yes—
326Dic11—28no—
327Dic11—24no—
328Dic11—24no—
329Dic11—42no—
330Dic11—44no—
331Dic11—42no—
332C25C22×D1111yescharacteristic
333C22×Q8—24no—
334C22×Q8—24no—
335C22×Q8C2×Dic1118yescomplemented
336C22×Q8C22×D1114yes—
337C23×C4—24no—
338C23×C4—24no—
339C23×C4—111no—
340C23×C4—111no—
341C23×C4—112no—
342C23×C4—111no—
343C23×C4—111no—
344C23×C4—112no—
345C23×C4—118no—
346C23×C4—228no—
347C23×C4—228no—
348C23×C4—444no—
349C23×C4C2×Dic1118yes—
350C23×C4C22×D1114yes—
351C23×C4C22×D1112yes—
352C23×C4C22×D1112yes—
353C23×C4C22×D1111yescharacteristic
354C23×C4C22×D1111yescharacteristic
355C22⋊Q8C2×Dic11164yescomplemented
356C4×Q8—2232no—
357C2×C4⋊C4—118no—
358C2×C4⋊C4—114no—
359C2×C4⋊C4—118no—
360C2×C4⋊C4—114no—
361C2×C4⋊C4—118no—
362C2×C4⋊C4—118no—
363C2×C4⋊C4—2232no—
364C2×C4⋊C4—228no—
365C2×C4⋊C4—228no—
366C2×C4⋊C4—224no—
367C2×C4⋊C4—444no—
368C2×C4⋊C4C2×Dic1118yescomplemented
369C2×C4⋊C4C2×Dic11116yescomplemented
370C2×C4⋊C4C22×D1114yes—
371C2×C4⋊C4C22×D1118yes—
372C2×C22⋊C4—118no—
373C2×C22⋊C4—114no—
374C2×C22⋊C4—114no—
375C2×C22⋊C4—224no—
376C2×C22⋊C4C2×Dic11116yes—
377C2×C22⋊C4C22×D1118yes—
378C2×C42—118no—
379C2×C42—114no—
380C2×C42—114no—
381C2×C42—118no—
382C2×C42—228no—
383C2×C42—224no—
384C2×C42—224no—
385C2×C42—444no—
386#95—114no—
387#95—118no—
388#95—114no—
389#95—118no—
390#95—118no—
391#95—118no—
392C2×C11—28no—
393C2×C11#236018yescomplemented
394C2×C11C22×C4×D411yescharacteristic
395C2×C11C22×C4×Q811yescharacteristic
396C2×C11C2×#52011yescharacteristic
397C2×C11C2×#54611yescharacteristic
398C2×C11C2×D4×Q811yescharacteristic
399C2×C11C2×#55211yescharacteristic
400C2×C11C2×#55411yescharacteristic
401C24—22no—
402C24—22no—
403C24—22no—
404C24—22no—
405C24Q8×D1114yes—
406C24#434314yes—
407C24C22×Dic1114yes—
408C24C23×D1112yes—
409C24C23×D1111yescharacteristic
410C2×Q8—232no—
411C2×Q8—28no—
412C2×Q8—28no—
413C2×Q8C22×Dic11116yescomplemented
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—232no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—118no—
424C22×C4—114no—
425C22×C4—114no—
426C22×C4—112no—
427C22×C4—112no—
428C22×C4—114no—
429C22×C4—112no—
430C22×C4—114no—
431C22×C4—112no—
432C22×C4—2216no—
433C22×C4—222no—
434C22×C4—222no—
435C22×C4—224no—
436C22×C4—222no—
437C22×C4—224no—
438C22×C4—222no—
439C22×C4—222no—
440C22×C4—224no—
441C22×C4—222no—
442C22×C4—222no—
443C22×C4—222no—
444C22×C4—2216no—
445C22×C4—224no—
446C22×C4—222no—
447C22×C4—224no—
448C22×C4—222no—
449C22×C4—448no—
450C22×C4—4432no—
451C22×C4D4×D1114yes—
452C22×C4#434114yes—
453C22×C4C22×Dic1118yes—
454C22×C4C22×Dic1118yes—
455C22×C4C22×Dic1114yes—
456C22×C4C22×Dic11116yes—
457C22×C4C22×Dic1114yes—
458C22×C4C23×D1114yes—
459C22×C4C23×D1114yes—
460C22×C4C23×D1112yes—
461C22×C4C23×D1112yes—
462C4⋊C4—1116no—
463C4⋊C4—2216no—
464C4⋊C4—2216no—
465C4⋊C4—228no—
466C4⋊C4—448no—
467C4⋊C4C22×Dic11116yescomplemented
468C4⋊C4C22×Dic11132yescomplemented
469C22⋊C4—1116no—
470C22⋊C4—228no—
471C22⋊C4C22×Dic11132yes—
472C42—2216no—
473C42—228no—
474C42—228no—
475C42—448no—
476C11C2×#236011yescharacteristic, complemented, Sylow
477C23—28no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23Q8×Dic11116yescomplemented
488C23C2×D4×D1112yes—
489C23C2×#434112yes—
490C23C2×Q8×D1114yes—
491C23C2×Q8×D1112yes—
492C23C2×#434314yes—
493C23C2×#434312yes—
494C23#6704212yes—
495C23#6704312yes—
496C23C23×Dic1114yes—
497C23C23×Dic1112yes—
498C23C24×D1111yescharacteristic
499Q8—232no—
500C2×C4—28no—
501C2×C4—216no—
502C2×C4—216no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—232no—
510C2×C4—118no—
511C2×C4—114no—
512C2×C4—114no—
513C2×C4—2216no—
514C2×C4—228no—
515C2×C4—228no—
516C2×C4—222no—
517C2×C4—222no—
518C2×C4—224no—
519C2×C4—222no—
520C2×C4—224no—
521C2×C4—222no—
522C2×C4—222no—
523C2×C4—224no—
524C2×C4—222no—
525C2×C4—222no—
526C2×C4—222no—
527C2×C4—224no—
528C2×C4—222no—
529C2×C4—224no—
530C2×C4—222no—
531C2×C4—4432no—
532C2×C4—448no—
533C2×C4—444no—
534C2×C4—444no—
535C2×C4D4×Dic11116yes—
536C2×C4C2×D4×D1114yes—
537C2×C4C2×#434114yes—
538C2×C4C23×Dic1118yes—
539C2×C4C23×Dic1118yes—
540C2×C4C23×Dic1114yes—
541C2×C4C23×Dic1114yes—
542C22—232no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—24no—
549C22#1062316114yes—
550C22#1062316314yes—
551C22#1062316614yes—
552C22#1062317314yes—
553C22C2×D4×Dic1114yes—
554C22C2×Q8×Dic1114yes—
555C22C2×Q8×Dic1118yescomplemented
556C22#1062335814yes—
557C22C22×D4×D1111yescharacteristic
558C22C22×#434111yescharacteristic
559C22C22×Q8×D1111yescharacteristic
560C22C22×#434311yescharacteristic
561C22C2×#6704211yescharacteristic
562C22C2×#6704311yescharacteristic
563C22C24×Dic1111yescharacteristic
564C4—24no—
565C4—28no—
566C4—28no—
567C4—224no—
568C4—228no—
569C4—224no—
570C4—442no—
571C4—444no—
572C4—442no—
573C4C2×D4×Dic1118yes—
574C2—28no—
575C2#1307162518yesdirect product factor
576C2C22×D4×Dic1111yescharacteristic
577C2C22×Q8×Dic1111yescharacteristic
578C2C2×#1062335811yescharacteristic
579C2C2×#1062316111yescharacteristic
580C2C2×#1062316311yescharacteristic
581C2C2×#1062317311yescharacteristic
582C2C2×#1062316611yescharacteristic
subgroupG/H
centerC24C23×D11
commutatorC22×C11C24×C4
FrattiniC23C24×D11
FittingC22×C11×C22⋊Q8C2
socleC24×C11C24
prime radicalC25×C11C23
cosocleC23×C11C25
hypercenterC22×C22⋊Q8D11
hypocenterC11C2×#2360
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C24 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×Q8×Dic11C2semidirect
3. #13071625C2direct (N × H)
4. C2×#10622843C2semidirect
5. C2×#10622598C2semidirect
6. C2×#10622598C2semidirect
7. C2×#10622594C2semidirect
9. C2×C4⋊C4×Dic11C2semidirect
16. C2×C11×C22⋊Q8C4semidirect
24. C2×Q8×Dic11C22semidirect
36. #10622843C22semidirect
44. #10622598C22semidirect
45. #10622598C22semidirect
46. #10622594C22semidirect
47. C4⋊C4×Dic11C22semidirect
53. C22×Q8×C11C2×C4semidirect
73. C11×C22⋊Q8C2×C4semidirect
74. C2×C11×C4⋊C4C2×C4semidirect
75. C2×C11×C4⋊C4C2×C4semidirect
81. C2×C22⋊Dic11Q8semidirect
123. C2×Q8×C11C22×C4semidirect
172. C11×C4⋊C4C22×C4semidirect
173. C11×C4⋊C4C22×C4semidirect
176. C22⋊Dic11C2×Q8semidirect
208. C23×C11C4×Q8semidirect
266. C2×C22⋊Q8Dic11semidirect
309. C22×C11C2×C4×Q8semidirect
335. C22×Q8C2×Dic11semidirect
355. C22⋊Q8C2×Dic11semidirect
368. C2×C4⋊C4C2×Dic11semidirect
369. C2×C4⋊C4C2×Dic11semidirect
393. C2×C11#2360semidirect
413. C2×Q8C22×Dic11semidirect
467. C4⋊C4C22×Dic11semidirect
468. C4⋊C4C22×Dic11semidirect
476. C11C2×#2360semidirect
487. C23Q8×Dic11semidirect
555. C22C2×Q8×Dic11semidirect
575. C2#13071625direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×Dic112
C23×C4⋊Dic112
#130716252
C2×#106228432
C2×#106225982
C2×#106225942
C2×#106225632
C2×C4⋊C4×Dic112
C2×C22⋊C4×Dic112
C2×C4×C22⋊Dic112
C22×C11×C22⋊Q82
C2×#236011