Finite Group Databasebeta

← C2×D4×#6101 #16022201

Subgroups — C2×D4×#6101 #16022201

Order 3328.

4718 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic13C211yesmaximal, characteristic
2C22×D4×Dic13C211yesmaximal, characteristic
3C23×#6101C212yesmaximal, complemented
4D4×#6101C2116yesmaximal, direct product factor
5C2×#11718589C211yesmaximal, characteristic, complemented
6C2×#11718634C211yesmaximal, characteristic, complemented
7C2×#11718871C212yesmaximal, complemented
8C2×#11718607C212yesmaximal, complemented
9C2×#11718637C211yesmaximal, characteristic, complemented
10C2×#11718603C212yesmaximal, complemented
11C2×C4×#6101C211yesmaximal, characteristic, complemented
12C22×C4×D4×C13C211yesmaximal, characteristic
13C23×D4×C13C2211yescharacteristic
14C24×C4×C13C2212yes—
15C24×Dic13C2212yes—
16C24×Dic13C2212yes—
17C2×C4×D4×C13—28no—
18C2×C4×D4×C13C2218yes—
19C22×C13×C4⋊C4C2211yescharacteristic
20C22×C13×C22⋊C4C2212yes—
21C22×C42×C13C2211yescharacteristic
22C22×C22⋊Dic13C2212yes—
23C22×C22⋊Dic13C2212yes—
24C2×D4×Dic13—28no—
25C2×D4×Dic13C4116yescomplemented
26C2×D4×Dic13C2218yes—
27C2×D4×Dic13C2218yes—
28C22×C4⋊Dic13C2211yescharacteristic
29C22×C4⋊Dic13C2211yescharacteristic
30C22×#6101—216no—
31C22×#6101C2211yescharacteristic
32C22×#6101C2212yes—
33C22×#6101C2212yes—
34C22×#6101C2212yes—
35C22×#6101C2212yes—
36C22×#6101C2211yescharacteristic
37C22×#6101C22116yescomplemented
38C22×C4×Dic13C2211yescharacteristic
39C22×C4×Dic13C2211yescharacteristic
40#11718871C22116yescomplemented
41C2×#88071C2212yes—
42C2×#88071C2212yes—
43C2×#88071C2211yescharacteristic
44C2×#88071C2211yescharacteristic
45#11718637C2218yescomplemented
46#11718634C2218yescomplemented
47#11718607C22116yescomplemented
48#11718603C22116yescomplemented
49C4×#6101C2218yescomplemented
50#11718589C2218yescomplemented
51C25×C13C2312yes—
52C22×D4×C13C2×C418yes—
53C22×D4×C13D418yes—
54C22×D4×C13Q818yes—
55C22×D4×C13C2314yes—
56C23×C4×C13—28no—
57C23×C4×C13—28no—
58C23×C4×C13—44no—
59C23×C4×C13C2312yes—
60C23×C4×C13C2311yescharacteristic
61C23×C4×C13C2318yes—
62C23×C4×C13C2311yescharacteristic
63C23×C4×C13C2311yescharacteristic
64C23×Dic13—28no—
65C23×Dic13—28no—
66C23×Dic13—28no—
67C23×Dic13—28no—
68C23×Dic13—44no—
69C23×Dic13C2×C4116yescomplemented
70C23×Dic13C2311yescharacteristic
71C23×Dic13C2312yes—
72C23×Dic13C2318yes—
73C23×Dic13C2311yescharacteristic
74C23×Dic13C2312yes—
75C23×Dic13C2318yes—
76C23×Dic13C2311yescharacteristic
77C23×Dic13C2311yescharacteristic
78C4×D4×C13—232no—
79C2×C13×C4⋊C4—24no—
80C2×C13×C4⋊C4C2314yes—
81C2×C13×C22⋊C4—28no—
82C2×C13×C22⋊C4C2318yes—
83C2×C42×C13—24no—
84C2×C42×C13C2314yes—
85C2×C22⋊Dic13—28no—
86C2×C22⋊Dic13C2×C4116yescomplemented
87C2×C22⋊Dic13C2318yes—
88C2×C22⋊Dic13C2318yes—
89D4×Dic13—232no—
90D4×Dic13C2×C4164yescomplemented
91C2×C4⋊Dic13—24no—
92C2×C4⋊Dic13C2×C418yescomplemented
93C2×C4⋊Dic13C2314yes—
94C2×C4⋊Dic13C2314yes—
95C2×#6101—28no—
96C2×#6101—28no—
97C2×#6101—28no—
98C2×#6101—24no—
99C2×#6101—264no—
100C2×#6101—44no—
101C2×#6101D418yesdirect product factor
102C2×#6101C2318yes—
103C2×#6101C2318yes—
104C2×#6101C2318yes—
105C2×#6101C2314yes—
106C2×#6101C2318yes—
107C2×#6101C2314yes—
108C2×C4×Dic13—24no—
109C2×C4×Dic13C2×C418yescomplemented
110C2×C4×Dic13C2314yes—
111C2×C4×Dic13C2314yes—
112#88071C2318yes—
113#88071C2314yes—
114#88071C2314yes—
115#88071C2318yes—
116C2×D4×C4⋊C4—131nomaximal, Sylow
117C24×C13—24no—
118C24×C13—24no—
119C24×C13—24no—
120C24×C13—24no—
121C24×C13C22×C418yes—
122C24×C13C2×D418yes—
123C24×C13C2×Q818yes—
124C24×C13C2414yes—
125C24×C13C2411yescharacteristic
126C2×D4×C13C4⋊C4164yescomplemented
127C2×D4×C13C22×C4116yes—
128C2×D4×C13C2×D4116yes—
129C2×D4×C13C2×Q8116yes—
130C22×C4×C13—22no—
131C22×C4×C13—22no—
132C22×C4×C13—24no—
133C22×C4×C13—22no—
134C22×C4×C13—24no—
135C22×C4×C13—216no—
136C22×C4×C13—22no—
137C22×C4×C13—24no—
138C22×C4×C13—22no—
139C22×C4×C13—216no—
140C22×C4×C13—48no—
141C22×C4×C13—432no—
142C22×C4×C13C22×C414yes—
143C22×C4×C13C2×D414yes—
144C22×C4×C13C2×D414yes—
145C22×C4×C13C2×Q814yes—
146C22×C4×C13C2412yes—
147C22×C4×C13C2414yes—
148C22×C4×C13C2412yes—
149C22×C4×C13C2412yes—
150C22×Dic13—22no—
151C22×Dic13—24no—
152C22×Dic13—24no—
153C22×Dic13—22no—
154C22×Dic13—22no—
155C22×Dic13—22no—
156C22×Dic13—264no—
157C22×Dic13—216no—
158C22×Dic13—216no—
159C22×Dic13—22no—
160C22×Dic13—24no—
161C22×Dic13—24no—
162C22×Dic13—216no—
163C22×Dic13—22no—
164C22×Dic13—24no—
165C22×Dic13—22no—
166C22×Dic13—216no—
167C22×Dic13—48no—
168C22×Dic13—432no—
169C22×Dic13C22×C414yes—
170C22×Dic13C22×C418yes—
171C22×Dic13C22×C414yes—
172C22×Dic13C22×C4132yescomplemented
173C22×Dic13C2×D414yes—
174C22×Dic13C2×D414yes—
175C22×Dic13#4114yes—
176C22×Dic13C2412yes—
177C22×Dic13C2414yes—
178C22×Dic13C2412yes—
179C22×Dic13C2414yes—
180C22×Dic13C2412yes—
181C22×Dic13C2412yes—
182C13×C4⋊C4—28no—
183C13×C22⋊C4—216no—
184C42×C13—28no—
185C22⋊Dic13—216no—
186C22⋊Dic13C22×C4132yescomplemented
187C4⋊Dic13—28no—
188C4⋊Dic13C22×C4116yescomplemented
189#6101—216no—
190#6101—216no—
191#6101—216no—
192#6101—28no—
193#6101—48no—
194#6101C2×D4116yesdirect product factor
195C4×Dic13—28no—
196C4×Dic13C22×C4116yescomplemented
197C22×C4×D4—131no—
198C22×C4×D4—131no—
199C22×C4×D4D1311yescharacteristic
200C23×C4⋊C4—132no—
201D4×C4⋊C4—1316no—
202C2×#389—131no—
203C2×#389—131no—
204C2×#385—132no—
205C2×#385—132no—
206C2×#384—131no—
207C2×#380—132no—
208C2×C4×C4⋊C4—131no—
209C23×C13—216no—
210C23×C13—216no—
211C23×C13—216no—
212C23×C13—216no—
213C23×C13—216no—
214C23×C13—216no—
215C23×C13—24no—
216C23×C13—24no—
217C23×C13—24no—
218C23×C13—24no—
219C23×C13C2×C4⋊C4132yescomplemented
220C23×C13C23×C418yes—
221C23×C13C23×C412yes—
222C23×C13C22×D418yes—
223C23×C13C22×D412yes—
224C23×C13C22×D412yes—
225C23×C13C22×Q818yes—
226C23×C13C22×Q812yes—
227C23×C13C2×#4112yes—
228C23×C13H(5,𝔽2)12yes—
229C23×C132−1+412yes—
230C23×C13C2511yescharacteristic
231D4×C13C2×C4⋊C4164yescomplemented
232C2×C4×C13—28no—
233C2×C4×C13—22no—
234C2×C4×C13—22no—
235C2×C4×C13—24no—
236C2×C4×C13—22no—
237C2×C4×C13—24no—
238C2×C4×C13—22no—
239C2×C4×C13—24no—
240C2×C4×C13—22no—
241C2×C4×C13—432no—
242C2×C4×C13—48no—
243C2×C4×C13—44no—
244C2×C4×C13C2×C4⋊C4116yescomplemented
245C2×C4×C13C23×C414yes—
246C2×C4×C13C22×D414yes—
247C2×C4×C13C22×D414yes—
248C2×C4×C13C22×Q814yes—
249C2×Dic13—216no—
250C2×Dic13—28no—
251C2×Dic13—28no—
252C2×Dic13—22no—
253C2×Dic13—24no—
254C2×Dic13—24no—
255C2×Dic13—22no—
256C2×Dic13—22no—
257C2×Dic13—22no—
258C2×Dic13—264no—
259C2×Dic13—22no—
260C2×Dic13—24no—
261C2×Dic13—24no—
262C2×Dic13—22no—
263C2×Dic13—24no—
264C2×Dic13—22no—
265C2×Dic13—432no—
266C2×Dic13—48no—
267C2×Dic13—44no—
268C2×Dic13C4×D4116yescomplemented
269C2×Dic13C23×C414yes—
270C2×Dic13C23×C418yes—
271C2×Dic13C23×C414yes—
272C2×Dic13C22×D414yes—
273C2×Dic13C22×D414yes—
274C2×Dic13C2×#4114yes—
275C23×D4C2×D1311yescharacteristic
276C24×C4—132no—
277C24×C4—132no—
278C24×C4C2×D1312yes—
279C2×C4×D4—28no—
280C2×C4×D4—1316no—
281C2×C4×D4—138no—
282C2×C4×D4—138no—
283C2×C4×D4—268no—
284C2×C4×D4C2×D1318yes—
285C22×C4⋊C4—131no—
286C22×C4⋊C4—131no—
287C22×C4⋊C4—1316no—
288C22×C4⋊C4—132no—
289C22×C4⋊C4—132no—
290C22×C4⋊C4—132no—
291C22×C4⋊C4—131no—
292C22×C4⋊C4—131no—
293C22×C4⋊C4—132no—
294C22×C4⋊C4—2616no—
295C22×C4⋊C4C2×D1311yescharacteristic
296C22×C22⋊C4—132no—
297C22×C22⋊C4—132no—
298C22×C22⋊C4C2×D1312yes—
299C22×C42—131no—
300C22×C42—131no—
301C22×C42C2×D1311yescharacteristic
302#389—138no—
303#389—138no—
304#385—1316no—
305#385—1316no—
306#384—138no—
307#380—1316no—
308C4×C4⋊C4—138no—
309C2×#95—131no—
310C2×#95—131no—
311C2×#95—132no—
312C2×#95—132no—
313C22×C13—264no—
314C22×C13—28no—
315C22×C13—28no—
316C22×C13—28no—
317C22×C13—28no—
318C22×C13—28no—
319C22×C13—28no—
320C22×C13C22×C4⋊C414yes—
321C22×C13C22×C4⋊C4116yescomplemented
322C22×C13C2×C4×D414yes—
323C22×C13#52014yes—
324C22×C13D4214yes—
325C22×C13#54714yes—
326C22×C13D4×Q814yes—
327C22×C13#55414yes—
328C22×C13C24×C411yescharacteristic
329C22×C13C23×D411yescharacteristic
330C22×C13C23×D411yescharacteristic
331C22×C13C23×Q811yescharacteristic
332C22×C13C22×#4111yescharacteristic
333C22×C13C2×H(5,𝔽2)11yescharacteristic
334C22×C13C2×2−1+411yescharacteristic
335C4×C13—24no—
336C4×C13—44no—
337C4×C13—42no—
338C4×C13C22×C4⋊C418yescomplemented
339Dic13—28no—
340Dic13—24no—
341Dic13—24no—
342Dic13—44no—
343Dic13—42no—
344Dic13C2×C4×D418yescomplemented
345C25C22×D1312yes—
346C22×D4Dic2618yes—
347C22×D4C4×D1318yes—
348C22×D4C22⋊D1318yes—
349C22×D4C22×D1314yes—
350C23×C4—28no—
351C23×C4—28no—
352C23×C4—44no—
353C23×C4—131no—
354C23×C4—1316no—
355C23×C4—138no—
356C23×C4—131no—
357C23×C4—132no—
358C23×C4—131no—
359C23×C4—131no—
360C23×C4—132no—
361C23×C4—138no—
362C23×C4—268no—
363C23×C4—268no—
364C23×C4—268no—
365C23×C4—268no—
366C23×C4—524no—
367C23×C4C22×D1311yescharacteristic
368C23×C4C22×D1312yes—
369C23×C4C22×D1318yes—
370C23×C4C22×D1311yescharacteristic
371C23×C4C22×D1311yescharacteristic
372C4×D4—232no—
373C4×D4—1364no—
374C4×D4—2632no—
375C2×C4⋊C4—24no—
376C2×C4⋊C4—138no—
377C2×C4⋊C4—138no—
378C2×C4⋊C4—134no—
379C2×C4⋊C4—134no—
380C2×C4⋊C4—138no—
381C2×C4⋊C4—138no—
382C2×C4⋊C4—138no—
383C2×C4⋊C4—134no—
384C2×C4⋊C4—134no—
385C2×C4⋊C4—138no—
386C2×C4⋊C4—2664no—
387C2×C4⋊C4—268no—
388C2×C4⋊C4—268no—
389C2×C4⋊C4—264no—
390C2×C4⋊C4—268no—
391C2×C4⋊C4—264no—
392C2×C4⋊C4—524no—
393C2×C4⋊C4C22×D1314yes—
394C2×C22⋊C4—28no—
395C2×C22⋊C4—1316no—
396C2×C22⋊C4—138no—
397C2×C22⋊C4—138no—
398C2×C22⋊C4—268no—
399C2×C22⋊C4C22×D1318yes—
400C2×C42—24no—
401C2×C42—138no—
402C2×C42—134no—
403C2×C42—134no—
404C2×C42—264no—
405C2×C42C22×D1314yes—
406#95—134no—
407#95—134no—
408#95—138no—
409#95—138no—
410C2×C13—216no—
411C2×C13D4×C4⋊C418yescomplemented
412C2×C13C23×C4⋊C411yescharacteristic
413C2×C13C22×C4×D411yescharacteristic
414C2×C13C2×#52011yescharacteristic
415C2×C13C2×D4211yescharacteristic
416C2×C13C2×#54711yescharacteristic
417C2×C13C2×D4×Q811yescharacteristic
418C2×C13C2×#55411yescharacteristic
419C24—24no—
420C24—24no—
421C24—24no—
422C24—24no—
423C24C2×Dic2618yes—
424C24C2×C4×D1318yes—
425C24C2×C22⋊D1318yes—
426C24C23×D1314yes—
427C24C23×D1311yescharacteristic
428C2×D4#6101164yesdirect product factor
429C2×D4C2×Dic26116yes—
430C2×D4C2×C4×D13116yes—
431C2×D4C2×C22⋊D13116yes—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—216no—
441C22×C4—216no—
442C22×C4—48no—
443C22×C4—432no—
444C22×C4—134no—
445C22×C4—134no—
446C22×C4—134no—
447C22×C4—132no—
448C22×C4—134no—
449C22×C4—138no—
450C22×C4—1332no—
451C22×C4—134no—
452C22×C4—132no—
453C22×C4—134no—
454C22×C4—132no—
455C22×C4—132no—
456C22×C4—134no—
457C22×C4—2616no—
458C22×C4—2664no—
459C22×C4—2616no—
460C22×C4—2616no—
461C22×C4—264no—
462C22×C4—262no—
463C22×C4—262no—
464C22×C4—264no—
465C22×C4—262no—
466C22×C4—262no—
467C22×C4—264no—
468C22×C4—264no—
469C22×C4—262no—
470C22×C4—262no—
471C22×C4—262no—
472C22×C4—264no—
473C22×C4—2616no—
474C22×C4—528no—
475C22×C4—5232no—
476C22×C4C2×Dic2614yes—
477C22×C4C2×C4×D1314yes—
478C22×C4D4×D1314yes—
479C22×C4C2×C22⋊D1314yes—
480C22×C4C23×D1312yes—
481C22×C4C23×D1314yes—
482C22×C4C23×D1312yes—
483C22×C4C23×D1312yes—
484C4⋊C4—28no—
485C4⋊C4—1316no—
486C4⋊C4—1316no—
487C4⋊C4—2616no—
488C4⋊C4—2616no—
489C4⋊C4—268no—
490C4⋊C4—2616no—
491C4⋊C4—268no—
492C4⋊C4—528no—
493C22⋊C4—216no—
494C22⋊C4—1332no—
495C22⋊C4—2616no—
496C42—28no—
497C42—1316no—
498C42—268no—
499C13C2×D4×C4⋊C411yescharacteristic, complemented, Sylow
500C23—216no—
501C23—216no—
502C23—216no—
503C23—216no—
504C23—216no—
505C23—216no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23C2×#6101132yescomplemented
511C23C22×Dic2618yes—
512C23C22×Dic2612yes—
513C23C22×C4×D1318yes—
514C23C22×C4×D1312yes—
515C23C2×D4×D1312yes—
516C23#8825112yes—
517C23#41×D1312yes—
518C23#8825712yes—
519C23C22×C22⋊D1318yes—
520C23C22×C22⋊D1312yes—
521C23C24×D1311yescharacteristic
522D4C2×#6101164yesdirect product factor
523C2×C4—28no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—432no—
533C2×C4—48no—
534C2×C4—44no—
535C2×C4—1316no—
536C2×C4—134no—
537C2×C4—134no—
538C2×C4—134no—
539C2×C4—134no—
540C2×C4—138no—
541C2×C4—134no—
542C2×C4—2664no—
543C2×C4—2616no—
544C2×C4—268no—
545C2×C4—268no—
546C2×C4—264no—
547C2×C4—262no—
548C2×C4—262no—
549C2×C4—264no—
550C2×C4—262no—
551C2×C4—262no—
552C2×C4—264no—
553C2×C4—264no—
554C2×C4—262no—
555C2×C4—262no—
556C2×C4—262no—
557C2×C4—264no—
558C2×C4—5232no—
559C2×C4—528no—
560C2×C4—524no—
561C2×C4C2×#6101116yescomplemented
562C2×C4C22×Dic2614yes—
563C2×C4C22×C4×D1314yes—
564C2×C4C2×D4×D1314yes—
565C2×C4C22×C22⋊D1314yes—
566C22—264no—
567C22—28no—
568C22—28no—
569C22—28no—
570C22—28no—
571C22—28no—
572C22—28no—
573C22D4×Dic2614yes—
574C22#1171924414yes—
575C22C4×D4×D1314yes—
576C22#1171925114yes—
577C22C22×#610114yes—
578C22C22×#6101116yescomplemented
579C22D4×C22⋊D1314yes—
580C22#1171952914yes—
581C22C23×Dic2611yescharacteristic
582C22C23×C4×D1311yescharacteristic
583C22C22×D4×D1311yescharacteristic
584C22C2×#8825111yescharacteristic
585C22C2×#41×D1311yescharacteristic
586C22C2×#8825711yescharacteristic
587C22C23×C22⋊D1311yescharacteristic
588C4—24no—
589C4—44no—
590C4—42no—
591C4—138no—
592C4—268no—
593C4—264no—
594C4—264no—
595C4—524no—
596C4—522no—
597C4C22×#610118yescomplemented
598C2—216no—
599C2D4×#610118yesdirect product factor
600C2C23×#610111yescharacteristic
601C2C2×C4×D4×D1311yescharacteristic
602C2C2×#1171925111yescharacteristic
603C2C2×D4×C22⋊D1311yescharacteristic
604C2C2×#1171952911yescharacteristic
605C2C2×D4×Dic2611yescharacteristic
606C2C2×#1171924411yescharacteristic
subgroupG/H
centerC24C23×D13
commutatorC22×C13C24×C4
FrattiniC23C24×D13
FittingC22×C4×D4×C13C2
socleC24×C13C24
prime radicalC23×D4×C13C22
cosocleC23×C13C25
hypercenterC22×C4×D4D13
hypocenterC13C2×D4×C4⋊C4
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
3. C23×#6101C2semidirect
4. D4×#6101C2direct (N × H)
5. C2×#11718589C2semidirect
6. C2×#11718634C2semidirect
7. C2×#11718871C2semidirect
8. C2×#11718607C2semidirect
9. C2×#11718637C2semidirect
10. C2×#11718603C2semidirect
11. C2×C4×#6101C2semidirect
25. C2×D4×Dic13C4semidirect
37. C22×#6101C22semidirect
40. #11718871C22semidirect
45. #11718637C22semidirect
46. #11718634C22semidirect
47. #11718607C22semidirect
48. #11718603C22semidirect
49. C4×#6101C22semidirect
50. #11718589C22semidirect
69. C23×Dic13C2×C4semidirect
86. C2×C22⋊Dic13C2×C4semidirect
90. D4×Dic13C2×C4semidirect
92. C2×C4⋊Dic13C2×C4semidirect
101. C2×#6101D4direct (N × H)
109. C2×C4×Dic13C2×C4semidirect
126. C2×D4×C13C4⋊C4semidirect
172. C22×Dic13C22×C4semidirect
186. C22⋊Dic13C22×C4semidirect
188. C4⋊Dic13C22×C4semidirect
194. #6101C2×D4direct (N × H)
196. C4×Dic13C22×C4semidirect
219. C23×C13C2×C4⋊C4semidirect
231. D4×C13C2×C4⋊C4semidirect
244. C2×C4×C13C2×C4⋊C4semidirect
268. C2×Dic13C4×D4semidirect
321. C22×C13C22×C4⋊C4semidirect
338. C4×C13C22×C4⋊C4semidirect
344. Dic13C2×C4×D4semidirect
411. C2×C13D4×C4⋊C4semidirect
428. C2×D4#6101direct (N × H)
499. C13C2×D4×C4⋊C4semidirect
510. C23C2×#6101semidirect
522. D4C2×#6101direct (N × H)
561. C2×C4C2×#6101semidirect
578. C22C22×#6101semidirect
597. C4C22×#6101semidirect
599. C2D4×#6101direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×Dic132
C23×#61012
D4×#61012
C2×#117185892
C2×#117186342
C2×#117188712
C2×#117186072
C2×#117186372
C2×#117186032
C2×C4×#61012
C22×C4×D4×C132
C2×D4×C4⋊C413