Finite Group Databasebeta

← C2×#24921173 #36312346

Subgroups — C2×#24921173 #36312346

Order 21248.

7910 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
1#24921173C2132yesmaximal, direct product factor
2C2×#19128412C214yesmaximal, complemented
3C2×#19128404C214yesmaximal, complemented
4C2×#19128389C214yesmaximal, complemented
5C2×#19128354C214yesmaximal, complemented
6C2×#19128300C214yesmaximal, complemented
7C2×#19128282C211yesmaximal, characteristic, complemented
8C2×#19128214C214yesmaximal, complemented
9C2×#19128213C212yesmaximal, complemented
10C2×#19128181C212yesmaximal, complemented
11C22×C4⋊C4×D83C211yesmaximal, characteristic, complemented
12C2×#544×C83C211yesmaximal, characteristic, complemented
13C22×#13050591C2214yes—
14C22×#13050590C2212yes—
15C23×C4×D83C2211yescharacteristic
16C23×C4×D83C2212yes—
17#19128412C22164yescomplemented
18#19128404C22164yescomplemented
19#19128389C22164yescomplemented
20#19128354C22164yescomplemented
21#19128300C22164yescomplemented
22#19128282C22116yescomplemented
23C2×#14581787C2218yescomplemented
24C2×#14581783C2218yescomplemented
25#19128214C22164yescomplemented
26#19128213C22132yescomplemented
27C2×#14581777C2218yes—
28C2×#14581777C2218yescomplemented
29C2×#14581776C2214yes—
30C2×#14581772C2212yes—
31C2×#14581772C2212yes—
32C2×#14581772C2214yescomplemented
33C2×#14581768C2214yes—
34C2×#14581768C2218yescomplemented
35C2×#14581767C2212yes—
36C2×#14581767C2212yes—
37C2×#14581767C2214yes—
38C2×#14581767C2214yescomplemented
39C2×#14581766C2214yes—
40C2×#14581762C2214yes—
41#19128181C22132yescomplemented
42C2×C4×Dic166C2214yescomplemented
43C2×C4×D332C2214yes—
44C2×C4×D332C2214yes—
45C2×C4×D332C2214yescomplemented
46C2×#14581754C2218yes—
47C2×(C22⋊D83)⋊C4C2218yes—
48C2×#14581750C2214yes—
49C2×#14581750C2214yescomplemented
50C22×C4⋊Dic83C2211yescharacteristic
51C22×Dic83⋊C4C2212yes—
52C2×C4⋊C4×D83C2214yes—
53C2×C4⋊C4×D83C2214yes—
54C2×C4⋊C4×D83C22116yescomplemented
55C2×C4⋊C4×D83C2214yescomplemented
56#544×C83C22116yescomplemented
57C2×C4⋊Q8×C83C2214yescomplemented
58C2×#125×C83C2214yescomplemented
59C2×C22⋊Q8×C83C2214yes—
60C2×#117×C83C2212yes—
61C22×C4⋊C4×C83C2211yescharacteristic
62C24×D83C2311yescharacteristic
63C2×#13050591—416no—
64C2×#13050591C23132yes—
65C2×#13050590—216no—
66C2×#13050590C23116yes—
67C22×Dic166C2312yes—
68C22×D332C2314yes—
69C22×D332C2314yes—
70C22×D332C2312yes—
71C22×D332C2314yes—
72C22×C22⋊D83C2314yes—
73C23×Dic83C2311yescharacteristic
74C23×Dic83C2312yes—
75C22×C4×D83—28no—
76C22×C4×D83—48no—
77C22×C4×D83C2314yes—
78C22×C4×D83C2312yes—
79C22×C4×D83C2312yes—
80C22×C4×D83C2318yes—
81C22×C4×D83C2314yes—
82C22×C4×D83C2314yes—
83C22×C4×D83C23116yes—
84C22×C4×D83C2318yes—
85#14581787C23164yescomplemented
86#14581783C23164yescomplemented
87#14581777C23164yes—
88#14581777C23164yescomplemented
89#14581776C23132yes—
90#14581772C23116yes—
91#14581772C23116yes—
92#14581772C23132yescomplemented
93#14581768C23132yes—
94#14581768C23164yescomplemented
95#14581767C23116yes—
96#14581767C23116yes—
97#14581767C23132yes—
98#14581767C23132yescomplemented
99#14581766C23132yes—
100#14581762C23132yes—
101C4×Dic166C23132yescomplemented
102C4×D332C23132yes—
103C4×D332C23132yes—
104C4×D332C23132yescomplemented
105#14581754C23164yes—
106(C22⋊D83)⋊C4C23164yes—
107#14581750C23132yes—
108#14581750C23132yescomplemented
109C2×C4⋊Dic83C2314yes—
110C2×C4⋊Dic83C2318yes—
111C2×C4⋊Dic83C2314yes—
112C2×C4⋊Dic83C2314yes—
113C2×C4⋊Dic83C2314yes—
114C2×Dic83⋊C4C2318yes—
115C2×Dic83⋊C4C23116yes—
116C2×Dic83⋊C4C2318yes—
117C4⋊C4×D83C23132yes—
118C4⋊C4×D83C23132yes—
119C4⋊C4×D83C23132yescomplemented
120C2×(C2×D83)⋊C4C2314yes—
121C2×(C2×D83)⋊C4C2318yes—
122C2×(C2×D83)⋊C4C2314yes—
123C2×(C2×D83)⋊C4C2318yes—
124C2×(C2×D83)⋊C4C2314yes—
125C2×(C2×D83)⋊C4C2314yes—
126C2×C4×Dic83C2318yes—
127C22×Q8×C83C2314yes—
128C23×C4×C83C2312yes—
129C23×C4×C83C2311yescharacteristic
130C4⋊Q8×C83C23132yescomplemented
131#125×C83C23132yescomplemented
132C22⋊Q8×C83C23132yes—
133#117×C83C23116yes—
134C2×C4⋊C4×C83C2314yes—
135C2×C4⋊C4×C83C2318yes—
136C2×C4⋊C4×C83C2312yes—
137C2×C4⋊C4×C83C2312yes—
138C2×C4⋊C4×C83C2314yes—
139C2×C4⋊C4×C83C2318yes—
140C2×C22⋊C4×C83C2314yes—
141C2×C42×C83C2314yes—
142C23×D83—24no—
143C23×D83—24no—
144C23×D83—24no—
145C23×D83C2414yes—
146C23×D83C2414yes—
147C23×D83C2411yescharacteristic
148C23×D83C2411yescharacteristic
149#13050591—464no—
150#13050590—264no—
151C2×Dic166—28no—
152C2×Dic166C2418yes—
153C2×D332—216no—
154C2×D332—216no—
155C2×D332—28no—
156C2×D332—48no—
157C2×D332C24116yes—
158C2×D332C24116yes—
159C2×D332C2418yes—
160C2×D332C24116yes—
161C2×C22⋊D83—216no—
162C2×C22⋊D83C24116yes—
163C22×Dic83—24no—
164C22×Dic83—44no—
165C22×Dic83C2414yes—
166C22×Dic83C2414yes—
167C22×Dic83C2412yes—
168C22×Dic83C2414yes—
169C22×Dic83C2418yes—
170C2×C4×D83—216no—
171C2×C4×D83—28no—
172C2×C4×D83—28no—
173C2×C4×D83—216no—
174C2×C4×D83—232no—
175C2×C4×D83—432no—
176C2×C4×D83—48no—
177C2×C4×D83—416no—
178C2×C4×D83C2418yes—
179C2×C4×D83C2418yes—
180C2×C4×D83C24116yes—
181C2×C4×D83C24116yes—
182C2×C4×D83C24116yes—
183C2×C4×D83C24132yes—
184C4⋊Dic83C24116yes—
185C4⋊Dic83C24116yes—
186C4⋊Dic83C24116yes—
187C4⋊Dic83C24116yes—
188Dic83⋊C4C24132yes—
189Dic83⋊C4C24132yes—
190(C2×D83)⋊C4C24116yes—
191(C2×D83)⋊C4C24116yes—
192(C2×D83)⋊C4C24132yes—
193(C2×D83)⋊C4C24116yes—
194(C2×D83)⋊C4C24116yes—
195(C2×D83)⋊C4C24132yes—
196C4×Dic83C24132yes—
197C24×C83C2411yescharacteristic
198C2×Q8×C83—48no—
199C2×Q8×C83C24116yes—
200C22×C4×C83—44no—
201C22×C4×C83#4118yescomplemented
202C22×C4×C83C2414yes—
203C22×C4×C83C2414yes—
204C22×C4×C83C2414yes—
205C22×C4×C83C2418yes—
206C22×C4×C83C2412yes—
207C22×C4×C83C2414yes—
208C4⋊C4×C83C2418yes—
209C4⋊C4×C83C2418yes—
210C4⋊C4×C83C24116yes—
211C4⋊C4×C83C24116yes—
212C4⋊C4×C83C24132yes—
213C22⋊C4×C83C24116yes—
214C42×C83C24116yes—
215C23×C83—22no—
216C23×C83—22no—
217C23×C83C2×#4114yes—
218C23×C83C2512yes—
219C23×C83C2511yescharacteristic
220C22×D83—232no—
221C22×D83—28no—
222C22×D83—22no—
223C22×D83—22no—
224C22×D83—22no—
225C22×D83—22no—
226C22×D83—28no—
227C22×D83—28no—
228C22×D83—28no—
229C22×D83—28no—
230C22×D83—28no—
231C22×D83H(5,𝔽2)14yes—
232C22×D832−1+414yes—
233C22×D83C2518yes—
234C22×D83C2512yes—
235C22×D83C2512yes—
236Q8×C83—416no—
237C2×C4×C83—28no—
238C2×C4×C83—28no—
239C2×C4×C83—28no—
240C2×C4×C83—48no—
241C2×C4×C83C2×#4118yes—
242C2×C4×C83C2×#41116yescomplemented
243C2×C4×C83C2518yes—
244C2×C4×C83C2518yes—
245C2×C4×C83C2518yes—
246C2×C4×C83C2514yes—
247C22⋊D83—232no—
248C2×Dic83—28no—
249C2×Dic83—24no—
250C2×Dic83—28no—
251C2×Dic83—28no—
252C2×Dic83—48no—
253C2×Dic83C2518yes—
254C2×Dic83C2518yes—
255C2×Dic83C2514yes—
256D332—232no—
257D332—232no—
258D332—216no—
259D332—416no—
260C4×D83—232no—
261C4×D83—216no—
262C4×D83—216no—
263C4×D83—232no—
264C4×D83—416no—
265C4×D83—432no—
266Dic166—216no—
267C22×C83—22no—
268C22×C83—22no—
269C22×C83—28no—
270C22×C83C22×#4114yes—
271C22×C83C22×#4112yes—
272C22×C83C2×H(5,𝔽2)12yes—
273C22×C83C2×2−1+412yes—
274C22×C83C2611yescharacteristic
275C2×D83—232no—
276C2×D83—22no—
277C2×D83—22no—
278C2×D83—22no—
279C2×D83—22no—
280C2×D83—28no—
281C2×D83—28no—
282C2×D83—28no—
283C2×D83—28no—
284C2×D83—28no—
285C2×D83—416no—
286C2×D83C2×H(5,𝔽2)14yes—
287C2×D83C2×2−1+414yes—
288C4×C83—28no—
289C4×C83—28no—
290C4×C83—28no—
291C4×C83C22×#4118yes—
292Dic83—28no—
293Dic83—24no—
294Dic83—28no—
295C2×#3492—831nomaximal, Sylow
296C2×C83—24no—
297C2×C83#349214yescomplemented
298C2×C83C23×#4111yescharacteristic
299C2×C83C22×H(5,𝔽2)11yescharacteristic
300C2×C83C22×2−1+411yescharacteristic
301D83—24no—
302D83—24no—
303D83—48no—
304#3492—8332no—
305C2×#554—834no—
306C2×#553—834no—
307C2×#552—834no—
308C2×#547—834no—
309C2×#544D8311yescharacteristic, complemented
310C2×#539—834no—
311C2×#537—831no—
312C2×#525—834no—
313C2×#525—832no—
314C2×#520—832no—
315C23×C4⋊C4—831no—
316C83C2×#349211yescharacteristic, complemented, Sylow
317C22×#41—834no—
318C22×#41—832no—
319C24×C4—832no—
320C24×C4—831no—
321#554—8364no—
322#553—8364no—
323#552—8364no—
324#547—8364no—
325#544C2×D83116yescomplemented
326#539—8364no—
327#537—8316no—
328C2×C4⋊Q8C2×D8314yescomplemented
329C2×#126—838no—
330C2×#125—838no—
331C2×#125C2×D8314yescomplemented
332#525—8364no—
333#525—8332no—
334C2×#123—838no—
335C2×#123—838no—
336C2×#123—834no—
337C2×C22⋊Q8—834no—
338C2×C22⋊Q8—838no—
339C2×C22⋊Q8—832no—
340C2×C22⋊Q8—832no—
341C2×C22⋊Q8—834no—
342C2×C22⋊Q8C2×D8314yes—
343C2×#121—834no—
344C2×#121—832no—
345C2×#121—834no—
346C2×#121—832no—
347C2×#121—834no—
348C2×#120—834no—
349#520—8332no—
350C2×C4×Q8—834no—
351C2×C4×D4—838no—
352C2×C4×D4—838no—
353C2×C4×D4—834no—
354C2×C4×D4—834no—
355C2×C4×D4—834no—
356C2×#117—834no—
357C2×#117—834no—
358C2×#117C2×D8312yes—
359C22×C4⋊C4—832no—
360C22×C4⋊C4—831no—
361C22×C4⋊C4—834no—
362C22×C4⋊C4—834no—
363C22×C4⋊C4—8316no—
364C22×C4⋊C4—834no—
365C22×C4⋊C4C2×D8311yescharacteristic
366C25—831no—
367C2×#41—33216no—
368C2×#41—8332no—
369C2×#41—8316no—
370C2×#41—16616no—
371C22×Q8—832no—
372C22×Q8C22×D8314yes—
373C22×D4—834no—
374C22×D4—834no—
375C22×D4—834no—
376C22×D4—832no—
377C22×D4—834no—
378C23×C4—3328no—
379C23×C4—832no—
380C23×C4—834no—
381C23×C4—8316no—
382C23×C4—838no—
383C23×C4—831no—
384C23×C4—834no—
385C23×C4—832no—
386C23×C4—832no—
387C23×C4—838no—
388C23×C4—834no—
389C23×C4—1668no—
390C23×C4C22×D8312yes—
391C23×C4C22×D8311yescharacteristic
392C4⋊Q8C22×D83132yescomplemented
393#126—8364no—
394#125—8364no—
395#125C22×D83132yescomplemented
396#123—8364no—
397#123—8364no—
398#123—8332no—
399C22⋊Q8—8332no—
400C22⋊Q8—8364no—
401C22⋊Q8—8316no—
402C22⋊Q8—8316no—
403C22⋊Q8—8332no—
404C22⋊Q8C22×D83132yes—
405#121—8332no—
406#121—8316no—
407#121—8332no—
408#121—8316no—
409#121—8332no—
410#120—8332no—
411C4×Q8—8332no—
412C4×D4—8364no—
413C4×D4—8364no—
414C4×D4—8332no—
415C4×D4—8332no—
416C4×D4—8332no—
417#117—8332no—
418#117—8332no—
419#117C22×D83116yes—
420C2×C4⋊C4—834no—
421C2×C4⋊C4—838no—
422C2×C4⋊C4—838no—
423C2×C4⋊C4—8316no—
424C2×C4⋊C4—838no—
425C2×C4⋊C4—834no—
426C2×C4⋊C4—834no—
427C2×C4⋊C4—834no—
428C2×C4⋊C4—8332no—
429C2×C4⋊C4—8332no—
430C2×C4⋊C4—8332no—
431C2×C4⋊C4C22×D8314yes—
432C2×C4⋊C4C22×D8318yes—
433C2×C4⋊C4C22×D8312yes—
434C2×C4⋊C4C22×D8312yes—
435C2×C4⋊C4C22×D8314yes—
436C2×C4⋊C4C22×D8318yes—
437C2×C22⋊C4—834no—
438C2×C22⋊C4—838no—
439C2×C22⋊C4—834no—
440C2×C22⋊C4—838no—
441C2×C22⋊C4—834no—
442C2×C22⋊C4—834no—
443C2×C22⋊C4C22×D8314yes—
444C2×C42—838no—
445C2×C42C22×D8314yes—
446C24—831no—
447C24—831no—
448C24—834no—
449C24—834no—
450C24—1664no—
451C24—1664no—
452C24—1664no—
453C24C23×D8311yescharacteristic
454#41—33264no—
455#41—16664no—
456C2×Q8—48no—
457C2×Q8—838no—
458C2×Q8—1668no—
459C2×Q8C23×D83116yes—
460C2×D4—3328no—
461C2×D4—8316no—
462C2×D4—8316no—
463C2×D4—8316no—
464C2×D4—838no—
465C2×D4—8316no—
466C2×D4—16616no—
467C2×D4—16616no—
468C2×D4—16616no—
469C2×D4—1668no—
470C22×C4—44no—
471C22×C4—3328no—
472C22×C4—33216no—
473C22×C4—33232no—
474C22×C4—3324no—
475C22×C4—838no—
476C22×C4—8316no—
477C22×C4—8332no—
478C22×C4—834no—
479C22×C4—834no—
480C22×C4—834no—
481C22×C4—832no—
482C22×C4—8316no—
483C22×C4—838no—
484C22×C4—838no—
485C22×C4—8316no—
486C22×C4—16616no—
487C22×C4—1664no—
488C22×C4—1668no—
489C22×C4—1668no—
490C22×C4—16616no—
491C22×C4—16632no—
492C22×C4#1305059118yescomplemented
493C22×C4C23×D8314yes—
494C22×C4C23×D8314yes—
495C22×C4C23×D8314yes—
496C22×C4C23×D8318yes—
497C22×C4C23×D8312yes—
498C22×C4C23×D8314yes—
499C4⋊C4—8316no—
500C4⋊C4—8332no—
501C4⋊C4—8332no—
502C4⋊C4—8316no—
503C4⋊C4—8316no—
504C4⋊C4—8316no—
505C4⋊C4C23×D83116yes—
506C4⋊C4C23×D8318yes—
507C4⋊C4C23×D8318yes—
508C4⋊C4C23×D83116yes—
509C4⋊C4C23×D83132yes—
510C22⋊C4—8316no—
511C22⋊C4—8332no—
512C22⋊C4—8316no—
513C22⋊C4—8332no—
514C22⋊C4—8316no—
515C22⋊C4—8316no—
516C22⋊C4C23×D83116yes—
517C42—8332no—
518C42C23×D83116yes—
519C23—22no—
520C23—22no—
521C23—834no—
522C23—834no—
523C23—832no—
524C23—832no—
525C23—838no—
526C23—1668no—
527C23—1668no—
528C23—16632no—
529C23—1668no—
530C23—1668no—
531C23—1668no—
532C23—1668no—
533C23—1662no—
534C23—1662no—
535C23—1662no—
536C23—1662no—
537C23C2×#1305059114yes—
538C23C24×D8312yes—
539C23C24×D8311yescharacteristic
540Q8—416no—
541Q8—16616no—
542D4—33216no—
543D4—16632no—
544D4—16632no—
545D4—16632no—
546D4—16616no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—48no—
551C2×C4—33216no—
552C2×C4—33232no—
553C2×C4—3328no—
554C2×C4—838no—
555C2×C4—838no—
556C2×C4—834no—
557C2×C4—16632no—
558C2×C4—1668no—
559C2×C4—1664no—
560C2×C4—1668no—
561C2×C4—1668no—
562C2×C4—16616no—
563C2×C4—16616no—
564C2×C4—16632no—
565C2×C4C2×#1305059118yes—
566C2×C4C2×#13050591116yescomplemented
567C2×C4C24×D8318yes—
568C2×C4C24×D8318yes—
569C2×C4C24×D8318yes—
570C2×C4C24×D8314yes—
571C22—28no—
572C22—22no—
573C22—22no—
574C22—33216no—
575C22—834no—
576C22—834no—
577C22—16632no—
578C22—1668no—
579C22—1668no—
580C22—1668no—
581C22—1668no—
582C22—1668no—
583C22—1662no—
584C22—1662no—
585C22—1662no—
586C22—1662no—
587C22C22×#1305059114yes—
588C22C22×#1305059112yes—
589C22H(5,𝔽2)×D8312yes—
590C222−1+4×D8312yes—
591C22C25×D8311yescharacteristic
592C4—28no—
593C4—28no—
594C4—28no—
595C4—1664no—
596C4—1668no—
597C4—1668no—
598C4C22×#1305059118yes—
599C2—24no—
600C2—3328no—
601C2—1664no—
602C2—1664no—
603C2#2492117314yesdirect product factor
604C2C23×#1305059111yescharacteristic
605C2C2×H(5,𝔽2)×D8311yescharacteristic
606C2C2×2−1+4×D8311yescharacteristic
subgroupG/H
centerC23C24×D83
commutator, cosocleC22×C83C26
FrattiniC22C25×D83
FittingC2×#544×C83C2
socleC23×C83C25
hypercenterC2×#544D83
hypocenterC83C2×#3492
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
1. #24921173C2direct (N × H)
2. C2×#19128412C2semidirect
3. C2×#19128404C2semidirect
4. C2×#19128389C2semidirect
5. C2×#19128354C2semidirect
6. C2×#19128300C2semidirect
7. C2×#19128282C2semidirect
8. C2×#19128214C2semidirect
9. C2×#19128213C2semidirect
10. C2×#19128181C2semidirect
11. C22×C4⋊C4×D83C2semidirect
12. C2×#544×C83C2semidirect
17. #19128412C22semidirect
18. #19128404C22semidirect
19. #19128389C22semidirect
20. #19128354C22semidirect
21. #19128300C22semidirect
22. #19128282C22semidirect
23. C2×#14581787C22semidirect
24. C2×#14581783C22semidirect
25. #19128214C22semidirect
26. #19128213C22semidirect
28. C2×#14581777C22semidirect
32. C2×#14581772C22semidirect
34. C2×#14581768C22semidirect
38. C2×#14581767C22semidirect
41. #19128181C22semidirect
42. C2×C4×Dic166C22semidirect
45. C2×C4×D332C22semidirect
49. C2×#14581750C22semidirect
54. C2×C4⋊C4×D83C22semidirect
55. C2×C4⋊C4×D83C22semidirect
56. #544×C83C22semidirect
57. C2×C4⋊Q8×C83C22semidirect
58. C2×#125×C83C22semidirect
85. #14581787C23semidirect
86. #14581783C23semidirect
88. #14581777C23semidirect
92. #14581772C23semidirect
94. #14581768C23semidirect
98. #14581767C23semidirect
101. C4×Dic166C23semidirect
104. C4×D332C23semidirect
108. #14581750C23semidirect
119. C4⋊C4×D83C23semidirect
130. C4⋊Q8×C83C23semidirect
131. #125×C83C23semidirect
201. C22×C4×C83#41semidirect
242. C2×C4×C83C2×#41semidirect
297. C2×C83#3492semidirect
309. C2×#544D83semidirect
316. C83C2×#3492semidirect
325. #544C2×D83semidirect
328. C2×C4⋊Q8C2×D83semidirect
331. C2×#125C2×D83semidirect
392. C4⋊Q8C22×D83semidirect
395. #125C22×D83semidirect
492. C22×C4#13050591semidirect
566. C2×C4C2×#13050591semidirect
603. C2#24921173direct (N × H)

Maximal subgroups

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

subgroupindex
#249211732
C2×#191284122
C2×#191284042
C2×#191283892
C2×#191283542
C2×#191283002
C2×#191282822
C2×#191282142
C2×#191282132
C2×#191281812
C22×C4⋊C4×D832
C2×#544×C832
C2×#349283