Finite Group Databasebeta

← C2×#24921170 #36312344

Subgroups — C2×#24921170 #36312344

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#24921170C2132yesmaximal, direct product factor
2C2×#19128409C214yesmaximal, complemented
3C2×#19128398C214yesmaximal, complemented
4C2×#19128385C214yesmaximal, complemented
5C2×#19128351C214yesmaximal, complemented
6C2×#19128332C211yesmaximal, characteristic, complemented
7C2×#19128296C214yesmaximal, complemented
8C2×#19128211C214yesmaximal, complemented
9C2×#19128208C212yesmaximal, complemented
10C2×#19128184C212yesmaximal, complemented
11C22×C4⋊C4×D83C211yesmaximal, characteristic, complemented
12C2×#537×C83C211yesmaximal, characteristic, complemented
13C22×#13050589C2214yes—
14C23×C4×D83C2211yescharacteristic
15C23×C4×D83C2212yes—
16#19128409C22164yescomplemented
17#19128398C22164yescomplemented
18#19128385C22164yescomplemented
19#19128351C22164yescomplemented
20#19128332C22116yescomplemented
21#19128296C22164yescomplemented
22C2×#14581791C2214yescomplemented
23C2×#14581785C2218yescomplemented
24C2×#14581783C2214yescomplemented
25C2×#14581782C2218yescomplemented
26#19128211C22164yescomplemented
27#19128208C22132yescomplemented
28C2×#14581774C2218yes—
29C2×#14581773C2214yes—
30C2×#14581773C2218yescomplemented
31C2×#14581772C2214yes—
32C2×#14581771C2214yes—
33C2×#14581769C2218yescomplemented
34C2×#14581768C2212yes—
35C2×#14581768C2212yes—
36C2×#14581763C2212yes—
37C2×#14581763C2212yes—
38C2×#14581763C2214yes—
39C2×#14581760C2214yes—
40#19128184C22132yescomplemented
41C2×Q8×Dic83C2214yescomplemented
42C2×D4×Dic83C2218yes—
43C2×D4×Dic83C2214yescomplemented
44C2×C4×C22⋊D83C2214yes—
45C2×C4×C22⋊D83C2214yes—
46C2×(C22⋊D83)⋊C4C2218yes—
47C2×#14581751C2212yes—
48C2×#14581751C2214yescomplemented
49C2×#14581750C2214yes—
50C22×C4⋊Dic83C2211yescharacteristic
51C22×C4⋊Dic83C2214yescomplemented
52C22×Dic83⋊C4C2212yes—
53C2×C4⋊C4×D83C2214yes—
54C2×C4⋊C4×D83C2214yes—
55C2×C4⋊C4×D83C22116yescomplemented
56C22×#41×C83C2212yes—
57#537×C83C22116yescomplemented
58C2×C22⋊Q8×C83C2214yescomplemented
59C2×#121×C83C2214yes—
60C2×#121×C83C2214yescomplemented
61C22×C4⋊C4×C83C2211yescharacteristic
62C24×D83C2311yescharacteristic
63C2×#13050589—416no—
64C2×#13050589C23132yes—
65C22×Dic166C2314yes—
66C22×C22⋊D83C2314yes—
67C22×C22⋊D83C2314yes—
68C23×Dic83C2311yescharacteristic
69C23×Dic83C2312yes—
70C23×Dic83C2318yes—
71C22×C4×D83—28no—
72C22×C4×D83—48no—
73C22×C4×D83C2314yes—
74C22×C4×D83C2312yes—
75C22×C4×D83C2312yes—
76C22×C4×D83C2318yes—
77C22×C4×D83C2314yes—
78C22×C4×D83C23116yes—
79#14581791C23132yescomplemented
80#14581785C23164yescomplemented
81#14581783C23132yescomplemented
82#14581782C23164yescomplemented
83#14581774C23164yes—
84#14581773C23132yes—
85#14581773C23164yescomplemented
86#14581772C23132yes—
87#14581771C23132yes—
88#14581769C23164yescomplemented
89#14581768C23116yes—
90#14581768C23116yes—
91#14581763C23116yes—
92#14581763C23116yes—
93#14581763C23132yes—
94#14581760C23132yes—
95Q8×Dic83C23132yescomplemented
96D4×Dic83C23164yes—
97D4×Dic83C23132yescomplemented
98C4×C22⋊D83C23132yes—
99C4×C22⋊D83C23132yes—
100(C22⋊D83)⋊C4C23164yes—
101#14581751C23116yes—
102#14581751C23132yescomplemented
103#14581750C23132yes—
104C2×C4⋊Dic83C2314yes—
105C2×C4⋊Dic83C2318yes—
106C2×C4⋊Dic83C2312yes—
107C2×C4⋊Dic83C2318yes—
108C2×C4⋊Dic83C2314yes—
109C2×C4⋊Dic83C2312yes—
110C2×C4⋊Dic83C23132yescomplemented
111C2×Dic83⋊C4C2318yes—
112C2×Dic83⋊C4C23116yes—
113C2×Dic83⋊C4C2318yes—
114C2×Dic83⋊C4C2314yes—
115C2×Dic83⋊C4C2314yes—
116C4⋊C4×D83C23132yes—
117C4⋊C4×D83C23132yes—
118C2×C22⋊Dic83C2314yes—
119C2×C22⋊Dic83C2318yes—
120C2×C22⋊Dic83C2314yes—
121C2×C22⋊Dic83C2314yes—
122C2×(C2×D83)⋊C4C2314yes—
123C2×(C2×D83)⋊C4C2314yes—
124C2×C4×Dic83C2318yes—
125C2×C4×Dic83C2314yes—
126C2×#41×C83—216no—
127C2×#41×C83C23116yes—
128C22×Q8×C83C2312yes—
129C22×D4×C83C2314yes—
130C22×D4×C83C2312yes—
131C22×D4×C83C2314yes—
132C23×C4×C83C2314yes—
133C23×C4×C83C2312yes—
134C23×C4×C83C2311yescharacteristic
135C22⋊Q8×C83C23132yescomplemented
136#121×C83C23132yes—
137#121×C83C23132yescomplemented
138C2×C4⋊C4×C83C2318yes—
139C2×C4⋊C4×C83C2314yes—
140C2×C4⋊C4×C83C2314yes—
141C2×C22⋊C4×C83C2318yes—
142C23×D83—24no—
143C23×D83—24no—
144C23×D83—24no—
145C23×D83C2414yes—
146C23×D83C2411yescharacteristic
147C23×D83C2411yescharacteristic
148#13050589—464no—
149C2×Dic166—48no—
150C2×Dic166C24116yes—
151C2×C22⋊D83—216no—
152C2×C22⋊D83—216no—
153C2×C22⋊D83C24116yes—
154C2×C22⋊D83C24116yes—
155C22×Dic83—24no—
156C22×Dic83—44no—
157C22×Dic83—416no—
158C22×Dic83C2414yes—
159C22×Dic83C2414yes—
160C22×Dic83C2412yes—
161C22×Dic83C2414yes—
162C22×Dic83C2414yes—
163C22×Dic83C24132yes—
164C22×Dic83C2418yes—
165C2×C4×D83—216no—
166C2×C4×D83—28no—
167C2×C4×D83—28no—
168C2×C4×D83—232no—
169C2×C4×D83—432no—
170C2×C4×D83—48no—
171C2×C4×D83C2418yes—
172C2×C4×D83C2418yes—
173C2×C4×D83C24116yes—
174C2×C4×D83C24116yes—
175C4⋊Dic83C2418yes—
176C4⋊Dic83C2418yes—
177C4⋊Dic83C24116yes—
178C4⋊Dic83C24132yes—
179C4⋊Dic83C24116yes—
180Dic83⋊C4C24116yes—
181Dic83⋊C4C24116yes—
182Dic83⋊C4C24132yes—
183Dic83⋊C4C24132yes—
184C22⋊Dic83C24116yes—
185C22⋊Dic83C24116yes—
186C22⋊Dic83C24116yes—
187C22⋊Dic83C24132yes—
188(C2×D83)⋊C4C24116yes—
189(C2×D83)⋊C4C24116yes—
190C4×Dic83C24132yes—
191C4×Dic83C24116yes—
192C24×C83C2414yes—
193C24×C83C2411yescharacteristic
194#41×C83—264no—
195C2×Q8×C83—28no—
196C2×Q8×C83C2418yes—
197C2×D4×C83—216no—
198C2×D4×C83—28no—
199C2×D4×C83—48no—
200C2×D4×C83C24116yes—
201C2×D4×C83C2418yes—
202C2×D4×C83C24116yes—
203C22×C4×C83—216no—
204C22×C4×C83—44no—
205C22×C4×C83#4118yescomplemented
206C22×C4×C83C2414yes—
207C22×C4×C83C2414yes—
208C22×C4×C83C24116yes—
209C22×C4×C83C2418yes—
210C22×C4×C83C2412yes—
211C22×C4×C83C2414yes—
212C4⋊C4×C83C24116yes—
213C4⋊C4×C83C24116yes—
214C22⋊C4×C83C24132yes—
215C23×C83—22no—
216C23×C83—22no—
217C23×C83—28no—
218C23×C83—28no—
219C23×C83—28no—
220C23×C83C2×#4114yes—
221C23×C83C2518yes—
222C23×C83C2512yes—
223C23×C83C2511yescharacteristic
224C22×D83—232no—
225C22×D83—28no—
226C22×D83—22no—
227C22×D83—22no—
228C22×D83—22no—
229C22×D83—22no—
230C22×D83—28no—
231C22×D83—28no—
232C22×D83H(5,𝔽2)14yes—
233C22×D832−1+414yes—
234C22×D83C2512yes—
235C22×D83C2512yes—
236Q8×C83—216no—
237D4×C83—232no—
238D4×C83—216no—
239D4×C83—416no—
240C2×C4×C83—28no—
241C2×C4×C83—28no—
242C2×C4×C83—232no—
243C2×C4×C83—48no—
244C2×C4×C83C2×#4118yes—
245C2×C4×C83C2×#41116yescomplemented
246C2×C4×C83C2518yes—
247C2×C4×C83C2518yes—
248C2×C4×C83C2514yes—
249C22⋊D83—232no—
250C22⋊D83—232no—
251C2×Dic83—28no—
252C2×Dic83—24no—
253C2×Dic83—28no—
254C2×Dic83—28no—
255C2×Dic83—28no—
256C2×Dic83—48no—
257C2×Dic83—432no—
258C2×Dic83C2518yes—
259C2×Dic83C2518yes—
260C2×Dic83C2518yes—
261C2×Dic83C2514yes—
262C4×D83—232no—
263C4×D83—216no—
264C4×D83—216no—
265C4×D83—416no—
266Dic166—416no—
267C22×C83—22no—
268C22×C83—22no—
269C22×C83—28no—
270C22×C83—28no—
271C22×C83—28no—
272C22×C83—28no—
273C22×C83—416no—
274C22×C83C22×#4114yes—
275C22×C83C22×#4112yes—
276C22×C83C2×H(5,𝔽2)12yes—
277C22×C83C2×2−1+412yes—
278C22×C83C2611yescharacteristic
279C2×D83—232no—
280C2×D83—22no—
281C2×D83—22no—
282C2×D83—22no—
283C2×D83—22no—
284C2×D83—28no—
285C2×D83—28no—
286C2×D83C2×H(5,𝔽2)14yes—
287C2×D83C2×2−1+414yes—
288C4×C83—28no—
289C4×C83—28no—
290C4×C83C22×#4118yes—
291Dic83—24no—
292Dic83—28no—
293Dic83—28no—
294Dic83—28no—
295C2×#3492—831nomaximal, Sylow
296C2×C83—24no—
297C2×C83—48no—
298C2×C83#349214yescomplemented
299C2×C83C23×#4111yescharacteristic
300C2×C83C22×H(5,𝔽2)11yescharacteristic
301C2×C83C22×2−1+411yescharacteristic
302D83—24no—
303D83—24no—
304#3492—8332no—
305C2×#554—834no—
306C2×#553—834no—
307C2×#552—834no—
308C2×#547—834no—
309C2×#544—831no—
310C2×#539—834no—
311C2×#537D8311yescharacteristic, complemented
312C2×#525—832no—
313C2×#525—834no—
314C2×#520—832no—
315C23×C4⋊C4—831no—
316C83C2×#349211yescharacteristic, complemented, Sylow
317C22×#41—834no—
318C22×#41C2×D8312yes—
319C24×C4—832no—
320C24×C4—831no—
321#554—8364no—
322#553—8364no—
323#552—8364no—
324#547—8364no—
325#544—8316no—
326#539—8364no—
327#537C2×D83116yescomplemented
328C2×C4⋊Q8—834no—
329C2×#126—838no—
330C2×#125—838no—
331C2×#125—834no—
332#525—8332no—
333#525—8364no—
334C2×#123—834no—
335C2×#123—838no—
336C2×#123—838no—
337C2×C22⋊Q8—832no—
338C2×C22⋊Q8—832no—
339C2×C22⋊Q8—834no—
340C2×C22⋊Q8—834no—
341C2×C22⋊Q8—838no—
342C2×C22⋊Q8C2×D8314yescomplemented
343C2×#121—832no—
344C2×#121—834no—
345C2×#121—832no—
346C2×#121C2×D8314yes—
347C2×#121C2×D8314yescomplemented
348C2×#120—834no—
349#520—8332no—
350C2×C4×Q8—834no—
351C2×C4×D4—838no—
352C2×C4×D4—834no—
353C2×C4×D4—834no—
354C2×C4×D4—834no—
355C2×C4×D4—838no—
356C2×#117—832no—
357C2×#117—834no—
358C2×#117—834no—
359C22×C4⋊C4—832no—
360C22×C4⋊C4—831no—
361C22×C4⋊C4—834no—
362C22×C4⋊C4—834no—
363C22×C4⋊C4—834no—
364C22×C4⋊C4—8316no—
365C22×C4⋊C4C2×D8311yescharacteristic
366C25—831no—
367C2×#41—216no—
368C2×#41—33216no—
369C2×#41—8332no—
370C2×#41C22×D83116yes—
371C22×Q8—834no—
372C22×Q8C22×D8312yes—
373C22×D4—834no—
374C22×D4—834no—
375C22×D4C22×D8314yes—
376C22×D4C22×D8312yes—
377C22×D4C22×D8314yes—
378C23×C4—3328no—
379C23×C4—832no—
380C23×C4—838no—
381C23×C4—834no—
382C23×C4—8316no—
383C23×C4—831no—
384C23×C4—834no—
385C23×C4—832no—
386C23×C4—832no—
387C23×C4—838no—
388C23×C4—1668no—
389C23×C4C22×D8314yes—
390C23×C4C22×D8312yes—
391C23×C4C22×D8311yescharacteristic
392C4⋊Q8—8332no—
393#126—8364no—
394#125—8364no—
395#125—8332no—
396#123—8332no—
397#123—8364no—
398#123—8364no—
399C22⋊Q8—8316no—
400C22⋊Q8—8316no—
401C22⋊Q8—8332no—
402C22⋊Q8—8332no—
403C22⋊Q8—8364no—
404C22⋊Q8C22×D83132yescomplemented
405#121—8316no—
406#121—8332no—
407#121—8316no—
408#121C22×D83132yes—
409#121C22×D83132yescomplemented
410#120—8332no—
411C4×Q8—8332no—
412C4×D4—8364no—
413C4×D4—8332no—
414C4×D4—8332no—
415C4×D4—8332no—
416C4×D4—8364no—
417#117—8316no—
418#117—8332no—
419#117—8332no—
420C2×C4⋊C4—834no—
421C2×C4⋊C4—838no—
422C2×C4⋊C4—838no—
423C2×C4⋊C4—8316no—
424C2×C4⋊C4—832no—
425C2×C4⋊C4—838no—
426C2×C4⋊C4—8332no—
427C2×C4⋊C4—832no—
428C2×C4⋊C4—834no—
429C2×C4⋊C4—834no—
430C2×C4⋊C4—838no—
431C2×C4⋊C4—834no—
432C2×C4⋊C4—8332no—
433C2×C4⋊C4—8332no—
434C2×C4⋊C4C22×D8314yes—
435C2×C4⋊C4C22×D8318yes—
436C2×C4⋊C4C22×D8314yes—
437C2×C22⋊C4—834no—
438C2×C22⋊C4—838no—
439C2×C22⋊C4—834no—
440C2×C22⋊C4—834no—
441C2×C22⋊C4—834no—
442C2×C22⋊C4—834no—
443C2×C22⋊C4C22×D8318yes—
444C2×C42—838no—
445C2×C42—834no—
446C24—831no—
447C24—831no—
448C24—834no—
449C24—1664no—
450C24—1664no—
451C24—1664no—
452C24C23×D8314yes—
453C24C23×D8311yescharacteristic
454#41—264no—
455#41—33264no—
456C2×Q8—28no—
457C2×Q8—3328no—
458C2×Q8—8316no—
459C2×Q8C23×D8318yes—
460C2×D4—216no—
461C2×D4—28no—
462C2×D4—48no—
463C2×D4—8316no—
464C2×D4—8316no—
465C2×D4—16616no—
466C2×D4—16616no—
467C2×D4C23×D83116yes—
468C2×D4C23×D8318yes—
469C2×D4C23×D83116yes—
470C22×C4—216no—
471C22×C4—44no—
472C22×C4—3328no—
473C22×C4—33232no—
474C22×C4—3324no—
475C22×C4—33216no—
476C22×C4—838no—
477C22×C4—8332no—
478C22×C4—8316no—
479C22×C4—834no—
480C22×C4—834no—
481C22×C4—834no—
482C22×C4—834no—
483C22×C4—832no—
484C22×C4—8316no—
485C22×C4—838no—
486C22×C4—838no—
487C22×C4—16616no—
488C22×C4—1664no—
489C22×C4—1668no—
490C22×C4—1668no—
491C22×C4—16632no—
492C22×C4#1305058918yescomplemented
493C22×C4C23×D83116yes—
494C22×C4C23×D8314yes—
495C22×C4C23×D8314yes—
496C22×C4C23×D8318yes—
497C22×C4C23×D8312yes—
498C22×C4C23×D8314yes—
499C4⋊C4—8316no—
500C4⋊C4—8332no—
501C4⋊C4—838no—
502C4⋊C4—8332no—
503C4⋊C4—838no—
504C4⋊C4—8316no—
505C4⋊C4—8316no—
506C4⋊C4—8332no—
507C4⋊C4—8316no—
508C4⋊C4C23×D83116yes—
509C4⋊C4C23×D83116yes—
510C22⋊C4—8316no—
511C22⋊C4—8332no—
512C22⋊C4—8316no—
513C22⋊C4—8316no—
514C22⋊C4—8316no—
515C22⋊C4—8316no—
516C22⋊C4C23×D83132yes—
517C42—8332no—
518C42—8316no—
519C23—22no—
520C23—22no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—834no—
525C23—834no—
526C23—832no—
527C23—832no—
528C23—1668no—
529C23—1668no—
530C23—16632no—
531C23—1668no—
532C23—1662no—
533C23—1662no—
534C23—1662no—
535C23—1662no—
536C23C2×#1305058914yes—
537C23C24×D8318yes—
538C23C24×D8312yes—
539C23C24×D8311yescharacteristic
540Q8—216no—
541Q8—33216no—
542D4—232no—
543D4—216no—
544D4—416no—
545D4—16632no—
546D4—16632no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—232no—
550C2×C4—48no—
551C2×C4—33216no—
552C2×C4—3328no—
553C2×C4—33232no—
554C2×C4—838no—
555C2×C4—838no—
556C2×C4—838no—
557C2×C4—834no—
558C2×C4—16632no—
559C2×C4—1668no—
560C2×C4—1664no—
561C2×C4—1668no—
562C2×C4—1668no—
563C2×C4—1668no—
564C2×C4—16616no—
565C2×C4—16616no—
566C2×C4C2×#1305058918yes—
567C2×C4C2×#13050589116yescomplemented
568C2×C4C24×D8318yes—
569C2×C4C24×D8318yes—
570C2×C4C24×D8314yes—
571C22—28no—
572C22—22no—
573C22—22no—
574C22—28no—
575C22—28no—
576C22—28no—
577C22—416no—
578C22—834no—
579C22—834no—
580C22—16632no—
581C22—1668no—
582C22—1668no—
583C22—1662no—
584C22—1662no—
585C22—1662no—
586C22—1662no—
587C22C22×#1305058914yes—
588C22C22×#1305058912yes—
589C22H(5,𝔽2)×D8312yes—
590C222−1+4×D8312yes—
591C22C25×D8311yescharacteristic
592C4—28no—
593C4—28no—
594C4—1664no—
595C4—1668no—
596C4—1668no—
597C4—1668no—
598C4C22×#1305058918yes—
599C2—24no—
600C2—48no—
601C2—1664no—
602C2—1664no—
603C2#2492117014yesdirect product factor
604C2C23×#1305058911yescharacteristic
605C2C2×H(5,𝔽2)×D8311yescharacteristic
606C2C2×2−1+4×D8311yescharacteristic
subgroupG/H
centerC23C24×D83
commutator, cosocleC22×C83C26
FrattiniC22C25×D83
FittingC2×#537×C83C2
socleC23×C83C25
hypercenterC2×#537D83
hypocenterC83C2×#3492
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #24921170C2direct (N × H)
2. C2×#19128409C2semidirect
3. C2×#19128398C2semidirect
4. C2×#19128385C2semidirect
5. C2×#19128351C2semidirect
6. C2×#19128332C2semidirect
7. C2×#19128296C2semidirect
8. C2×#19128211C2semidirect
9. C2×#19128208C2semidirect
10. C2×#19128184C2semidirect
11. C22×C4⋊C4×D83C2semidirect
12. C2×#537×C83C2semidirect
16. #19128409C22semidirect
17. #19128398C22semidirect
18. #19128385C22semidirect
19. #19128351C22semidirect
20. #19128332C22semidirect
21. #19128296C22semidirect
22. C2×#14581791C22semidirect
23. C2×#14581785C22semidirect
24. C2×#14581783C22semidirect
25. C2×#14581782C22semidirect
26. #19128211C22semidirect
27. #19128208C22semidirect
30. C2×#14581773C22semidirect
33. C2×#14581769C22semidirect
40. #19128184C22semidirect
41. C2×Q8×Dic83C22semidirect
43. C2×D4×Dic83C22semidirect
48. C2×#14581751C22semidirect
51. C22×C4⋊Dic83C22semidirect
55. C2×C4⋊C4×D83C22semidirect
57. #537×C83C22semidirect
58. C2×C22⋊Q8×C83C22semidirect
60. C2×#121×C83C22semidirect
79. #14581791C23semidirect
80. #14581785C23semidirect
81. #14581783C23semidirect
82. #14581782C23semidirect
85. #14581773C23semidirect
88. #14581769C23semidirect
95. Q8×Dic83C23semidirect
97. D4×Dic83C23semidirect
102. #14581751C23semidirect
110. C2×C4⋊Dic83C23semidirect
135. C22⋊Q8×C83C23semidirect
137. #121×C83C23semidirect
205. C22×C4×C83#41semidirect
245. C2×C4×C83C2×#41semidirect
298. C2×C83#3492semidirect
311. C2×#537D83semidirect
316. C83C2×#3492semidirect
327. #537C2×D83semidirect
342. C2×C22⋊Q8C2×D83semidirect
347. C2×#121C2×D83semidirect
404. C22⋊Q8C22×D83semidirect
409. #121C22×D83semidirect
492. C22×C4#13050589semidirect
567. C2×C4C2×#13050589semidirect
603. C2#24921170direct (N × H)

Maximal subgroups

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

subgroupindex
#249211702
C2×#191284092
C2×#191283982
C2×#191283852
C2×#191283512
C2×#191283322
C2×#191282962
C2×#191282112
C2×#191282082
C2×#191281842
C22×C4⋊C4×D832
C2×#537×C832
C2×#349283