Finite Group Databasebeta

← C2×#24921859 #36312196

Subgroups — C2×#24921859 #36312196

Order 21248.

7810 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24921859C2132yesmaximal, direct product factor
2C2×#19128449C211yesmaximal, characteristic, complemented
3C2×#19128445C212yesmaximal, complemented
4C2×#19128405C214yesmaximal, complemented
5C2×#19128369C214yesmaximal, complemented
6C2×#19128349C214yesmaximal, complemented
7C2×#19128303C214yesmaximal, complemented
8C2×#19128281C214yesmaximal, complemented
9C2×#19128278C214yesmaximal, complemented
10C2×#19128257C211yesmaximal, characteristic, complemented
11C2×#19128250C212yesmaximal, complemented
12C2×#562×C83C211yesmaximal, characteristic, complemented
13C22×#13050591C2212yes—
14C22×#13050590C2214yes—
15C22×#13050589C2214yes—
16C22×#13050589C2212yes—
17#19128449C22116yescomplemented
18#19128445C22132yescomplemented
19#19128405C22164yescomplemented
20#19128369C22164yescomplemented
21#19128349C22164yescomplemented
22#19128303C22164yescomplemented
23#19128281C22164yescomplemented
24#19128278C22164yescomplemented
25#19128257C22116yescomplemented
26#19128250C22132yescomplemented
27C2×#14581791C2212yes—
28C2×#14581790C2212yes—
29C2×#14581788C2212yes—
30C2×#14581788C2214yes—
31C2×#14581784C2211yescharacteristic
32C2×#14581782C2214yes—
33C2×#14581780C2212yes—
34C2×#14581777C2214yes—
35C2×#14581776C2214yes—
36C2×#14581774C2214yes—
37C2×#14581770C2214yes—
38C2×#14581770C2214yes—
39C2×#14581770C2214yes—
40C2×#14581769C2214yes—
41C2×#14581769C2214yes—
42C2×#14581768C2214yes—
43C2×#14581767C2214yes—
44C2×#14581767C2214yes—
45C2×#14581766C2214yes—
46C2×#14581765C2214yes—
47C2×#14581765C2214yes—
48C2×#14581765C2214yes—
49C2×#14581765C2214yes—
50C2×#14581765C2214yes—
51C2×#14581764C2214yes—
52C2×#14581764C2214yes—
53C2×#14581764C2214yes—
54C2×#14581757C2214yes—
55C2×C4×C22⋊D83C2214yes—
56C2×(C22⋊D83)⋊C4C2214yes—
57C2×(C22⋊D83)⋊C4C2214yes—
58C2×(C22⋊D83)⋊C4C2214yes—
59C2×#14581750C2212yes—
60C22×Dic83⋊C4C2214yes—
61C2×C4⋊C4×D83C2214yes—
62C2×C42×D83C2211yescharacteristic
63#562×C83C22116yescomplemented
64C2×#125×C83C2211yescharacteristic
65C2×#124×C83C2212yes—
66C2×#123×C83C2214yes—
67C2×C22⋊Q8×C83C2214yes—
68C2×#121×C83C2214yes—
69C2×#13050591—48no—
70C2×#13050591C23116yes—
71C2×#13050590—416no—
72C2×#13050590C23132yes—
73C2×#13050589—48no—
74C2×#13050589—416no—
75C2×#13050589C23132yes—
76C2×#13050589C23116yes—
77C22×Dic166C2314yes—
78C22×Dic166C2312yes—
79C22×D332C2314yes—
80C22×D332C2312yes—
81C22×D332C2314yes—
82C22×C22⋊D83C2314yes—
83C22×C22⋊D83C2314yes—
84C22×C22⋊D83C2314yes—
85C22×C22⋊D83C2314yes—
86C22×C22⋊D83C2314yes—
87C23×Dic83C2314yes—
88C23×Dic83C2314yes—
89C22×C4×D83C2311yescharacteristic
90C22×C4×D83C2312yes—
91C22×C4×D83C2314yes—
92C22×C4×D83C2314yes—
93C22×C4×D83C2314yes—
94#14581791C23116yes—
95#14581790C23116yes—
96#14581788C23116yes—
97#14581788C23132yes—
98#14581784C2318yes—
99#14581782C23132yes—
100#14581780C23116yes—
101#14581777C23132yes—
102#14581776C23132yes—
103#14581774C23132yes—
104#14581770C23132yes—
105#14581770C23132yes—
106#14581770C23132yes—
107#14581769C23132yes—
108#14581769C23132yes—
109#14581768C23132yes—
110#14581767C23132yes—
111#14581767C23132yes—
112#14581766C23132yes—
113#14581765C23132yes—
114#14581765C23132yes—
115#14581765C23132yes—
116#14581765C23132yes—
117#14581765C23132yes—
118#14581764C23132yes—
119#14581764C23132yes—
120#14581764C23132yes—
121#14581757C23132yes—
122C4×C22⋊D83C23132yes—
123(C22⋊D83)⋊C4C23132yes—
124(C22⋊D83)⋊C4C23132yes—
125(C22⋊D83)⋊C4C23132yes—
126#14581750C23116yes—
127C2×C4⋊Dic83C2314yes—
128C2×C4⋊Dic83C2312yes—
129C2×Dic83⋊C4C23132yes—
130C2×Dic83⋊C4C2314yes—
131C2×Dic83⋊C4C2314yes—
132C2×Dic83⋊C4C2314yes—
133C2×Dic83⋊C4C2314yes—
134C2×Dic83⋊C4C2314yes—
135C2×Dic83⋊C4C2314yes—
136C4⋊C4×D83C23132yes—
137C2×C22⋊Dic83C2314yes—
138C2×(C2×D83)⋊C4C2314yes—
139C2×(C2×D83)⋊C4C2314yes—
140C2×(C2×D83)⋊C4C2314yes—
141C2×(C2×D83)⋊C4C2314yes—
142C2×(C2×D83)⋊C4C2314yes—
143C2×C4×Dic83C2311yescharacteristic
144C2×C4×Dic83C2312yes—
145C2×C4×Dic83C2314yes—
146C42×D83C2318yes—
147C22×Q8×C83C2312yes—
148C22×D4×C83C2314yes—
149C22×D4×C83C2312yes—
150C23×C4×C83C2314yes—
151#125×C83C2318yes—
152#124×C83C23116yes—
153#123×C83C23132yes—
154C22⋊Q8×C83C23132yes—
155#121×C83C23132yes—
156C2×C4⋊C4×C83C2314yes—
157C2×C4⋊C4×C83C2314yes—
158C2×C4⋊C4×C83C2312yes—
159C2×C22⋊C4×C83C2314yes—
160C2×C22⋊C4×C83C2314yes—
161C2×C22⋊C4×C83C2314yes—
162C2×C42×C83C2311yescharacteristic
163C23×D83C2411yescharacteristic
164C23×D83C2414yes—
165#13050591—432no—
166#13050590—464no—
167#13050589—464no—
168#13050589—432no—
169C2×Dic166—216no—
170C2×Dic166—44no—
171C2×Dic166C24116yes—
172C2×Dic166C2418yes—
173C2×D332—48no—
174C2×D332—48no—
175C2×D332—44no—
176C2×D332C24116yes—
177C2×D332C2418yes—
178C2×D332C24116yes—
179C2×C22⋊D83—216no—
180C2×C22⋊D83—216no—
181C2×C22⋊D83—48no—
182C2×C22⋊D83—48no—
183C2×C22⋊D83—48no—
184C2×C22⋊D83C24116yes—
185C2×C22⋊D83C24116yes—
186C2×C22⋊D83C24116yes—
187C2×C22⋊D83C24116yes—
188C2×C22⋊D83C24116yes—
189C22×Dic83—216no—
190C22×Dic83—48no—
191C22×Dic83C24116yes—
192C22×Dic83C2414yes—
193C22×Dic83C2414yes—
194C22×Dic83C24116yes—
195C22×Dic83C2412yes—
196C22×Dic83C2411yescharacteristic
197C2×C4×D83—28no—
198C2×C4×D83—216no—
199C2×C4×D83—24no—
200C2×C4×D83—48no—
201C2×C4×D83—48no—
202C2×C4×D83C2414yes—
203C2×C4×D83C2418yes—
204C2×C4×D83C24116yes—
205C2×C4×D83C24116yes—
206C2×C4×D83C24116yes—
207C4⋊Dic83C24116yes—
208C4⋊Dic83C2418yes—
209Dic83⋊C4C24116yes—
210Dic83⋊C4C24116yes—
211Dic83⋊C4C24116yes—
212Dic83⋊C4C24116yes—
213Dic83⋊C4C24116yes—
214Dic83⋊C4C24116yes—
215C22⋊Dic83C24116yes—
216(C2×D83)⋊C4C24116yes—
217(C2×D83)⋊C4C24116yes—
218(C2×D83)⋊C4C24116yes—
219(C2×D83)⋊C4C24116yes—
220(C2×D83)⋊C4C24116yes—
221C4×Dic83C24116yes—
222C4×Dic83C2418yes—
223C4×Dic83C2414yes—
224C24×C83C2414yes—
225C2×Q8×C83—44no—
226C2×Q8×C83C2418yes—
227C2×D4×C83—48no—
228C2×D4×C83—44no—
229C2×D4×C83C24116yes—
230C2×D4×C83C2418yes—
231C22×C4×C83—48no—
232C22×C4×C83C2414yes—
233C22×C4×C83C2412yes—
234C22×C4×C83C2414yes—
235C22×C4×C83C2411yescharacteristic
236C22×C4×C83C24116yes—
237C4⋊C4×C83C24116yes—
238C4⋊C4×C83C24116yes—
239C4⋊C4×C83C2418yes—
240C22⋊C4×C83C24116yes—
241C22⋊C4×C83C24116yes—
242C22⋊C4×C83C24116yes—
243C42×C83C2414yes—
244C23×C83—28no—
245C23×C83—28no—
246C23×C83—28no—
247C23×C83C2518yes—
248C23×C83C2511yescharacteristic
249C22×D83—24no—
250C22×D83—28no—
251C22×D83—28no—
252C22×D83—22no—
253C22×D83—28no—
254C22×D83C2518yes—
255C22×D83C2512yes—
256Q8×C83—48no—
257D4×C83—416no—
258D4×C83—48no—
259C2×C4×C83—28no—
260C2×C4×C83—28no—
261C2×C4×C83—24no—
262C2×C4×C83—22no—
263C2×C4×C83—416no—
264C2×C4×C83C2512yes—
265C2×C4×C83C2514yes—
266C2×C4×C83C2518yes—
267C2×C4×C83C2518yes—
268C22⋊D83—232no—
269C22⋊D83—232no—
270C22⋊D83—416no—
271C22⋊D83—416no—
272C22⋊D83—416no—
273C2×Dic83—28no—
274C2×Dic83—28no—
275C2×Dic83—232no—
276C2×Dic83—416no—
277C2×Dic83H(5,𝔽2)18yes—
278C2×Dic832−1+414yes—
279C2×Dic83C2518yes—
280C2×Dic83C2518yes—
281C2×Dic83C2514yes—
282C2×Dic83C2512yes—
283D332—416no—
284D332—48no—
285D332—416no—
286C4×D83—216no—
287C4×D83—232no—
288C4×D83—28no—
289C4×D83—416no—
290C4×D83—416no—
291Dic166—232no—
292Dic166—48no—
293C22×C83—28no—
294C22×C83—28no—
295C22×C83—28no—
296C22×C83—416no—
297C22×C83C2×H(5,𝔽2)14yes—
298C22×C83C2×2−1+412yes—
299C22×C83C2611yescharacteristic
300C2×D83—28no—
301C2×D83—24no—
302C2×D83—28no—
303C2×D83—22no—
304C2×D83—28no—
305C2×D83—44no—
306C2×D83—416no—
307C4×C83—28no—
308C4×C83—24no—
309C4×C83—28no—
310C4×C83—22no—
311Dic83—28no—
312Dic83—28no—
313Dic83C2×H(5,𝔽2)18yes—
314Dic83C2×2−1+414yes—
315C2×#3548—831nomaximal, Sylow
316C2×C83—48no—
317C2×C83#354814yescomplemented
318C2×C83C22×H(5,𝔽2)12yes—
319C2×C83C22×2−1+411yescharacteristic
320D83—48no—
321D83—42no—
322#3548—8332no—
323C2×#562—831no—
324C2×#562—832no—
325C2×#562D8311yescharacteristic, complemented
326C2×#554—834no—
327C2×#550—834no—
328C2×#547—834no—
329C2×#540—834no—
330C2×#537—834no—
331C2×#537—834no—
332C2×#533—831no—
333C2×#532—832no—
334C83C2×#354811yescharacteristic, complemented, Sylow
335C22×#41—832no—
336C22×#41—832no—
337C22×#41—834no—
338C22×#41—834no—
339#562—8316no—
340#562—8332no—
341#562C2×D83116yescomplemented
342#554—8364no—
343#550—8364no—
344#547—8364no—
345#540—8364no—
346#537—8364no—
347#537—8364no—
348#533—8316no—
349#532—8332no—
350C2×C4⋊Q8—832no—
351C2×C4⋊Q8—832no—
352C2×#127—832no—
353C2×#127—834no—
354C2×#125—834no—
355C2×#125—831no—
356C2×#125C2×D8311yescharacteristic
357C2×#124—832no—
358C2×#124C2×D8312yes—
359C2×#123—834no—
360C2×#123—834no—
361C2×#123—834no—
362C2×#123C2×D8314yes—
363C2×C22⋊Q8—834no—
364C2×C22⋊Q8—834no—
365C2×C22⋊Q8—834no—
366C2×C22⋊Q8—834no—
367C2×C22⋊Q8—834no—
368C2×C22⋊Q8—834no—
369C2×C22⋊Q8C2×D8314yes—
370C2×#121—834no—
371C2×#121—834no—
372C2×#121—834no—
373C2×#121—834no—
374C2×#121—834no—
375C2×#121—834no—
376C2×#121—834no—
377C2×#121—834no—
378C2×#121—834no—
379C2×#121—834no—
380C2×#121—834no—
381C2×#121C2×D8314yes—
382C2×C4×Q8—834no—
383C2×C4×D4—834no—
384C2×C4×D4—834no—
385C2×C4×D4—834no—
386C2×C4×D4—834no—
387C2×#117—832no—
388C22×C4⋊C4—834no—
389C22×C4⋊C4—834no—
390C22×C42—831no—
391C2×#41—33216no—
392C2×#41—33216no—
393C2×#41—3328no—
394C2×#41—3328no—
395C2×#41—8316no—
396C2×#41—8316no—
397C2×#41—8332no—
398C2×#41—8332no—
399C22×Q8—832no—
400C22×Q8—834no—
401C22×Q8C22×D8312yes—
402C22×D4—834no—
403C22×D4—834no—
404C22×D4—834no—
405C22×D4—834no—
406C22×D4—834no—
407C22×D4—834no—
408C22×D4—834no—
409C22×D4—832no—
410C22×D4C22×D8314yes—
411C22×D4C22×D8312yes—
412C23×C4—834no—
413C23×C4—834no—
414C23×C4—834no—
415C23×C4—834no—
416C23×C4—834no—
417C23×C4—831no—
418C23×C4—832no—
419C23×C4C22×D8314yes—
420C4⋊Q8—8316no—
421C4⋊Q8—8316no—
422#127—8316no—
423#127—8332no—
424#125—8332no—
425#125—838no—
426#125C22×D8318yes—
427#124—8316no—
428#124C22×D83116yes—
429#123—8332no—
430#123—8332no—
431#123—8332no—
432#123C22×D83132yes—
433C22⋊Q8—8332no—
434C22⋊Q8—8332no—
435C22⋊Q8—8332no—
436C22⋊Q8—8332no—
437C22⋊Q8—8332no—
438C22⋊Q8—8332no—
439C22⋊Q8C22×D83132yes—
440#121—8332no—
441#121—8332no—
442#121—8332no—
443#121—8332no—
444#121—8332no—
445#121—8332no—
446#121—8332no—
447#121—8332no—
448#121—8332no—
449#121—8332no—
450#121—8332no—
451#121C22×D83132yes—
452C4×Q8—8332no—
453C4×D4—8332no—
454C4×D4—8332no—
455C4×D4—8332no—
456C4×D4—8332no—
457#117—8316no—
458C2×C4⋊C4—834no—
459C2×C4⋊C4—834no—
460C2×C4⋊C4—834no—
461C2×C4⋊C4—834no—
462C2×C4⋊C4—834no—
463C2×C4⋊C4—8332no—
464C2×C4⋊C4—832no—
465C2×C4⋊C4—834no—
466C2×C4⋊C4—834no—
467C2×C4⋊C4—8332no—
468C2×C4⋊C4C22×D8314yes—
469C2×C4⋊C4C22×D8312yes—
470C2×C4⋊C4C22×D8314yes—
471C2×C22⋊C4—834no—
472C2×C22⋊C4—834no—
473C2×C22⋊C4—834no—
474C2×C22⋊C4—834no—
475C2×C22⋊C4—834no—
476C2×C22⋊C4—834no—
477C2×C22⋊C4C22×D8314yes—
478C2×C22⋊C4C22×D8314yes—
479C2×C22⋊C4C22×D8314yes—
480C2×C42—834no—
481C2×C42—831no—
482C2×C42—832no—
483C2×C42—838no—
484C2×C42C22×D8311yescharacteristic
485C24—834no—
486C24—831no—
487C24C23×D8314yes—
488#41—33264no—
489#41—33264no—
490#41—33232no—
491#41—33232no—
492C2×Q8—44no—
493C2×Q8—3324no—
494C2×Q8—838no—
495C2×Q8—8316no—
496C2×Q8—16616no—
497C2×Q8C23×D8318yes—
498C2×D4—48no—
499C2×D4—44no—
500C2×D4—3328no—
501C2×D4—3328no—
502C2×D4—3324no—
503C2×D4—3328no—
504C2×D4—3328no—
505C2×D4—3328no—
506C2×D4—8316no—
507C2×D4—8316no—
508C2×D4—8316no—
509C2×D4—8316no—
510C2×D4—8316no—
511C2×D4—8316no—
512C2×D4—8316no—
513C2×D4—838no—
514C2×D4—16616no—
515C2×D4—16616no—
516C2×D4C23×D83116yes—
517C2×D4C23×D8318yes—
518C22×C4—48no—
519C22×C4—3328no—
520C22×C4—3328no—
521C22×C4—3328no—
522C22×C4—8316no—
523C22×C4—8316no—
524C22×C4—8316no—
525C22×C4—831no—
526C22×C4—832no—
527C22×C4—8316no—
528C22×C4—834no—
529C22×C4—834no—
530C22×C4—834no—
531C22×C4—838no—
532C22×C4—8316no—
533C22×C4—16616no—
534C22×C4—1668no—
535C22×C4—1664no—
536C22×C4—16616no—
537C22×C4C23×D8311yescharacteristic
538C22×C4C23×D8312yes—
539C22×C4C23×D8314yes—
540C22×C4C23×D8314yes—
541C22×C4C23×D83116yes—
542C4⋊C4—8316no—
543C4⋊C4—8316no—
544C4⋊C4—8316no—
545C4⋊C4—8316no—
546C4⋊C4—8316no—
547C4⋊C4—838no—
548C4⋊C4—8316no—
549C4⋊C4—8316no—
550C4⋊C4C23×D83116yes—
551C4⋊C4C23×D8318yes—
552C4⋊C4C23×D83116yes—
553C22⋊C4—8316no—
554C22⋊C4—8316no—
555C22⋊C4—8316no—
556C22⋊C4—8316no—
557C22⋊C4—8316no—
558C22⋊C4—8316no—
559C22⋊C4C23×D83116yes—
560C22⋊C4C23×D83116yes—
561C22⋊C4C23×D83116yes—
562C42—834no—
563C42—838no—
564C42—8316no—
565C42C23×D8314yes—
566C23—28no—
567C23—28no—
568C23—28no—
569C23—838no—
570C23—832no—
571C23—1668no—
572C23—1668no—
573C23—1664no—
574C23—1662no—
575C23—1668no—
576C23C24×D8318yes—
577C23C24×D8311yescharacteristic
578Q8—48no—
579Q8—3328no—
580Q8—16632no—
581D4—416no—
582D4—48no—
583D4—33216no—
584D4—33216no—
585D4—3328no—
586D4—33216no—
587D4—33216no—
588D4—33216no—
589D4—16632no—
590D4—16632no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—416no—
596C2×C4—33216no—
597C2×C4—33216no—
598C2×C4—33216no—
599C2×C4—838no—
600C2×C4—834no—
601C2×C4—832no—
602C2×C4—834no—
603C2×C4—838no—
604C2×C4—838no—
605C2×C4—16632no—
606C2×C4—16616no—
607C2×C4—1668no—
608C2×C4—16632no—
609C2×C4—1668no—
610C2×C4—1668no—
611C2×C4C24×D8312yes—
612C2×C4C24×D8314yes—
613C2×C4C24×D8318yes—
614C2×C4C24×D8318yes—
615C22—28no—
616C22—28no—
617C22—28no—
618C22—416no—
619C22—33216no—
620C22—3324no—
621C22—1668no—
622C22—1668no—
623C22—1664no—
624C22—1662no—
625C22—1668no—
626C22H(5,𝔽2)×D8314yes—
627C222−1+4×D8312yes—
628C22C25×D8311yescharacteristic
629C4—28no—
630C4—28no—
631C4—24no—
632C4—22no—
633C4—838no—
634C4—834no—
635C4—1668no—
636C4—1668no—
637C2—48no—
638C2—3328no—
639C2—3322no—
640C2#2492185914yesdirect product factor
641C2C2×H(5,𝔽2)×D8312yes—
642C2C2×2−1+4×D8311yescharacteristic
subgroupG/H
centerC23C24×D83
commutator, cosocleC22×C83C26
FrattiniC22C25×D83
FittingC2×#562×C83C2
socleC23×C83C25
hypercenterC2×#562D83
hypocenterC83C2×#3548
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#562

Complements

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

NH ≅ G/Ntype
1. #24921859C2direct (N × H)
2. C2×#19128449C2semidirect
3. C2×#19128445C2semidirect
4. C2×#19128405C2semidirect
5. C2×#19128369C2semidirect
6. C2×#19128349C2semidirect
7. C2×#19128303C2semidirect
8. C2×#19128281C2semidirect
9. C2×#19128278C2semidirect
10. C2×#19128257C2semidirect
11. C2×#19128250C2semidirect
12. C2×#562×C83C2semidirect
17. #19128449C22semidirect
18. #19128445C22semidirect
19. #19128405C22semidirect
20. #19128369C22semidirect
21. #19128349C22semidirect
22. #19128303C22semidirect
23. #19128281C22semidirect
24. #19128278C22semidirect
25. #19128257C22semidirect
26. #19128250C22semidirect
63. #562×C83C22semidirect
317. C2×C83#3548semidirect
325. C2×#562D83semidirect
334. C83C2×#3548semidirect
341. #562C2×D83semidirect
640. C2#24921859direct (N × H)

Maximal subgroups

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

subgroupindex
#249218592
C2×#191284492
C2×#191284452
C2×#191284052
C2×#191283692
C2×#191283492
C2×#191283032
C2×#191282812
C2×#191282782
C2×#191282572
C2×#191282502
C2×#562×C832
C2×#354883