Finite Group Databasebeta

← C2×#24911311 #35572314

Subgroups — C2×#24911311 #35572314

Order 21248.

3614 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581791C211yesmaximal, characteristic, complemented
2C22×#14581772C212yesmaximal, complemented
3C22×#14581754C211yesmaximal, characteristic
4#24911311C2116yesmaximal, direct product factor
5C2×#19127623C212yesmaximal, complemented
6C2×#19127614C212yesmaximal, complemented
7C2×#19127603C211yesmaximal, characteristic
8C2×#19127602C211yesmaximal, characteristic
9C2×#19127600C211yesmaximal, characteristic, complemented
10C2×#19127575C212yesmaximal, complemented
11C2×C4×C4⋊Dic83C211yesmaximal, characteristic, complemented
12C2×#389×C83C211yesmaximal, characteristic, complemented
13C23×Dic166C2212yes—
14C23×D332C2211yescharacteristic
15C23×C4×D83C2212yes—
16C2×#14581791—28no—
17C2×#14581791C2218yescomplemented
18C2×#14581772—216no—
19C2×#14581772C22116yescomplemented
20C2×#14581754—28no—
21C2×#14581754C2218yes—
22C22×C4⋊Dic83C2212yes—
23C22×C4⋊Dic83C2211yescharacteristic
24C22×C4⋊Dic83C2211yescharacteristic
25C22×C4⋊Dic83C2211yescharacteristic
26C22×Dic83⋊C4C2212yes—
27C22×(C2×D83)⋊C4C2212yes—
28C22×(C2×D83)⋊C4C2212yes—
29C22×(C2×D83)⋊C4C2212yes—
30C22×(C2×D83)⋊C4C2212yes—
31C22×(C2×D83)⋊C4C2212yes—
32C22×C4×Dic83C2211yescharacteristic
33C22×C4×Dic83C2211yescharacteristic
34#19127623C22116yescomplemented
35#19127614C22116yescomplemented
36#19127603C2218yes—
37#19127602C2218yes—
38#19127600C2218yescomplemented
39#19127575C22116yescomplemented
40C4×C4⋊Dic83C2218yescomplemented
41C2×C83⋊#95C2211yescharacteristic
42C2×C83⋊#95C2211yescharacteristic
43C2×C83⋊#95C2212yes—
44C22×C4⋊C4×C83C2211yescharacteristic
45C22×C4⋊C4×C83C2212yes—
46C22×C4⋊C4×C83C2211yescharacteristic
47C22×C42×C83C2211yescharacteristic
48#389×C83C2218yescomplemented
49C2×#95×C83C2212yes—
50C24×D83C2312yes—
51C22×Dic166—28no—
52C22×Dic166—28no—
53C22×Dic166—44no—
54C22×Dic166C2318yes—
55C22×D332—24no—
56C22×D332—24no—
57C22×D332Q818yescomplemented
58C22×D332C2314yes—
59C23×Dic83C2311yescharacteristic
60C23×Dic83C2312yes—
61C23×Dic83C2311yescharacteristic
62C23×Dic83C2311yescharacteristic
63C23×Dic83C2311yescharacteristic
64C22×C4×D83—28no—
65C22×C4×D83—28no—
66C22×C4×D83—44no—
67C22×C4×D83C2318yes—
68#14581791—232no—
69#14581772—264no—
70#14581754—232no—
71C2×C4⋊Dic83—28no—
72C2×C4⋊Dic83—24no—
73C2×C4⋊Dic83—24no—
74C2×C4⋊Dic83D418yescomplemented
75C2×C4⋊Dic83C2314yes—
76C2×C4⋊Dic83C2318yes—
77C2×C4⋊Dic83C2314yes—
78C2×C4⋊Dic83C2314yes—
79C2×Dic83⋊C4—28no—
80C2×Dic83⋊C4C2318yes—
81C2×(C2×D83)⋊C4—28no—
82C2×(C2×D83)⋊C4—28no—
83C2×(C2×D83)⋊C4—28no—
84C2×(C2×D83)⋊C4—28no—
85C2×(C2×D83)⋊C4—44no—
86C2×(C2×D83)⋊C4C2318yes—
87C2×(C2×D83)⋊C4C2318yes—
88C2×(C2×D83)⋊C4C2318yes—
89C2×(C2×D83)⋊C4C2318yes—
90C2×(C2×D83)⋊C4C2318yes—
91C2×C4×Dic83—24no—
92C2×C4×Dic83—24no—
93C2×C4×Dic83C2314yes—
94C2×C4×Dic83C2314yes—
95C83⋊#95C2314yes—
96C83⋊#95C2314yes—
97C83⋊#95C2318yes—
98C23×C4×C83C2311yescharacteristic
99C23×C4×C83C2312yes—
100C23×C4×C83C2311yescharacteristic
101C23×C4×C83C2312yes—
102C23×C4×C83C2311yescharacteristic
103C2×C4⋊C4×C83—28no—
104C2×C4⋊C4×C83—24no—
105C2×C4⋊C4×C83D418yes—
106C2×C4⋊C4×C83C2314yes—
107C2×C4⋊C4×C83C2318yes—
108C2×C4⋊C4×C83C2314yes—
109C2×C42×C83—24no—
110C2×C42×C83C2314yes—
111#95×C83C2318yes—
112C23×D83—24no—
113C23×D83—24no—
114C23×D83—24no—
115C23×D83—24no—
116C23×D83—24no—
117C23×D83—24no—
118C23×D83C2×Q818yes—
119C23×D83C2414yes—
120C2×Dic166—216no—
121C2×Dic166—216no—
122C2×Dic166—48no—
123C2×Dic166—432no—
124C2×D332—232no—
125C2×D332—28no—
126C2×D332—28no—
127C2×D332C2×Q8116yescomplemented
128C22×Dic83—22no—
129C22×Dic83—22no—
130C22×Dic83—22no—
131C22×Dic83—22no—
132C22×Dic83—22no—
133C22×Dic83—22no—
134C22×Dic83—24no—
135C22×Dic83—24no—
136C22×Dic83—22no—
137C22×Dic83—22no—
138C22×Dic83—24no—
139C22×Dic83—22no—
140C22×Dic83C2×D414yes—
141C22×Dic83C2×D414yes—
142C22×Dic83#4114yes—
143C22×Dic83C2414yes—
144C22×Dic83C2412yes—
145C22×Dic83C2412yes—
146C22×Dic83C2412yes—
147C22×Dic83C2412yes—
148C2×C4×D83—216no—
149C2×C4×D83—216no—
150C2×C4×D83—48no—
151C2×C4×D83—432no—
152C4⋊Dic83—28no—
153C4⋊Dic83—216no—
154C4⋊Dic83—28no—
155C4⋊Dic83C2×D4116yescomplemented
156Dic83⋊C4—216no—
157(C2×D83)⋊C4—216no—
158(C2×D83)⋊C4—216no—
159(C2×D83)⋊C4—216no—
160(C2×D83)⋊C4—216no—
161(C2×D83)⋊C4—48no—
162C4×Dic83—28no—
163C4×Dic83—28no—
164C24×C83C2411yescharacteristic
165C22×C4×C83—24no—
166C22×C4×C83—22no—
167C22×C4×C83—22no—
168C22×C4×C83—24no—
169C22×C4×C83—24no—
170C22×C4×C83—22no—
171C22×C4×C83—24no—
172C22×C4×C83—24no—
173C22×C4×C83—22no—
174C22×C4×C83—22no—
175C22×C4×C83C2×D414yes—
176C22×C4×C83C2×D418yes—
177C22×C4×C83C2×D414yes—
178C22×C4×C83C2×Q814yes—
179C22×C4×C83#4114yes—
180C22×C4×C83C2412yes—
181C22×C4×C83C2414yes—
182C22×C4×C83C2414yes—
183C22×C4×C83C2412yes—
184C22×C4×C83C2412yes—
185C4⋊C4×C83—216no—
186C4⋊C4×C83—28no—
187C4⋊C4×C83C2×D4116yes—
188C42×C83—28no—
189C23×C83C22×D412yes—
190C23×C83C22×D412yes—
191C23×C83C22×Q812yes—
192C23×C83C2×#4112yes—
193C23×C83C2×#4112yes—
194C23×C83H(5,𝔽2)12yes—
195C23×C832−1+412yes—
196C23×C83C2511yescharacteristic
197C22×D83—216no—
198C22×D83—216no—
199C22×D83—216no—
200C22×D83—24no—
201C22×D83—24no—
202C22×D83—24no—
203C22×D83—24no—
204C22×D83—24no—
205C22×D83—24no—
206C22×D83—48no—
207C22×D83—48no—
208C22×D83—48no—
209C22×D83—48no—
210C22×D83C22×Q818yes—
211C2×C4×C83—28no—
212C2×C4×C83—216no—
213C2×C4×C83—24no—
214C2×C4×C83—22no—
215C2×C4×C83—22no—
216C2×C4×C83—24no—
217C2×C4×C83—24no—
218C2×C4×C83—22no—
219C2×C4×C83—24no—
220C2×C4×C83—24no—
221C2×C4×C83—22no—
222C2×C4×C83—22no—
223C2×C4×C83—48no—
224C2×C4×C83—44no—
225C2×C4×C83C22⋊Q8116yescomplemented
226C2×C4×C83C22×D414yes—
227C2×C4×C83C22×D418yes—
228C2×C4×C83C22×D414yes—
229C2×C4×C83C22×Q814yes—
230C2×C4×C83C2×#4114yes—
231C2×Dic83—28no—
232C2×Dic83—28no—
233C2×Dic83—28no—
234C2×Dic83—22no—
235C2×Dic83—22no—
236C2×Dic83—22no—
237C2×Dic83—22no—
238C2×Dic83—22no—
239C2×Dic83—22no—
240C2×Dic83—22no—
241C2×Dic83—24no—
242C2×Dic83—24no—
243C2×Dic83—22no—
244C2×Dic83—24no—
245C2×Dic83—22no—
246C2×Dic83—48no—
247C2×Dic83—44no—
248C2×Dic83C22×D414yes—
249C2×Dic83C22×D414yes—
250C2×Dic83C2×#4114yes—
251D332—232no—
252C4×D83—432no—
253Dic166—432no—
254C22×C83C2×C22⋊Q814yes—
255C22×C83#53214yes—
256C22×C83#54214yes—
257C22×C83D4214yes—
258C22×C83#54814yes—
259C22×C83D4×Q814yes—
260C22×C83#55414yes—
261C22×C83C23×D411yescharacteristic
262C22×C83C23×D411yescharacteristic
263C22×C83C23×Q811yescharacteristic
264C22×C83C22×#4111yescharacteristic
265C22×C83C22×#4111yescharacteristic
266C22×C83C2×H(5,𝔽2)11yescharacteristic
267C22×C83C2×2−1+411yescharacteristic
268C2×D83—28no—
269C2×D83—28no—
270C2×D83—28no—
271C2×D83—44no—
272C2×D83—44no—
273C2×D83—44no—
274C2×D83—44no—
275C2×D83—432no—
276C4×C83—24no—
277C4×C83—28no—
278C4×C83—44no—
279C4×C83—42no—
280C4×C83C2×C22⋊Q818yescomplemented
281Dic83—24no—
282Dic83—24no—
283Dic83—24no—
284Dic83—44no—
285Dic83—42no—
286C2×#2452—831nomaximal, Sylow
287C2×C83#245218yescomplemented
288C2×C83C22×C22⋊Q811yescharacteristic
289C2×C83C2×#53211yescharacteristic
290C2×C83C2×#54211yescharacteristic
291C2×C83C2×D4211yescharacteristic
292C2×C83C2×#54811yescharacteristic
293C2×C83C2×D4×Q811yescharacteristic
294C2×C83C2×#55411yescharacteristic
295D83—48no—
296C22×C4⋊Q8—831no—
297C22×C22⋊Q8—832no—
298C22×C4×D4—831no—
299#2452—8316no—
300C2×#396—832no—
301C2×#393—832no—
302C2×#390—831no—
303C2×#390—831no—
304C2×#389—831no—
305C2×#389D8311yescharacteristic, complemented
306C2×#385—832no—
307C2×C4×C4⋊C4—831no—
308C83C2×#245211yescharacteristic, complemented, Sylow
309C23×Q8—832no—
310C23×D4—831no—
311C24×C4—832no—
312C2×C4⋊Q8—838no—
313C2×C4⋊Q8—1668no—
314C2×C22⋊Q8—8316no—
315C2×C22⋊Q8—16616no—
316C2×C4×D4—838no—
317C2×C4×D4—1668no—
318C22×C4⋊C4—831no—
319C22×C4⋊C4—831no—
320C22×C4⋊C4—832no—
321C22×C4⋊C4—832no—
322C22×C4⋊C4—831no—
323C22×C4⋊C4C2×D8312yes—
324C22×C4⋊C4C2×D8311yescharacteristic
325C22×C4⋊C4C2×D8311yescharacteristic
326C22×C22⋊C4—832no—
327C22×C22⋊C4—832no—
328C22×C22⋊C4—832no—
329C22×C22⋊C4—832no—
330C22×C22⋊C4—832no—
331C22×C42—831no—
332C22×C42—831no—
333C22×C42C2×D8311yescharacteristic
334#396—8316no—
335#393—8316no—
336#390—838no—
337#390—838no—
338#389—838no—
339#389C2×D8318yescomplemented
340#385—8316no—
341C4×C4⋊C4—838no—
342C2×#95—831no—
343C2×#95—832no—
344C2×#95—831no—
345C2×#95C2×D8312yes—
346C25—832no—
347C22×Q8—3324no—
348C22×Q8—838no—
349C22×Q8—1668no—
350C22×Q8—1668no—
351C22×D4—838no—
352C22×D4—834no—
353C22×D4—1664no—
354C22×D4—1664no—
355C23×C4—3324no—
356C23×C4—838no—
357C23×C4—831no—
358C23×C4—831no—
359C23×C4—831no—
360C23×C4—832no—
361C23×C4—831no—
362C23×C4—1668no—
363C23×C4—1668no—
364C23×C4C22×D8312yes—
365C23×C4C22×D8311yescharacteristic
366C23×C4C22×D8311yescharacteristic
367C23×C4C22×D8312yes—
368C23×C4C22×D8311yescharacteristic
369C4⋊Q8—16632no—
370C22⋊Q8—16664no—
371C4×D4—16632no—
372C2×C4⋊C4—28no—
373C2×C4⋊C4—24no—
374C2×C4⋊C4—838no—
375C2×C4⋊C4—834no—
376C2×C4⋊C4—838no—
377C2×C4⋊C4—838no—
378C2×C4⋊C4—834no—
379C2×C4⋊C4—834no—
380C2×C4⋊C4—1668no—
381C2×C4⋊C4—1664no—
382C2×C4⋊C4—1668no—
383C2×C4⋊C4—1664no—
384C2×C4⋊C4D33218yes—
385C2×C4⋊C4C22×D8318yes—
386C2×C4⋊C4C22×D8314yes—
387C2×C4⋊C4C22×D8314yes—
388C2×C22⋊C4—3324no—
389C2×C22⋊C4—838no—
390C2×C22⋊C4—838no—
391C2×C22⋊C4—838no—
392C2×C22⋊C4—838no—
393C2×C22⋊C4—838no—
394C2×C22⋊C4—1668no—
395C2×C22⋊C4—1668no—
396C2×C22⋊C4—1668no—
397C2×C22⋊C4—1668no—
398C2×C42—24no—
399C2×C42—834no—
400C2×C42—834no—
401C2×C42—1664no—
402C2×C42—1664no—
403C2×C42C22×D8314yes—
404#95—834no—
405#95—834no—
406#95—838no—
407#95C22×D8318yes—
408C24—838no—
409C24—834no—
410C24—1664no—
411C24—1664no—
412C24—1664no—
413C24—1664no—
414C24—1664no—
415C24—1664no—
416C24C23×D8311yescharacteristic
417C2×Q8—33232no—
418C2×Q8—3328no—
419C2×Q8—16616no—
420C2×Q8—16616no—
421C2×D4—8316no—
422C2×D4—16632no—
423C2×D4—1668no—
424C2×D4—1668no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—33232no—
436C22×C4—3328no—
437C22×C4—834no—
438C22×C4—834no—
439C22×C4—834no—
440C22×C4—832no—
441C22×C4—832no—
442C22×C4—832no—
443C22×C4—832no—
444C22×C4—834no—
445C22×C4—16616no—
446C22×C4—1664no—
447C22×C4—1662no—
448C22×C4—1664no—
449C22×C4—1662no—
450C22×C4—1664no—
451C22×C4—1662no—
452C22×C4—16616no—
453C22×C4—1662no—
454C22×C4—1662no—
455C22×C4—1662no—
456C22×C4—1662no—
457C22×C4—1662no—
458C22×C4—1662no—
459C22×C4D4×D8314yes—
460C22×C4C2×D33218yes—
461C22×C4C2×D33214yes—
462C22×C4Q8×D8314yes—
463C22×C4#1305058914yes—
464C22×C4C23×D8312yes—
465C22×C4C23×D8314yes—
466C22×C4C23×D8312yes—
467C22×C4C23×D8314yes—
468C22×C4C23×D8312yes—
469C4⋊C4—216no—
470C4⋊C4—28no—
471C4⋊C4—8316no—
472C4⋊C4—16616no—
473C4⋊C4—1668no—
474C4⋊C4—16616no—
475C4⋊C4—1668no—
476C4⋊C4C2×D332116yes—
477C22⋊C4—3328no—
478C22⋊C4—16616no—
479C22⋊C4—16616no—
480C22⋊C4—16616no—
481C22⋊C4—16616no—
482C42—28no—
483C42—1668no—
484C42—1668no—
485C23—3328no—
486C23—3328no—
487C23—3328no—
488C23—3328no—
489C23—838no—
490C23—16616no—
491C23—16616no—
492C23—16616no—
493C23—1664no—
494C23—1664no—
495C23—1664no—
496C23—1664no—
497C23—1664no—
498C23—1664no—
499C23C2×D4×D8312yes—
500C23C22×D33212yes—
501C23C2×Q8×D8312yes—
502C23#41×D8312yes—
503C23C2×#1305058912yes—
504C23#1458183812yes—
505C23#1458184012yes—
506C23C24×D8311yescharacteristic
507Q8—33232no—
508D4—16632no—
509C2×C4—216no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—44no—
522C2×C4—48no—
523C2×C4—33232no—
524C2×C4—3328no—
525C2×C4—3324no—
526C2×C4—834no—
527C2×C4—834no—
528C2×C4—834no—
529C2×C4—1664no—
530C2×C4—1662no—
531C2×C4—1664no—
532C2×C4—1662no—
533C2×C4—1664no—
534C2×C4—1662no—
535C2×C4—1668no—
536C2×C4—1668no—
537C2×C4—1668no—
538C2×C4—1662no—
539C2×C4—1662no—
540C2×C4—1662no—
541C2×C4—1662no—
542C2×C4—1662no—
543C2×C4—1662no—
544C2×C4#14581772116yescomplemented
545C2×C4C2×D4×D8314yes—
546C2×C4C22×D33218yes—
547C2×C4C22×D33214yes—
548C2×C4C2×Q8×D8314yes—
549C2×C4C2×#1305058914yes—
550C22—3324no—
551C22—3324no—
552C22—3324no—
553C22—3324no—
554C22—33232no—
555C22—1668no—
556C22—1668no—
557C22—1668no—
558C22C2×#1458177214yes—
559C22#1912824614yes—
560C22#1912832414yes—
561C22D4×D33214yes—
562C22#1912836314yes—
563C22#1912836614yes—
564C22#1912840714yes—
565C22C22×D4×D8311yescharacteristic
566C22C23×D33211yescharacteristic
567C22C22×Q8×D8311yescharacteristic
568C22C2×#41×D8311yescharacteristic
569C22C22×#1305058911yescharacteristic
570C22C2×#1458183811yescharacteristic
571C22C2×#1458184011yescharacteristic
572C4—28no—
573C4—24no—
574C4—42no—
575C4—44no—
576C4—3324no—
577C4—3322no—
578C4—1664no—
579C4—1664no—
580C4—1664no—
581C4C2×#1458177218yescomplemented
582C2—3328no—
583C2#2491131118yesdirect product factor
584C2C22×#1458177211yescharacteristic
585C2C2×#1912824611yescharacteristic
586C2C2×#1912832411yescharacteristic
587C2C2×D4×D33211yescharacteristic
588C2C2×#1912836311yescharacteristic
589C2C2×#1912836611yescharacteristic
590C2C2×#1912840711yescharacteristic
subgroupG/H
centerC24C23×D83
commutator, cosocleC23×C83C25
FrattiniC23C24×D83
FittingC2×#389×C83C2
socleC24×C83C24
prime radicalC23×D332C22
hypercenterC2×#389D83
hypocenterC83C2×#2452
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C22×#14581791C2semidirect
2. C22×#14581772C2semidirect
4. #24911311C2direct (N × H)
5. C2×#19127623C2semidirect
6. C2×#19127614C2semidirect
9. C2×#19127600C2semidirect
10. C2×#19127575C2semidirect
11. C2×C4×C4⋊Dic83C2semidirect
12. C2×#389×C83C2semidirect
17. C2×#14581791C22semidirect
19. C2×#14581772C22semidirect
34. #19127623C22semidirect
35. #19127614C22semidirect
38. #19127600C22semidirect
39. #19127575C22semidirect
40. C4×C4⋊Dic83C22semidirect
48. #389×C83C22semidirect
57. C22×D332Q8semidirect
74. C2×C4⋊Dic83D4semidirect
127. C2×D332C2×Q8semidirect
155. C4⋊Dic83C2×D4semidirect
225. C2×C4×C83C22⋊Q8semidirect
280. C4×C83C2×C22⋊Q8semidirect
287. C2×C83#2452semidirect
305. C2×#389D83semidirect
308. C83C2×#2452semidirect
339. #389C2×D83semidirect
544. C2×C4#14581772semidirect
581. C4C2×#14581772semidirect
583. C2#24911311direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145817912
C22×#145817722
C22×#145817542
#249113112
C2×#191276232
C2×#191276142
C2×#191276032
C2×#191276022
C2×#191276002
C2×#191275752
C2×C4×C4⋊Dic832
C2×#389×C832
C2×#245283