Finite Group Databasebeta

← C2×#22589529 #32024916

Subgroups — C2×#22589529 #32024916

Order 18176.

2006 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556102C211yesmaximal, characteristic
2C22×#14556059C211yesmaximal, characteristic, complemented
3C22×#14556048C211yesmaximal, characteristic, complemented
4C22×#14556048C211yesmaximal, characteristic, complemented
5C22×C22⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
6#22589529C218yesmaximal, direct product factor
7C2×#17985761C211yesmaximal, characteristic, complemented
8C2×#380×C71C211yesmaximal, characteristic, complemented
9C23×D284C2211yescharacteristic
10C23×C22⋊D71—21no—
11C23×C71⋊C8C2211yescharacteristic
12C23×C71⋊C8C2211yescharacteristic
13C2×#14556102—24no—
14C2×#14556102—22no—
15C2×#14556102—24no—
16C2×#14556102C2214yes—
17C22×C4⋊Dic71C2211yescharacteristic
18C22×(C2×D71)⋊C4—21no—
19C2×#14556059—24no—
20C2×#14556059C2214yescomplemented
21C2×#14556048—24no—
22C2×#14556048—24no—
23C2×#14556048C2214yescomplemented
24C2×#14556048C2214yescomplemented
25C2×C22⋊(C71⋊C8)—24no—
26C2×C22⋊(C71⋊C8)C2214yescomplemented
27#17985761C2214yescomplemented
28C24×C4×C71C2211yescharacteristic
29C22×C4⋊C4×C71C2211yescharacteristic
30C22×C4⋊C4×C71C2211yescharacteristic
31C22×C22⋊C4×C71—21no—
32#380×C71—22no—
33#380×C71C2214yescomplemented
34C2×#95×C71—21no—
35C24×D71—21no—
36C22×D284—22no—
37C22×D284—22no—
38C22×D284—21no—
39C22×D284—22no—
40C22×D284—22no—
41C22×D284—22no—
42C22×D284D414yes—
43C22×D284C2312yes—
44C22×C22⋊D71—28no—
45C22×C22⋊D71—24no—
46C22×C22⋊D71—44no—
47C22×C22⋊D71—44no—
48C23×Dic71—21no—
49C22×C71⋊C8—22no—
50C22×C71⋊C8—22no—
51C22×C71⋊C8—22no—
52C22×C71⋊C8—22no—
53C22×C71⋊C8—22no—
54C22×C71⋊C8D414yescomplemented
55C22×C71⋊C8C2312yes—
56C22×C71⋊C8C2312yes—
57#14556102—216no—
58#14556102—28no—
59#14556102—28no—
60C2×C4⋊Dic71—22no—
61C2×C4⋊Dic71—21no—
62C2×C4⋊Dic71D414yes—
63C2×C4⋊Dic71C2312yes—
64C2×(C2×D71)⋊C4—28no—
65C2×(C2×D71)⋊C4—24no—
66#14556059—28no—
67#14556048—28no—
68#14556048—28no—
69C22⋊(C71⋊C8)—28no—
70C25×C71D411yescharacteristic
71C23×C4×C71—22no—
72C23×C4×C71—22no—
73C23×C4×C71—22no—
74C23×C4×C71—22no—
75C23×C4×C71—22no—
76C23×C4×C71—22no—
77C23×C4×C71—21no—
78C23×C4×C71—21no—
79C23×C4×C71D411yescharacteristic
80C23×C4×C71D412yes—
81C23×C4×C71C2312yes—
82C23×C4×C71C2311yescharacteristic
83C2×C4⋊C4×C71—22no—
84C2×C4⋊C4×C71—22no—
85C2×C4⋊C4×C71—22no—
86C2×C4⋊C4×C71—22no—
87C2×C4⋊C4×C71—21no—
88C2×C4⋊C4×C71—21no—
89C2×C4⋊C4×C71C2312yes—
90C2×C4⋊C4×C71C2312yes—
91C2×C22⋊C4×C71—24no—
92C2×C22⋊C4×C71—44no—
93#95×C71—24no—
94C23×D71—24no—
95C23×D71—22no—
96C23×D71—42no—
97C23×D71—42no—
98C23×D71—42no—
99C23×D71—42no—
100C23×D71—42no—
101C23×D71—42no—
102C2×D284—28no—
103C2×D284—28no—
104C2×D284—22no—
105C2×D284—22no—
106C2×D284—22no—
107C2×D284—22no—
108C2×D284—24no—
109C2×D284—22no—
110C2×D284—42no—
111C2×D284—44no—
112C2×D284—42no—
113C2×D284—44no—
114C2×D284C2×D414yes—
115C2×C22⋊D71—216no—
116C2×C22⋊D71—48no—
117C2×C22⋊D71—432no—
118C2×C22⋊D71—48no—
119C22×Dic71—24no—
120C22×Dic71—22no—
121C22×Dic71—42no—
122C22×Dic71—42no—
123C2×C71⋊C8—28no—
124C2×C71⋊C8—22no—
125C2×C71⋊C8—22no—
126C2×C71⋊C8—22no—
127C2×C71⋊C8—22no—
128C2×C71⋊C8—22no—
129C2×C71⋊C8—44no—
130C2×C71⋊C8C2×D414yescomplemented
131C4⋊Dic71—24no—
132C4⋊Dic71—22no—
133C4⋊Dic71C2×D414yes—
134(C2×D71)⋊C4—216no—
135C24×C71—22no—
136C24×C71—22no—
137C24×C71—22no—
138C24×C71—22no—
139C24×C71—22no—
140C24×C71—22no—
141C24×C71SD1614yes—
142C24×C71C2×D412yes—
143C24×C71C2×D411yescharacteristic
144C22×C4×C71—22no—
145C22×C4×C71—22no—
146C22×C4×C71—22no—
147C22×C4×C71—24no—
148C22×C4×C71—24no—
149C22×C4×C71—24no—
150C22×C4×C71—22no—
151C22×C4×C71—22no—
152C22×C4×C71—22no—
153C22×C4×C71—22no—
154C22×C4×C71—22no—
155C22×C4×C71—22no—
156C22×C4×C71—22no—
157C22×C4×C71—22no—
158C22×C4×C71—44no—
159C22×C4×C71—44no—
160C22×C4×C71—44no—
161C22×C4×C71—44no—
162C22×C4×C71—42no—
163C22×C4×C71—42no—
164C22×C4×C71—42no—
165C22×C4×C71—42no—
166C22×C4×C71—42no—
167C22×C4×C71—42no—
168C22×C4×C71C2×D412yes—
169C22×C4×C71C2×D412yes—
170C22×C4×C71C2×D412yes—
171C22×C4×C71C2×D412yes—
172C22×C4×C71C2×D411yescharacteristic
173C22×C4×C71#4112yes—
174C22×C4×C71#4112yes—
175C22×C4×C71#4112yes—
176C22×C4×C71C2411yescharacteristic
177C4⋊C4×C71—22no—
178C4⋊C4×C71—22no—
179C4⋊C4×C71—22no—
180C4⋊C4×C71—22no—
181C4⋊C4×C71—42no—
182C4⋊C4×C71—42no—
183C22⋊C4×C71—48no—
184C23×C71—22no—
185C23×C71—22no—
186C23×C71—22no—
187C23×C71—22no—
188C23×C71—22no—
189C23×C71—22no—
190C23×C71—28no—
191C23×C71—28no—
192C23×C71—28no—
193C23×C71—44no—
194C23×C71—44no—
195C23×C71—44no—
196C23×C71—44no—
197C23×C71#12012yes—
198C23×C71#12112yes—
199C23×C71#12312yes—
200C23×C71C2×SD1614yes—
201C23×C71C2×SD1612yes—
202C23×C71#13512yes—
203C23×C71AGL(1,ℤ/8)12yes—
204C23×C71AGL(1,ℤ/8)12yes—
205C23×C71C22×D411yescharacteristic
206C22×D71—24no—
207C22×D71—42no—
208C22×D71—42no—
209C22×D71—42no—
210C22×D71—42no—
211C22×D71—42no—
212C22×D71—42no—
213C22×D71—48no—
214C22×D71—48no—
215C22×D71—48no—
216C22×D71—84no—
217C22×D71—84no—
218C22×D71—84no—
219C22×D71—84no—
220C2×C4×C71—22no—
221C2×C4×C71—22no—
222C2×C4×C71—22no—
223C2×C4×C71—44no—
224C2×C4×C71—42no—
225C2×C4×C71—42no—
226C2×C4×C71—42no—
227C2×C4×C71—42no—
228C2×C4×C71—48no—
229C2×C4×C71—42no—
230C2×C4×C71—42no—
231C2×C4×C71—42no—
232C2×C4×C71—42no—
233C2×C4×C71—42no—
234C2×C4×C71—42no—
235C2×C4×C71—84no—
236C2×C4×C71—84no—
237C2×C4×C71#12012yes—
238C2×C4×C71#12112yes—
239C2×C4×C71#12114yes—
240C2×C4×C71#12114yes—
241C2×C4×C71#12314yes—
242C2×C4×C71#12312yes—
243C2×C4×C71#12414yes—
244C2×C4×C71C22×D411yescharacteristic
245C2×C4×C71C22×D411yescharacteristic
246C2×C4×C71C22×D411yescharacteristic
247C2×C4×C71C2×#4111yescharacteristic
248C2×C4×C71C2×#4111yescharacteristic
249C2×C4×C71C2×#4111yescharacteristic
250C22⋊D71—432no—
251C2×Dic71—24no—
252C2×Dic71—42no—
253C2×Dic71—42no—
254C2×Dic71—48no—
255D284—24no—
256D284—24no—
257D284—42no—
258D284—42no—
259D284—48no—
260C71⋊C8—24no—
261C71⋊C8—42no—
262C22×C71—24no—
263C22×C71—24no—
264C22×C71—24no—
265C22×C71—42no—
266C22×C71—42no—
267C22×C71—42no—
268C22×C71—42no—
269C22×C71—416no—
270C22×C71#39414yes—
271C22×C71#44914yes—
272C22×C71#45014yes—
273C22×C71#46514yes—
274C22×C71#46914yes—
275C22×C71#48114yes—
276C22×C71#48214yes—
277C22×C71C2×#12011yescharacteristic
278C22×C71C2×#12111yescharacteristic
279C22×C71C2×#12311yescharacteristic
280C22×C71C22×SD1611yescharacteristic
281C22×C71C2×#13511yescharacteristic
282C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
284C2×D71—44no—
285C2×D71—44no—
286C2×D71—44no—
287C2×D71—82no—
288C2×D71—82no—
289C2×D71—82no—
290C2×D71—82no—
291C2×D71—816no—
292C4×C71—42no—
293C4×C71—82no—
294C4×C71—82no—
295C4×C71#39414yes—
296C4×C71C2×#12111yescharacteristic
297C4×C71C2×#12111yescharacteristic
298C4×C71C2×#12311yescharacteristic
299C4×C71C2×#12411yescharacteristic
300Dic71—44no—
301C2×#2034—711nomaximal, Sylow
302C2×C71—44no—
303C2×C71#203418yescomplemented
304C2×C71C2×#39411yescharacteristic
305C2×C71C2×#45011yescharacteristic
306C2×C71C2×#44911yescharacteristic
307C2×C71C2×#46511yescharacteristic
308C2×C71C2×#46911yescharacteristic
309C2×C71C2×#48211yescharacteristic
310C2×C71C2×#48111yescharacteristic
311D71—84no—
312C22×#121—711no—
313C22×#106—711no—
314C22×#102—711no—
315C22×#102—711no—
316C22×C22⋊C8—711no—
317C2×#380D7111yescharacteristic, complemented
318#2034—718no—
319C2×#340—711no—
320C71C2×#203411yescharacteristic, complemented, Sylow
321C23×D4—711no—
322C23×D4—1421no—
323C24×C4C2×D7111yescharacteristic
324C23×C8—711no—
325C23×C8—711no—
326C2×#121—714no—
327C2×#121—1424no—
328C2×#121—1424no—
329C2×#121—1422no—
330C22×C4⋊C4—711no—
331C22×C4⋊C4C2×D7111yescharacteristic
332C22×C4⋊C4C2×D7111yescharacteristic
333C22×C22⋊C4—21no—
334C22×C22⋊C4—1421no—
335C2×#106—714no—
336C2×#106—1424no—
337C2×#102—714no—
338C2×#102—714no—
339C2×#102—1424no—
340C2×#102—1424no—
341C2×C22⋊C8—714no—
342C2×C22⋊C8—1424no—
343#380—22no—
344#380C2×D7114yescomplemented
345C2×#95—21no—
346#340—714no—
347C25—1421no—
348C25C22⋊D7111yescharacteristic
349C22×D4—2844no—
350C22×D4—2844no—
351C22×D4—714no—
352C22×D4—712no—
353C22×D4—1428no—
354C22×D4—1424no—
355C22×D4—1422no—
356C22×D4—1422no—
357C22×D4—1421no—
358C22×D4—1422no—
359C22×D4—1422no—
360C22×D4—1422no—
361C23×C4—22no—
362C23×C4—22no—
363C23×C4—22no—
364C23×C4—22no—
365C23×C4—22no—
366C23×C4—22no—
367C23×C4—21no—
368C23×C4—21no—
369C23×C4—1421no—
370C23×C4C22⋊D7112yes—
371C23×C4C22⋊D7111yescharacteristic
372C23×C4C22×D7112yes—
373C23×C4C22×D7111yescharacteristic
374C22×C8—714no—
375C22×C8—712no—
376C22×C8—712no—
377C22×C8—1422no—
378C22×C8—1422no—
379C22×C8—1422no—
380C22×C8—1422no—
381C22×C8—1422no—
382#121—1428no—
383#121—1428no—
384#121—14216no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—714no—
392C2×C4⋊C4—712no—
393C2×C4⋊C4—1422no—
394C2×C4⋊C4—1421no—
395C2×C4⋊C4C22×D7112yes—
396C2×C4⋊C4C22×D7112yes—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—44no—
399C2×C22⋊C4—1428no—
400C2×C22⋊C4—1424no—
401#106—1428no—
402#102—1428no—
403#102—1428no—
404C22⋊C8—1428no—
405#95—24no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—2842no—
413C24—2842no—
414C24—2842no—
415C24—2842no—
416C24—2842no—
417C24—2842no—
418C24—1422no—
419C24—1424no—
420C24#1176585214yes—
421C24C2×C22⋊D7112yes—
422C24C2×C22⋊D7111yescharacteristic
423C2×D4—28432no—
424C2×D4—2848no—
425C2×D4—2848no—
426C2×D4—2842no—
427C2×D4—2844no—
428C2×D4—2842no—
429C2×D4—2844no—
430C2×D4—714no—
431C2×D4—14216no—
432C2×D4—1422no—
433C2×D4—1422no—
434C2×D4—1422no—
435C2×D4—1422no—
436C2×D4—1422no—
437C2×D4—1424no—
438C2×D4—1428no—
439C2×D4—1428no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—2842no—
465C22×C4—2842no—
466C22×C4—1422no—
467C22×C4—1424no—
468C22×C4D4×D7112yes—
469C22×C4D4×D7112yes—
470C22×C4D4×D7112yes—
471C22×C4C2×C22⋊D7112yes—
472C22×C4C2×C22⋊D7111yescharacteristic
473C22×C4#1176586412yes—
474C22×C4#1176586512yes—
475C22×C4#1176586512yes—
476C22×C4C23×D7111yescharacteristic
477C2×C8—2844no—
478C2×C8—714no—
479C2×C8—1422no—
480C2×C8—1422no—
481C2×C8—1422no—
482C2×C8—1422no—
483C2×C8—1422no—
484C2×C8—1428no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—714no—
492C4⋊C4—1422no—
493C4⋊C4—1424no—
494C22⋊C4—48no—
495C22⋊C4—14216no—
496C23—28no—
497C23—28no—
498C23—22no—
499C23—28no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—2848no—
510C23—2848no—
511C23—2848no—
512C23—2842no—
513C23—2842no—
514C23—2842no—
515C23—2842no—
516C23—2842no—
517C23—2842no—
518C23—5684no—
519C23—5684no—
520C23—5684no—
521C23—5684no—
522C23—1424no—
523C23#1455609612yes—
524C23#1455610012yes—
525C23#1455611112yes—
526C23C2×#1176585214yes—
527C23C2×#1176585212yes—
528C23#1455614812yes—
529C23#1455615312yes—
530C23#1455615412yes—
531C23C22×C22⋊D7111yescharacteristic
532D4—28432no—
533D4—2848no—
534D4—2842no—
535D4—2842no—
536D4—1424no—
537D4—1424no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—44no—
542C2×C4—48no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—2848no—
556C2×C4—2842no—
557C2×C4—2842no—
558C2×C4—1424no—
559C2×C4#1455609612yes—
560C2×C4#1455610012yes—
561C2×C4#1455610114yes—
562C2×C4#1455610114yes—
563C2×C4#1455611014yes—
564C2×C4#1455611112yes—
565C2×C4#1455611414yes—
566C2×C4C2×D4×D7111yescharacteristic
567C2×C4C2×D4×D7111yescharacteristic
568C2×C4C2×D4×D7111yescharacteristic
569C2×C4C2×#1176586411yescharacteristic
570C2×C4C2×#1176586511yescharacteristic
571C2×C4C2×#1176586511yescharacteristic
572C8—2842no—
573C8—1424no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—416no—
581C22—42no—
582C22—2844no—
583C22—2844no—
584C22—2844no—
585C22—5682no—
586C22—5682no—
587C22—5682no—
588C22—56816no—
589C22—5682no—
590C22#1798591414yes—
591C22#1798614214yes—
592C22#1798614914yes—
593C22#1798623014yes—
594C22#1798625014yes—
595C22#1798631714yes—
596C22#1798632314yes—
597C22C2×#1455609611yescharacteristic
598C22C2×#1455610011yescharacteristic
599C22C2×#1455611111yescharacteristic
600C22C22×#1176585211yescharacteristic
601C22C2×#1455614811yescharacteristic
602C22C2×#1455615311yescharacteristic
603C22C2×#1455615411yescharacteristic
604C4—42no—
605C4—82no—
606C4—82no—
607C4—2844no—
608C4#1798591414yes—
609C4C2×#1455610111yescharacteristic
610C4C2×#1455610111yescharacteristic
611C4C2×#1455611011yescharacteristic
612C4C2×#1455611411yescharacteristic
613C2—44no—
614C2—5684no—
615C2#2258952918yesdirect product factor
616C2C2×#1798591411yescharacteristic
617C2C2×#1798614911yescharacteristic
618C2C2×#1798614211yescharacteristic
619C2C2×#1798623011yescharacteristic
620C2C2×#1798625011yescharacteristic
621C2C2×#1798632311yescharacteristic
622C2C2×#1798631711yescharacteristic
subgroupG/H
centerC24C2×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC2×#380×C71C2
socleC24×C71C2×D4
prime radicalC22×#14556102C2
hypercenterC2×#380D71
hypocenterC71C2×#2034
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C24 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×#14556059C2semidirect
3. C22×#14556048C2semidirect
4. C22×#14556048C2semidirect
5. C22×C22⋊(C71⋊C8)C2semidirect
6. #22589529C2direct (N × H)
7. C2×#17985761C2semidirect
8. C2×#380×C71C2semidirect
20. C2×#14556059C22semidirect
23. C2×#14556048C22semidirect
24. C2×#14556048C22semidirect
26. C2×C22⋊(C71⋊C8)C22semidirect
27. #17985761C22semidirect
33. #380×C71C22semidirect
54. C22×C71⋊C8D4semidirect
130. C2×C71⋊C8C2×D4semidirect
303. C2×C71#2034semidirect
317. C2×#380D71semidirect
320. C71C2×#2034semidirect
344. #380C2×D71semidirect
615. C2#22589529direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561022
C22×#145560592
C22×#145560482
C22×C22⋊(C71⋊C8)2
#225895292
C2×#179857612
C2×#380×C712
C2×#203471