Finite Group Databasebeta

← C2×#22589520 #32024898

Subgroups — C2×#22589520 #32024898

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×#14556099C211yesmaximal, characteristic
2C22×#14556061C211yesmaximal, characteristic, complemented
3C22×#14556049C211yesmaximal, characteristic, complemented
4C22×(C2×D71)⋊C8C211yesmaximal, characteristic, complemented
5C2×#17985856C211yesmaximal, characteristic, complemented
6#22589520C218yesmaximal, direct product factor
7C2×#17985762C211yesmaximal, characteristic, complemented
8C22×#102×C71C211yesmaximal, characteristic, complemented
9C23×C22⋊D71—21no—
10C23×C4×D71C2211yescharacteristic
11C23×C71⋊C8C2211yescharacteristic
12C2×#14556099—24no—
13C2×#14556099—24no—
14C2×#14556099—22no—
15C2×#14556099C2214yes—
16C22×C4⋊Dic71C2211yescharacteristic
17C22×C4⋊Dic71C2211yescharacteristic
18C22×C22⋊Dic71—21no—
19C22×(C2×D71)⋊C4—21no—
20C2×#14556061—24no—
21C2×#14556061C2214yescomplemented
22C2×#14556049—24no—
23C2×#14556049C2214yescomplemented
24C2×(C2×D71)⋊C8—24no—
25C2×(C2×D71)⋊C8C2214yescomplemented
26#17985856—22no—
27#17985856C2214yescomplemented
28C2×C71⋊#95—21no—
29#17985762C2214yescomplemented
30C23×D4×C71C2211yescharacteristic
31C23×C8×C71C2211yescharacteristic
32C22×C4⋊C4×C71C2211yescharacteristic
33C2×#102×C71—24no—
34C2×#102×C71C2214yescomplemented
35C24×D71D411yescharacteristic
36C22×C22⋊D71—28no—
37C22×C22⋊D71—24no—
38C22×C22⋊D71—44no—
39C22×C22⋊D71—44no—
40C23×Dic71—21no—
41C23×Dic71—21no—
42C23×Dic71D411yescharacteristic
43C22×C4×D71—22no—
44C22×C4×D71—22no—
45C22×C4×D71—22no—
46C22×C4×D71—22no—
47C22×C4×D71—22no—
48C22×C4×D71—22no—
49C22×C4×D71D412yes—
50C22×C4×D71C2312yes—
51C22×C71⋊C8—22no—
52C22×C71⋊C8—22no—
53C22×C71⋊C8D414yescomplemented
54C22×C71⋊C8C2312yes—
55#14556099—216no—
56#14556099—28no—
57#14556099—28no—
58C2×C4⋊Dic71—21no—
59C2×C4⋊Dic71—22no—
60C2×C4⋊Dic71—22no—
61C2×C4⋊Dic71—21no—
62C2×C4⋊Dic71—22no—
63C2×C4⋊Dic71D414yes—
64C2×C4⋊Dic71C2312yes—
65C2×C4⋊Dic71C2312yes—
66C2×C22⋊Dic71—28no—
67C2×C22⋊Dic71—24no—
68C2×(C2×D71)⋊C4—24no—
69C2×(C2×D71)⋊C4—44no—
70#14556061—28no—
71#14556049—28no—
72(C2×D71)⋊C8—28no—
73C71⋊#95—24no—
74C25×C71—21no—
75C22×D4×C71—22no—
76C22×D4×C71—21no—
77C22×D4×C71—22no—
78C22×D4×C71—22no—
79C22×D4×C71—22no—
80C22×D4×C71—22no—
81C22×D4×C71D414yes—
82C22×D4×C71C2312yes—
83C23×C4×C71—21no—
84C23×C4×C71C2311yescharacteristic
85C22×C8×C71—22no—
86C22×C8×C71—22no—
87C22×C8×C71—22no—
88C22×C8×C71C2312yes—
89C2×C4⋊C4×C71—22no—
90C2×C4⋊C4×C71—22no—
91C2×C4⋊C4×C71—21no—
92C2×C4⋊C4×C71C2312yes—
93#102×C71—28no—
94C23×D71—22no—
95C23×D71—22no—
96C23×D71—22no—
97C23×D71—22no—
98C23×D71—22no—
99C23×D71—22no—
100C23×D71D814yes—
101C23×D71C2×D412yes—
102C2×C22⋊D71—216no—
103C2×C22⋊D71—48no—
104C2×C22⋊D71—432no—
105C2×C22⋊D71—48no—
106C22×Dic71—22no—
107C22×Dic71—24no—
108C22×Dic71—22no—
109C22×Dic71—22no—
110C22×Dic71—22no—
111C22×Dic71—22no—
112C22×Dic71—42no—
113C22×Dic71—42no—
114C22×Dic71—42no—
115C22×Dic71—42no—
116C22×Dic71—42no—
117C22×Dic71C2×D412yes—
118C2×C4×D71—24no—
119C2×C4×D71—24no—
120C2×C4×D71—24no—
121C2×C4×D71—22no—
122C2×C4×D71—22no—
123C2×C4×D71—22no—
124C2×C4×D71—22no—
125C2×C4×D71—22no—
126C2×C4×D71—22no—
127C2×C4×D71—44no—
128C2×C4×D71—44no—
129C2×C4×D71—44no—
130C2×C4×D71—44no—
131C2×C71⋊C8—28no—
132C2×C71⋊C8—22no—
133C2×C71⋊C8—22no—
134C2×C71⋊C8C2×D414yescomplemented
135C4⋊Dic71—24no—
136C4⋊Dic71—22no—
137C4⋊Dic71—22no—
138C4⋊Dic71—22no—
139C4⋊Dic71—42no—
140C4⋊Dic71C2×D414yes—
141C22⋊Dic71—216no—
142(C2×D71)⋊C4—48no—
143C24×C71—24no—
144C24×C71—22no—
145C24×C71—42no—
146C24×C71—42no—
147C24×C71—42no—
148C24×C71—42no—
149C24×C71—42no—
150C24×C71—42no—
151C24×C71C2×D411yescharacteristic
152C2×D4×C71—24no—
153C2×D4×C71—28no—
154C2×D4×C71—28no—
155C2×D4×C71—22no—
156C2×D4×C71—22no—
157C2×D4×C71—22no—
158C2×D4×C71—22no—
159C2×D4×C71—22no—
160C2×D4×C71—44no—
161C2×D4×C71—42no—
162C2×D4×C71—42no—
163C2×D4×C71—44no—
164C2×D4×C71C2×D414yes—
165C22×C4×C71—22no—
166C22×C4×C71—42no—
167C22×C4×C71—42no—
168C22×C4×C71—42no—
169C22×C4×C71C2×D411yescharacteristic
170C22×C4×C71C2×D412yes—
171C22×C4×C71C2×D412yes—
172C22×C4×C71C2×D412yes—
173C22×C4×C71#4112yes—
174C22×C4×C71#4112yes—
175C22×C4×C71#4112yes—
176C22×C4×C71C2411yescharacteristic
177C2×C8×C71—22no—
178C2×C8×C71—22no—
179C2×C8×C71—22no—
180C2×C8×C71—44no—
181C4⋊C4×C71—22no—
182C4⋊C4×C71—22no—
183C4⋊C4×C71—42no—
184C23×C71—24no—
185C23×C71—42no—
186C23×C71—42no—
187C23×C71—42no—
188C23×C71—42no—
189C23×C71—42no—
190C23×C71—42no—
191C23×C71—48no—
192C23×C71—48no—
193C23×C71—48no—
194C23×C71—84no—
195C23×C71—84no—
196C23×C71—84no—
197C23×C71—84no—
198C23×C71#12012yes—
199C23×C71#12112yes—
200C23×C71#12312yes—
201C23×C71C2×D812yes—
202C23×C71#13512yes—
203C23×C71AGL(1,ℤ/8)12yes—
204C23×C71#13712yes—
205C23×C71C22×D411yescharacteristic
206C22×D71—22no—
207C22×D71—22no—
208C22×D71—22no—
209C22×D71—22no—
210C22×D71—22no—
211C22×D71—22no—
212C22×D71—28no—
213C22×D71—28no—
214C22×D71—28no—
215C22×D71—44no—
216C22×D71—44no—
217C22×D71—44no—
218C22×D71—44no—
219C22×D71C2×D814yes—
220D4×C71—24no—
221D4×C71—24no—
222D4×C71—42no—
223D4×C71—42no—
224D4×C71—48no—
225C2×C4×C71—42no—
226C2×C4×C71—42no—
227C2×C4×C71—42no—
228C2×C4×C71—84no—
229C2×C4×C71#12012yes—
230C2×C4×C71#12114yes—
231C2×C4×C71#12114yes—
232C2×C4×C71#12112yes—
233C2×C4×C71#12312yes—
234C2×C4×C71#12314yes—
235C2×C4×C71#12414yes—
236C2×C4×C71C22×D411yescharacteristic
237C2×C4×C71C22×D411yescharacteristic
238C2×C4×C71C22×D411yescharacteristic
239C2×C4×C71C2×#4111yescharacteristic
240C2×C4×C71C2×#4111yescharacteristic
241C2×C4×C71C2×#4111yescharacteristic
242C22⋊D71—432no—
243C2×Dic71—22no—
244C2×Dic71—24no—
245C2×Dic71—22no—
246C2×Dic71—22no—
247C2×Dic71—42no—
248C2×Dic71—42no—
249C2×Dic71—44no—
250C2×Dic71—48no—
251C2×Dic71—42no—
252C2×Dic71—42no—
253C2×Dic71—42no—
254C2×Dic71—84no—
255C4×D71—42no—
256C4×D71—42no—
257C4×D71—42no—
258C4×D71—42no—
259C4×D71—48no—
260C8×C71—42no—
261C71⋊C8—24no—
262C22×C71—44no—
263C22×C71—44no—
264C22×C71—44no—
265C22×C71—82no—
266C22×C71—82no—
267C22×C71—82no—
268C22×C71—82no—
269C22×C71—816no—
270C22×C71#39414yes—
271C22×C71#44714yes—
272C22×C71#45214yes—
273C22×C71#46614yes—
274C22×C71#46814yes—
275C22×C71#48014yes—
276C22×C71#48214yes—
277C22×C71C2×#12011yescharacteristic
278C22×C71C2×#12111yescharacteristic
279C22×C71C2×#12311yescharacteristic
280C22×C71C22×D811yescharacteristic
281C22×C71C2×#13511yescharacteristic
282C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C71C2×#13711yescharacteristic
284C2×D71—24no—
285C2×D71—24no—
286C2×D71—24no—
287C2×D71—42no—
288C2×D71—42no—
289C2×D71—42no—
290C2×D71—42no—
291C2×D71—416no—
292C4×C71—82no—
293C4×C71#39414yes—
294C4×C71C2×#12111yescharacteristic
295C4×C71C2×#12111yescharacteristic
296C4×C71C2×#12311yescharacteristic
297C4×C71C2×#12411yescharacteristic
298Dic71—42no—
299Dic71—44no—
300Dic71—82no—
301C2×#2033—711nomaximal, Sylow
302C2×C71—84no—
303C2×C71#203318yescomplemented
304C2×C71C2×#39411yescharacteristic
305C2×C71C2×#44711yescharacteristic
306C2×C71C2×#45211yescharacteristic
307C2×C71C2×#46611yescharacteristic
308C2×C71C2×#46811yescharacteristic
309C2×C71C2×#48011yescharacteristic
310C2×C71C2×#48211yescharacteristic
311D71—44no—
312C22×#121—711no—
313C22×#107—711no—
314C22×#102—711no—
315C22×#102D7111yescharacteristic, complemented
316C22×C22⋊C8—711no—
317C2×#380—711no—
318#2033—718no—
319C2×#340—711no—
320C71C2×#203311yescharacteristic, complemented, Sylow
321C23×D4—1421no—
322C23×D4C2×D7111yescharacteristic
323C24×C4—711no—
324C23×C8—711no—
325C23×C8C2×D7111yescharacteristic
326C2×#121—714no—
327C2×#121—1424no—
328C2×#121—1422no—
329C2×#121—1424no—
330C22×C4⋊C4—711no—
331C22×C4⋊C4—711no—
332C22×C4⋊C4C2×D7111yescharacteristic
333C22×C22⋊C4—1421no—
334C22×C22⋊C4—1421no—
335C2×#107—714no—
336C2×#107—1424no—
337C2×#102—24no—
338C2×#102—714no—
339C2×#102—1424no—
340C2×#102C2×D7114yescomplemented
341C2×C22⋊C8—714no—
342C2×C22⋊C8—1424no—
343#380—714no—
344#380—1422no—
345C2×#95—1421no—
346#340—714no—
347C25—21no—
348C25—711no—
349C22×D4—22no—
350C22×D4—22no—
351C22×D4—21no—
352C22×D4—22no—
353C22×D4—22no—
354C22×D4—22no—
355C22×D4—2844no—
356C22×D4—2844no—
357C22×D4—1424no—
358C22×D4—1428no—
359C22×D4C22⋊D7114yes—
360C22×D4C22×D7112yes—
361C23×C4—21no—
362C23×C4—711no—
363C23×C4—712no—
364C23×C4—712no—
365C23×C4—1421no—
366C23×C4—1421no—
367C23×C4—1422no—
368C23×C4—1422no—
369C23×C4—1422no—
370C23×C4—1422no—
371C23×C4—1422no—
372C23×C4—1422no—
373C23×C4C22×D7111yescharacteristic
374C22×C8—22no—
375C22×C8—22no—
376C22×C8—22no—
377C22×C8—714no—
378C22×C8—712no—
379C22×C8—1422no—
380C22×C8—1422no—
381C22×C8C22×D7112yes—
382#121—1428no—
383#121—14216no—
384#121—1428no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—714no—
389C2×C4⋊C4—712no—
390C2×C4⋊C4—712no—
391C2×C4⋊C4—1421no—
392C2×C4⋊C4—1422no—
393C2×C4⋊C4—1422no—
394C2×C4⋊C4—1421no—
395C2×C4⋊C4—1422no—
396C2×C4⋊C4C22×D7112yes—
397C2×C22⋊C4—2844no—
398C2×C22⋊C4—1424no—
399C2×C22⋊C4—1424no—
400C2×C22⋊C4—1428no—
401#107—1428no—
402#102—28no—
403#102—1428no—
404C22⋊C8—1428no—
405#95—1424no—
406C24—22no—
407C24—24no—
408C24—42no—
409C24—42no—
410C24—42no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24—712no—
415C24—714no—
416C24—1422no—
417C24—1422no—
418C24—1422no—
419C24—1422no—
420C24—1422no—
421C24—1422no—
422C24D4×D7111yescharacteristic
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—28no—
430C2×D4—28no—
431C2×D4—42no—
432C2×D4—44no—
433C2×D4—42no—
434C2×D4—44no—
435C2×D4—2848no—
436C2×D4—28432no—
437C2×D4—2848no—
438C2×D4—14216no—
439C2×D4C2×C22⋊D7114yes—
440C22×C4—22no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—2842no—
445C22×C4—2842no—
446C22×C4—2844no—
447C22×C4—2844no—
448C22×C4—2844no—
449C22×C4—2844no—
450C22×C4—2842no—
451C22×C4—2842no—
452C22×C4—2842no—
453C22×C4—712no—
454C22×C4—1422no—
455C22×C4—1422no—
456C22×C4—1422no—
457C22×C4—1424no—
458C22×C4—1422no—
459C22×C4—1422no—
460C22×C4—1422no—
461C22×C4—1422no—
462C22×C4—1422no—
463C22×C4—1422no—
464C22×C4—1422no—
465C22×C4—1422no—
466C22×C4—1424no—
467C22×C4—1424no—
468C22×C4—1424no—
469C22×C4D4×D7112yes—
470C22×C4D4×D7112yes—
471C22×C4D4×D7111yescharacteristic
472C22×C4C2×C22⋊D7112yes—
473C22×C4#1176586412yes—
474C22×C4#1176586512yes—
475C22×C4#1176586512yes—
476C22×C4C23×D7111yescharacteristic
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—44no—
481C2×C8—714no—
482C2×C8—1422no—
483C2×C8—1422no—
484C2×C8—1428no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—42no—
488C4⋊C4—2842no—
489C4⋊C4—714no—
490C4⋊C4—1422no—
491C4⋊C4—1422no—
492C4⋊C4—1422no—
493C4⋊C4—1424no—
494C22⋊C4—2848no—
495C22⋊C4—14216no—
496C23—24no—
497C23—48no—
498C23—48no—
499C23—48no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—84no—
507C23—84no—
508C23—84no—
509C23—84no—
510C23—2844no—
511C23—2844no—
512C23—2844no—
513C23—2844no—
514C23—714no—
515C23—1428no—
516C23—1428no—
517C23—1422no—
518C23—1428no—
519C23—1422no—
520C23—1422no—
521C23—1422no—
522C23—1422no—
523C23—1422no—
524C23#1455609612yes—
525C23#1455610112yes—
526C23#1455611012yes—
527C23D8×D7112yes—
528C23#1455614912yes—
529C23#1455615212yes—
530C23#1455615712yes—
531C23C2×D4×D7111yescharacteristic
532D4—24no—
533D4—24no—
534D4—48no—
535D4—42no—
536D4—42no—
537D4—28432no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—84no—
542C2×C4—2848no—
543C2×C4—2844no—
544C2×C4—2842no—
545C2×C4—2842no—
546C2×C4—2848no—
547C2×C4—2842no—
548C2×C4—2842no—
549C2×C4—2842no—
550C2×C4—2842no—
551C2×C4—2842no—
552C2×C4—2842no—
553C2×C4—2842no—
554C2×C4—5684no—
555C2×C4—1422no—
556C2×C4—1424no—
557C2×C4—1422no—
558C2×C4—1422no—
559C2×C4#1455609612yes—
560C2×C4#1455610014yes—
561C2×C4#1455610112yes—
562C2×C4#1455610114yes—
563C2×C4#1455611012yes—
564C2×C4#1455611114yes—
565C2×C4#1455611414yes—
566C2×C4C2×D4×D7111yescharacteristic
567C2×C4C2×D4×D7111yescharacteristic
568C2×C4C22×C22⋊D7111yescharacteristic
569C2×C4C2×#1176586411yescharacteristic
570C2×C4C2×#1176586511yescharacteristic
571C2×C4C2×#1176586511yescharacteristic
572C8—42no—
573C8—1424no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—816no—
582C22—2842no—
583C22—2842no—
584C22—2842no—
585C22—28416no—
586C22—2842no—
587C22—1424no—
588C22—1424no—
589C22—1424no—
590C22#1798591414yes—
591C22#1798613114yes—
592C22#1798615814yes—
593C22#1798623414yes—
594C22#1798624414yes—
595C22#1798631114yes—
596C22#1798632214yes—
597C22C2×#1455609611yescharacteristic
598C22C2×#1455610111yescharacteristic
599C22C2×#1455611011yescharacteristic
600C22C2×D8×D7111yescharacteristic
601C22C2×#1455614911yescharacteristic
602C22C2×#1455615211yescharacteristic
603C22C2×#1455615711yescharacteristic
604C4—82no—
605C4—2842no—
606C4—2844no—
607C4—5682no—
608C4#1798591414yes—
609C4C2×#1455610011yescharacteristic
610C4C2×#1455610111yescharacteristic
611C4C2×#1455611111yescharacteristic
612C4C2×#1455611411yescharacteristic
613C2—84no—
614C2—2844no—
615C2#2258952018yesdirect product factor
616C2C2×#1798591411yescharacteristic
617C2C2×#1798613111yescharacteristic
618C2C2×#1798615811yescharacteristic
619C2C2×#1798623411yescharacteristic
620C2C2×#1798624411yescharacteristic
621C2C2×#1798631111yescharacteristic
622C2C2×#1798632211yescharacteristic
subgroupG/H
centerC24D4×D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC22×#102×C71C2
socleC24×C71C2×D4
prime radicalC22×#14556099C2
hypercenterC22×#102D71
hypocenterC71C2×#2033
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
2. C22×#14556061C2semidirect
3. C22×#14556049C2semidirect
4. C22×(C2×D71)⋊C8C2semidirect
5. C2×#17985856C2semidirect
6. #22589520C2direct (N × H)
7. C2×#17985762C2semidirect
8. C22×#102×C71C2semidirect
21. C2×#14556061C22semidirect
23. C2×#14556049C22semidirect
25. C2×(C2×D71)⋊C8C22semidirect
27. #17985856C22semidirect
29. #17985762C22semidirect
34. C2×#102×C71C22semidirect
53. C22×C71⋊C8D4semidirect
134. C2×C71⋊C8C2×D4semidirect
303. C2×C71#2033semidirect
315. C22×#102D71semidirect
320. C71C2×#2033semidirect
340. C2×#102C2×D71semidirect
615. C2#22589520direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145560992
C22×#145560612
C22×#145560492
C22×(C2×D71)⋊C82
C2×#179858562
#225895202
C2×#179857622
C22×#102×C712
C2×#203371