Finite Group Databasebeta

← C2×#22588777 #31999139

Subgroups — C2×#22588777 #31999139

Order 18176.

2310 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11765848C211yesmaximal, characteristic, complemented
2C22×#14556103C211yesmaximal, characteristic
3C22×#14556050C211yesmaximal, characteristic, complemented
4C22×#14556048C211yesmaximal, characteristic, complemented
5C2×#17985856C211yesmaximal, characteristic, complemented
6#22588777C218yesmaximal, direct product factor
7C2×#17985761C211yesmaximal, characteristic, complemented
8C2×#340×C71C211yesmaximal, characteristic, complemented
9C23×D284—21no—
10C23×D284C2211yescharacteristic
11C23×C4×D71C2211yescharacteristic
12C22×#11765848—24no—
13C22×#11765848—24no—
14C22×#11765848—24no—
15C22×#11765848C2214yescomplemented
16C23×C71⋊C8C2211yescharacteristic
17C2×#14556103—22no—
18C2×#14556103—24no—
19C2×#14556103C418yescomplemented
20C2×#14556103C2214yes—
21C22×C4⋊Dic71C2211yescharacteristic
22C22×(C2×D71)⋊C4—21no—
23C22×(C2×D71)⋊C4—21no—
24C2×#14556050—24no—
25C2×#14556050C2214yescomplemented
26C2×#14556048—24no—
27C2×#14556048C2214yescomplemented
28#17985856—22no—
29#17985856C2214yescomplemented
30C2×C71⋊#95—21no—
31#17985761C2214yescomplemented
32C23×C8×C71C2211yescharacteristic
33C22×C4⋊C4×C71C2211yescharacteristic
34C22×C4⋊C4×C71C2211yescharacteristic
35#340×C71C2214yescomplemented
36C24×D71—21no—
37C24×D71D411yescharacteristic
38C22×D284—28no—
39C22×D284—24no—
40C22×D284—22no—
41C22×D284—22no—
42C22×D284—21no—
43C22×D284—22no—
44C22×D284—22no—
45C22×D284—22no—
46C22×D284—44no—
47C22×D284—44no—
48C22×D284C2×C414yes—
49C22×D284C2312yes—
50C23×Dic71—21no—
51C23×Dic71D411yescharacteristic
52C22×C4×D71—22no—
53C22×C4×D71—22no—
54C22×C4×D71—22no—
55C22×C4×D71—22no—
56C22×C4×D71C2×C414yes—
57C22×C4×D71D412yes—
58C22×C4×D71D414yes—
59C22×C4×D71C2312yes—
60C2×#11765848—232no—
61C2×#11765848—28no—
62C2×#11765848—28no—
63C2×#11765848—28no—
64C22×C71⋊C8—22no—
65C22×C71⋊C8—22no—
66C22×C71⋊C8D414yescomplemented
67C22×C71⋊C8C2312yes—
68#14556103—216no—
69#14556103—28no—
70#14556103C2×C4116yescomplemented
71C2×C4⋊Dic71—22no—
72C2×C4⋊Dic71—21no—
73C2×C4⋊Dic71—22no—
74C2×C4⋊Dic71C2312yes—
75C2×(C2×D71)⋊C4—28no—
76C2×(C2×D71)⋊C4—24no—
77C2×(C2×D71)⋊C4—24no—
78C2×(C2×D71)⋊C4—44no—
79#14556050—28no—
80#14556048—28no—
81C71⋊#95—24no—
82C23×C4×C71—21no—
83C23×C4×C71—21no—
84C23×C4×C71C2311yescharacteristic
85C22×C8×C71—22no—
86C22×C8×C71—22no—
87C22×C8×C71D414yescomplemented
88C22×C8×C71C2312yes—
89C2×C4⋊C4×C71—21no—
90C2×C4⋊C4×C71—22no—
91C2×C4⋊C4×C71—22no—
92C2×C4⋊C4×C71—22no—
93C2×C4⋊C4×C71—21no—
94C2×C4⋊C4×C71C2×C414yes—
95C2×C4⋊C4×C71C2312yes—
96C2×C4⋊C4×C71C2312yes—
97C23×D71—22no—
98C23×D71—22no—
99C23×D71—22no—
100C23×D71—22no—
101C23×D71—22no—
102C23×D71—22no—
103C23×D71—24no—
104C23×D71—22no—
105C23×D71—42no—
106C23×D71—42no—
107C23×D71—42no—
108C23×D71—42no—
109C23×D71—42no—
110C23×D71—42no—
111C23×D71C22⋊C414yes—
112C23×D71C2×D412yes—
113C2×D284—28no—
114C2×D284—28no—
115C2×D284—22no—
116C2×D284—22no—
117C2×D284—22no—
118C2×D284—22no—
119C2×D284—216no—
120C2×D284—24no—
121C2×D284—22no—
122C2×D284—48no—
123C2×D284—48no—
124C2×D284—432no—
125C2×D284—44no—
126C2×D284—42no—
127C2×D284—42no—
128C2×D284—44no—
129C2×D284C22×C414yes—
130C22×Dic71—22no—
131C22×Dic71—22no—
132C22×Dic71—22no—
133C22×Dic71—42no—
134C22×Dic71—42no—
135C22×Dic71—42no—
136C22×Dic71C22⋊C414yes—
137C22×Dic71C2×D412yes—
138C2×C4×D71—28no—
139C2×C4×D71—28no—
140C2×C4×D71—28no—
141C2×C4×D71—24no—
142C2×C4×D71—24no—
143C2×C4×D71—28no—
144C2×C4×D71—22no—
145C2×C4×D71—22no—
146C2×C4×D71—22no—
147C2×C4×D71—22no—
148C2×C4×D71C22⋊C418yes—
149C2×C4×D71C22×C414yes—
150C2×C4×D71C2×D414yes—
151#11765848—232no—
152C2×C71⋊C8—28no—
153C2×C71⋊C8—22no—
154C2×C71⋊C8—22no—
155C2×C71⋊C8C2×D414yescomplemented
156C4⋊Dic71—22no—
157C4⋊Dic71—22no—
158C4⋊Dic71—42no—
159(C2×D71)⋊C4—216no—
160(C2×D71)⋊C4—48no—
161C24×C71C2×D411yescharacteristic
162C22×C4×C71—24no—
163C22×C4×C71—22no—
164C22×C4×C71—22no—
165C22×C4×C71—42no—
166C22×C4×C71—42no—
167C22×C4×C71—42no—
168C22×C4×C71—42no—
169C22×C4×C71—42no—
170C22×C4×C71C22×C412yes—
171C22×C4×C71C2×D411yescharacteristic
172C22×C4×C71C2×D412yes—
173C22×C4×C71C2×D412yes—
174C22×C4×C71C2×D412yes—
175C22×C4×C71#4112yes—
176C22×C4×C71#4112yes—
177C22×C4×C71C2411yescharacteristic
178C2×C8×C71—28no—
179C2×C8×C71—22no—
180C2×C8×C71—22no—
181C2×C8×C71C2×D414yescomplemented
182C4⋊C4×C71—24no—
183C4⋊C4×C71—22no—
184C4⋊C4×C71—22no—
185C4⋊C4×C71—22no—
186C4⋊C4×C71—42no—
187C4⋊C4×C71C22×C414yes—
188C23×C71C2×C22⋊C412yes—
189C23×C71#12112yes—
190C23×C71#12112yes—
191C23×C71AGL(1,ℤ/8)12yes—
192C23×C71AGL(1,ℤ/8)12yes—
193C23×C71AGL(1,ℤ/8)12yes—
194C23×C71#13712yes—
195C23×C71C22×D411yescharacteristic
196C22×D71—22no—
197C22×D71—22no—
198C22×D71—22no—
199C22×D71—22no—
200C22×D71—22no—
201C22×D71—22no—
202C22×D71—28no—
203C22×D71—28no—
204C22×D71—28no—
205C22×D71—24no—
206C22×D71—44no—
207C22×D71—44no—
208C22×D71—44no—
209C22×D71—44no—
210C22×D71—42no—
211C22×D71—42no—
212C22×D71—42no—
213C22×D71—42no—
214C22×D71—42no—
215C22×D71—42no—
216C22×D71—48no—
217C22×D71—48no—
218C22×D71—48no—
219C22×D71—84no—
220C22×D71—84no—
221C22×D71—84no—
222C22×D71—84no—
223C22×D71C2×C22⋊C414yes—
224C2×C4×C71—24no—
225C2×C4×C71—42no—
226C2×C4×C71—42no—
227C2×C4×C71—48no—
228C2×C4×C71—42no—
229C2×C4×C71—42no—
230C2×C4×C71—42no—
231C2×C4×C71—84no—
232C2×C4×C71C2×C22⋊C412yes—
233C2×C4×C71C4×D414yes—
234C2×C4×C71C4×D414yes—
235C2×C4×C71#12014yes—
236C2×C4×C71#12112yes—
237C2×C4×C71#12112yes—
238C2×C4×C71#12314yes—
239C2×C4×C71C23×C411yescharacteristic
240C2×C4×C71C22×D411yescharacteristic
241C2×C4×C71C22×D411yescharacteristic
242C2×C4×C71C22×D411yescharacteristic
243C2×C4×C71C2×#4111yescharacteristic
244C2×C4×C71C2×#4111yescharacteristic
245C2×Dic71—22no—
246C2×Dic71—22no—
247C2×Dic71—28no—
248C2×Dic71—42no—
249C2×Dic71—42no—
250C2×Dic71—42no—
251C2×Dic71—84no—
252C2×Dic71C2×C22⋊C414yes—
253D284—24no—
254D284—24no—
255D284—432no—
256D284—42no—
257D284—42no—
258D284—48no—
259C4×D71—24no—
260C4×D71—24no—
261C4×D71—24no—
262C4×D71—24no—
263C4×D71—216no—
264C8×C71—24no—
265C71⋊C8—24no—
266C22×C71#38614yes—
267C22×C71#41714yes—
268C22×C71#41814yes—
269C22×C71#46814yes—
270C22×C71#46814yes—
271C22×C71#46914yes—
272C22×C71#46914yes—
273C22×C71C22×C22⋊C411yescharacteristic
274C22×C71C2×#12111yescharacteristic
275C22×C71C2×#12111yescharacteristic
276C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C71C2×#13711yescharacteristic
280C2×D71—24no—
281C2×D71—24no—
282C2×D71—24no—
283C2×D71—416no—
284C2×D71—42no—
285C2×D71—42no—
286C2×D71—42no—
287C2×D71—42no—
288C2×D71—44no—
289C2×D71—44no—
290C2×D71—44no—
291C2×D71—816no—
292C2×D71—82no—
293C2×D71—82no—
294C2×D71—82no—
295C2×D71—82no—
296C4×C71—44no—
297C4×C71—82no—
298C4×C71#38614yes—
299C4×C71C2×C4×D411yescharacteristic
300C4×C71C2×C4×D411yescharacteristic
301C4×C71C2×#12011yescharacteristic
302C4×C71C2×#12311yescharacteristic
303Dic71—24no—
304Dic71—82no—
305C2×#1895—711nomaximal, Sylow
306C2×C71#189518yescomplemented
307C2×C71C2×#38611yescharacteristic
308C2×C71C2×#41811yescharacteristic
309C2×C71C2×#41711yescharacteristic
310C2×C71C2×#46811yescharacteristic
311C2×C71C2×#46811yescharacteristic
312C2×C71C2×#46911yescharacteristic
313C2×C71C2×#46911yescharacteristic
314D71—44no—
315D71—84no—
316C23×M16—711no—
317C22×#121—711no—
318C22×#102—711no—
319C22×#102—711no—
320C2×#380—711no—
321#1895—718no—
322C2×#340—711no—
323C2×#340D7111yescharacteristic, complemented
324C71C2×#189511yescharacteristic, complemented, Sylow
325C23×D4—711no—
326C23×D4—1421no—
327C24×C4—711no—
328C22×M16—714no—
329C22×M16—1424no—
330C22×M16—1424no—
331C22×M16—1424no—
332C23×C8—711no—
333C23×C8C2×D7111yescharacteristic
334C2×#121—718no—
335C2×#121—714no—
336C2×#121—1424no—
337C2×#121—1422no—
338C22×C4⋊C4—711no—
339C22×C4⋊C4C2×D7111yescharacteristic
340C22×C4⋊C4C2×D7111yescharacteristic
341C22×C22⋊C4—1421no—
342C22×C22⋊C4—1421no—
343C2×#102—714no—
344C2×#102—714no—
345C2×#102—1424no—
346C2×#102—1424no—
347#380—714no—
348#380—1422no—
349C2×#95—1421no—
350#340—714no—
351#340C2×D7114yescomplemented
352C25—711no—
353C25—1421no—
354C22×D4—2844no—
355C22×D4—2844no—
356C22×D4—714no—
357C22×D4—712no—
358C22×D4—1424no—
359C22×D4—1428no—
360C22×D4—1422no—
361C22×D4—1421no—
362C22×D4—1422no—
363C22×D4—1422no—
364C22×D4—1422no—
365C22×D4—1422no—
366C23×C4—21no—
367C23×C4—21no—
368C23×C4—711no—
369C23×C4—714no—
370C23×C4—714no—
371C23×C4—712no—
372C23×C4—712no—
373C23×C4—1421no—
374C23×C4—1422no—
375C23×C4—1422no—
376C23×C4—1422no—
377C23×C4—1422no—
378C23×C4C22×D7111yescharacteristic
379C2×M16—1428no—
380C2×M16—1428no—
381C2×M16—14232no—
382C2×M16—1428no—
383C22×C8—22no—
384C22×C8—22no—
385C22×C8—714no—
386C22×C8—712no—
387C22×C8—1422no—
388C22×C8—1422no—
389C22×C8D28414yescomplemented
390C22×C8C22×D7112yes—
391#121—7116no—
392#121—1428no—
393#121—14216no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—712no—
400C2×C4⋊C4—1421no—
401C2×C4⋊C4—1422no—
402C2×C4⋊C4—1422no—
403C2×C4⋊C4C4×D7114yes—
404C2×C4⋊C4C22×D7112yes—
405C2×C4⋊C4C22×D7112yes—
406C2×C22⋊C4—2844no—
407C2×C22⋊C4—1424no—
408C2×C22⋊C4—1424no—
409C2×C22⋊C4—1428no—
410#102—1428no—
411#102—1428no—
412#95—1424no—
413C24—2842no—
414C24—2842no—
415C24—2842no—
416C24—2842no—
417C24—2842no—
418C24—2842no—
419C24—712no—
420C24—714no—
421C24—1422no—
422C24—1424no—
423C24—1422no—
424C24—1422no—
425C24—1422no—
426C24—1422no—
427C24—1422no—
428C24—1422no—
429C24D4×D7111yescharacteristic
430C2×D4—28432no—
431C2×D4—2848no—
432C2×D4—2848no—
433C2×D4—2844no—
434C2×D4—2842no—
435C2×D4—2842no—
436C2×D4—2844no—
437C2×D4—714no—
438C2×D4—14216no—
439C2×D4—1422no—
440C2×D4—1422no—
441C2×D4—1422no—
442C2×D4—1422no—
443C2×D4—1422no—
444C2×D4—1424no—
445C2×D4—1428no—
446C2×D4—1428no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—2842no—
456C22×C4—2842no—
457C22×C4—2842no—
458C22×C4—712no—
459C22×C4—714no—
460C22×C4—714no—
461C22×C4—714no—
462C22×C4—718no—
463C22×C4—1422no—
464C22×C4—1422no—
465C22×C4—1422no—
466C22×C4—1422no—
467C22×C4—1422no—
468C22×C4—1422no—
469C22×C4—1422no—
470C22×C4—1428no—
471C22×C4—1428no—
472C22×C4—1428no—
473C22×C4—1424no—
474C22×C4—1424no—
475C22×C4—1428no—
476C22×C4C2×C4×D7112yes—
477C22×C4D4×D7112yes—
478C22×C4D4×D7112yes—
479C22×C4D4×D7111yescharacteristic
480C22×C4C2×D28412yes—
481C22×C4#1176586512yes—
482C22×C4#1176586612yes—
483C22×C4C23×D7111yescharacteristic
484M16—14232no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—28no—
488C2×C8—714no—
489C2×C8—1422no—
490C2×C8—1422no—
491C2×C8—1428no—
492C2×C8C2×D28414yescomplemented
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—42no—
498C4⋊C4—2842no—
499C4⋊C4—1422no—
500C4⋊C4—1422no—
501C4⋊C4C2×C4×D7114yes—
502C22⋊C4—2848no—
503C22⋊C4—14216no—
504C23—2844no—
505C23—2844no—
506C23—2844no—
507C23—2844no—
508C23—2848no—
509C23—2842no—
510C23—2842no—
511C23—2848no—
512C23—2848no—
513C23—2842no—
514C23—2842no—
515C23—2842no—
516C23—2842no—
517C23—5684no—
518C23—5684no—
519C23—5684no—
520C23—5684no—
521C23—714no—
522C23—1424no—
523C23—1428no—
524C23—1422no—
525C23—1422no—
526C23—1428no—
527C23—1428no—
528C23—1422no—
529C23—1422no—
530C23—1422no—
531C23—1422no—
532C23C22⋊C4×D7112yes—
533C23#1455610112yes—
534C23#1455610312yes—
535C23#1455615212yes—
536C23#1455615512yes—
537C23#1455615512yes—
538C23#1455616012yes—
539C23C2×D4×D7111yescharacteristic
540D4—28432no—
541D4—2848no—
542D4—2842no—
543D4—2842no—
544D4—1424no—
545D4—1424no—
546C2×C4—24no—
547C2×C4—48no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—84no—
554C2×C4—2842no—
555C2×C4—2842no—
556C2×C4—2842no—
557C2×C4—5684no—
558C2×C4—714no—
559C2×C4—1428no—
560C2×C4—1422no—
561C2×C4—1422no—
562C2×C4—14216no—
563C2×C4—1424no—
564C2×C4—1424no—
565C2×C4—1424no—
566C2×C4—1424no—
567C2×C4C22⋊C4×D7112yes—
568C2×C4#1455609014yes—
569C2×C4C4×D28414yes—
570C2×C4#1455609814yes—
571C2×C4#1455610112yes—
572C2×C4#1455610312yes—
573C2×C4#1455611314yes—
574C2×C4C22×C4×D7111yescharacteristic
575C2×C4C2×D4×D7111yescharacteristic
576C2×C4C2×D4×D7111yescharacteristic
577C2×C4C22×D28411yescharacteristic
578C2×C4C2×#1176586511yescharacteristic
579C2×C4C2×#1176586611yescharacteristic
580C8—24no—
581C8—1424no—
582C22—2842no—
583C22—2842no—
584C22—2842no—
585C22—2842no—
586C22—28416no—
587C22—2844no—
588C22—2844no—
589C22—2844no—
590C22—5682no—
591C22—5682no—
592C22—5682no—
593C22—5682no—
594C22—56816no—
595C22—1424no—
596C22—1424no—
597C22—1424no—
598C22#1798587714yes—
599C22#1798599914yes—
600C22#1798600214yes—
601C22#1798624614yes—
602C22#1798624814yes—
603C22#1798625114yes—
604C22#1798625314yes—
605C22C2×C22⋊C4×D7111yescharacteristic
606C22C2×#1455610111yescharacteristic
607C22C2×#1455610311yescharacteristic
608C22C2×#1455615211yescharacteristic
609C22C2×#1455615511yescharacteristic
610C22C2×#1455615511yescharacteristic
611C22C2×#1455616011yescharacteristic
612C4—44no—
613C4—82no—
614C4—5682no—
615C4—1424no—
616C4#1798587714yes—
617C4C2×#1455609011yescharacteristic
618C4C2×C4×D28411yescharacteristic
619C4C2×#1455609811yescharacteristic
620C4C2×#1455611311yescharacteristic
621C2—2844no—
622C2—5684no—
623C2#2258877718yesdirect product factor
624C2C2×#1798587711yescharacteristic
625C2C2×#1798600211yescharacteristic
626C2C2×#1798599911yescharacteristic
627C2C2×#1798624611yescharacteristic
628C2C2×#1798624811yescharacteristic
629C2C2×#1798625111yescharacteristic
630C2C2×#1798625311yescharacteristic
subgroupG/H
centerC24D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#340×C71C2
socleC24×C71C2×D4
prime radicalC22×#14556103C2
cosocleC22×C4×C71C24
hypercenterC2×#340D71
hypocenterC71C2×#1895
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
1. C23×#11765848C2semidirect
3. C22×#14556050C2semidirect
4. C22×#14556048C2semidirect
5. C2×#17985856C2semidirect
6. #22588777C2direct (N × H)
7. C2×#17985761C2semidirect
8. C2×#340×C71C2semidirect
15. C22×#11765848C22semidirect
19. C2×#14556103C4semidirect
25. C2×#14556050C22semidirect
27. C2×#14556048C22semidirect
29. #17985856C22semidirect
31. #17985761C22semidirect
35. #340×C71C22semidirect
66. C22×C71⋊C8D4semidirect
70. #14556103C2×C4semidirect
87. C22×C8×C71D4semidirect
155. C2×C71⋊C8C2×D4semidirect
181. C2×C8×C71C2×D4semidirect
306. C2×C71#1895semidirect
323. C2×#340D71semidirect
324. C71C2×#1895semidirect
351. #340C2×D71semidirect
389. C22×C8D284semidirect
492. C2×C8C2×D284semidirect
623. C2#22588777direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117658482
C22×#145561032
C22×#145560502
C22×#145560482
C2×#179858562
#225887772
C2×#179857612
C2×#340×C712
C2×#189571