Finite Group Databasebeta

← C2×#22588760 #31999112

Subgroups — C2×#22588760 #31999112

Order 18176.

2094 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C8×D71C211yesmaximal, characteristic
2C22×#14556104C211yesmaximal, characteristic
3C22×#14556052C211yesmaximal, characteristic, complemented
4C2×#17985856C211yesmaximal, characteristic
5#22588760C218yesmaximal, direct product factor
6C2×#17985762C211yesmaximal, characteristic, complemented
7C2×#17985761C211yesmaximal, characteristic, complemented
8C22×#103×C71C211yesmaximal, characteristic, complemented
9C23×C4×D71C2211yescharacteristic
10C23×C71⋊C8C2211yescharacteristic
11C22×C8×D71—24no—
12C22×C8×D71—24no—
13C22×C8×D71—24no—
14C22×C8×D71C2214yes—
15C2×#14556104—22no—
16C2×#14556104—24no—
17C2×#14556104C418yescomplemented
18C2×#14556104C2214yes—
19C22×C4⋊Dic71C2211yescharacteristic
20C22×C4⋊Dic71C2211yescharacteristic
21C22×Dic71⋊C4—21no—
22C22×(C2×D71)⋊C4—21no—
23C22×(C2×D71)⋊C4—21no—
24C2×#14556052—24no—
25C2×#14556052C2214yescomplemented
26#17985856—22no—
27#17985856C2214yes—
28C2×C71⋊#95—21no—
29#17985762C2214yescomplemented
30#17985761C2214yescomplemented
31C23×Q8×C71C2211yescharacteristic
32C23×C8×C71C2211yescharacteristic
33C22×C4⋊C4×C71C2211yescharacteristic
34C2×#103×C71—24no—
35C2×#103×C71C2214yescomplemented
36C24×D71D411yescharacteristic
37C23×Dic71—21no—
38C23×Dic71—21no—
39C23×Dic71D411yescharacteristic
40C22×C4×D71—22no—
41C22×C4×D71—22no—
42C22×C4×D71—22no—
43C22×C4×D71—22no—
44C22×C4×D71C2×C414yes—
45C22×C4×D71D412yes—
46C22×C4×D71D414yes—
47C22×C4×D71C2312yes—
48C22×C71⋊C8—22no—
49C22×C71⋊C8—22no—
50C22×C71⋊C8D414yes—
51C22×C71⋊C8C2312yes—
52C2×C8×D71—232no—
53C2×C8×D71—28no—
54C2×C8×D71—28no—
55C2×C8×D71—28no—
56#14556104—216no—
57#14556104—28no—
58#14556104C2×C4116yescomplemented
59C2×C4⋊Dic71—21no—
60C2×C4⋊Dic71—22no—
61C2×C4⋊Dic71—22no—
62C2×C4⋊Dic71—21no—
63C2×C4⋊Dic71—22no—
64C2×C4⋊Dic71C2×C414yes—
65C2×C4⋊Dic71C2312yes—
66C2×C4⋊Dic71C2312yes—
67C2×Dic71⋊C4—28no—
68C2×Dic71⋊C4—24no—
69C2×(C2×D71)⋊C4—28no—
70C2×(C2×D71)⋊C4—24no—
71C2×(C2×D71)⋊C4—24no—
72C2×(C2×D71)⋊C4—44no—
73#14556052—28no—
74C71⋊#95—24no—
75C22×Q8×C71—22no—
76C22×Q8×C71—21no—
77C22×Q8×C71—22no—
78C22×Q8×C71—22no—
79C22×Q8×C71—22no—
80C22×Q8×C71—22no—
81C22×Q8×C71C2×C414yes—
82C22×Q8×C71C2312yes—
83C23×C4×C71—21no—
84C23×C4×C71—21no—
85C23×C4×C71C2311yescharacteristic
86C22×C8×C71—22no—
87C22×C8×C71—22no—
88C22×C8×C71D414yes—
89C22×C8×C71C2312yes—
90C2×C4⋊C4×C71—22no—
91C2×C4⋊C4×C71—22no—
92C2×C4⋊C4×C71—21no—
93C2×C4⋊C4×C71C2312yes—
94#103×C71—28no—
95C23×D71—22no—
96C23×D71—22no—
97C23×D71—22no—
98C23×D71—22no—
99C23×D71C22⋊C414yes—
100C23×D71SD1614yes—
101C23×D71Q1614yes—
102C23×D71C2×D412yes—
103C22×Dic71—22no—
104C22×Dic71—22no—
105C22×Dic71—24no—
106C22×Dic71—22no—
107C22×Dic71—22no—
108C22×Dic71—42no—
109C22×Dic71—42no—
110C22×Dic71—42no—
111C22×Dic71—42no—
112C22×Dic71—42no—
113C22×Dic71C22⋊C414yes—
114C22×Dic71C2×D412yes—
115C2×C4×D71—28no—
116C2×C4×D71—28no—
117C2×C4×D71—28no—
118C2×C4×D71—24no—
119C2×C4×D71—24no—
120C2×C4×D71—28no—
121C2×C4×D71—22no—
122C2×C4×D71—22no—
123C2×C4×D71—22no—
124C2×C4×D71—22no—
125C2×C4×D71C22⋊C418yes—
126C2×C4×D71C22×C414yes—
127C2×C4×D71C2×D414yes—
128C2×C71⋊C8—28no—
129C2×C71⋊C8—22no—
130C2×C71⋊C8—22no—
131C2×C71⋊C8C2×D414yes—
132C8×D71—232no—
133C4⋊Dic71—24no—
134C4⋊Dic71—22no—
135C4⋊Dic71—22no—
136C4⋊Dic71—22no—
137C4⋊Dic71—42no—
138C4⋊Dic71C22×C414yes—
139Dic71⋊C4—216no—
140(C2×D71)⋊C4—216no—
141(C2×D71)⋊C4—48no—
142C24×C71C2×D411yescharacteristic
143C2×Q8×C71—24no—
144C2×Q8×C71—22no—
145C2×Q8×C71—28no—
146C2×Q8×C71—28no—
147C2×Q8×C71—22no—
148C2×Q8×C71—22no—
149C2×Q8×C71—22no—
150C2×Q8×C71—22no—
151C2×Q8×C71—44no—
152C2×Q8×C71—42no—
153C2×Q8×C71—42no—
154C2×Q8×C71—44no—
155C2×Q8×C71C22×C414yes—
156C22×C4×C71—24no—
157C22×C4×C71—22no—
158C22×C4×C71—22no—
159C22×C4×C71—42no—
160C22×C4×C71—42no—
161C22×C4×C71—42no—
162C22×C4×C71—42no—
163C22×C4×C71—42no—
164C22×C4×C71C22×C412yes—
165C22×C4×C71C2×D411yescharacteristic
166C22×C4×C71C2×D412yes—
167C22×C4×C71C2×D412yes—
168C22×C4×C71C2×D412yes—
169C22×C4×C71#4112yes—
170C22×C4×C71#4112yes—
171C22×C4×C71C2411yescharacteristic
172C2×C8×C71—28no—
173C2×C8×C71—22no—
174C2×C8×C71—22no—
175C2×C8×C71C2×D414yes—
176C4⋊C4×C71—22no—
177C4⋊C4×C71—22no—
178C4⋊C4×C71—42no—
179C23×C71C2×C22⋊C412yes—
180C23×C71#12112yes—
181C23×C71#12112yes—
182C23×C71C2×SD1612yes—
183C23×C71C2×Q1612yes—
184C23×C71#13512yes—
185C23×C71#13512yes—
186C23×C71C22×D411yescharacteristic
187C22×D71—22no—
188C22×D71—22no—
189C22×D71—22no—
190C22×D71—22no—
191C22×D71—28no—
192C22×D71—28no—
193C22×D71—28no—
194C22×D71—28no—
195C22×D71—28no—
196C22×D71—28no—
197C22×D71#103116yescomplemented
198C22×D71C2×C22⋊C414yes—
199C22×D71C2×SD1614yes—
200C22×D71C2×Q1614yes—
201Q8×C71—24no—
202Q8×C71—24no—
203Q8×C71—42no—
204Q8×C71—42no—
205Q8×C71—48no—
206C2×C4×C71—24no—
207C2×C4×C71—42no—
208C2×C4×C71—42no—
209C2×C4×C71—48no—
210C2×C4×C71—42no—
211C2×C4×C71—42no—
212C2×C4×C71—42no—
213C2×C4×C71—84no—
214C2×C4×C71C2×C22⋊C412yes—
215C2×C4×C71C4×D414yes—
216C2×C4×C71C4×D414yes—
217C2×C4×C71#12014yes—
218C2×C4×C71#12112yes—
219C2×C4×C71#12112yes—
220C2×C4×C71#12314yes—
221C2×C4×C71C23×C411yescharacteristic
222C2×C4×C71C22×D411yescharacteristic
223C2×C4×C71C22×D411yescharacteristic
224C2×C4×C71C22×D411yescharacteristic
225C2×C4×C71C2×#4111yescharacteristic
226C2×C4×C71C2×#4111yescharacteristic
227C2×Dic71—22no—
228C2×Dic71—22no—
229C2×Dic71—28no—
230C2×Dic71—24no—
231C2×Dic71—42no—
232C2×Dic71—42no—
233C2×Dic71—48no—
234C2×Dic71—42no—
235C2×Dic71—42no—
236C2×Dic71—42no—
237C2×Dic71—84no—
238C2×Dic71C2×C22⋊C414yes—
239C4×D71—24no—
240C4×D71—24no—
241C4×D71—24no—
242C4×D71—24no—
243C4×D71—216no—
244C8×C71—24no—
245C71⋊C8—24no—
246C22×C71#38614yes—
247C22×C71C2×#10314yes—
248C22×C71#41614yes—
249C22×C71#46514yes—
250C22×C71#46514yes—
251C22×C71#46714yes—
252C22×C71#46714yes—
253C22×C71C22×C22⋊C411yescharacteristic
254C22×C71C2×#12111yescharacteristic
255C22×C71C2×#12111yescharacteristic
256C22×C71C22×SD1611yescharacteristic
257C22×C71C22×Q1611yescharacteristic
258C22×C71C2×#13511yescharacteristic
259C22×C71C2×#13511yescharacteristic
260C2×D71—24no—
261C2×D71—232no—
262C2×D71—24no—
263C2×D71—24no—
264C2×D71—24no—
265C2×D71—24no—
266C2×D71—24no—
267C2×D71C2×#10318yescomplemented
268C4×C71—44no—
269C4×C71—82no—
270C4×C71#38614yes—
271C4×C71C2×C4×D411yescharacteristic
272C4×C71C2×C4×D411yescharacteristic
273C4×C71C2×#12011yescharacteristic
274C4×C71C2×#12311yescharacteristic
275Dic71—24no—
276Dic71—44no—
277Dic71—82no—
278C2×#1892—711nomaximal, Sylow
279C2×C71#189218yescomplemented
280C2×C71C2×#38611yescharacteristic
281C2×C71C22×#10311yescharacteristic
282C2×C71C2×#41611yescharacteristic
283C2×C71C2×#46511yescharacteristic
284C2×C71C2×#46511yescharacteristic
285C2×C71C2×#46711yescharacteristic
286C2×C71C2×#46711yescharacteristic
287D71—28no—
288C24×C8—711no—
289C22×C22⋊Q8—711no—
290C22×#103—711no—
291C22×#103D7111yescharacteristic, complemented
292C2×#380—711no—
293#1892—718no—
294C2×#340—711no—
295C2×#340—711no—
296C71C2×#189211yescharacteristic, complemented, Sylow
297C23×Q8C2×D7111yescharacteristic
298C24×C4—711no—
299C23×C8—711no—
300C23×C8—714no—
301C23×C8—1424no—
302C23×C8—1424no—
303C23×C8—1424no—
304C23×C8C2×D7111yescharacteristic
305C2×C22⋊Q8—718no—
306C2×C22⋊Q8—714no—
307C2×C22⋊Q8—1424no—
308C2×C22⋊Q8—1422no—
309C22×C4⋊C4—711no—
310C22×C4⋊C4—711no—
311C22×C4⋊C4—1421no—
312C22×C4⋊C4C2×D7111yescharacteristic
313C22×C22⋊C4—1421no—
314C22×C22⋊C4—1421no—
315C2×#103—24no—
316C2×#103—714no—
317C2×#103—1424no—
318C2×#103C2×D7114yescomplemented
319#380—714no—
320#380—1422no—
321C2×#95—1421no—
322#340—714no—
323#340—714no—
324C25—711no—
325C22×Q8—21no—
326C22×Q8—22no—
327C22×Q8—22no—
328C22×Q8—22no—
329C22×Q8—22no—
330C22×Q8—22no—
331C22×Q8C4×D7114yes—
332C22×Q8C22×D7112yes—
333C23×C4—21no—
334C23×C4—21no—
335C23×C4—711no—
336C23×C4—714no—
337C23×C4—714no—
338C23×C4—712no—
339C23×C4—712no—
340C23×C4—1421no—
341C23×C4—1421no—
342C23×C4—1422no—
343C23×C4—1422no—
344C23×C4—1422no—
345C23×C4—1422no—
346C23×C4C22×D7111yescharacteristic
347C22×C8—22no—
348C22×C8—22no—
349C22×C8—714no—
350C22×C8—712no—
351C22×C8—1428no—
352C22×C8—1428no—
353C22×C8—14232no—
354C22×C8—1422no—
355C22×C8—1422no—
356C22×C8—1428no—
357C22×C8C22⋊D7114yes—
358C22×C8C22×D7112yes—
359C22⋊Q8—7116no—
360C22⋊Q8—1428no—
361C22⋊Q8—14216no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—714no—
366C2×C4⋊C4—712no—
367C2×C4⋊C4—712no—
368C2×C4⋊C4—1421no—
369C2×C4⋊C4—1422no—
370C2×C4⋊C4—1422no—
371C2×C4⋊C4—1424no—
372C2×C4⋊C4—1428no—
373C2×C4⋊C4—1422no—
374C2×C4⋊C4—1421no—
375C2×C4⋊C4C22×D7112yes—
376C2×C22⋊C4—2844no—
377C2×C22⋊C4—1424no—
378C2×C22⋊C4—1424no—
379C2×C22⋊C4—1428no—
380#103—28no—
381#103—1428no—
382#95—1424no—
383C24—712no—
384C24—714no—
385C24—714no—
386C24—714no—
387C24—1422no—
388C24—1422no—
389C24—1422no—
390C24—1422no—
391C24D4×D7111yescharacteristic
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—28no—
399C2×Q8—28no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8C2×C4×D7114yes—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—2842no—
414C22×C4—2842no—
415C22×C4—2842no—
416C22×C4—2842no—
417C22×C4—2842no—
418C22×C4—712no—
419C22×C4—714no—
420C22×C4—714no—
421C22×C4—714no—
422C22×C4—718no—
423C22×C4—1422no—
424C22×C4—1422no—
425C22×C4—1424no—
426C22×C4—1422no—
427C22×C4—1422no—
428C22×C4—1422no—
429C22×C4—1422no—
430C22×C4—1422no—
431C22×C4—1422no—
432C22×C4—1428no—
433C22×C4—1428no—
434C22×C4—1428no—
435C22×C4—1424no—
436C22×C4—1424no—
437C22×C4—1428no—
438C22×C4C2×C4×D7112yes—
439C22×C4D4×D7112yes—
440C22×C4D4×D7112yes—
441C22×C4D4×D7111yescharacteristic
442C22×C4C2×C22⋊D7112yes—
443C22×C4#1176586412yes—
444C22×C4#1176586512yes—
445C22×C4C23×D7111yescharacteristic
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—28no—
449C2×C8—714no—
450C2×C8—14232no—
451C2×C8—1422no—
452C2×C8—1422no—
453C2×C8—1428no—
454C2×C8C2×C22⋊D7114yes—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—42no—
458C4⋊C4—2842no—
459C4⋊C4—714no—
460C4⋊C4—14216no—
461C4⋊C4—1422no—
462C4⋊C4—1422no—
463C4⋊C4—1422no—
464C4⋊C4—1424no—
465C22⋊C4—2848no—
466C22⋊C4—14216no—
467C23—714no—
468C23—7116no—
469C23—714no—
470C23—714no—
471C23—1428no—
472C23—1428no—
473C23—1428no—
474C23—1422no—
475C23—1422no—
476C23—1428no—
477C23—1428no—
478C23—1428no—
479C23—1422no—
480C23—1422no—
481C23C22⋊C4×D7112yes—
482C23#1455609912yes—
483C23#1455610112yes—
484C23SD16×D7112yes—
485C23Q16×D7112yes—
486C23#1455614912yes—
487C23#1455615112yes—
488C23C2×D4×D7111yescharacteristic
489Q8—24no—
490Q8—24no—
491Q8—48no—
492Q8—42no—
493Q8—42no—
494C2×C4—24no—
495C2×C4—48no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—84no—
502C2×C4—2848no—
503C2×C4—2842no—
504C2×C4—2842no—
505C2×C4—2842no—
506C2×C4—2842no—
507C2×C4—2842no—
508C2×C4—5684no—
509C2×C4—714no—
510C2×C4—1424no—
511C2×C4—1428no—
512C2×C4—1422no—
513C2×C4—1422no—
514C2×C4—14216no—
515C2×C4—1424no—
516C2×C4—1424no—
517C2×C4—1424no—
518C2×C4—1424no—
519C2×C4C22⋊C4×D7112yes—
520C2×C4(C22⋊D71)⋊C414yes—
521C2×C4C4×C22⋊D7114yes—
522C2×C4#1455609614yes—
523C2×C4#1455609912yes—
524C2×C4#1455610112yes—
525C2×C4#1455611014yes—
526C2×C4C22×C4×D7111yescharacteristic
527C2×C4C2×D4×D7111yescharacteristic
528C2×C4C2×D4×D7111yescharacteristic
529C2×C4C22×C22⋊D7111yescharacteristic
530C2×C4C2×#1176586411yescharacteristic
531C2×C4C2×#1176586511yescharacteristic
532C8—24no—
533C8—1424no—
534C22—718no—
535C22—1424no—
536C22—1424no—
537C22—1424no—
538C22—1424no—
539C22—1424no—
540C22—14232no—
541C22—1424no—
542C22#1798587514yes—
543C22#103×D7114yes—
544C22#1798599414yes—
545C22#1798622514yes—
546C22#1798623114yes—
547C22#1798623714yes—
548C22#1798623914yes—
549C22C2×C22⋊C4×D7111yescharacteristic
550C22C2×#1455609911yescharacteristic
551C22C2×#1455610111yescharacteristic
552C22C2×SD16×D7111yescharacteristic
553C22C2×Q16×D7111yescharacteristic
554C22C2×#1455614911yescharacteristic
555C22C2×#1455615111yescharacteristic
556C4—44no—
557C4—82no—
558C4—2844no—
559C4—5682no—
560C4—1424no—
561C4#1798587514yes—
562C4C2×(C22⋊D71)⋊C411yescharacteristic
563C4C2×C4×C22⋊D7111yescharacteristic
564C4C2×#1455609611yescharacteristic
565C4C2×#1455611011yescharacteristic
566C2—1428no—
567C2#2258876018yesdirect product factor
568C2C2×#1798587511yescharacteristic
569C2C2×#103×D7111yescharacteristic
570C2C2×#1798599411yescharacteristic
571C2C2×#1798622511yescharacteristic
572C2C2×#1798623111yescharacteristic
573C2C2×#1798623711yescharacteristic
574C2C2×#1798623911yescharacteristic
subgroupG/H
centerC24D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC22×#103×C71C2
socleC24×C71C2×D4
prime radicalC24×D71D4
cosocleC22×C4×C71C24
hypercenterC22×#103D71
hypocenterC71C2×#1892
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
3. C22×#14556052C2semidirect
5. #22588760C2direct (N × H)
6. C2×#17985762C2semidirect
7. C2×#17985761C2semidirect
8. C22×#103×C71C2semidirect
17. C2×#14556104C4semidirect
25. C2×#14556052C22semidirect
29. #17985762C22semidirect
30. #17985761C22semidirect
35. C2×#103×C71C22semidirect
58. #14556104C2×C4semidirect
197. C22×D71#103semidirect
267. C2×D71C2×#103semidirect
279. C2×C71#1892semidirect
291. C22×#103D71semidirect
296. C71C2×#1892semidirect
318. C2×#103C2×D71semidirect
567. C2#22588760direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C8×D712
C22×#145561042
C22×#145560522
C2×#179858562
#225887602
C2×#179857622
C2×#179857612
C22×#103×C712
C2×#189271