Finite Group Databasebeta

← C2×#22591918 #32165597

Subgroups — C2×#22591918 #32165597

Order 18176.

3894 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556117C211yesmaximal, characteristic, complemented
2C22×#14556104C211yesmaximal, characteristic, complemented
3C22×#14556103C211yesmaximal, characteristic
4C22×C4×D284C211yesmaximal, characteristic, complemented
5#22591918C2116yesmaximal, direct product factor
6C2×#17985928C212yesmaximal, complemented
7C2×#17985917C212yesmaximal, complemented
8C2×#17985895C211yesmaximal, characteristic, complemented
9C2×#17985877C212yesmaximal, complemented
10C2×#17985858C212yesmaximal, complemented
11C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic, complemented
12C2×#391×C71C211yesmaximal, characteristic, complemented
13C23×D284C2212yes—
14C23×D284C2211yescharacteristic
15C23×C4×D71C2211yescharacteristic
16C23×C4×D71C2212yes—
17C2×#14556117—28no—
18C2×#14556117C2218yescomplemented
19C2×#14556104—28no—
20C2×#14556104C2218yescomplemented
21C2×#14556103—28no—
22C2×#14556103C2218yes—
23C2×C4×D284—28no—
24C2×C4×D284C2218yescomplemented
25C22×C4⋊Dic71C2211yescharacteristic
26C22×Dic71⋊C4C2212yes—
27C22×Dic71⋊C4C2212yes—
28C22×(C2×D71)⋊C4C2211yescharacteristic
29C22×(C2×D71)⋊C4C2212yes—
30C22×(C2×D71)⋊C4C2212yes—
31C22×(C2×D71)⋊C4C2212yes—
32C22×(C2×D71)⋊C4C2211yescharacteristic
33C22×(C2×D71)⋊C4C2212yes—
34C22×(C2×D71)⋊C4C2212yes—
35C22×C4×Dic71C2211yescharacteristic
36#17985928C22116yescomplemented
37#17985917C22116yescomplemented
38#17985895C2218yescomplemented
39#17985877C22116yescomplemented
40#17985858C22116yescomplemented
41C4×(C2×D71)⋊C4C2218yescomplemented
42C2×C71⋊#95C2212yes—
43C2×C71⋊#95C2211yescharacteristic
44C2×C71⋊#95C2211yescharacteristic
45C23×Q8×C71C2211yescharacteristic
46C22×C4⋊C4×C71C2211yescharacteristic
47C22×C42×C71C2211yescharacteristic
48#391×C71C2218yescomplemented
49C2×#95×C71C2212yes—
50C2×#95×C71C2212yes—
51C24×D71C2312yes—
52C24×D71C2311yescharacteristic
53C22×D284—28no—
54C22×D284—28no—
55C22×D284—24no—
56C22×D284—24no—
57C22×D284—44no—
58C22×D284D418yes—
59C22×D284C2318yes—
60C22×D284C2314yes—
61C23×Dic71C2311yescharacteristic
62C23×Dic71C2311yescharacteristic
63C23×Dic71C2311yescharacteristic
64C23×Dic71C2312yes—
65C22×C4×D71—28no—
66C22×C4×D71—24no—
67C22×C4×D71—24no—
68C22×C4×D71—24no—
69C22×C4×D71—28no—
70C22×C4×D71—44no—
71C22×C4×D71C2314yes—
72C22×C4×D71C2318yes—
73#14556117—232no—
74#14556104—232no—
75#14556103—232no—
76C4×D284—232no—
77C2×C4⋊Dic71D418yescomplemented
78C2×C4⋊Dic71C2314yes—
79C2×Dic71⋊C4—28no—
80C2×Dic71⋊C4—44no—
81C2×Dic71⋊C4C2318yes—
82C2×Dic71⋊C4C2318yes—
83C2×(C2×D71)⋊C4—28no—
84C2×(C2×D71)⋊C4—28no—
85C2×(C2×D71)⋊C4—28no—
86C2×(C2×D71)⋊C4—24no—
87C2×(C2×D71)⋊C4—28no—
88C2×(C2×D71)⋊C4—28no—
89C2×(C2×D71)⋊C4—24no—
90C2×(C2×D71)⋊C4C2314yes—
91C2×(C2×D71)⋊C4C2314yes—
92C2×(C2×D71)⋊C4C2318yes—
93C2×(C2×D71)⋊C4C2318yes—
94C2×(C2×D71)⋊C4C2318yes—
95C2×(C2×D71)⋊C4C2318yes—
96C2×(C2×D71)⋊C4C2318yes—
97C2×C4×Dic71—24no—
98C2×C4×Dic71C2314yes—
99C71⋊#95C2314yes—
100C71⋊#95C2314yes—
101C71⋊#95C2318yes—
102C22×Q8×C71—24no—
103C22×Q8×C71—24no—
104C22×Q8×C71—24no—
105C22×Q8×C71C2314yes—
106C23×C4×C71C2312yes—
107C23×C4×C71C2311yescharacteristic
108C23×C4×C71C2312yes—
109C23×C4×C71C2311yescharacteristic
110C23×C4×C71C2311yescharacteristic
111C2×C4⋊C4×C71—24no—
112C2×C4⋊C4×C71C2314yes—
113C2×C42×C71—24no—
114C2×C42×C71C2314yes—
115#95×C71C2318yes—
116#95×C71C2318yes—
117C23×D71—22no—
118C23×D71—24no—
119C23×D71—22no—
120C23×D71—22no—
121C23×D71—24no—
122C23×D71—24no—
123C23×D71—24no—
124C23×D71—22no—
125C23×D71—24no—
126C23×D71—22no—
127C23×D71—24no—
128C23×D71—22no—
129C23×D71C2×D418yes—
130C23×D71#4114yes—
131C23×D71C2414yes—
132C23×D71C2412yes—
133C2×D284—232no—
134C2×D284—28no—
135C2×D284—28no—
136C2×D284—216no—
137C2×D284—216no—
138C2×D284—48no—
139C2×D284—432no—
140C2×D284C2×D4116yes—
141C22×Dic71—24no—
142C22×Dic71—24no—
143C22×Dic71—22no—
144C22×Dic71—22no—
145C22×Dic71—22no—
146C22×Dic71—22no—
147C22×Dic71—22no—
148C22×Dic71—22no—
149C22×Dic71—22no—
150C22×Dic71—22no—
151C22×Dic71—22no—
152C22×Dic71C2×D418yes—
153C22×Dic71C2412yes—
154C22×Dic71C2412yes—
155C22×Dic71C2412yes—
156C22×Dic71C2414yes—
157C2×C4×D71—28no—
158C2×C4×D71—28no—
159C2×C4×D71—28no—
160C2×C4×D71—216no—
161C2×C4×D71—216no—
162C2×C4×D71—48no—
163C2×C4×D71—432no—
164C2×C4×D71—416no—
165C4⋊Dic71C2×D4116yescomplemented
166Dic71⋊C4—216no—
167Dic71⋊C4—48no—
168(C2×D71)⋊C4—216no—
169(C2×D71)⋊C4—216no—
170(C2×D71)⋊C4—216no—
171(C2×D71)⋊C4—28no—
172(C2×D71)⋊C4—28no—
173(C2×D71)⋊C4—216no—
174(C2×D71)⋊C4—216no—
175C4×Dic71—28no—
176C24×C71C2411yescharacteristic
177C2×Q8×C71—28no—
178C2×Q8×C71—28no—
179C2×Q8×C71—28no—
180C2×Q8×C71—416no—
181C22×C4×C71—22no—
182C22×C4×C71—22no—
183C22×C4×C71—24no—
184C22×C4×C71—22no—
185C22×C4×C71—24no—
186C22×C4×C71—24no—
187C22×C4×C71—24no—
188C22×C4×C71—24no—
189C22×C4×C71—22no—
190C22×C4×C71—24no—
191C22×C4×C71—22no—
192C22×C4×C71C2×D414yes—
193C22×C4×C71C2×D414yes—
194C22×C4×C71#4114yes—
195C22×C4×C71#4114yes—
196C22×C4×C71C2414yes—
197C22×C4×C71C2412yes—
198C22×C4×C71C2414yes—
199C22×C4×C71C2412yes—
200C22×C4×C71C2412yes—
201C4⋊C4×C71—28no—
202C42×C71—28no—
203C23×C71C22×D412yes—
204C23×C71C22×D412yes—
205C23×C71C2×#4112yes—
206C23×C71C2×#4112yes—
207C23×C71C2×#4112yes—
208C23×C71H(5,𝔽2)12yes—
209C23×C712−1+412yes—
210C23×C71C2511yescharacteristic
211C22×D71—216no—
212C22×D71—28no—
213C22×D71—28no—
214C22×D71—216no—
215C22×D71—216no—
216C22×D71—28no—
217C22×D71—22no—
218C22×D71—24no—
219C22×D71—22no—
220C22×D71—22no—
221C22×D71—22no—
222C22×D71—24no—
223C22×D71—24no—
224C22×D71—24no—
225C22×D71—24no—
226C22×D71—22no—
227C22×D71—24no—
228C22×D71—22no—
229C22×D71—44no—
230C22×D71—48no—
231C22×D71—44no—
232C22×D71—48no—
233C22×D71—48no—
234C22×D71—44no—
235C22×D71—44no—
236C22×D71—48no—
237C22×D71C22×D418yes—
238C22×D71C2×#4114yes—
239Q8×C71—416no—
240C2×C4×C71—28no—
241C2×C4×C71—22no—
242C2×C4×C71—22no—
243C2×C4×C71—22no—
244C2×C4×C71—24no—
245C2×C4×C71—24no—
246C2×C4×C71—24no—
247C2×C4×C71—24no—
248C2×C4×C71—22no—
249C2×C4×C71—24no—
250C2×C4×C71—24no—
251C2×C4×C71—22no—
252C2×C4×C71—48no—
253C2×C4×C71—48no—
254C2×C4×C71—44no—
255C2×C4×C71#121116yes—
256C2×C4×C71C22×D414yes—
257C2×C4×C71C22×D414yes—
258C2×C4×C71C2×#4114yes—
259C2×C4×C71C2×#4114yes—
260C2×Dic71—216no—
261C2×Dic71—24no—
262C2×Dic71—24no—
263C2×Dic71—22no—
264C2×Dic71—22no—
265C2×Dic71—22no—
266C2×Dic71—22no—
267C2×Dic71—22no—
268C2×Dic71—22no—
269C2×Dic71—22no—
270C2×Dic71—22no—
271C2×Dic71—22no—
272C2×Dic71—44no—
273C2×Dic71—44no—
274C2×Dic71—44no—
275C2×Dic71C22×D418yes—
276D284—232no—
277D284—432no—
278C4×D71—432no—
279C4×D71—416no—
280C22×C71C2×#12114yes—
281C22×C71#52914yes—
282C22×C71#54214yes—
283C22×C71#54614yes—
284C22×C71#54614yes—
285C22×C71#54714yes—
286C22×C71#54814yes—
287C22×C71C23×D411yescharacteristic
288C22×C71C23×D411yescharacteristic
289C22×C71C22×#4111yescharacteristic
290C22×C71C22×#4111yescharacteristic
291C22×C71C22×#4111yescharacteristic
292C22×C71C2×H(5,𝔽2)11yescharacteristic
293C22×C71C2×2−1+411yescharacteristic
294C2×D71—28no—
295C2×D71—24no—
296C2×D71—24no—
297C2×D71—28no—
298C2×D71—28no—
299C2×D71—24no—
300C2×D71—42no—
301C2×D71—44no—
302C2×D71—432no—
303C2×D71—416no—
304C2×D71—42no—
305C2×D71—44no—
306C2×D71—44no—
307C2×D71—42no—
308C2×D71—42no—
309C2×D71—44no—
310C4×C71—24no—
311C4×C71—44no—
312C4×C71—44no—
313C4×C71—42no—
314C4×C71C2×#12118yes—
315Dic71—28no—
316Dic71—42no—
317Dic71—42no—
318Dic71—42no—
319C2×#2427—711nomaximal, Sylow
320C2×C71#242718yescomplemented
321C2×C71C22×#12111yescharacteristic
322C2×C71C2×#52911yescharacteristic
323C2×C71C2×#54211yescharacteristic
324C2×C71C2×#54611yescharacteristic
325C2×C71C2×#54611yescharacteristic
326C2×C71C2×#54711yescharacteristic
327C2×C71C2×#54811yescharacteristic
328D71—48no—
329D71—44no—
330C22×#124—711no—
331C22×C22⋊Q8—711no—
332C22×#121—711no—
333C22×C4×D4—711no—
334#2427—7116no—
335C2×#397—712no—
336C2×#394—712no—
337C2×#391D7111yescharacteristic, complemented
338C2×#389—711no—
339C2×#386—712no—
340C2×#380—712no—
341C2×C4×C22⋊C4—711no—
342C71C2×#242711yescharacteristic, complemented, Sylow
343C23×Q8C2×D7111yescharacteristic
344C23×D4—712no—
345C23×D4—711no—
346C24×C4—712no—
347C24×C4—711no—
348C2×#124—718no—
349C2×#124—1428no—
350C2×C22⋊Q8—718no—
351C2×C22⋊Q8—1428no—
352C2×#121—718no—
353C2×#121—1428no—
354C2×C4×D4—718no—
355C2×C4×D4—1428no—
356C22×C4⋊C4—712no—
357C22×C4⋊C4—711no—
358C22×C4⋊C4—712no—
359C22×C4⋊C4C2×D7111yescharacteristic
360C22×C22⋊C4—711no—
361C22×C22⋊C4—712no—
362C22×C22⋊C4—712no—
363C22×C22⋊C4—712no—
364C22×C22⋊C4—712no—
365C22×C22⋊C4—712no—
366C22×C22⋊C4—711no—
367C22×C42—711no—
368C22×C42C2×D7111yescharacteristic
369#397—7116no—
370#394—7116no—
371#391C2×D7118yescomplemented
372#389—718no—
373#386—7116no—
374#380—7116no—
375C4×C22⋊C4—718no—
376C2×#95—711no—
377C2×#95—712no—
378C2×#95—711no—
379C2×#95C2×D7112yes—
380C2×#95C2×D7112yes—
381C25—712no—
382C25—711no—
383C22×Q8—24no—
384C22×Q8—24no—
385C22×Q8—24no—
386C22×Q8C22×D7114yes—
387C22×D4—2844no—
388C22×D4—718no—
389C22×D4—718no—
390C22×D4—714no—
391C22×D4—1428no—
392C22×D4—1428no—
393C22×D4—1424no—
394C22×D4—1424no—
395C23×C4—2844no—
396C23×C4—718no—
397C23×C4—712no—
398C23×C4—711no—
399C23×C4—711no—
400C23×C4—711no—
401C23×C4—714no—
402C23×C4—1428no—
403C23×C4—1424no—
404C23×C4—1424no—
405C23×C4—1424no—
406C23×C4—1428no—
407C23×C4C22×D7112yes—
408C23×C4C22×D7111yescharacteristic
409C23×C4C22×D7112yes—
410C23×C4C22×D7111yescharacteristic
411C23×C4C22×D7111yescharacteristic
412#124—14232no—
413C22⋊Q8—14232no—
414#121—14232no—
415C4×D4—14232no—
416C2×C4⋊C4—24no—
417C2×C4⋊C4—2844no—
418C2×C4⋊C4—718no—
419C2×C4⋊C4—718no—
420C2×C4⋊C4—714no—
421C2×C4⋊C4—718no—
422C2×C4⋊C4—1428no—
423C2×C4⋊C4C22×D7114yes—
424C2×C22⋊C4—718no—
425C2×C22⋊C4—718no—
426C2×C22⋊C4—718no—
427C2×C22⋊C4—718no—
428C2×C22⋊C4—718no—
429C2×C22⋊C4—714no—
430C2×C22⋊C4—714no—
431C2×C22⋊C4—1428no—
432C2×C22⋊C4—1428no—
433C2×C22⋊C4—1428no—
434C2×C22⋊C4—1428no—
435C2×C22⋊C4—1428no—
436C2×C22⋊C4—1424no—
437C2×C22⋊C4—1424no—
438C2×C42—24no—
439C2×C42—714no—
440C2×C42—1424no—
441C2×C42C22×D7114yes—
442#95—714no—
443#95—714no—
444#95—718no—
445#95C22×D7118yes—
446#95C22×D7118yes—
447C24—718no—
448C24—714no—
449C24—714no—
450C24—712no—
451C24—1422no—
452C24—1424no—
453C24—1422no—
454C24—1424no—
455C24—1422no—
456C24—1424no—
457C24—1422no—
458C24—1424no—
459C24—1424no—
460C24—1422no—
461C24—1422no—
462C24—1424no—
463C24C23×D7111yescharacteristic
464C2×Q8—28no—
465C2×Q8—28no—
466C2×Q8—28no—
467C2×Q8—416no—
468C2×D4—2848no—
469C2×D4—28432no—
470C2×D4—7116no—
471C2×D4—14216no—
472C2×D4—14216no—
473C2×D4—1428no—
474C2×D4—1428no—
475C2×D4—14232no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—28432no—
488C22×C4—2848no—
489C22×C4—28416no—
490C22×C4—718no—
491C22×C4—714no—
492C22×C4—712no—
493C22×C4—712no—
494C22×C4—712no—
495C22×C4—14216no—
496C22×C4—1422no—
497C22×C4—1422no—
498C22×C4—1422no—
499C22×C4—1422no—
500C22×C4—1422no—
501C22×C4—1422no—
502C22×C4—1422no—
503C22×C4—1422no—
504C22×C4—1422no—
505C22×C4—1428no—
506C22×C4—1428no—
507C22×C4—1428no—
508C22×C4—14216no—
509C22×C4—1424no—
510C22×C4—1424no—
511C22×C4D4×D7114yes—
512C22×C4D4×D7114yes—
513C22×C4#1176586514yes—
514C22×C4#1176586614yes—
515C22×C4C23×D7112yes—
516C22×C4C23×D7114yes—
517C22×C4C23×D7114yes—
518C22×C4C23×D7112yes—
519C22×C4C23×D7112yes—
520C4⋊C4—28no—
521C4⋊C4—2848no—
522C4⋊C4—7116no—
523C4⋊C4—14216no—
524C22⋊C4—14216no—
525C22⋊C4—14216no—
526C22⋊C4—1428no—
527C22⋊C4—14216no—
528C22⋊C4—14216no—
529C22⋊C4—14216no—
530C22⋊C4—1428no—
531C42—28no—
532C42—1428no—
533C23—2844no—
534C23—2848no—
535C23—2848no—
536C23—2844no—
537C23—2844no—
538C23—2848no—
539C23—2844no—
540C23—2848no—
541C23—718no—
542C23—714no—
543C23—1422no—
544C23—1424no—
545C23—1422no—
546C23—1424no—
547C23—1422no—
548C23—1424no—
549C23—1422no—
550C23—1424no—
551C23—1424no—
552C23—1422no—
553C23—1422no—
554C23—1424no—
555C23—1428no—
556C23—14216no—
557C23—14216no—
558C23—1428no—
559C23—14216no—
560C23—1428no—
561C23C2×D4×D7112yes—
562C23C2×D4×D7112yes—
563C23#41×D7112yes—
564C23C2×#1176586512yes—
565C23C2×#1176586612yes—
566C23#1455617412yes—
567C23#1455617612yes—
568C23C24×D7111yescharacteristic
569Q8—416no—
570D4—28432no—
571D4—14232no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—48no—
585C2×C4—44no—
586C2×C4—48no—
587C2×C4—28432no—
588C2×C4—2844no—
589C2×C4—2844no—
590C2×C4—2844no—
591C2×C4—28416no—
592C2×C4—718no—
593C2×C4—1422no—
594C2×C4—1422no—
595C2×C4—1422no—
596C2×C4—1422no—
597C2×C4—1422no—
598C2×C4—1422no—
599C2×C4—1422no—
600C2×C4—1422no—
601C2×C4—1422no—
602C2×C4—1424no—
603C2×C4—1424no—
604C2×C4—14216no—
605C2×C4#14556101116yes—
606C2×C4C2×D4×D7114yes—
607C2×C4C2×D4×D7114yes—
608C2×C4C2×#1176586514yes—
609C2×C4C2×#1176586614yes—
610C22—2844no—
611C22—2842no—
612C22—2842no—
613C22—2844no—
614C22—2842no—
615C22—2844no—
616C22—28432no—
617C22—28416no—
618C22—2842no—
619C22—2844no—
620C22—1424no—
621C22—1428no—
622C22—1428no—
623C22—1424no—
624C22—1428no—
625C22—1424no—
626C22C2×#1455610114yes—
627C22#1798653614yes—
628C22#1798662114yes—
629C22#1798664014yes—
630C22#1798664214yes—
631C22#1798665214yes—
632C22#1798665914yes—
633C22C22×D4×D7111yescharacteristic
634C22C22×D4×D7111yescharacteristic
635C22C2×#41×D7111yescharacteristic
636C22C22×#1176586511yescharacteristic
637C22C22×#1176586611yescharacteristic
638C22C2×#1455617411yescharacteristic
639C22C2×#1455617611yescharacteristic
640C4—24no—
641C4—44no—
642C4—42no—
643C4—44no—
644C4—2842no—
645C4—2842no—
646C4—2842no—
647C4—1428no—
648C4C2×#1455610118yes—
649C2—2848no—
650C2—2844no—
651C2#2259191818yesdirect product factor
652C2C22×#1455610111yescharacteristic
653C2C2×#1798653611yescharacteristic
654C2C2×#1798662111yescharacteristic
655C2C2×#1798664011yescharacteristic
656C2C2×#1798664211yescharacteristic
657C2C2×#1798665211yescharacteristic
658C2C2×#1798665911yescharacteristic
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#391×C71C2
socleC24×C71C24
prime radicalC22×#14556103C2
hypercenterC2×#391D71
hypocenterC71C2×#2427
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
1. C22×#14556117C2semidirect
2. C22×#14556104C2semidirect
4. C22×C4×D284C2semidirect
5. #22591918C2direct (N × H)
6. C2×#17985928C2semidirect
7. C2×#17985917C2semidirect
8. C2×#17985895C2semidirect
9. C2×#17985877C2semidirect
10. C2×#17985858C2semidirect
11. C2×C4×(C2×D71)⋊C4C2semidirect
12. C2×#391×C71C2semidirect
18. C2×#14556117C22semidirect
20. C2×#14556104C22semidirect
24. C2×C4×D284C22semidirect
36. #17985928C22semidirect
37. #17985917C22semidirect
38. #17985895C22semidirect
39. #17985877C22semidirect
40. #17985858C22semidirect
41. C4×(C2×D71)⋊C4C22semidirect
48. #391×C71C22semidirect
77. C2×C4⋊Dic71D4semidirect
165. C4⋊Dic71C2×D4semidirect
320. C2×C71#2427semidirect
337. C2×#391D71semidirect
342. C71C2×#2427semidirect
371. #391C2×D71semidirect
651. C2#22591918direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561172
C22×#145561042
C22×#145561032
C22×C4×D2842
#225919182
C2×#179859282
C2×#179859172
C2×#179858952
C2×#179858772
C2×#179858582
C2×C4×(C2×D71)⋊C42
C2×#391×C712
C2×#242771