Finite Group Databasebeta

← C2×#22592040 #32165282

Subgroups — C2×#22592040 #32165282

Order 18176.

3550 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556127C211yesmaximal, characteristic, complemented
2C22×#14556108C212yesmaximal
3C22×C4⋊C4×D71C211yesmaximal, characteristic
4#22592040C2116yesmaximal, direct product factor
5C2×#17985929C212yesmaximal, complemented
6C2×#17985920C212yesmaximal, complemented
7C2×#17985898C211yesmaximal, characteristic, complemented
8C2×#17985873C212yesmaximal
9C2×#17985869C211yesmaximal, characteristic, complemented
10C2×#17985856C211yesmaximal, characteristic
11C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic
12C2×#389×C71C211yesmaximal, characteristic, complemented
13C23×Dic142C2212yes—
14C23×C4×D71C2212yes—
15C23×C4×D71C2211yescharacteristic
16C2×#14556127—28no—
17C2×#14556127C2218yescomplemented
18C2×#14556108—216no—
19C2×#14556108C22116yes—
20C22×C4⋊Dic71C2211yescharacteristic
21C22×C4⋊Dic71C2212yes—
22C22×C4⋊Dic71C2211yescharacteristic
23C22×C4⋊Dic71C2212yes—
24C22×C4⋊Dic71C2212yes—
25C22×Dic71⋊C4C2212yes—
26C22×Dic71⋊C4C2212yes—
27C2×C4⋊C4×D71—28no—
28C2×C4⋊C4×D71C2218yes—
29C22×(C2×D71)⋊C4C2212yes—
30C22×(C2×D71)⋊C4C2211yescharacteristic
31C22×(C2×D71)⋊C4C2211yescharacteristic
32C22×C4×Dic71C2211yescharacteristic
33#17985929C22116yescomplemented
34#17985920C22116yescomplemented
35#17985898C2218yescomplemented
36#17985873C22116yes—
37#17985869C2218yescomplemented
38#17985856C2218yes—
39C4×(C2×D71)⋊C4C2218yes—
40C2×C71⋊#95C2212yes—
41C2×C71⋊#95C2211yescharacteristic
42C2×C71⋊#95C2212yes—
43C2×C71⋊#95C2211yescharacteristic
44C22×C4⋊C4×C71C2211yescharacteristic
45C22×C4⋊C4×C71C2212yes—
46C22×C4⋊C4×C71C2211yescharacteristic
47C22×C42×C71C2211yescharacteristic
48#389×C71C2218yescomplemented
49C2×#95×C71C2212yes—
50C24×D71C2311yescharacteristic
51C22×Dic142—28no—
52C22×Dic142—28no—
53C22×Dic142—44no—
54C22×Dic142C2318yes—
55C23×Dic71C2312yes—
56C23×Dic71C2312yes—
57C23×Dic71C2311yescharacteristic
58C23×Dic71C2311yescharacteristic
59C23×Dic71C2311yescharacteristic
60C22×C4×D71—28no—
61C22×C4×D71—24no—
62C22×C4×D71—28no—
63C22×C4×D71—24no—
64C22×C4×D71—44no—
65C22×C4×D71Q818yescomplemented
66C22×C4×D71C2318yes—
67C22×C4×D71C2314yes—
68#14556127—232no—
69#14556108—264no—
70C2×C4⋊Dic71—28no—
71C2×C4⋊Dic71—24no—
72C2×C4⋊Dic71—28no—
73C2×C4⋊Dic71—24no—
74C2×C4⋊Dic71—44no—
75C2×C4⋊Dic71C2314yes—
76C2×C4⋊Dic71C2314yes—
77C2×C4⋊Dic71C2318yes—
78C2×C4⋊Dic71C2318yes—
79C2×C4⋊Dic71C2318yes—
80C2×Dic71⋊C4—28no—
81C2×Dic71⋊C4—28no—
82C2×Dic71⋊C4C2318yes—
83C2×Dic71⋊C4C2318yes—
84C4⋊C4×D71—232no—
85C2×(C2×D71)⋊C4—28no—
86C2×(C2×D71)⋊C4D418yes—
87C2×(C2×D71)⋊C4Q818yescomplemented
88C2×(C2×D71)⋊C4C2318yes—
89C2×(C2×D71)⋊C4C2314yes—
90C2×(C2×D71)⋊C4C2314yes—
91C2×C4×Dic71—24no—
92C2×C4×Dic71C2314yes—
93C71⋊#95C2314yes—
94C71⋊#95C2314yes—
95C71⋊#95C2318yes—
96C71⋊#95C2318yes—
97C23×C4×C71C2312yes—
98C23×C4×C71C2312yes—
99C23×C4×C71C2311yescharacteristic
100C23×C4×C71C2311yescharacteristic
101C23×C4×C71C2311yescharacteristic
102C2×C4⋊C4×C71—28no—
103C2×C4⋊C4×C71—24no—
104C2×C4⋊C4×C71D418yes—
105C2×C4⋊C4×C71C2314yes—
106C2×C4⋊C4×C71C2318yes—
107C2×C4⋊C4×C71C2314yes—
108C2×C42×C71—24no—
109C2×C42×C71C2314yes—
110#95×C71C2318yes—
111C23×D71—22no—
112C23×D71—22no—
113C23×D71—22no—
114C23×D71—22no—
115C23×D71C2×D414yes—
116C23×D71C2×Q814yes—
117C23×D71C2×Q814yes—
118C23×D71C2412yes—
119C2×Dic142—216no—
120C2×Dic142—216no—
121C2×Dic142—48no—
122C2×Dic142—432no—
123C22×Dic71—22no—
124C22×Dic71—22no—
125C22×Dic71—22no—
126C22×Dic71—22no—
127C22×Dic71—22no—
128C22×Dic71—22no—
129C22×Dic71—24no—
130C22×Dic71—24no—
131C22×Dic71—24no—
132C22×Dic71—24no—
133C22×Dic71—24no—
134C22×Dic71—24no—
135C22×Dic71C2×D414yes—
136C22×Dic71C2×Q814yes—
137C22×Dic71C2×Q814yes—
138C22×Dic71C2414yes—
139C22×Dic71C2414yes—
140C22×Dic71C2412yes—
141C22×Dic71C2412yes—
142C22×Dic71C2412yes—
143C2×C4×D71—232no—
144C2×C4×D71—28no—
145C2×C4×D71—28no—
146C2×C4×D71—216no—
147C2×C4×D71—216no—
148C2×C4×D71—48no—
149C2×C4×D71—432no—
150C2×C4×D71C2×Q8116yescomplemented
151C4⋊Dic71—28no—
152C4⋊Dic71—216no—
153C4⋊Dic71—28no—
154C4⋊Dic71—216no—
155C4⋊Dic71—48no—
156Dic71⋊C4—216no—
157Dic71⋊C4—216no—
158(C2×D71)⋊C4—216no—
159(C2×D71)⋊C4C2×D4116yes—
160(C2×D71)⋊C4C2×Q8116yescomplemented
161C4×Dic71—28no—
162C24×C71C2411yescharacteristic
163C22×C4×C71—24no—
164C22×C4×C71—22no—
165C22×C4×C71—22no—
166C22×C4×C71—24no—
167C22×C4×C71—22no—
168C22×C4×C71—22no—
169C22×C4×C71—24no—
170C22×C4×C71—24no—
171C22×C4×C71—24no—
172C22×C4×C71C2×D414yes—
173C22×C4×C71C2×D418yes—
174C22×C4×C71C2×D414yes—
175C22×C4×C71C2×Q814yes—
176C22×C4×C71C2×Q814yes—
177C22×C4×C71#4114yes—
178C22×C4×C71C2414yes—
179C22×C4×C71C2414yes—
180C22×C4×C71C2412yes—
181C22×C4×C71C2412yes—
182C22×C4×C71C2412yes—
183C4⋊C4×C71—216no—
184C4⋊C4×C71—28no—
185C4⋊C4×C71C2×D4116yes—
186C42×C71—28no—
187C23×C71C22×D412yes—
188C23×C71C22×D412yes—
189C23×C71C22×Q812yes—
190C23×C71C22×Q812yes—
191C23×C71C2×#4112yes—
192C23×C71H(5,𝔽2)12yes—
193C23×C712−1+412yes—
194C23×C71C2511yescharacteristic
195C22×D71—28no—
196C22×D71—28no—
197C22×D71—28no—
198C22×D71—28no—
199C22×D71—28no—
200C22×D71—28no—
201C22×D71—22no—
202C22×D71—22no—
203C22×D71—22no—
204C22×D71—22no—
205C22×D71C4⋊Q8116yescomplemented
206C22×D71C22×D414yes—
207C22×D71C22×Q814yes—
208C22×D71C22×Q814yes—
209C2×C4×C71—216no—
210C2×C4×C71—28no—
211C2×C4×C71—28no—
212C2×C4×C71—24no—
213C2×C4×C71—22no—
214C2×C4×C71—22no—
215C2×C4×C71—22no—
216C2×C4×C71—24no—
217C2×C4×C71—22no—
218C2×C4×C71—24no—
219C2×C4×C71—24no—
220C2×C4×C71—24no—
221C2×C4×C71—48no—
222C2×C4×C71C22⋊Q8116yescomplemented
223C2×C4×C71C22×D414yes—
224C2×C4×C71C22×D418yes—
225C2×C4×C71C22×D414yes—
226C2×C4×C71C22×Q814yes—
227C2×C4×C71C22×Q814yes—
228C2×C4×C71C2×#4114yes—
229C2×Dic71—28no—
230C2×Dic71—28no—
231C2×Dic71—28no—
232C2×Dic71—22no—
233C2×Dic71—22no—
234C2×Dic71—22no—
235C2×Dic71—22no—
236C2×Dic71—22no—
237C2×Dic71—22no—
238C2×Dic71—24no—
239C2×Dic71—24no—
240C2×Dic71—24no—
241C2×Dic71—24no—
242C2×Dic71—24no—
243C2×Dic71—24no—
244C2×Dic71—48no—
245C2×Dic71—48no—
246C2×Dic71C22×D414yes—
247C2×Dic71C22×Q814yes—
248C2×Dic71C22×Q814yes—
249C4×D71—232no—
250C4×D71—432no—
251Dic142—432no—
252C22×C71C2×C22⋊Q814yes—
253C22×C71C2×C4⋊Q814yes—
254C22×C71#54414yes—
255C22×C71D4214yes—
256C22×C71#54714yes—
257C22×C71D4×Q814yes—
258C22×C71#55414yes—
259C22×C71C23×D411yescharacteristic
260C22×C71C23×D411yescharacteristic
261C22×C71C23×Q811yescharacteristic
262C22×C71C23×Q811yescharacteristic
263C22×C71C22×#4111yescharacteristic
264C22×C71C2×H(5,𝔽2)11yescharacteristic
265C22×C71C2×2−1+411yescharacteristic
266C2×D71—24no—
267C2×D71—24no—
268C2×D71—24no—
269C2×D71—232no—
270C2×D71—24no—
271C2×D71—24no—
272C2×D71—24no—
273C2×D71C2×C4⋊Q818yescomplemented
274C4×C71—28no—
275C4×C71—24no—
276C4×C71—24no—
277C4×C71—44no—
278C4×C71C2×C22⋊Q818yescomplemented
279Dic71—24no—
280Dic71—24no—
281Dic71—24no—
282Dic71—44no—
283Dic71—44no—
284C2×#2440—711nomaximal, Sylow
285C2×C71#244018yescomplemented
286C2×C71C22×C22⋊Q811yescharacteristic
287C2×C71C22×C4⋊Q811yescharacteristic
288C2×C71C2×#54411yescharacteristic
289C2×C71C2×D4211yescharacteristic
290C2×C71C2×#54711yescharacteristic
291C2×C71C2×D4×Q811yescharacteristic
292C2×C71C2×#55411yescharacteristic
293D71—28no—
294C22×C4⋊Q8—711no—
295C22×C22⋊Q8—712no—
296C23×C4⋊C4—711no—
297#2440—7116no—
298C2×#398—712no—
299C2×#395—712no—
300C2×#389—711no—
301C2×#389D7111yescharacteristic, complemented
302C2×#385—712no—
303C2×#384—711no—
304C2×#380—711no—
305C2×C4×C22⋊C4—711no—
306C71C2×#244011yescharacteristic, complemented, Sylow
307C23×Q8—712no—
308C24×C4—712no—
309C24×C4—711no—
310C2×C4⋊Q8—718no—
311C2×C4⋊Q8—1428no—
312C2×C22⋊Q8—7116no—
313C2×C22⋊Q8—14216no—
314C22×C4⋊C4—712no—
315C22×C4⋊C4—712no—
316C22×C4⋊C4—712no—
317C22×C4⋊C4—712no—
318C22×C4⋊C4—712no—
319C22×C4⋊C4—711no—
320C22×C4⋊C4—711no—
321C22×C4⋊C4—718no—
322C22×C4⋊C4—1428no—
323C22×C4⋊C4C2×D7112yes—
324C22×C4⋊C4C2×D7111yescharacteristic
325C22×C4⋊C4C2×D7111yescharacteristic
326C22×C22⋊C4—711no—
327C22×C22⋊C4—711no—
328C22×C22⋊C4—712no—
329C22×C42—711no—
330C22×C42C2×D7111yescharacteristic
331#398—7116no—
332#395—7116no—
333#389—718no—
334#389C2×D7118yescomplemented
335#385—7116no—
336#384—718no—
337#380—718no—
338C4×C22⋊C4—718no—
339C2×#95—711no—
340C2×#95—712no—
341C2×#95—711no—
342C2×#95—712no—
343C2×#95C2×D7112yes—
344C25—711no—
345C22×Q8—2844no—
346C22×Q8—718no—
347C22×Q8—1428no—
348C22×Q8—1428no—
349C23×C4—2844no—
350C23×C4—718no—
351C23×C4—711no—
352C23×C4—711no—
353C23×C4—711no—
354C23×C4—718no—
355C23×C4—714no—
356C23×C4—712no—
357C23×C4—712no—
358C23×C4—1428no—
359C23×C4—1428no—
360C23×C4—1424no—
361C23×C4—1424no—
362C23×C4C22×D7112yes—
363C23×C4C22×D7111yescharacteristic
364C23×C4C22×D7112yes—
365C23×C4C22×D7111yescharacteristic
366C23×C4C22×D7111yescharacteristic
367C4⋊Q8—14232no—
368C22⋊Q8—14264no—
369C2×C4⋊C4—28no—
370C2×C4⋊C4—24no—
371C2×C4⋊C4—2844no—
372C2×C4⋊C4—718no—
373C2×C4⋊C4—714no—
374C2×C4⋊C4—718no—
375C2×C4⋊C4—718no—
376C2×C4⋊C4—718no—
377C2×C4⋊C4—714no—
378C2×C4⋊C4—718no—
379C2×C4⋊C4—1428no—
380C2×C4⋊C4—1428no—
381C2×C4⋊C4—1428no—
382C2×C4⋊C4—1424no—
383C2×C4⋊C4—1428no—
384C2×C4⋊C4—1424no—
385C2×C4⋊C4—14232no—
386C2×C4⋊C4D28418yes—
387C2×C4⋊C4C22×D7118yes—
388C2×C4⋊C4C22×D7114yes—
389C2×C4⋊C4C22×D7114yes—
390C2×C22⋊C4—718no—
391C2×C22⋊C4—718no—
392C2×C22⋊C4—714no—
393C2×C22⋊C4—714no—
394C2×C22⋊C4—718no—
395C2×C22⋊C4—1428no—
396C2×C42—24no—
397C2×C42—714no—
398C2×C42—1424no—
399C2×C42C22×D7114yes—
400#95—714no—
401#95—714no—
402#95—718no—
403#95—718no—
404#95C22×D7118yes—
405C24—714no—
406C24—714no—
407C24—714no—
408C24—712no—
409C24—1422no—
410C24—1422no—
411C24—1422no—
412C24—1422no—
413C24C23×D7111yescharacteristic
414C2×Q8—28432no—
415C2×Q8—2848no—
416C2×Q8—14216no—
417C2×Q8—14216no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—28432no—
428C22×C4—2848no—
429C22×C4—714no—
430C22×C4—714no—
431C22×C4—714no—
432C22×C4—712no—
433C22×C4—712no—
434C22×C4—712no—
435C22×C4—7116no—
436C22×C4—714no—
437C22×C4—714no—
438C22×C4—14216no—
439C22×C4—1424no—
440C22×C4—1424no—
441C22×C4—1424no—
442C22×C4—1424no—
443C22×C4—1424no—
444C22×C4—1424no—
445C22×C4—14216no—
446C22×C4—1422no—
447C22×C4—1422no—
448C22×C4—1422no—
449C22×C4—1422no—
450C22×C4—1422no—
451C22×C4—1422no—
452C22×C4—1428no—
453C22×C4—1428no—
454C22×C4—14232no—
455C22×C4D4×D7114yes—
456C22×C4C2×D28418yes—
457C22×C4C2×D28414yes—
458C22×C4Q8×D7114yes—
459C22×C4Q8×D7114yes—
460C22×C4#1176586414yes—
461C22×C4C23×D7114yes—
462C22×C4C23×D7112yes—
463C22×C4C23×D7114yes—
464C22×C4C23×D7112yes—
465C22×C4C23×D7112yes—
466C4⋊C4—216no—
467C4⋊C4—28no—
468C4⋊C4—2848no—
469C4⋊C4—14216no—
470C4⋊C4—14216no—
471C4⋊C4—14216no—
472C4⋊C4—14216no—
473C4⋊C4—1428no—
474C4⋊C4—1428no—
475C4⋊C4C2×D284116yes—
476C22⋊C4—7116no—
477C22⋊C4—7116no—
478C22⋊C4—14216no—
479C42—28no—
480C42—1428no—
481C23—714no—
482C23—714no—
483C23—714no—
484C23—7116no—
485C23—1422no—
486C23—1422no—
487C23—1422no—
488C23—1422no—
489C23—1428no—
490C23—1428no—
491C23—1428no—
492C23—1428no—
493C23—1428no—
494C23—1428no—
495C23C2×D4×D7112yes—
496C23C22×D28412yes—
497C23C2×Q8×D7112yes—
498C23C2×Q8×D7112yes—
499C23C2×#1176586412yes—
500C23#1455617412yes—
501C23#1455617512yes—
502C23C24×D7111yescharacteristic
503Q8—28432no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—216no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—48no—
517C2×C4—28432no—
518C2×C4—2848no—
519C2×C4—2848no—
520C2×C4—714no—
521C2×C4—714no—
522C2×C4—714no—
523C2×C4—1424no—
524C2×C4—1424no—
525C2×C4—1424no—
526C2×C4—1424no—
527C2×C4—1424no—
528C2×C4—1424no—
529C2×C4—1422no—
530C2×C4—1422no—
531C2×C4—1422no—
532C2×C4—1422no—
533C2×C4—1422no—
534C2×C4—1422no—
535C2×C4—1428no—
536C2×C4—1428no—
537C2×C4—1428no—
538C2×C4—14232no—
539C2×C4#14556108116yescomplemented
540C2×C4C2×D4×D7114yes—
541C2×C4C22×D28418yes—
542C2×C4C22×D28414yes—
543C2×C4C2×Q8×D7114yes—
544C2×C4C2×Q8×D7114yes—
545C2×C4C2×#1176586414yes—
546C22—718no—
547C22—1424no—
548C22—14232no—
549C22—1424no—
550C22—1424no—
551C22—1424no—
552C22—1424no—
553C22—1424no—
554C22C2×#1455610814yes—
555C22C4⋊Q8×D7114yes—
556C22#1798662914yes—
557C22D4×D28414yes—
558C22#1798664914yes—
559C22Q8×D28414yes—
560C22#1798670714yes—
561C22C22×D4×D7111yescharacteristic
562C22C23×D28411yescharacteristic
563C22C22×Q8×D7111yescharacteristic
564C22C22×Q8×D7111yescharacteristic
565C22C22×#1176586411yescharacteristic
566C22C2×#1455617411yescharacteristic
567C22C2×#1455617511yescharacteristic
568C4—28no—
569C4—24no—
570C4—24no—
571C4—44no—
572C4—2844no—
573C4—2844no—
574C4—1424no—
575C4—1424no—
576C4—1424no—
577C4C2×#1455610818yescomplemented
578C2—1428no—
579C2#2259204018yesdirect product factor
580C2C22×#1455610811yescharacteristic
581C2C2×C4⋊Q8×D7111yescharacteristic
582C2C2×#1798662911yescharacteristic
583C2C2×D4×D28411yescharacteristic
584C2C2×#1798664911yescharacteristic
585C2C2×Q8×D28411yescharacteristic
586C2C2×#1798670711yescharacteristic
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#389×C71C2
socleC24×C71C24
prime radicalC24×D71C23
hypercenterC2×#389D71
hypocenterC71C2×#2440
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C22×#14556127C2semidirect
4. #22592040C2direct (N × H)
5. C2×#17985929C2semidirect
6. C2×#17985920C2semidirect
7. C2×#17985898C2semidirect
9. C2×#17985869C2semidirect
12. C2×#389×C71C2semidirect
17. C2×#14556127C22semidirect
33. #17985929C22semidirect
34. #17985920C22semidirect
35. #17985898C22semidirect
37. #17985869C22semidirect
48. #389×C71C22semidirect
65. C22×C4×D71Q8semidirect
87. C2×(C2×D71)⋊C4Q8semidirect
150. C2×C4×D71C2×Q8semidirect
160. (C2×D71)⋊C4C2×Q8semidirect
205. C22×D71C4⋊Q8semidirect
222. C2×C4×C71C22⋊Q8semidirect
273. C2×D71C2×C4⋊Q8semidirect
278. C4×C71C2×C22⋊Q8semidirect
285. C2×C71#2440semidirect
301. C2×#389D71semidirect
306. C71C2×#2440semidirect
334. #389C2×D71semidirect
539. C2×C4#14556108semidirect
577. C4C2×#14556108semidirect
579. C2#22592040direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561272
C22×#145561082
C22×C4⋊C4×D712
#225920402
C2×#179859292
C2×#179859202
C2×#179858982
C2×#179858732
C2×#179858692
C2×#179858562
C2×C4×(C2×D71)⋊C42
C2×#389×C712
C2×#244071