Finite Group Databasebeta

← C2×#22593492 #32167823

Subgroups — C2×#22593492 #32167823

Order 18176.

4166 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556108C211yesmaximal, characteristic, complemented
2C22×C22⋊C4×D71C211yesmaximal, characteristic
3#22593492C2116yesmaximal, direct product factor
4C2×#17985925C211yesmaximal, characteristic, complemented
5C2×#17985912C211yesmaximal, characteristic, complemented
6C2×#17985911C211yesmaximal, characteristic, complemented
7C2×#17985892C212yesmaximal, complemented
8C2×#17985891C212yesmaximal, complemented
9C2×#17985861C212yesmaximal, complemented
10C2×#17985854C212yesmaximal
11C2×#17985846C211yesmaximal, characteristic, complemented
12C2×#397×C71C211yesmaximal, characteristic, complemented
13C25×D71C2211yescharacteristic
14C23×Dic142C2211yescharacteristic, complemented
15C23×C4×D71C2211yescharacteristic
16C23×C4×D71C2211yescharacteristic
17C2×#14556108—28no—
18C2×#14556108C2218yescomplemented
19C22×C4⋊Dic71C2212yescomplemented
20C22×C22⋊Dic71C2211yescharacteristic
21C22×C22⋊Dic71C2212yes—
22C22×(C2×D71)⋊C4C2211yescharacteristic
23C22×(C2×D71)⋊C4C2211yescharacteristic
24C22×(C2×D71)⋊C4C2212yes—
25C22×(C2×D71)⋊C4C2212yes—
26C22×(C2×D71)⋊C4C2212yes—
27C22×(C2×D71)⋊C4C2212yes—
28C2×C22⋊C4×D71—28no—
29C2×C22⋊C4×D71C2218yes—
30C22×C4×Dic71C2212yescomplemented
31#17985925C2218yescomplemented
32#17985912C2218yescomplemented
33#17985911C2218yescomplemented
34#17985892C22116yescomplemented
35#17985891C22116yescomplemented
36#17985861C22116yescomplemented
37#17985854C22116yes—
38#17985846C2218yescomplemented
39C2×C71⋊#95C2212yescomplemented
40C2×C71⋊#95C2211yescharacteristic, complemented
41C2×C71⋊#95C2212yescomplemented
42C2×C71⋊#95C2212yescomplemented
43C22×C4⋊C4×C71C2211yescharacteristic, complemented
44C22×C22⋊C4×C71C2211yescharacteristic
45C22×C22⋊C4×C71C2212yes—
46#397×C71C2218yescomplemented
47C2×#95×C71C2212yescomplemented
48C2×#95×C71C2211yescharacteristic, complemented
49C24×D71—24no—
50C24×D71—24no—
51C24×D71—28no—
52C24×D71—24no—
53C24×D71—44no—
54C24×D71C2311yescharacteristic
55C24×D71C2314yes—
56C24×D71C2311yescharacteristic
57C22×Dic142—24no—
58C22×Dic142—44no—
59C22×Dic142C2314yescomplemented
60C23×Dic71C2311yescharacteristic
61C23×Dic71C2312yes—
62C23×Dic71C2312yes—
63C23×Dic71C2311yescharacteristic
64C22×C4×D71—24no—
65C22×C4×D71—28no—
66C22×C4×D71—44no—
67C22×C4×D71D418yes—
68C22×C4×D71C2314yes—
69C22×C4×D71C2314yes—
70#14556108—232no—
71C2×C4⋊Dic71—28no—
72C2×C4⋊Dic71C2318yescomplemented
73C2×C22⋊Dic71—24no—
74C2×C22⋊Dic71—44no—
75C2×C22⋊Dic71C2314yes—
76C2×C22⋊Dic71C2318yes—
77C2×(C2×D71)⋊C4—28no—
78C2×(C2×D71)⋊C4—28no—
79C2×(C2×D71)⋊C4—28no—
80C2×(C2×D71)⋊C4—24no—
81C2×(C2×D71)⋊C4—24no—
82C2×(C2×D71)⋊C4—44no—
83C2×(C2×D71)⋊C4C2318yes—
84C2×(C2×D71)⋊C4C2318yes—
85C2×(C2×D71)⋊C4C2318yes—
86C2×(C2×D71)⋊C4C2314yes—
87C2×(C2×D71)⋊C4C2314yes—
88C2×(C2×D71)⋊C4C2318yes—
89C22⋊C4×D71—232no—
90C2×C4×Dic71—44no—
91C2×C4×Dic71C2318yescomplemented
92C71⋊#95C2314yescomplemented
93C71⋊#95C2318yescomplemented
94C71⋊#95C2318yescomplemented
95C71⋊#95C2318yescomplemented
96C25×C71C2311yescharacteristic
97C23×C4×C71C2311yescharacteristic
98C23×C4×C71C2312yes—
99C23×C4×C71C2312yes—
100C23×C4×C71C2311yescharacteristic
101C2×C4⋊C4×C71—24no—
102C2×C4⋊C4×C71C2314yescomplemented
103C2×C22⋊C4×C71—28no—
104C2×C22⋊C4×C71—24no—
105C2×C22⋊C4×C71C2318yes—
106C2×C22⋊C4×C71C2314yes—
107#95×C71C2314yescomplemented
108#95×C71C2318yescomplemented
109C23×D71—22no—
110C23×D71—22no—
111C23×D71—24no—
112C23×D71—232no—
113C23×D71—22no—
114C23×D71—24no—
115C23×D71—28no—
116C23×D71—22no—
117C23×D71—24no—
118C23×D71—28no—
119C23×D71—28no—
120C23×D71—216no—
121C23×D71—48no—
122C23×D71—432no—
123C23×D71—432no—
124C23×D71—416no—
125C23×D71—416no—
126C23×D71C2×D414yes—
127C23×D71#4114yes—
128C23×D71#4118yes—
129C23×D71C2412yes—
130C23×D71C2412yes—
131C2×Dic142—28no—
132C2×Dic142—48no—
133C2×Dic142—416no—
134C22×Dic71—24no—
135C22×Dic71—22no—
136C22×Dic71—24no—
137C22×Dic71—24no—
138C22×Dic71—24no—
139C22×Dic71—24no—
140C22×Dic71—24no—
141C22×Dic71—24no—
142C22×Dic71—24no—
143C22×Dic71C2×D414yes—
144C22×Dic71C2412yes—
145C22×Dic71C2414yes—
146C22×Dic71C2414yes—
147C22×Dic71C2412yes—
148C2×C4×D71—28no—
149C2×C4×D71—216no—
150C2×C4×D71—416no—
151C2×C4×D71—48no—
152C2×C4×D71—416no—
153C2×C4×D71C2×D4116yes—
154C4⋊Dic71—216no—
155C22⋊Dic71—28no—
156C22⋊Dic71—48no—
157(C2×D71)⋊C4—216no—
158(C2×D71)⋊C4—216no—
159(C2×D71)⋊C4—216no—
160(C2×D71)⋊C4—28no—
161(C2×D71)⋊C4—28no—
162(C2×D71)⋊C4—48no—
163C4×Dic71—48no—
164C24×C71—22no—
165C24×C71—22no—
166C24×C71—24no—
167C24×C71—24no—
168C24×C71#4114yes—
169C24×C71C2412yes—
170C24×C71C2411yescharacteristic
171C22×C4×C71—24no—
172C22×C4×C71—22no—
173C22×C4×C71—24no—
174C22×C4×C71—24no—
175C22×C4×C71—24no—
176C22×C4×C71—24no—
177C22×C4×C71—24no—
178C22×C4×C71—24no—
179C22×C4×C71—24no—
180C22×C4×C71C2×D414yes—
181C22×C4×C71C2412yes—
182C22×C4×C71C2414yes—
183C22×C4×C71C2414yes—
184C22×C4×C71C2412yes—
185C4⋊C4×C71—28no—
186C22⋊C4×C71—216no—
187C22⋊C4×C71—28no—
188C23×C71—28no—
189C23×C71—216no—
190C23×C71—22no—
191C23×C71—22no—
192C23×C71—24no—
193C23×C71—24no—
194C23×C71—44no—
195C23×C71—44no—
196C23×C71—48no—
197C23×C71C22×D412yes—
198C23×C71C2×#4112yes—
199C23×C71C2×#4114yes—
200C23×C71C2×#4114yes—
201C23×C71C2×#4112yes—
202C23×C71H(5,𝔽2)14yes—
203C23×C71C2511yescharacteristic
204C22×D71—28no—
205C22×D71—28no—
206C22×D71—216no—
207C22×D71—216no—
208C22×D71—22no—
209C22×D71—22no—
210C22×D71—24no—
211C22×D71—232no—
212C22×D71—28no—
213C22×D71—216no—
214C22×D71—22no—
215C22×D71—24no—
216C22×D71—22no—
217C22×D71—24no—
218C22×D71—432no—
219C22×D71—432no—
220C22×D71—4128no—
221C22×D71—416no—
222C22×D71—44no—
223C22×D71—44no—
224C22×D71—48no—
225C22×D71—416no—
226C22×D71#124116yes—
227C22×D71C22×D414yes—
228C22×D71C2×#4114yes—
229C22×D71C2×#4118yes—
230C2×C4×C71—28no—
231C2×C4×C71—24no—
232C2×C4×C71—22no—
233C2×C4×C71—24no—
234C2×C4×C71—24no—
235C2×C4×C71—24no—
236C2×C4×C71—24no—
237C2×C4×C71—24no—
238C2×C4×C71—24no—
239C2×C4×C71—24no—
240C2×C4×C71—48no—
241C2×C4×C71—44no—
242C2×C4×C71—48no—
243C2×C4×C71C22×D414yes—
244C2×Dic71—28no—
245C2×Dic71—24no—
246C2×Dic71—22no—
247C2×Dic71—24no—
248C2×Dic71—24no—
249C2×Dic71—24no—
250C2×Dic71—24no—
251C2×Dic71—24no—
252C2×Dic71—24no—
253C2×Dic71—24no—
254C2×Dic71—44no—
255C2×Dic71—48no—
256C2×Dic71—48no—
257C2×Dic71C22×D414yes—
258C4×D71—416no—
259C4×D71—416no—
260Dic142—416no—
261C22×C71—24no—
262C22×C71—28no—
263C22×C71—416no—
264C22×C71—42no—
265C22×C71—42no—
266C22×C71—44no—
267C22×C71#52514yes—
268C22×C71C2×#12414yes—
269C22×C71#53414yes—
270C22×C71#55118yes—
271C22×C71#55118yes—
272C22×C71C23×D411yescharacteristic
273C22×C71C22×#4111yescharacteristic
274C22×C71C22×#4112yes—
275C22×C71C22×#4111yescharacteristic
276C22×C71C2×H(5,𝔽2)12yes—
277C2×D71—232no—
278C2×D71—24no—
279C2×D71—24no—
280C2×D71—28no—
281C2×D71—28no—
282C2×D71—24no—
283C2×D71—28no—
284C2×D71—464no—
285C2×D71—416no—
286C2×D71—42no—
287C2×D71—42no—
288C2×D71—44no—
289C2×D71C2×#12418yes—
290C4×C71—24no—
291C4×C71—44no—
292C4×C71—42no—
293C4×C71—44no—
294Dic71—24no—
295Dic71—42no—
296Dic71—44no—
297Dic71—44no—
298C2×#2561—711nomaximal, Sylow
299C2×C71—44no—
300C2×C71#256118yescomplemented
301C2×C71C2×#52511yescharacteristic
302C2×C71C22×#12411yescharacteristic
303C2×C71C2×#53411yescharacteristic
304C2×C71C2×#55112yes—
305C2×C71C2×#55112yes—
306D71—28no—
307D71—44no—
308C22×C22⋊Q8—711no—
309C23×C22⋊C4—711no—
310#2561—7116no—
311C2×#397—711no—
312C2×#397D7111yescharacteristic, complemented
313C2×#393—711no—
314C2×#393—711no—
315C2×#388—712no—
316C2×#388—712no—
317C2×#381—712no—
318C2×#379—712no—
319C2×C4×C22⋊C4—711no—
320C71C2×#256111yescharacteristic, complemented, Sylow
321C26—711no—
322C23×Q8—711no—
323C24×C4—711no—
324C24×C4—711no—
325C2×C22⋊Q8—718no—
326C2×C22⋊Q8—1428no—
327C22×C4⋊C4—712no—
328C22×C4⋊C4C2×D7111yescharacteristic, complemented
329C22×C22⋊C4—712no—
330C22×C22⋊C4—712no—
331C22×C22⋊C4—711no—
332C22×C22⋊C4—711no—
333C22×C22⋊C4—712no—
334C22×C22⋊C4—711no—
335C22×C22⋊C4—712no—
336C22×C22⋊C4—712no—
337C22×C22⋊C4—718no—
338C22×C22⋊C4—1428no—
339C22×C22⋊C4C2×D7111yescharacteristic
340C22×C22⋊C4C2×D7112yes—
341C22×C42—712no—
342#397—718no—
343#397C2×D7118yescomplemented
344#393—718no—
345#393—718no—
346#388—7116no—
347#388—7116no—
348#381—7116no—
349#379—7116no—
350C4×C22⋊C4—718no—
351C2×#95—711no—
352C2×#95—712no—
353C2×#95—712no—
354C2×#95—712no—
355C2×#95C2×D7111yescharacteristic, complemented
356C2×#95C2×D7112yescomplemented
357C25—2844no—
358C25—714no—
359C25—711no—
360C25—711no—
361C25—1428no—
362C25—1424no—
363C25—1424no—
364C25—1424no—
365C25C22×D7111yescharacteristic
366C22×Q8—2844no—
367C22×Q8—714no—
368C22×Q8—1424no—
369C23×C4—2844no—
370C23×C4—718no—
371C23×C4—714no—
372C23×C4—711no—
373C23×C4—712no—
374C23×C4—712no—
375C23×C4—711no—
376C23×C4—714no—
377C23×C4—1424no—
378C23×C4—1428no—
379C23×C4C22×D7112yes—
380C23×C4C22×D7112yes—
381C23×C4C22×D7111yescharacteristic
382C23×C4C22×D7111yescharacteristic
383C22⋊Q8—14232no—
384C2×C4⋊C4—24no—
385C2×C4⋊C4—718no—
386C2×C4⋊C4—1428no—
387C2×C4⋊C4C22×D7114yescomplemented
388C2×C22⋊C4—28no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—2844no—
391C2×C22⋊C4—2844no—
392C2×C22⋊C4—718no—
393C2×C22⋊C4—718no—
394C2×C22⋊C4—714no—
395C2×C22⋊C4—714no—
396C2×C22⋊C4—714no—
397C2×C22⋊C4—718no—
398C2×C22⋊C4—718no—
399C2×C22⋊C4—718no—
400C2×C22⋊C4—14232no—
401C2×C22⋊C4—1424no—
402C2×C22⋊C4—1428no—
403C2×C22⋊C4—1428no—
404C2×C22⋊C4—1428no—
405C2×C22⋊C4—1424no—
406C2×C22⋊C4—1424no—
407C2×C22⋊C4C22×D7114yes—
408C2×C22⋊C4C22×D7118yes—
409C2×C42—2844no—
410C2×C42—718no—
411#95—714no—
412#95—718no—
413#95—718no—
414#95—718no—
415#95C22×D7114yescomplemented
416#95C22×D7118yescomplemented
417C24—22no—
418C24—24no—
419C24—22no—
420C24—24no—
421C24—28432no—
422C24—2848no—
423C24—28416no—
424C24—28416no—
425C24—28432no—
426C24—714no—
427C24—718no—
428C24—712no—
429C24—714no—
430C24—712no—
431C24—14216no—
432C24—1428no—
433C24—1428no—
434C24—14232no—
435C24—1422no—
436C24—1424no—
437C24—1428no—
438C24—1422no—
439C24—1424no—
440C24—1422no—
441C24—1422no—
442C24—1424no—
443C24#1176586414yes—
444C24C23×D7112yes—
445C24C23×D7111yescharacteristic
446C2×Q8—28416no—
447C2×Q8—2848no—
448C2×Q8—1428no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—28416no—
459C22×C4—2848no—
460C22×C4—28416no—
461C22×C4—7116no—
462C22×C4—714no—
463C22×C4—712no—
464C22×C4—712no—
465C22×C4—714no—
466C22×C4—714no—
467C22×C4—1428no—
468C22×C4—14216no—
469C22×C4—1422no—
470C22×C4—1424no—
471C22×C4—1424no—
472C22×C4—1424no—
473C22×C4—1424no—
474C22×C4—1424no—
475C22×C4—1424no—
476C22×C4—1424no—
477C22×C4—1424no—
478C22×C4D4×D7114yes—
479C22×C4C23×D7112yes—
480C22×C4C23×D7114yes—
481C22×C4C23×D7114yes—
482C22×C4C23×D7112yes—
483C4⋊C4—28no—
484C4⋊C4—14216no—
485C22⋊C4—216no—
486C22⋊C4—28no—
487C22⋊C4—2848no—
488C22⋊C4—2848no—
489C22⋊C4—1428no—
490C22⋊C4—14216no—
491C22⋊C4—14216no—
492C22⋊C4—14216no—
493C22⋊C4—1428no—
494C22⋊C4—1428no—
495C42—2848no—
496C23—22no—
497C23—24no—
498C23—22no—
499C23—24no—
500C23—216no—
501C23—28no—
502C23—44no—
503C23—48no—
504C23—44no—
505C23—28432no—
506C23—28416no—
507C23—28416no—
508C23—284128no—
509C23—28432no—
510C23—2844no—
511C23—2848no—
512C23—2844no—
513C23—714no—
514C23—718no—
515C23—7116no—
516C23—714no—
517C23—14232no—
518C23—1422no—
519C23—1424no—
520C23—1422no—
521C23—1424no—
522C23—14216no—
523C23—1428no—
524C23—1428no—
525C23—1422no—
526C23—1422no—
527C23—1424no—
528C23—14216no—
529C23—14216no—
530C23—1428no—
531C23C2×D4×D7112yes—
532C23#41×D7114yes—
533C23#41×D7112yes—
534C23C2×#1176586414yes—
535C23C2×#1176586412yes—
536C23#1455617414yes—
537C23C24×D7111yescharacteristic
538Q8—28416no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—48no—
550C2×C4—48no—
551C2×C4—44no—
552C2×C4—28416no—
553C2×C4—28416no—
554C2×C4—2848no—
555C2×C4—2848no—
556C2×C4—2844no—
557C2×C4—714no—
558C2×C4—1422no—
559C2×C4—1424no—
560C2×C4—1424no—
561C2×C4—1424no—
562C2×C4—1424no—
563C2×C4—1424no—
564C2×C4—1424no—
565C2×C4—1424no—
566C2×C4—1428no—
567C2×C4—1424no—
568C2×C4C2×D4×D7114yes—
569C22—28no—
570C22—24no—
571C22—44no—
572C22—42no—
573C22—42no—
574C22—416no—
575C22—28464no—
576C22—2844no—
577C22—2842no—
578C22—2842no—
579C22—28416no—
580C22—718no—
581C22—1428no—
582C22—1424no—
583C22—1428no—
584C22—1428no—
585C22—1424no—
586C22—1424no—
587C22—14232no—
588C22#1798650914yes—
589C22#124×D7114yes—
590C22#1798656014yes—
591C22#1798667218yes—
592C22#1798667418yes—
593C22C22×D4×D7111yescharacteristic
594C22C2×#41×D7112yes—
595C22C2×#41×D7111yescharacteristic
596C22C22×#1176586411yescharacteristic
597C22C2×#1455617412yes—
598C4—24no—
599C4—44no—
600C4—44no—
601C4—42no—
602C4—2844no—
603C4—2842no—
604C4—2844no—
605C4—1424no—
606C2—44no—
607C2—2844no—
608C2—1428no—
609C2#2259349218yesdirect product factor
610C2C2×#1798650911yescharacteristic
611C2C2×#124×D7111yescharacteristic
612C2C2×#1798656011yescharacteristic
613C2C2×#1798667212yes—
614C2C2×#1798667412yes—
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#397×C71C2
socleC24×C71C24
prime radicalC25×D71C22
hypercenterC2×#397D71
hypocenterC71C2×#2561
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
1. C22×#14556108C2semidirect
3. #22593492C2direct (N × H)
4. C2×#17985925C2semidirect
5. C2×#17985912C2semidirect
6. C2×#17985911C2semidirect
7. C2×#17985892C2semidirect
8. C2×#17985891C2semidirect
9. C2×#17985861C2semidirect
11. C2×#17985846C2semidirect
12. C2×#397×C71C2semidirect
14. C23×Dic142C22semidirect
18. C2×#14556108C22semidirect
19. C22×C4⋊Dic71C22semidirect
30. C22×C4×Dic71C22semidirect
31. #17985925C22semidirect
32. #17985912C22semidirect
33. #17985911C22semidirect
34. #17985892C22semidirect
35. #17985891C22semidirect
36. #17985861C22semidirect
38. #17985846C22semidirect
39. C2×C71⋊#95C22semidirect
40. C2×C71⋊#95C22semidirect
41. C2×C71⋊#95C22semidirect
42. C2×C71⋊#95C22semidirect
43. C22×C4⋊C4×C71C22semidirect
46. #397×C71C22semidirect
47. C2×#95×C71C22semidirect
48. C2×#95×C71C22semidirect
59. C22×Dic142C23semidirect
72. C2×C4⋊Dic71C23semidirect
91. C2×C4×Dic71C23semidirect
92. C71⋊#95C23semidirect
93. C71⋊#95C23semidirect
94. C71⋊#95C23semidirect
95. C71⋊#95C23semidirect
102. C2×C4⋊C4×C71C23semidirect
107. #95×C71C23semidirect
108. #95×C71C23semidirect
300. C2×C71#2561semidirect
312. C2×#397D71semidirect
320. C71C2×#2561semidirect
328. C22×C4⋊C4C2×D71semidirect
343. #397C2×D71semidirect
355. C2×#95C2×D71semidirect
356. C2×#95C2×D71semidirect
387. C2×C4⋊C4C22×D71semidirect
415. #95C22×D71semidirect
416. #95C22×D71semidirect
609. C2#22593492direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561082
C22×C22⋊C4×D712
#225934922
C2×#179859252
C2×#179859122
C2×#179859112
C2×#179858922
C2×#179858912
C2×#179858612
C2×#179858542
C2×#179858462
C2×#397×C712
C2×#256171